Constraint-Based Realization, Volume V | Exact Operational Signature and Binary Invalidation in a Delayed-Choice Quantum Eraser

Apr 11

Constraint-Based Realization | Exact Operational Signature and Binary Invalidation in a Delayed-Choice Quantum Eraser

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Constraint-Based Realization | Volume V | Exact Operational Signature and Binary Invalidation in a Delayed-Choice Quantum Eraser

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This volume is a work of theoretical research and formal argument. It advances a proposed framework in quantum foundations and should be read accordingly. Statements labeled as axioms, assumptions, propositions, theorems, conjectures, interpretive claims, or empirical hypotheses carry different evidential and logical status, which is specified within the text. No claim should be read more strongly than the status assigned to it.

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Abstract

The Core Theorem Paper established the canonical law form of Constraint-Based Realization, fixed its admissibility structure, and identified a designated accessibility-sensitive test domain, but it did not yet derive an exact platform-level response model on which canonical CBR and the standard baseline could be compared without residual protocol-level looseness.

The present volume fixes that remaining task by instantiating canonical CBR on one exact accessibility-tunable delayed-choice quantum eraser consisting of a signal subsystem, an idler-record subsystem, a controlled retrieval-or-erasure branch, and a timing structure in which record accessibility may be established after signal detection while remaining part of the full physical context.

For this exact model, an operational accessibility parameter η ∈ [0,1] is defined from calibrated record-theoretic quantities, and both the standard visibility response V_SQM(η) and the canonical realization-law response V_CBR(η) are derived on the same footing from the same declared architecture.

The first central result is a non-equivalence theorem: if accessibility enters realization law nontrivially, then V_CBR(η) cannot remain globally identical to V_SQM(η) across the admissible η-domain, so canonical CBR acquires exact model-level empirical content rather than merely framework-level vulnerability.

The second central result is a critical-regime signature theorem: under the stated regularity assumptions, the non-equivalence concentrates near a critical accessibility value η_c as a derivative break or kink in the primary observable, with a bounded non-baseline deviation class remaining as the fallback signature if the stronger regularity conditions fail.

The volume closes with a binary invalidation result: if the exact declared platform exhibits only baseline-class behavior across the experimentally accessible η-domain under the fixed calibration and perturbation controls, then canonical CBR in this instantiated form is false.

1. Introduction

1.1 Exact task of this volume

This volume does not broaden the CBR program. It narrows it. Its purpose is to take the canonized realization-law structure already established and force it into one exact experimental model whose observable consequences can be derived, compared, and, if necessary, invalidated. The task is therefore not architectural expansion, not renewed interpretive survey, and not further generalized program defense. It is exact implementation.

Concretely, the work of this volume consists of five linked steps. First, it declares one exact model: an accessibility-tunable delayed-choice quantum eraser with fixed signal, idler, timing, and reconstruction structure. Second, it derives the standard baseline response on that model rather than appealing only to a general smooth-response class. Third, it derives the canonical CBR response on the same model using the already frozen realization-law form. Fourth, it proves an exact discrimination theorem establishing that the two responses cannot remain globally equivalent if accessibility is physically realization-relevant. Fifth, it states exact invalidation logic specifying what experimental outcome would kill the instantiated law form. The resulting object is not a general framework statement, but an exact theory-to-protocol bridge.

1.2 What this volume claims

The claims of this volume are intentionally narrow and correspondingly stronger. The first claim is that canonical CBR can be instantiated without remainder on one exact delayed-choice accessibility-tunable platform whose physical structure is sufficiently explicit to support direct model comparison. The second claim is that, on that exact platform, the induced CBR response cannot remain globally equivalent to the standard baseline if operational accessibility enters realization law nontrivially. The third claim is that the separation is not expected to be uniformly dramatic across all accessibility regimes, but is concentrated in a critical accessibility region near η_c. The fourth claim is that the resulting signature remains detectable under bounded detector, erasure, environmental, and calibration perturbations satisfying the declared tolerances. The fifth claim is that a clean null result on the exact model, under those same declared controls, invalidates the instantiated canonical law form.

These claims are sufficient for scientific legibility. They do not yet establish broad empirical dominance, but they do establish that canonical CBR, once instantiated, cannot retreat into pure formal ambiguity. The model either produces an exact non-baseline signature under the stated conditions or fails in the stated domain.

1.3 What this volume does not claim

This volume does not prove universal deviation from standard quantum theory across all measurement settings. The present results are locked to one exact architecture and one exact accessibility construction. Nor does it prove that every delayed-choice or quantum eraser implementation will display the same signal with equal clarity. Platform engineering matters, calibration matters, and the theory is not entitled to empirical visibility in arbitrarily weak or poorly aligned realizations.

The volume also does not provide a final universal Born-neutrality closure. The deeper burden of deriving full probabilistic structure without hidden circularity remains distinct from the task undertaken here. Finally, this work is not a closure proof against every conceivable rival realization-law theory. What it establishes is narrower and more useful: one exact canonical instantiation, one exact baseline comparison, one exact signature claim, and one exact invalidation criterion. That is enough to convert the theory from a sharpened program into a vulnerable model.

1.4 Why one exact model is enough

A realization-law theory becomes scientifically legible when it survives or fails on one exact discriminator. It does not first need universal extension across every architecture, every measurement class, and every interpretive rival. What it needs is one fully declared model on which the law either produces nontrivial observable content or does not. That threshold is what this volume is designed to cross. By fixing one exact platform and one exact observable burden, the present work makes canonical CBR assessable in the ordinary scientific sense: finite, public, and at risk.

2. Exact Protocol Declaration

The purpose of this section is to remove the residual looseness that attends discussion of broad protocol families. The present volume does not operate at that level. It fixes one exact experimental architecture and uses it throughout. Every later derivation of η, V_SQM(η), V_CBR(η), non-equivalence, critical-regime structure, robustness, and invalidation refers back to the platform declared here. No subsequent theorem is to be interpreted as ranging over an indefinite family of loosely similar arrangements.

2.1 Exact platform statement

The exact platform studied in this volume is an accessibility-tunable delayed-choice quantum eraser consisting of a signal subsystem, an idler-record subsystem, a controlled branch in which which-path information is either rendered operationally accessible or erased at the retrieval level, and a signal reconstruction procedure based primarily on measured interference visibility. The defining feature of the platform is that record accessibility is not treated as a vague conceptual label but as a tunable physical property of the idler-record structure, and that the retrieval-or-erasure decision is permitted to occur after signal detection while remaining part of the full context C relevant to realization. The platform is thus designed to isolate the distinction between mere signal–idler correlation and physically operative accessibility of which-path information.

2.2 Signal subsystem

Let ℋ_s denote the Hilbert space of the signal subsystem. For the exact model adopted here, ℋ_s is two-dimensional and is spanned by orthonormal path states |u⟩ and |d⟩ corresponding to the two interferometric alternatives available to the signal. The signal subsystem is prepared so that, absent record-bearing distinguishability, the two paths interfere coherently at the signal detection stage.

The primary signal observable is the reconstructed interference visibility V extracted from the signal detection statistics. In the standard operational form used throughout this volume,

V ＝ (I_max − I_min)/(I_max ＋ I_min),

where I_max and I_min denote the extremal intensities or count rates obtained in the appropriate signal reconstruction basis. This quantity is the principal observable burden carried by both the baseline and the CBR model. Secondary observables, such as conditional visibility under retrieval or delayed-reconstruction asymmetry, may be introduced only where necessary to clarify the exact source of the separation theorem, but visibility remains primary.

2.3 Idler / record subsystem

Let ℋ_i denote the Hilbert space of the idler-record subsystem. This subsystem carries the path-defining record degrees of freedom. It is not treated as a passive ancilla but as the physical carrier of which-path information whose accessibility is later tuned. Let |r_u⟩ and |r_d⟩ denote record states correlated respectively with the signal alternatives |u⟩ and |d⟩. These states need not be fully orthogonal in all regimes, since the accessibility-tuning mechanism may alter their retrievability, effective distinguishability, or stable recoverability without requiring a trivial binary switch between complete overlap and complete orthogonality.

The essential role of the idler-record subsystem is therefore twofold. First, it stores the path-correlated record structure generated by the signal–idler interaction. Second, it serves as the locus at which accessibility is tuned through controlled retrieval, erasure, stability management, dissemination, or readout burden. The idler subsystem is thus where the distinction between raw correlation and operational record accessibility becomes exact.

2.4 Joint entangled state preparation

The exact starting state for the model is a signal–idler entangled state of the form

|Ψ₀⟩ ＝ (1/√2)(|u⟩ ⊗ |r_u⟩ ＋ e^{iφ}|d⟩ ⊗ |r_d⟩),

where φ is the controllable relative phase associated with the interferometric geometry. This state is the fixed initial state used throughout the volume. It captures the basic structure required for the problem: coherent path alternatives in the signal subsystem and path-correlated record structure in the idler subsystem.

The reduced signal behavior depends on the overlap and operational role of the record states. In the baseline account, this determines the degree of interference recovery or suppression through ordinary distinguishability logic. In the canonical CBR account, the same starting state is retained, but realization is governed by the context-indexed admissible class and realization functional already canonized in the earlier work. The exact comparison later in the volume is therefore not between different initial models, but between two different law-level interpretations of one and the same declared architecture.

2.5 Retrieval branch and erasure branch

The exact protocol contains two operational branches. In the retrieval branch, the idler-record subsystem is interrogated or stabilized in such a way that which-path information becomes operationally accessible according to the declared accessibility construction. The point is not merely that path information exists somewhere in principle, but that it is made physically retrievable, recoverable, or publicly available in the sense later quantified by η.

In the erasure branch, the idler-record subsystem is manipulated in a basis or channel structure that neutralizes or recombines the record at the retrieval level so that which-path information is not operationally available in the same sense. This erasure is not interpreted as metaphysical deletion of prior correlation, but as a controlled transformation of the retrieval structure relevant to both baseline visibility reconstruction and CBR accessibility dependence.

Both branches are part of the exact declared platform. They are not optional later add-ons. The retrieval branch supplies the high-accessibility regime, the erasure branch supplies the low-accessibility or accessibility-neutralized regime, and intermediate settings generated by controlled record tuning fill the continuum between them.

2.6 Delayed-choice timing structure

The timing structure of the exact model is fixed as follows. At time t₀, the joint state is prepared. At time t₁, the signal and idler subsystems undergo the entangling interaction that generates the state |Ψ₀⟩. At time t₂, the signal subsystem is detected in the signal interference channel. At time t₃ ≥ t₂, the idler subsystem is directed into either the retrieval branch or the erasure branch, or into an intermediate accessibility-tuned operation. At time t₄, the operational accessibility of the resulting record structure is established. At time t₅, if applicable, the record becomes publicly available or physically disseminated beyond the private retrieval channel.

This temporal ordering is exact for the present platform. The delayed-choice structure enters because the retrieval-or-erasure operation may be performed after signal detection while still remaining part of the full measurement context C. That is essential for two reasons. First, it prevents any reduction of the theory to naïve premeasurement observation language. Second, it allows accessibility to be treated as a context-level property of the complete protocol rather than as a local psychological or epistemic event.

2.7 Fixed observable declaration

The primary observable analyzed in this volume is the signal visibility V reconstructed from the exact platform defined above. All principal theorems — baseline derivation, canonical CBR derivation, non-equivalence, critical-regime signature, robustness, and invalidation — are stated relative to this observable unless otherwise noted. Where a secondary observable is introduced, it will be one of only two kinds: conditional visibility under branch-resolved reconstruction, or a delayed-retrieval reconstruction signal used to clarify the origin of the primary anomaly class. No broader observable expansion will be permitted in the main body.

The model is therefore fully frozen. The signal space, the idler-record space, the initial entangled state, the retrieval and erasure branches, the delayed-choice timing structure, and the primary observable class have all been declared exactly. All subsequent analysis in Volume V is to be read as analysis of this exact model and no other.

3. Exact Accessibility Construction

The present section performs one of the decisive technical moves of the volume: it converts accessibility from a high-level physical idea into an exact experimentally calibrated variable. Earlier stages of the CBR program were justified in treating accessibility as a realization-relevant structural feature whose importance had to be isolated before it could be operationalized. That stage is now over. In the present volume, η must be exact because the theory is no longer being defended at the level of conceptual architecture. It is being instantiated on one fixed platform and asked to survive model-level comparison against a declared standard baseline. That comparison is impossible if the variable through which accessibility enters the realization law remains elastic, metaphorical, or retrospectively adjustable.

Accordingly, η is not introduced here as a summary word for “how accessible a record feels” or as a placeholder for loosely correlated physical properties. It is a platform-specific scalar constructed from a fixed set of measured ingredients and calibrated before the discriminating comparison between V_SQM(η) and V_CBR(η) is performed. The role of this section is therefore threefold. First, it specifies the primitive measurable quantities from which accessibility is built. Second, it fixes one exact reduction rule and one exact calibration procedure for the declared delayed-choice quantum eraser platform. Third, it defines the η-regimes and the critical value η_c that will later organize the signature theorem. Once these steps are completed, the theory is no longer free to slide between incompatible notions of record accessibility. The experimental burden becomes exact.

3.1 Why η must now be exact

In the exact platform declared in Section 2, accessibility is the property of the idler-record structure that distinguishes a merely correlated record from a physically operative record whose information can be retrieved, stabilized, disseminated, and made available to downstream interaction. If canonical CBR is to claim that accessibility enters realization law nontrivially, then accessibility cannot remain an interpretive gloss layered on top of the experiment after the fact. It must become a declared control variable of the experiment itself.

That requirement is especially strict here because the central claim of Volume V is not merely that accessibility matters in principle, but that an exact model-level difference emerges between the baseline response and the CBR response when accessibility is varied over a controlled domain. A variable that is not fixed before comparison cannot carry such a claim. It could always be redefined to rescue either side of the comparison, at which point the experiment would no longer test the theory. The demand that η be exact is therefore not an optional refinement. It is the condition under which the later non-equivalence and invalidation theorems become scientifically meaningful.

The guiding principle is straightforward: η must be experimentally calibrated, dimensionless, normalized to [0,1], stable under η-equivalent realizations of the same physical regime, and defined in a way that permits uncertainty propagation. Those requirements are sufficient for the present volume. They do not imply that η is uniquely representable in every possible future platform, but they do imply that within the exact platform of this volume there is one declared and fixed construction of η and no moving target.

3.2 Primitive measurable ingredients

The accessibility variable is built from five primitive measurable quantities, each corresponding to a distinct physical aspect of the idler-record structure.

The first is retrieval fidelity, denoted R ∈ [0,1]. This quantity measures the degree to which the path-defining record encoded in the idler subsystem can be recovered in a way that reconstructs the relevant which-path information without ambiguity beyond declared instrumental tolerance. In practice, R is extracted from reconstruction success under a fixed retrieval protocol. A value R ＝ 0 corresponds to a record whose retrieval is effectively useless for path determination, whereas R ＝ 1 corresponds to ideal reconstruction fidelity within the platform’s declared error bars.

The second is public or intersubjective accessibility, denoted P ∈ [0,1]. This quantity measures the extent to which the record, once retrieved or stabilized, is available beyond a single fragile interaction channel. It is intended to capture whether the record remains confined to a private microscopic branch of the apparatus or whether it becomes available to multiple physically distinct readout pathways or observers without essential degradation of its content. In the present platform, P is operational rather than sociological: it is defined by the multiplicity and stability of independent accessible readout channels.

The third is temporal stability, denoted T ∈ [0,1]. This measures the persistence of the record over the physically relevant interval between record formation and the completion of the delayed-choice retrieval or erasure logic. A record that decoheres into inaccessible noise before retrieval has low T even if it briefly existed. A record that remains stable throughout the relevant window has high T.

The fourth is destructive burden of readout, denoted D ∈ [0,1]. This quantity measures the extent to which extracting the record destroys, scrambles, or irreversibly consumes the record-bearing degrees of freedom. Unlike the first three ingredients, D is a burden quantity rather than an enabling quantity. High D indicates that access to the record is available only at the price of strong destructive intervention; low D indicates comparatively non-destructive retrieval.

The fifth is redundancy spread, denoted S ∈ [0,1]. This measures the extent to which the record is replicated or distributed across multiple degrees of freedom in the record-bearing sector. S is included because accessibility is not exhausted by single-channel retrievability. A record redundantly encoded across multiple stable subsystems is more accessible, in the physical sense relevant here, than one confined to a single fragile carrier.

These five ingredients are sufficient for the exact platform of this volume. They are not introduced as an indefinitely extensible list. Their purpose is to make η reducible to measured quantities without introducing so many ingredients that the reduction becomes underdetermined.

3.3 Exact reduction map

For the exact declared platform, η is fixed by the following reduction rule:

η ＝ (R · P · T · (1 − D) · S)^(1/5).

This is the exact accessibility map used throughout the volume. It is a normalized geometric reduction of the five primitive ingredients, chosen for three reasons. First, it preserves the interval η ∈ [0,1]. Second, it enforces a strong conjunctive structure: a record cannot count as highly accessible if one core component of accessibility collapses, since the geometric mean penalizes near-zero factors more severely than an additive average would. Third, it avoids hidden dominance by any one factor in the absence of an independently justified weighting scheme.

The use of the geometric mean is not arbitrary. Accessibility in the present platform is not intended to represent a loose accumulation of partially compensating features. Rather, it is intended to represent the effective operational availability of the record as a whole. A record with high retrieval fidelity but vanishing temporal stability is not operationally accessible in the sense the realization law requires. A record with high redundancy but almost total destructive burden is likewise deficient. The chosen reduction map encodes precisely that logic.

No alternative reduction class will be used in the main body of this volume. Any discussion of other admissible reductions belongs in an appendix and cannot modify the meaning of η in the principal derivations. This removes a major source of ambiguity. From this point forward, η means exactly the quantity defined above.

3.4 Calibration map

The reduction rule becomes scientifically usable only if its primitive ingredients are themselves tied to measurable procedures. In the present platform, retrieval fidelity R is calibrated by repeated branch-resolved reconstruction of the idler-record state against the declared path labels. Operationally, one fixes the retrieval protocol, performs a sufficient number of runs, and defines R as the correctly reconstructed path fraction after baseline correction for detector inefficiency and declared apparatus asymmetry.

Public accessibility P is calibrated by the fraction of independent readout channels that can recover the record within a declared fidelity threshold relative to the total number of physically available readout channels in the platform. In the simplest implementation, this may reduce to the presence or absence of a second stable retrieval pathway. In more refined implementations, P may be constructed from weighted channel availability. The important point is that P is measured from declared access structure, not assigned impressionistically.

Temporal stability T is calibrated by the survival of the record under controlled waiting intervals relative to the delayed-choice window. Let τ denote the relevant protocol timescale between entanglement and effective retrieval completion. Then T is determined by normalized record survival over τ, corrected for ordinary instrumental drift already included in the baseline characterization.

The destructive burden D is calibrated by comparing the integrity of the record-bearing subsystem before and after retrieval under the declared readout procedure. High record degradation under retrieval pushes D toward 1. Minimal destructive disruption pushes D toward 0. Since D enters the reduction map through 1 − D, this convention preserves the interpretation that higher accessibility corresponds to lower destructive burden.

Redundancy spread S is calibrated by the extent to which the record is recoverable from multiple independent subcarriers within the record sector. In practice, S may be estimated by controlled partial tracing or selective suppression of subchannels and measuring whether path reconstruction remains robust. The more the record survives localized loss, the higher S.

The experimentally estimated accessibility is then

η̂ ＝ (R̂ · P̂ · T̂ · (1 − D̂) · Ŝ)^(1/5),

where hats denote calibrated estimates from the declared procedures. Uncertainty in η̂ is propagated from the uncertainties in the primitive estimates by standard error propagation. To first order,

δη/η ≈ (1/5)[δR/R ＋ δP/P ＋ δT/T ＋ δ(1 − D)/(1 − D) ＋ δS/S],

whenever all quantities are nonzero and the perturbative approximation is valid. The precise uncertainty treatment may be refined in the appendices, but this first-order form is sufficient to fix the calibration logic in the main text.

3.5 Accessibility-equivalence relation

The exact reduction map makes it possible to define accessibility-equivalence without vagueness. Two realizations of the platform, denoted C₁ and C₂, are said to be η-equivalent, written C₁ ≈_η C₂, if and only if three conditions hold.

First, they yield the same calibrated accessibility value within the declared tolerance, so that |η̂(C₁) − η̂(C₂)| ≤ ε_η for a fixed calibration tolerance ε_η. Second, they preserve the same signal–idler architecture, retrieval-or-erasure logic, and observable reconstruction procedure declared in Section 2. Third, any microscopic implementation differences between C₁ and C₂ affect the experiment only through quantities already captured in the calibration of R, P, T, D, and S.

This definition is strict enough to prevent arbitrary regrouping of unlike contexts, yet permissive enough to allow distinct laboratory realizations of the same accessibility regime to be treated as theoretically equivalent. That equivalence will matter later because the baseline/CBR comparison must not depend on superficial engineering details once the accessibility regime has been fixed.

3.6 Critical accessibility value η_c

The critical accessibility value η_c is not introduced abstractly. It is tied to the exact platform as the calibrated accessibility value at which the accessibility-sensitive term in the canonical realization law becomes competitive with the residual non-accessibility burden terms in the selected channel comparison.

In the present model, let ΔΞΩ(η) denote the effective combined differential burden between the leading admissible realization classes coming from representational invariance and record-structural coherence, and let ΔΛ(η) denote the corresponding accessibility-consistency differential. Then η_c is defined implicitly by the balance condition

βΔΩ(η_c) ＋ αΔΞ(η_c) ＝ γΔΛ(η_c),

with the understanding that the left-hand side collects the accessibility-insensitive burden differential and the right-hand side represents the accessibility-sensitive contribution. Operationally, η_c is therefore the accessibility value at which the minimizer is poised to change regime, or equivalently at which the accessibility term becomes large enough to alter the identity of the minimizing equivalence class.

This definition is platform-specific, as it must be. It does not presume a universal η_c for all quantum erasers or all realization-law models. In the exact platform of this volume, η_c is fixed by the declared burden structure and the exact calibration of η.

3.7 Accessibility regimes

With η and η_c fixed, the platform naturally decomposes into five exact regimes.

The low-accessibility regime is the region 0 ≤ η ≪ η_c in which the record exists only weakly as an operationally available structure. Retrieval fidelity, stability, dissemination, or non-destructive recoverability are sufficiently weak that accessibility does not dominate the minimization problem.

The precritical regime is the region 0 < η < η_c but sufficiently near η_c that accessibility-sensitive contributions are rising while the selected realization class has not yet changed. This is the approach region in which the theory begins to accumulate tension against the baseline without yet necessarily exhibiting the primary signature.

The critical regime is the neighborhood of η_c in which the accessibility-sensitive term becomes decisive in the minimization structure. This is the regime in which the strongest separation result will later be shown to concentrate. In the strongest regularity case, it is here that the derivative break or kink appears.

The postcritical regime is the region η > η_c in which the accessibility-sensitive ordering has already taken hold and the selected realization class reflects the high-accessibility branch of the canonical law. Depending on the exact derivation, the separation from the baseline may persist or may partially reconverge in this region.

The asymptotic high-accessibility regime is the region η → 1 in which the record is maximally retrievable, stable, and redundantly available within the declared platform. This regime is important because it determines whether the theory predicts saturation, residual deviation, or asymptotic reconvergence relative to the baseline.

The value of these regime definitions is not merely classificatory. They prepare the exact shape of the later separation theorem by showing where the theory is expected to differ most strongly from the baseline and where approximate agreement may remain.

4. Exact Standard Baseline Model

The present section derives the standard baseline response on the exact declared platform. This derivation is essential. The baseline cannot remain schematic if the volume is to support a true non-equivalence theorem. Canonical CBR acquires model-level meaning only when the ordinary quantum response is explicitly stated on the same architecture, with the same signal state, the same idler-record sector, the same delayed-choice timing, and the same accessibility calibration. The point of the present section is therefore to define the exact standard comparator, not an impressionistic baseline intuition.

4.1 Baseline assumptions

The baseline model is defined by five assumptions. First, the signal–idler system evolves according to ordinary quantum dynamics with no modification of the state evolution law. Second, the signal becomes entangled with the idler-record subsystem exactly as declared in Section 2. Third, coherence loss and path distinguishability are accounted for through standard decoherence and overlap logic. Fourth, conditional erasure and retrieval are treated through ordinary basis transformation and conditional reconstruction rules. Fifth, no realization-law augmentation is added; in particular, no context-indexed selection law beyond the standard formalism is introduced.

These assumptions are chosen to give the standard framework its strongest legitimate comparator form on the exact platform. The baseline is therefore not a caricature. It is the ordinary theory stated cleanly on the very model that CBR is required to confront.

4.2 Signal–idler evolution under the baseline

The exact initial state is

|Ψ₀⟩ ＝ (1/√2)(|u⟩ ⊗ |r_u⟩ ＋ e^{iφ}|d⟩ ⊗ |r_d⟩).

Under the standard baseline, the joint density operator is ρ_si ＝ |Ψ₀⟩⟨Ψ₀|. Tracing over the idler subsystem gives the reduced signal state

ρ_s ＝ Tr_i(ρ_si)
＝ (1/2)(|u⟩⟨u| ＋ |d⟩⟨d| ＋ e^{-iφ}⟨r_d|r_u⟩|u⟩⟨d| ＋ e^{iφ}⟨r_u|r_d⟩|d⟩⟨u|).

Thus the signal coherence is governed by the overlap μ ＝ ⟨r_d|r_u⟩ of the record states in the effective retrieval description. In the declared platform, accessibility tuning is implemented through operations on the idler-record sector that change the effective retrievable overlap entering signal reconstruction. Let μ(η) denote the overlap parameter after the accessibility-tuning logic has been applied in the baseline description. Then the reduced signal interference visibility is governed by |μ(η)|.

4.3 Baseline visibility function V_SQM(η)

For the exact platform of this volume, the baseline visibility is defined by

V_SQM(η) ＝ |μ(η)|.

To make this exact rather than schematic, the model now fixes μ(η) through the accessibility construction. Since η measures effective record availability, the simplest consistent baseline assumption is that growing accessibility corresponds to increasing effective path distinguishability and therefore decreasing overlap. In the exact baseline model adopted here,

μ(η) ＝ 1 − η,

so that

V_SQM(η) ＝ 1 − η.

This choice is not arbitrary. It is the simplest exact monotone mapping consistent with the platform design: η ＝ 0 corresponds to accessibility-neutralized or effectively erased record structure and therefore maximal visibility, while η ＝ 1 corresponds to maximally accessible which-path information and therefore vanishing unconditioned visibility. More refined overlap functions could be studied in appendices, but the main body of this volume uses this exact baseline form and no other.

4.4 Baseline smooth-response class

The exact baseline response class 𝒮_baseline for the present model is the class of functions on [0,1] that are globally continuous and piecewise C¹ with no intrinsic critical-regime singularity beyond that induced by declared experimental control discontinuities. For the exact main-text model, the representative member is simply

𝒮_baseline ＝ {V(η) : V(η) ＝ 1 − η}.

Where implementation noise or controlled nonidealities are included, the admissible baseline class expands to perturbative deformations of this representative that preserve global smoothness and do not introduce a localized derivative break or nonanalytic feature near η_c except through declared apparatus discontinuity already modeled independently. In plain terms, “smooth baseline behavior” for this exact platform means monotone visibility degradation with accessibility, without a sharp critical-regime change in the intrinsic response law.

4.5 Delayed retrieval baseline behavior

The delayed-choice structure does not alter the baseline law so long as the full experiment is treated consistently as one entangled protocol. Under ordinary theory, choosing retrieval or erasure after signal detection changes the conditional reconstruction logic but does not retrocausally alter the earlier signal statistics considered in isolation. If the erasure branch is selected and the appropriate conditional sorting is applied, visibility may be recovered at the conditional level according to the standard overlap logic. If the retrieval branch is selected and which-path information remains accessible, the conditional visibility remains suppressed accordingly.

Within the exact main-text baseline, the delay itself does not create a critical accessibility phenomenon. It only determines which branch-resolved conditional reconstruction applies. The presence of delayed retrieval therefore enriches the protocol architecture without, by itself, producing a non-smooth response in V_SQM(η).

4.6 Baseline invariance under η-equivalent realizations

Let C₁ and C₂ be two η-equivalent realizations of the exact platform. Then by construction they agree in their calibrated η value, preserve the same signal–idler architecture, and differ only in implementation details already absorbed into the primitive calibration variables. Since the baseline visibility function depends on accessibility only through η, it follows that

C₁ ≈_η C₂ ⇒ V_SQM^{(C₁)}(η) ＝ V_SQM^{(C₂)}(η).

Thus η-equivalent realizations do not alter the standard baseline beyond the declared η dependence. This fact will later matter decisively. If canonical CBR produces different behavior across the same η-equivalent class, that difference cannot be attributed to mere platform ambiguity at the level of the standard comparator.

5. Exact Instantiation of Canonical CBR on the Same Model

The preceding section fixed the exact standard comparator. The present section now instantiates canonical CBR on the same declared platform, with the same signal state, the same idler-record sector, the same delayed-choice timing, and the same accessibility variable. This is the core of the volume. If the theory is to claim exact model-level empirical content, then the realization law must be written down in exact platform-specific form and carried through to an explicit response function that can be compared directly with V_SQM(η).

5.1 Imported canonical law form

The present volume uses the already frozen canonical law

Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ),

with

ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ),

where α, β, γ ≥ 0 are fixed theory-level coefficients and C denotes the exact delayed-choice accessibility-tunable platform declared earlier. Nothing in the present section modifies that law. The task is not to redesign canonical CBR, but to instantiate it exactly on the current model.

5.2 Exact admissibility class for this platform

For the exact declared platform, 𝒜(C) consists of realization-compatible channels on the signal–idler state space satisfying five conditions. They must preserve standard pre-realization dynamical evolution. They must be invariant under physically irrelevant reparameterizations of the signal–idler description. They must respect the record structure encoded in the idler sector. They must treat η-equivalent realizations equivalently. And they must not import probability weighting by stipulation.

Excluded from 𝒜(C) are channels whose selectivity depends only on path relabeling, basis-choice artifacts, branch counting unconnected to record structure, accessibility-blind selection rules, or post hoc insertion of the desired response curve. In the present platform, this yields a restricted admissibility class with two leading equivalence classes of candidate realization behavior: one accessibility-subcritical class that treats the record as not yet realization-dominant, and one accessibility-supercritical class that treats the record as realization-effective.

5.3 Exact realization burden terms

The platform-specific representational invariance burden Ξ_C penalizes channels whose realization verdict changes under trivial reformulations of the path basis or idler encoding that leave all physical observables fixed. In the exact model, Ξ_C vanishes on the admissible equivalence classes that preserve the declared path–record architecture and is positive on representation-sensitive pseudo-selection channels.

The record-structural coherence burden Ω_C penalizes channels that fail to align the selected realization structure with the actual idler-record organization. In the exact model, Ω_C distinguishes channels that track the declared signal–idler record correlation from those that either ignore it or overinterpret inaccessible remnants as full realization-bearing records.

The accessibility-consistency burden Λ_C is the decisive term for the present volume. It penalizes channels whose realization verdict fails to co-vary coherently with the calibrated accessibility η. For the exact platform, the simplest admissible form is a two-branch burden differential in which the subcritical and supercritical realization classes exchange order near η_c. This can be represented by

ΔΛ(η) ＝ η − η_c,

so that the accessibility-sensitive contribution changes sign at η_c. This does not yet determine the full response by itself, but it fixes the platform-level mechanism through which accessibility enters the minimization problem.

5.4 Accessibility-sensitive minimization structure

Because ΔΛ(η) changes sign at η_c, the minimizer of ℛ_C is accessibility-sensitive. For η < η_c, the accessibility term favors the subcritical realization class. For η > η_c, it favors the supercritical realization class. At η ＝ η_c, the accessibility-sensitive ordering becomes marginal and the realized channel sits at the boundary between the two burden orderings.

Thus the selected channel equivalence class is piecewise defined by accessibility regime:

Φ★_C(η) ＝ Φ_sub for η < η_c,
Φ★_C(η) ＝ Φ_crit at η ＝ η_c,
Φ★_C(η) ＝ Φ_sup for η > η_c,

with the understanding that Φ_crit is the limiting equivalence class at the burden balance point. This regime-sensitive minimization is the exact source of the later signature theorem. The baseline model has no such change of realization class because it has no realization-law term at all.

5.5 Exact CBR response function

For the exact main-text model, the primary observable remains visibility. The canonical CBR response is defined as

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

where κ > 0 is a fixed theory-level signature coefficient determined by the strength of the accessibility-sensitive realization correction in the postcritical regime. This is the exact response used in the main text.

The structure of this formula is deliberate. For η ≤ η_c, the CBR response matches the baseline representative in its leading behavior. For η > η_c, the realization-law correction induces an additional slope change, yielding a derivative break at η_c. The function remains continuous, but its first derivative changes from −1 below η_c to −(1 ＋ κ) above η_c. That derivative break is the primary signature morphology of the volume.

If future refinements require a broader signature map S_CBR(η), that can be introduced in appendices. In the main body, the primary observable remains V_CBR(η), because the entire purpose of the present model-locking is to avoid diffusion into multiple parallel anomaly types.

5.6 Exact comparison form

The two theories now stand in exact comparison:

V_SQM(η) ＝ 1 − η,

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c}.

They are identical below η_c and diverge above η_c by the amount

ΔV(η) ＝ V_CBR(η) − V_SQM(η) ＝ −κ max{0, η − η_c}.

Thus the baseline predicts a globally smooth monotone degradation with accessibility, whereas the instantiated canonical CBR model predicts the same low-accessibility trend but a postcritical derivative shift anchored at η_c. This is the exact face-to-face comparison on which the next section’s theorem rests.

6. Baseline/CBR Non-Equivalence Theorem

The point of the preceding derivations was not merely to write down two formulas, but to place canonical CBR and the exact baseline in a relation sharp enough to sustain theorem-level comparison. The present section states the first flagship result of the volume: once accessibility enters realization law nontrivially, global response equivalence between the baseline and the instantiated CBR model is impossible on the exact declared platform.

6.1 Theorem 4 — Baseline/CBR Non-Equivalence

Theorem 4 (Baseline/CBR Non-Equivalence). For the exact declared accessibility-tunable delayed-choice quantum eraser of Sections 2–5, suppose that canonical CBR is instantiated by the law

Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ)

with a nontrivial accessibility-sensitive burden term Λ_C and corresponding critical accessibility value η_c ∈ (0,1). Then the induced primary observable V_CBR(η) is not globally identical to the exact standard baseline response V_SQM(η) across the admissible domain η ∈ [0,1]. Equivalently, if accessibility enters canonical realization law nontrivially, then

∃η ∈ [0,1] such that V_CBR(η) ≠ V_SQM(η).

6.2 Scope and assumptions

This theorem holds under the assumptions already fixed in the volume: the platform is exactly the declared signal–idler delayed-choice quantum eraser; η is given by the exact reduction map of Section 3; the standard baseline is the exact model of Section 4; canonical CBR is instantiated exactly as in Section 5; and the accessibility-sensitive burden term is genuinely nontrivial, in the sense that it changes the minimization structure across η-regimes rather than merely appearing symbolically in ℛ_C with no model-level consequence.

The theorem does not claim that the responses differ for every η. It claims only that global identity is impossible once accessibility has genuine realization-law force.

6.3 Proof architecture

The proof is immediate from the exact model structure. If η were realization-irrelevant, then Λ_C would either be constant or would not affect the minimizing channel equivalence class, and canonical CBR could collapse to the baseline response on the exact platform. That possibility is not denied in principle; it corresponds to a degenerate or empirically idle accessibility term.

But in the present instantiated model, η is realization-relevant by hypothesis. The accessibility-sensitive burden changes the ordering of admissible realization classes at η_c, and the derived response takes the exact form

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

with κ > 0. The baseline response is

V_SQM(η) ＝ 1 − η.

Subtracting gives

V_CBR(η) − V_SQM(η) ＝ −κ max{0, η − η_c}.

For η ≤ η_c, the difference vanishes. For any η > η_c, the difference is strictly negative. Therefore the two functions cannot be globally identical on [0,1]. Hence canonical CBR and the exact baseline are not globally response-equivalent.

The proof is not profound in its algebra. Its importance lies in what it eliminates. Once the law is instantiated exactly and η enters nontrivially, the theory can no longer hide behind the claim that accessibility matters only conceptually while leaving the platform response unchanged everywhere.

6.4 Corollary on empirical legibility

Corollary 4.1. Canonical CBR now has an exact model-level operational burden on the declared platform.

This follows directly from Theorem 4. Because the exact instantiated law and the exact baseline cannot remain globally response-identical, the theory now makes a finite, protocol-specific claim about observable behavior. That claim may later survive, weaken, or fail under perturbation analysis, but it is no longer absent. Canonical CBR, in this exact form, has become empirically legible.

7. Critical-Regime Signature Theorem

The preceding sections established four facts that now make a decisive theorem possible. First, the platform has been fixed exactly rather than treated as a broad protocol family. Second, accessibility has been reduced to a calibrated experimental variable η rather than left at the level of conceptual description. Third, the standard baseline response V_SQM(η) has been derived explicitly on the exact model. Fourth, canonical CBR has been instantiated on the same model and shown to be globally non-equivalent to the baseline whenever accessibility enters the realization law nontrivially. What remains is to identify the morphology of that non-equivalence.

The present section states the central empirical claim of Volume V. It does not distribute the signal across a diffuse family of anomaly types. It chooses one primary signature and one fallback only. The primary claim is that the exact instantiated CBR response develops a threshold-sensitive derivative break, or kink, near the critical accessibility value η_c. The fallback claim is weaker but still empirical: if the stronger regularity conditions fail, the model retains a bounded non-baseline deviation class concentrated in the same critical regime. This concentration is essential. It explains why the separation does not need to appear generically across the entire η-domain in order to be scientifically significant.

7.1 Primary predicted signature

The primary predicted signature of the instantiated CBR model is a threshold-sensitive derivative break at η_c. In the exact model already derived,

V_SQM(η) ＝ 1 − η,

whereas

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

with κ > 0 fixed and η_c ∈ (0,1) determined by the accessibility-sensitive minimization balance. The primary signature is therefore not merely that the CBR curve and the baseline curve differ somewhere. It is that the CBR curve changes local response class at η_c. Specifically, V_CBR(η) is continuous on [0,1] but exhibits a change in first derivative at η_c, whereas the declared baseline remains globally smooth in the same region.

This is the correct primary claim for the volume because it is both physically interpretable and mathematically exact. It localizes the non-equivalence in the regime where accessibility first becomes realization-dominant, rather than forcing the theory to predict anomalous behavior where its own structure does not require it. A kink is therefore not a cosmetic signal choice. It is the precise morphology expected when the minimizer of the realization functional changes regime at a finite critical accessibility threshold.

7.2 Secondary fallback signature

The volume permits only one fallback signature. If stronger regularity assumptions fail, or if small implementation refinements smooth the exact derivative discontinuity without restoring baseline equivalence, then the residual empirical signature is a bounded non-baseline deviation band concentrated near η_c. In that case the theory no longer claims a literal kink, but it still claims that the response cannot be absorbed into the declared smooth baseline class within a controlled neighborhood of the critical regime.

This fallback is weaker than the primary signature, but it is not empty. It preserves empirical distinctness while acknowledging that local morphology can be regularized by details that do not eliminate the underlying accessibility-sensitive regime change. The purpose of this fallback is not to protect the theory from risk by multiplying interpretive options. It is to distinguish the theorem’s strong form from its robust residual content.

7.3 Theorem 5 — Critical-Regime Signature Theorem

Theorem 5 (Critical-Regime Signature Theorem). For the exact declared accessibility-tunable delayed-choice quantum eraser of Sections 2–6, suppose canonical CBR is instantiated by

Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ),

with a nontrivial accessibility-sensitive burden term Λ_C, a critical accessibility value η_c ∈ (0,1), and the exact platform response

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

for some fixed κ > 0. Let 𝒮_baseline denote the declared baseline smooth-response class associated with

V_SQM(η) ＝ 1 − η.

Then, under the stated regularity assumptions, V_CBR develops a critical-regime structure at η_c not contained in 𝒮_baseline. In particular, the response exhibits a derivative break at η_c and therefore fails to belong to the baseline smooth-response class in any neighborhood of η_c. If the strong regularity assumptions required for literal derivative discontinuity are relaxed but the accessibility-sensitive minimization transition remains nontrivial, then there exists a nonempty neighborhood U of η_c in which V_CBR lies within a bounded non-baseline deviation class not reducible to any member of 𝒮_baseline.

The theorem is local by design. It does not claim that every η-regime must display the anomaly equally. It claims instead that the exact instantiated theory concentrates its empirical distinctness where its own internal structure says it should: at the accessibility threshold where realization order changes.

7.4 Strong form

The strong form of the theorem follows immediately from the exact response law. For η < η_c,

dV_CBR/dη ＝ −1,

whereas for η > η_c,

dV_CBR/dη ＝ −(1 ＋ κ).

Thus V_CBR is continuous at η_c, but its derivative is not. The response therefore exhibits a kink at η_c. Equivalently, the function is not globally C¹ on [0,1], even though the baseline representative V_SQM(η) ＝ 1 − η is globally smooth. In this exact main-text model, the strongest justified signature is therefore a slope discontinuity localized at the critical accessibility value.

This local nonanalyticity is not imposed arbitrarily. It is the direct mathematical image of a change in the minimizing realization class. Below η_c the accessibility-sensitive burden is not yet decisive; above η_c it is. When that shift is encoded through the exact piecewise minimization structure, the resulting response law acquires a derivative break. Thus the strong-form signature is theorem-driven, not decorative.

7.5 Weak form

If one relaxes the idealizing assumption that the minimization transition is represented by the exact max{0, η − η_c} term, or if one allows a microscopically smoothed transition in the platform-level instantiation while preserving accessibility-sensitive regime change, then the strongest literal kink may be regularized. In that case the weak form applies: there exists a neighborhood U of η_c and a positive constant ε_sig such that

|V_CBR(η) − V_SQM(η)| ≥ ε_sig

for some nonempty subset of U, and no member of the declared baseline class 𝒮_baseline reproduces the full local response within the declared tolerance without unauthorized parameter augmentation. The weak signature is therefore a bounded non-baseline deviation band near the critical regime.

This weaker result remains sufficient for empirical distinctness. It does not require a literal singularity. It requires only that the exact model continue to depart from the standard baseline in a way localized near η_c and not removable by ordinary baseline smoothing.

7.6 Physical interpretation

The signal concentrates near η_c because η_c is the point at which accessibility first becomes large enough to alter the realization ordering. Below η_c, the accessibility-sensitive burden term is present but subdominant. The selected channel equivalence class therefore tracks the same leading visibility trend as the baseline. Above η_c, accessibility becomes realization-effective and the selected channel changes regime. The strongest manifestation of that change is naturally localized at the transition itself.

This is physically appropriate. A realization-law theory need not predict dramatic deviations where the realization-relevant control parameter is either clearly irrelevant or already saturated. It should instead differ where a physical threshold is crossed. In the present platform, η_c marks exactly that threshold. The theory’s empirical signature is therefore strongest not because the platform has been engineered to manufacture drama, but because the law itself predicts that realization sensitivity becomes decisive there.

7.7 Why the signature is not a baseline artifact

The signature is not a baseline artifact for two reasons tied directly to the exact model. First, the baseline response has already been fixed on the same platform as

V_SQM(η) ＝ 1 − η,

with η defined by the exact reduction map of Section 3. The baseline therefore already incorporates the declared accessibility dependence of the platform through ordinary distinguishability logic. The anomaly is not being generated by comparing a refined CBR model to a caricatured standard model. It is generated by comparing two exact responses on one and the same declared architecture.

Second, the derivative break arises from the accessibility-sensitive minimization structure of the realization law, not from any undeclared change in the signal–idler dynamics. The signal state, idler architecture, timing, and observable remain fixed. What changes is the selected realization channel equivalence class once η crosses η_c. Since the baseline includes no such realization-law selection step, it has no mechanism for producing the same critical-regime morphology within the declared class 𝒮_baseline. The signature is therefore genuinely tied to the exact CBR instantiation rather than to a hidden baseline misspecification in the main model.

8. Noise, Perturbation, and Robust Detectability

The existence of a model-level signature is not yet enough. If the signal is destroyed or mimicked by ordinary perturbations, the theory may remain formally interesting but experimentally idle. The present section therefore introduces a full perturbation model, identifies the dominant error channels, propagates accessibility-calibration uncertainty, and states the detectability theorem that converts the signature from a mathematical curiosity into a technically serious empirical claim.

8.1 Full perturbation model

Let V_model(η) denote either V_SQM(η) or V_CBR(η), depending on which theoretical hypothesis is being tested. The experimentally observed response is written as

V_obs(η) ＝ V_model(η) ＋ δ_det(η) ＋ δ_erase(η) ＋ δ_env(η) ＋ δ_cal(η),

where the correction terms capture the dominant perturbations associated with the exact platform. This decomposition is not intended as an exact microscopic identity. Its purpose is to separate the principal perturbative channels that matter for detectability and invalidation logic.

The term δ_det(η) captures detector-originated perturbations. The term δ_erase(η) captures imperfections in the retrieval-or-erasure branch logic. The term δ_env(η) captures environmental and undeclared coupling effects. The term δ_cal(η) captures error arising from uncertainty in the accessibility calibration itself. The later theorem will impose explicit bounds on these terms sufficient to preserve the signature.

8.2 Detector perturbations

Detector perturbations include finite efficiency, detection-channel asymmetry, timing jitter, and dark counts. Finite efficiency reduces count totals and may distort visibility reconstruction if the loss is path-dependent. Channel asymmetry can shift I_max and I_min unequally, thereby biasing V_obs even in the absence of any true realization-law effect. Timing jitter can smear the phase reconstruction and soften a sharp local feature near η_c. Dark counts contribute additive background noise and can be especially damaging in the postcritical regime if the signal itself is already reduced.

For the present volume, detector perturbations are admissible provided they are bounded by a uniform envelope ε_det over the experimentally relevant η-domain and provided their η-dependence is independently characterized rather than absorbed into the theory fit. This independence requirement matters. Detector drift correlated with η tuning is one of the principal ways a false signature could be manufactured.

8.3 Erasure perturbations

Erasure perturbations arise from imperfect basis control in the idler subsystem, residual distinguishability after nominal erasure, and retrieval impurity in the nominal access branch. These perturbations are especially important because the exact platform’s meaning depends directly on whether the record is made operationally accessible or neutralized at retrieval. A small residual which-path trace in the erasure branch can reduce recovered visibility. Conversely, imperfect retrieval logic can blur the accessibility dependence that the theory is supposed to probe.

Let ε_erase be an upper bound on the effective visibility distortion induced by erasure and retrieval imperfection after independent calibration. Then any empirical assessment of the CBR signal must compare the predicted separation not merely against raw detector noise, but against the total branch-control uncertainty induced by ε_erase as well.

8.4 Environmental perturbations

Environmental perturbations include undeclared leakage of idler information, thermal drift, stray entanglement with uncontrolled degrees of freedom, and generic decoherence channels not included in the baseline model. These perturbations matter because they can either mimic additional visibility loss or wash out a localized derivative feature. In the present exact platform, the main requirement is that environmental contributions be independently modeled to the extent necessary to define a declared baseline class 𝒮_baseline^pert that incorporates ordinary smooth perturbative deformation without importing a critical-regime singularity by hand.

Let ε_env denote the effective upper bound on the environmental perturbation amplitude relevant to visibility reconstruction in the η-window of interest. As with detector and erasure perturbations, ε_env must be bounded independently of the theoretical comparison.

8.5 Accessibility calibration uncertainty

Because η is itself constructed from estimated quantities R, P, T, D, and S, uncertainty in those primitive measurements propagates into both the horizontal placement of the signal and the apparent vertical response if V is fitted as a function of η. If δη denotes the effective accessibility uncertainty, then to first order a model response V_model(η) acquires an induced calibration perturbation

δ_cal(η) ≈ (dV_model/dη) δη.

For the exact main-text model, this yields

|δ_cal(η)| ≤ max{|dV_SQM/dη|, |dV_CBR/dη|} δη
≤ (1 ＋ κ) δη,

since the largest slope magnitude occurs in the postcritical CBR regime. This bound is particularly important near η_c, where horizontal uncertainty can blur the apparent location of the derivative break. The calibrational burden is therefore not subordinate to ordinary detector noise. It is one of the principal technical conditions governing whether the critical-regime signature remains resolvable.

8.6 Theorem 6 — Robust Detectability Theorem

Theorem 6 (Robust Detectability Theorem). Let V_CBR(η) and V_SQM(η) be the exact responses of Sections 5 and 4 on the declared platform, and let

V_obs(η) ＝ V_model(η) ＋ δ_det(η) ＋ δ_erase(η) ＋ δ_env(η) ＋ δ_cal(η)

be the observed response under bounded perturbations. Suppose there exists a critical-regime neighborhood U around η_c and positive bounds ε_det, ε_erase, ε_env, ε_cal such that

for all η ∈ U, and suppose further that the exact CBR separation in U satisfies

sup_{η ∈ U} |V_CBR(η) − V_SQM(η)| > 2(ε_det ＋ ε_erase ＋ ε_env ＋ ε_cal).

Then the primary critical-regime signature, or its fallback bounded deviation class if the strong form is smoothed, remains statistically resolvable against the declared perturbed baseline.

Proof sketch. The observed difference between the two model classes in U is the exact separation minus the total perturbative envelope. If the exact separation exceeds twice the total envelope, then after perturbation the two admissible response bands remain disjoint on at least one nonempty subset of U. Therefore no member of the perturbed baseline class can absorb the entire local CBR signature without exceeding the declared tolerance. If the derivative break survives directly, the strong form is resolvable. If it is smoothed by perturbation while the amplitude separation remains above threshold, the weak-form bounded deviation class remains resolvable.

This theorem is intentionally comparative rather than absolute. It does not claim that every visible anomaly favors CBR. It claims only that under explicit tolerance conditions the exact CBR signal survives ordinary perturbation strongly enough to remain experimentally discriminable from the declared baseline.

8.7 Detectability region

The detectability region of the exact platform is the subset of parameter space for which the hypotheses of Theorem 6 hold. In practice, this means the region in which three conditions are simultaneously satisfied. First, η must be controllable with uncertainty small enough that the critical regime is not smeared beyond recognition. Second, the combined detector, erasure, and environmental perturbations must remain below the separation scale induced by κ in the vicinity of η_c. Third, the sampled η-domain must extend sufficiently across the critical regime that the slope change or bounded deviation can be resolved against the baseline trend.

In the exact main-text model, detectability is therefore concentrated in a strip around η_c wide enough to contain postcritical separation but narrow enough that the baseline remains locally simple. This is the natural target region for the later decision logic. The theory does not ask to be tested everywhere with equal power. It asks to be tested where it predicts the transition to matter.

9. Decision Logic: Support, Null Result, and Binary Invalidation

The preceding sections have fixed the exact platform, the exact accessibility variable, the exact baseline, the exact CBR response, the critical-regime signature, and the perturbative conditions under which that signature remains detectable. The present section turns those results into decision logic. This is the point at which the theory either becomes properly vulnerable or remains only rhetorically exposed. The goal here is therefore not diplomatic nuance. It is exact public risk.

9.1 Positive-result class

An observed dataset counts as evidence in favor of the instantiated canonical CBR model only if it satisfies all of the following conditions. First, the data are collected on the exact declared platform with the exact frozen accessibility construction η ＝ (R · P · T · (1 − D) · S)^(1/5), and the primitive calibration procedures have been completed independently of the final model comparison. Second, the observed visibility response exhibits either the primary critical-regime signature — a derivative break or kink near η_c — or the fallback bounded deviation class in a neighborhood of η_c. Third, the observed critical-regime structure cannot be absorbed into the declared perturbed baseline class without exceeding the calibrated tolerance bounds. Fourth, the effect survives the robustness controls implied by Theorem 6.

In practical terms, positive evidence requires more than “an anomaly exists.” It requires a non-baseline anomaly localized where the exact instantiated theory says it should be, of sufficient magnitude to outrun the declared perturbative envelope, and stable under the exact calibration logic of the platform.

9.2 Null-result class

A null result is not merely weak evidence. It is a precisely defined outcome class. The data count as a clean null result if, across the experimentally accessible η-domain and in particular across the critical neighborhood around η_c, the observed response remains within the declared perturbed baseline class 𝒮_baseline^pert and exhibits no statistically significant derivative break, kink, or bounded non-baseline deviation beyond the tolerance allowed by detector, erasure, environmental, and calibration uncertainty.

This standard is deliberately strict. It prevents the theory from being invalidated by poor calibration, insufficient η coverage, or unresolved perturbative structure. But once those ordinary conditions are satisfied, a clean null result means that the exact instantiated CBR model has failed in the very domain where it claimed distinct empirical content.

9.3 Theorem 7 — Binary Invalidation Theorem

Theorem 7 (Binary Invalidation Theorem). Let the exact declared platform, observable, accessibility construction, baseline model, and instantiated canonical CBR model be fixed as in Sections 2–8. If the experimentally observed response exhibits only baseline-class behavior across the experimentally accessible η-domain, including the declared critical neighborhood around η_c, under the stated calibration procedures and bounded perturbation controls, then canonical CBR in this instantiated form is false.

Proof sketch. By Theorem 4, once accessibility enters the realization law nontrivially, global response equivalence between the baseline and the instantiated CBR model is impossible. By Theorem 5, that non-equivalence must concentrate into either the primary critical-regime derivative break or the fallback bounded deviation class. By Theorem 6, if the perturbation bounds are satisfied and the critical regime is experimentally accessed, that signal remains resolvable. Therefore, if a properly controlled experiment on the exact declared platform nevertheless yields only baseline-class behavior across the accessible η-domain, the instantiated CBR model has not merely lacked support; it has failed to produce the exact model-level consequence that its own law structure entails. Hence it is false.

The force of this theorem is binary because the theory has already frozen the objects that matter. There is no further hidden freedom inside the main-text model with which to absorb a clean null result.

9.4 Why this is real invalidation

The invalidation result is genuine because the law is frozen, the platform is frozen, the observable is frozen, the η construction is frozen, and the failure class is finite and public. The law is fixed by

Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ),

with

ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ).

The platform is fixed by the exact delayed-choice accessibility-tunable quantum eraser declared in Section 2. The observable is fixed as visibility V, with only tightly constrained secondary use of conditional reconstruction where explicitly stated. The accessibility variable is fixed by one exact reduction rule and one exact calibration procedure. The failure class is fixed as baseline-class behavior across the experimentally accessed η-domain under the declared controls.

This closes the ordinary escape routes. A negative result cannot be reinterpreted away by changing the protocol family, redefining the observable, loosening the meaning of accessibility, or invoking an undeclared hidden signature. If the exact model is tested and the exact signal does not appear where it should, the exact model dies.

9.5 What survives invalidation

A clean invalidation would not show that every possible realization-law philosophy is false. Nor would it show that no accessibility-sensitive completion of quantum outcome selection can ever exist. What it would show is narrower and scientifically more valuable: the exact instantiated canonical form studied in this volume is false.

Broader realization-law reasoning may survive. Other noncanonical variants may survive. Other exact instantiations with different admissibility structure, different accessibility reduction, or different platform choice may remain logically possible. But this exact main-text model would not survive. That honesty is part of the strength of the volume. It converts the work from high-level seriousness into ordinary scientific seriousness.

10. Limits

The present volume applies to one exact declared platform and nothing more. Its theorems do not automatically extend to every delayed-choice quantum eraser, every interferometer, or every realization-sensitive measurement architecture. The exact model has been chosen precisely because one clean test is enough to make the theory vulnerable; broader generalization remains additional work, not an entitlement.

The accessibility variable η is fixed here by one exact reduction rule tied to this platform. That construction is sufficient for the present model, but it is not claimed to be the unique or universal reduction for all future architectures. A different platform may require a different exact accessibility map, and the present results should not be misread as proving otherwise.

The strongest signature morphology of this volume — a derivative break or kink at η_c — depends on the stated regularity assumptions of the exact main-text instantiation. If those assumptions are weakened, the strong form may soften into the fallback bounded deviation class. This does not erase empirical content, but it does narrow the force of the strongest version of the signature theorem.

Finally, the volume does not deliver full universal non-circular closure of the Born-structure problem. The exact empirical model constructed here is compatible with the canonical realization-law program and sharpens its vulnerability, but it does not by itself solve every deeper probabilistic burden associated with realization-law completion.

These limits do not trivialize the result. They identify exactly what the volume has achieved. One exact experimentally vulnerable realization-law model is enough to make the theory scientifically real. That is the threshold crossed here.

11. Conclusion

This volume fixed the CBR program to one exact platform: an accessibility-tunable delayed-choice quantum eraser with a two-path signal subsystem, a path-correlated idler-record subsystem, a controlled retrieval-or-erasure branch, an exact delayed-choice timing structure, and a primary visibility observable defined at the level of reconstructed signal interference. The theory was therefore not tested against a vague family resemblance among interferometric protocols, but against one declared architecture whose state preparation, accessibility control, and observational burden were frozen before theorem-level comparison began.

On that exact platform, the standard comparator was derived explicitly rather than invoked schematically. Starting from the declared entangled signal–idler state and using ordinary quantum evolution, ordinary entanglement accounting, standard decoherence logic, and standard conditional retrieval-or-erasure analysis with no realization-law augmentation, the baseline response was fixed as an exact visibility law V_SQM(η) together with its corresponding smooth-response class. In the main-text instantiation, the baseline visibility response was given by V_SQM(η) ＝ 1 − η, and the standard theory therefore predicted monotone accessibility-dependent visibility degradation with no intrinsic critical-regime singularity.

The same platform was then used to instantiate canonical CBR in exact form. The volume imported the already frozen law Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ), with ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ), and specified the admissibility class, the exact burden terms, the accessibility reduction η ＝ (R · P · T · (1 − D) · S)^(1/5), and the critical accessibility value η_c within the declared model. This yielded an exact CBR response law on the same footing as the baseline, namely V_CBR(η) ＝ 1 − η − κ max{0, η − η_c}, thereby placing the two theories into direct model-level comparison rather than leaving their relation at the level of conceptual contrast.

The central formal consequences then followed in the correct logical order. Theorem 4 established that once accessibility enters the canonical realization law nontrivially, V_CBR(η) cannot remain globally identical to V_SQM(η) across the admissible η-domain. Theorem 5 then identified the morphology of that non-equivalence: under the stated regularity assumptions, the separation concentrates into a critical-regime derivative break or kink at η_c, while a bounded non-baseline deviation class remains as the fallback result if the strongest local regularity assumptions are relaxed. Theorem 6 showed that this critical-regime signature, in strong or weak form, survives bounded detector, erasure, environmental, and calibration perturbations whenever the declared tolerance conditions are satisfied. The exact theory therefore did not merely acquire formal distinctness. It acquired an experimentally legible and perturbatively resilient signature structure.

Finally, the volume stated the binary logic under which the instantiated theory stands or fails. Because the law form, platform, observable, accessibility construction, perturbation structure, and failure class were all fixed in advance, the null-result consequence is exact rather than rhetorical: if the declared platform exhibits only baseline-class behavior across the experimentally accessible η-domain under the stated controls, then canonical CBR in this instantiated form is false. With this volume, CBR is no longer only a canonized realization-law proposal with a designated test domain, but an exact platform-level empirical theory whose central signature can be calculated, challenged, and in principle ruled out by a finite experimental program.

Appendix A-I

Appendix A. Exact Model Declaration

This appendix fixes the exact platform used throughout Volume V. Its function is to remove any residual ambiguity about the physical system, state space, timing structure, admissible measurement branches, and observable definitions. Every theorem in the main text is to be interpreted relative to the model declared here. No subsequent section is permitted to alter the signal space, the idler-record space, the branch logic, the accessibility control structure, or the primary observable without explicit notice. The purpose of this appendix is therefore not merely expository. It is constitutive of the theory’s exact test form.

A.1. Total Hilbert-space structure

Let the total Hilbert space be

ℋ ＝ ℋ_s ⊗ ℋ_i,

where ℋ_s is the signal Hilbert space and ℋ_i is the idler-record Hilbert space.

The signal subsystem is two-dimensional:

ℋ_s ＝ span{|u⟩, |d⟩},

where |u⟩ and |d⟩ denote the two interferometric path states. These basis states are taken to be orthonormal:

⟨u|u⟩ ＝ 1, ⟨d|d⟩ ＝ 1, ⟨u|d⟩ ＝ 0.

The idler-record subsystem is taken to contain at minimum a two-dimensional record-bearing subspace

ℋ_i^rec ⊂ ℋ_i,

spanned by states |r_u⟩ and |r_d⟩ that encode path-correlated record structure. These record states are not assumed a priori to be perfectly orthogonal in the effective retrieval description. Their operational role depends on the accessibility-tuning procedure specified below.

In addition to the record-bearing subspace, ℋ_i may contain auxiliary sectors representing retrieval channels, erasure channels, calibration structure, and environmental carriers relevant to record stability and dissemination. However, the main-text derivations depend only on the effective record-sector behavior, so these auxiliary sectors are treated through induced effective maps rather than through exhaustive microscopic decomposition unless otherwise required.

The state space of physical preparations is the density-operator space 𝒟(ℋ). All channels and realization maps considered in Volume V act on 𝒟(ℋ) or on relevant reduced spaces induced from it.

A.2. Initial state preparation

The exact initial state used throughout the volume is

|Ψ₀⟩ ＝ (1/√2)(|u⟩ ⊗ |r_u⟩ ＋ e^{iφ}|d⟩ ⊗ |r_d⟩),

where φ ∈ [0, 2π) is the controllable interferometric phase.

The corresponding density operator is

ρ₀ ＝ |Ψ₀⟩⟨Ψ₀|.

This state encodes the minimal structure required by the model:

coherent signal-path alternatives,
path-correlated idler-record structure,
a phase parameter whose variation supports visibility reconstruction.

No other initial state is used in the main text. Any variant state belongs, if at all, to the appendix on variant implementations and does not modify the principal theorems.

A.3. Signal observable and interference basis

The primary observable of Volume V is the signal visibility V reconstructed from detection statistics in the interference basis. Let the interference basis on ℋ_s be

|+⟩ ＝ (1/√2)(|u⟩ ＋ |d⟩),
|−⟩ ＝ (1/√2)(|u⟩ − |d⟩).

More generally, when the phase φ is explicitly scanned, one may use the phase-shifted basis

|χ(θ)⟩ ＝ (1/√2)(|u⟩ ＋ e^{iθ}|d⟩),

with θ the externally controlled reconstruction phase. The associated signal intensity is

I(θ) ＝ Tr[(|χ(θ)⟩⟨χ(θ)| ⊗ 𝟙_i) ρ],

where ρ is the relevant signal–idler state after the declared branch logic has been applied.

The visibility is then defined by

V ＝ (I_max − I_min)/(I_max ＋ I_min),

where I_max and I_min are the maximal and minimal values of I(θ) over the declared reconstruction scan.

This is the primary observable used in the main text. A secondary observable may be introduced only when explicitly stated, and only as one of the following:

branch-conditioned visibility,
delayed-retrieval reconstruction asymmetry,
an equivalent signature map S_CBR(η) whose primary component is still V.

The main theory is therefore visibility-led, not observable-diffuse.

A.4. Effective record overlap and reduced signal state

Given a joint state ρ on ℋ_s ⊗ ℋ_i, the reduced signal state is

ρ_s ＝ Tr_i(ρ).

For the initial state ρ₀, one obtains

ρ_s ＝ (1/2)(|u⟩⟨u| ＋ |d⟩⟨d| ＋ e^{-iφ}⟨r_d|r_u⟩|u⟩⟨d| ＋ e^{iφ}⟨r_u|r_d⟩|d⟩⟨u|).

Define the effective record overlap

μ ＝ ⟨r_d|r_u⟩.

In the standard baseline treatment, the signal coherence is governed by |μ|. In the exact model of Volume V, μ becomes an effective accessibility-dependent quantity after branch operations and accessibility calibration are imposed.

The appendices on baseline derivation and CBR instantiation will make this dependence exact. For present purposes, the important point is that the declared platform contains a well-defined route from idler-record structure to signal visibility through the reduced signal state.

A.5. Branch structure: retrieval and erasure

The platform contains two principal operational branches.

A.5.1. Retrieval branch

In the retrieval branch, the idler-record subsystem is subjected to a retrieval operation 𝒦_ret that makes which-path information operationally accessible in the sense later quantified by η. The role of 𝒦_ret is not merely to reveal an abstract preexisting label, but to transform the record-bearing sector into a physically retrievable and calibratable record state.

The retrieval branch may include:

readout-channel selection,
stabilization of the record,
dissemination to multiple accessible carriers,
calibration measurements of R, P, T, D, and S.

The resulting branch state is denoted

ρ_ret ＝ 𝒦_ret(ρ₀).

A.5.2. Erasure branch

In the erasure branch, the idler-record subsystem is subjected to an erasure operation 𝒦_era that neutralizes path distinguishability at the retrieval level. This does not mean that all prior correlations are metaphysically erased. It means that the record is transformed into a state that does not function as an operationally accessible which-path record in the sense relevant to the exact platform.

The resulting branch state is denoted

ρ_era ＝ 𝒦_era(ρ₀).

The distinction between 𝒦_ret and 𝒦_era is operational, fixed, and exhaustive for the main-text model. Intermediate accessibility settings are obtained by continuously varying the retrieval structure rather than by introducing new branch types.

A.6. Timing structure

The exact timing structure of the platform is fixed as follows.

At time t₀, the system is prepared in the declared initial state ρ₀.

At time t₁, the signal–idler entangling interaction is complete, and the path-correlated record structure is established.

At time t₂, the signal subsystem is detected in the interference reconstruction channel.

At time t₃, with t₃ ≥ t₂, the idler subsystem is directed into either the retrieval branch or the erasure branch, or into an intermediate branch corresponding to a tuned accessibility setting.

At time t₄, the operational accessibility of the resulting record structure is established through the declared retrieval, stabilization, and calibration procedure.

At time t₅, if relevant, the record becomes publicly available through dissemination to one or more independent readout pathways.

This timing structure is exact for the main-text model. The delayed-choice character of the experiment lies in the fact that the retrieval-or-erasure structure can be fixed after signal detection while still constituting part of the full physical context C relevant to realization.

No theorem in Volume V depends on human observation time. All dependence is on the physically declared context, of which the above timing relations are part.

A.7. Accessibility calibration primitives

The exact accessibility parameter η is built from five primitive quantities:

R: retrieval fidelity,
P: public or intersubjective accessibility,
T: temporal stability,
D: destructive burden of readout,
S: redundancy spread.

Their exact reduction into η is given in Appendix B and used throughout the main text. Here it is enough to state that each of these quantities is a measurable property of the idler-record branch structure and that they jointly determine the operational accessibility regime of the platform.

The exact main-text reduction is

η ＝ (R · P · T · (1 − D) · S)^(1/5).

This definition is fixed and not open to replacement in the body of the volume.

A.8. Exact baseline model objects

The standard comparator used throughout the volume is defined on this same platform with no realization-law augmentation. The baseline response is denoted

V_SQM(η).

In the exact main-text instantiation, the representative baseline response is

V_SQM(η) ＝ 1 − η.

The associated baseline smooth-response class is denoted 𝒮_baseline and is defined more fully in Appendix C. In the main-text model, it is the class generated by the exact linear accessibility response together with admissible smooth perturbative deformations.

A.9. Exact canonical CBR objects

The canonical realization law imported from the Core Theorem Paper is

Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ),

with

ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ),

where:

𝒜(C) is the admissible channel class for the declared platform,
Ξ_C is the representational invariance burden,
Ω_C is the record-structural coherence burden,
Λ_C is the accessibility-consistency burden,
α, β, γ ≥ 0 are fixed theory-level coefficients.

In the exact main-text instantiation, η enters Λ_C nontrivially and induces a critical accessibility value η_c ∈ (0,1) at which the minimizing realization class changes regime.

The resulting exact CBR response is

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

with κ > 0 fixed.

This response is the central model-level object of Volume V and is the basis of Theorems 4 through 7.

A.10. Operational equivalence

Two realizations C₁ and C₂ of the platform are operationally equivalent if and only if they satisfy all of the following:

they share the same signal Hilbert-space structure and primary observable definition,
they preserve the same timing logic and branch architecture,
they yield the same calibrated accessibility value η within the declared tolerance,
they differ only in implementation details that do not alter any realization-relevant observable beyond the declared perturbative envelope.

Operational equivalence matters because the theory’s claims are not about notation, engineering cosmetics, or basis relabeling. They are about exact model-level behavior up to physical equivalence.

A.11. Scope of the model declaration

This appendix completes the exact declaration of the platform used throughout Volume V. It fixes:

the Hilbert-space structure,
the initial state,
the branch logic,
the timing structure,
the accessibility ingredients,
the primary observable,
the baseline object,
the canonical CBR object,
the meaning of operational equivalence.

All later appendices inherit this declaration. Any derivation not consistent with the objects fixed here does not belong to the exact main-text model.

Appendix B. Accessibility Reduction and Calibration Mathematics

This appendix defines the exact mathematical construction of the accessibility parameter η used throughout Volume V and specifies the calibration logic by which η is estimated from experimentally measurable quantities. Its function is to eliminate ambiguity at the precise point where the model passes from formal realization-law structure to platform-level empirical comparison. In the main text, η was treated as the exact scalar control variable through which record accessibility enters both the baseline response V_SQM(η) and the instantiated canonical realization response V_CBR(η). The present appendix justifies that construction in full and fixes the associated uncertainty propagation, equivalence structure, and calibration tolerances. No alternative accessibility definition is used in the main body.

B.1. Purpose and scope of the accessibility construction

The accessibility parameter η is intended to measure, for the exact declared platform, the effective operational availability of the which-path record carried by the idler-record subsystem. It is not a purely semantic index and it is not a free phenomenological fit parameter. It is a reduced scalar summary of a structured physical condition: whether the record can be retrieved with fidelity, whether it persists over the relevant protocol timescale, whether it can be accessed across more than one effective readout channel, whether obtaining it is strongly destructive, and whether it is redundantly spread across record-bearing degrees of freedom.

The role of η in the main text is twofold. In the exact standard baseline model, η controls the effective visibility degradation through the relation V_SQM(η) ＝ 1 − η. In the exact instantiated CBR model, η enters the accessibility-sensitive burden term Λ_C and determines the critical regime near η_c at which the minimizing realization class changes. The same scalar must therefore be mathematically precise enough to support both ordinary visibility modeling and realization-sensitive threshold analysis. That is why the reduction and calibration are fixed here rather than left implicit.

The present appendix is restricted to the exact delayed-choice accessibility-tunable quantum eraser declared in Appendix A. It does not claim that the same reduction is uniquely correct for every conceivable future platform. It claims only that for this exact model there is one fixed accessibility construction sufficient to define, calibrate, and test the theory’s main empirical burden.

B.2. Primitive accessibility variables

The exact accessibility reduction uses five primitive variables:

R ∈ [0,1],
P ∈ [0,1],
T ∈ [0,1],
D ∈ [0,1],
S ∈ [0,1].

These variables are defined as follows.

B.2.1. Retrieval fidelity R

Retrieval fidelity R measures the extent to which the record-bearing idler subsystem can be interrogated so as to reconstruct the correct path label associated with the signal subsystem. It is defined relative to the declared retrieval protocol of the platform and therefore depends on the actual operational reconstruction channel rather than on abstract distinguishability in Hilbert space alone.

Let N_tot be the total number of retrieval attempts under the declared branch conditions, and let N_corr be the number of retrieval outcomes that correctly reconstruct the path label when benchmarked against the branch-resolved preparation logic. Then the raw retrieval estimate is

R_raw ＝ N_corr / N_tot.

After correcting for independently calibrated detector asymmetry, dark counts, and known retrieval-channel bias, one obtains the effective calibrated retrieval fidelity R̂. In the ideal limit, R̂ ＝ 1 corresponds to perfect recovery of which-path information, while R̂ ＝ 0 corresponds to total retrieval failure.

B.2.2. Public or intersubjective accessibility P

Public accessibility P measures whether the record is confined to a single fragile access route or is physically available through multiple independent readout channels. The language “public” is used in an operational, not sociological, sense. It refers to the ability of multiple physically distinct retrieval pathways to access the same record content without essential destruction or reconstruction from mutually exclusive measurement acts.

Let M_avail denote the number of declared readout pathways physically available in the platform, and let M_eff denote the number of those pathways capable of recovering the record above a declared fidelity threshold R_min. Then the simplest exact calibration is

P_raw ＝ M_eff / M_avail.

In the exact main-text model, P may be refined by weighting the pathways according to stability or independence, but the main point remains fixed: P is low when the record is effectively private to one fragile route and high when it is stably accessible through multiple effective routes. The calibrated value is denoted P̂.

B.2.3. Temporal stability T

Temporal stability T measures whether the record survives over the protocol window relevant to delayed retrieval and accessibility establishment. A record that exists only transiently but decays before retrieval is not operationally accessible in the sense required by the exact platform. Temporal stability therefore refers to persistence of retrievable record structure over the relevant interval.

Let τ_rel denote the declared relevant timescale between record formation and effective retrieval completion. Let C_rec(t) denote a normalized record-coherence or reconstruction-survival measure over time. Then a natural normalized stability measure is

T_raw ＝ (1/τ_rel) ∫₀^τ_rel C_rec(t) dt,

with T_raw ∈ [0,1] after normalization. In discrete implementations, T may instead be estimated by survival fractions across declared waiting times. The calibrated stability is denoted T̂.

B.2.4. Destructive burden of readout D

Destructive burden D measures the extent to which retrieving the record destroys, scrambles, or irreversibly consumes the record-bearing subsystem. Accessibility in the present platform is not merely about whether the record can be extracted once. It is about whether the record functions as a physically available carrier of information rather than a one-shot sacrificial trace.

Let I_pre denote a declared pre-readout integrity measure of the record sector and I_post the corresponding post-readout integrity measure under the declared retrieval protocol. Then a normalized destructive burden may be defined by

D_raw ＝ 1 − I_post/I_pre,

with D_raw clipped to [0,1] after calibration. Thus D_raw ＝ 0 corresponds to effectively non-destructive readout, while D_raw ＝ 1 corresponds to maximally destructive retrieval. The calibrated quantity is denoted D̂.

B.2.5. Redundancy spread S

Redundancy spread S measures the extent to which the record is distributed across more than one effective carrier or environmental degree of freedom. A record encoded redundantly is more operationally available than one confined to a single delicate subcarrier, even if both have similar immediate retrievability under one channel. Redundancy is therefore treated here as a genuine accessibility ingredient rather than as a secondary interpretive gloss.

Let K_tot denote the declared number of effective record-bearing subchannels and let K_stable denote the number of such subchannels from which the path label can still be reconstructed above threshold when others are selectively suppressed or ignored. Then the simplest normalized estimate is

S_raw ＝ K_stable / K_tot.

More refined redundancy measures could be entropy-based or mutual-information-based, but the exact platform of Volume V does not require that complexity in the main text. The calibrated value is denoted Ŝ.

B.3. Exact reduction rule for η

The exact accessibility parameter is defined by the geometric reduction

η ＝ (R · P · T · (1 − D) · S)^(1/5).

The experimentally calibrated estimate is therefore

η̂ ＝ (R̂ · P̂ · T̂ · (1 − D̂) · Ŝ)^(1/5).

This reduction has several properties that make it appropriate for the declared platform.

First, it is normalized. Since all five primitive factors lie in [0,1], η also lies in [0,1].

Second, it is monotone in each enabling ingredient R, P, T, and S, and monotone decreasing in the destructive burden D through the factor 1 − D.

Third, it is conjunctive rather than compensatory. A near-zero value in one essential ingredient strongly suppresses η, which reflects the intended physics: a record should not count as highly accessible if it is, for example, perfectly retrievable but temporally unstable, or highly redundant but only through maximally destructive readout.

Fourth, it is dimensionless and platform-calibratable. This makes it suitable as the horizontal control variable in the main-text visibility laws and in the definition of η-equivalence.

The fifth-root normalization is not incidental. It ensures that η remains on a comparable numerical scale across the declared five-factor structure and avoids artificial compression toward zero or one that would occur with a simple raw product.

B.4. Mathematical properties of the reduction map

The reduction map has several basic properties that will matter later.

B.4.1. Boundary conditions

If any one of R, P, T, 1 − D, or S vanishes, then η vanishes. Thus

R ＝ 0 ⇒ η ＝ 0,
P ＝ 0 ⇒ η ＝ 0,
T ＝ 0 ⇒ η ＝ 0,
D ＝ 1 ⇒ η ＝ 0,
S ＝ 0 ⇒ η ＝ 0.

This is physically appropriate: a record lacking any one indispensable accessibility component is not operationally accessible in the strong sense relevant to realization.

If all enabling factors are unity and D ＝ 0, then η ＝ 1.

B.4.2. Monotonicity

For strictly interior points of the admissible domain, η is strictly increasing in R, P, T, and S, and strictly decreasing in D. For example,

∂η/∂R ＝ (1/5)η/R,
∂η/∂P ＝ (1/5)η/P,
∂η/∂T ＝ (1/5)η/T,
∂η/∂S ＝ (1/5)η/S,
∂η/∂D ＝ −(1/5)η/(1 − D).

Thus calibration changes in any primitive variable induce a controlled and explicitly computable change in η.

B.4.3. Scale symmetry of the logarithmic form

Taking logarithms gives

ln η ＝ (1/5)[ln R ＋ ln P ＋ ln T ＋ ln(1 − D) ＋ ln S].

This is useful because it makes the contribution of each primitive factor additive in log-scale and simplifies uncertainty propagation in the regime where all factors are strictly positive.

B.5. Calibration of the primitive variables

The exact platform requires that each primitive quantity be obtained through an explicit calibration protocol. These calibrations are part of the experiment, not post hoc interpretation.

B.5.1. Calibration of R

R̂ is obtained by repeated path-reconstruction trials under the declared retrieval branch. Calibration includes correction for detector inefficiency and known readout asymmetry. Let p_corr be the corrected probability of successful reconstruction. Then

R̂ ＝ p_corr.

Confidence intervals on R̂ are computed by standard binomial or multinomial inference, depending on the retrieval-output structure.

B.5.2. Calibration of P

P̂ is obtained by evaluating how many independent declared retrieval channels can recover the path label above threshold. Let the platform define a set of channels {𝒞_j}. Then

P̂ ＝ (1/M_avail) ∑_j 𝟙[R_j ≥ R_min],

where R_j is the calibrated retrieval fidelity of channel 𝒞_j and 𝟙 denotes the indicator function. Weighted generalizations may be included in supplementary analysis, but the exact main-text calibration uses the threshold-count form unless explicitly stated otherwise.

B.5.3. Calibration of T

T̂ is measured by repeated retrieval after controlled waiting intervals covering the relevant delayed-choice window. Let {t_n} be the declared sampling times up to τ_rel, and let C_rec(t_n) be the calibrated record-survival measure at each point. Then a discrete stability estimator is

T̂ ＝ (1/N_t) ∑_n C_rec(t_n),

with N_t the number of sampling times. In the continuum limit this approaches the averaged integral definition above.

B.5.4. Calibration of D

D̂ is obtained by comparing record integrity before and after the declared readout. Let I_pre and I_post be calibrated integrity measures. Then

D̂ ＝ 1 − I_post/I_pre,

subject to clipping into [0,1] if necessary after correction for instrumental bias.

B.5.5. Calibration of S

Ŝ is obtained by testing whether the record remains reconstructible when subsets of the record-bearing channels are selectively ignored or suppressed. Let the declared subcarrier set be {𝒦_m}. Then

Ŝ ＝ K_stable/K_tot,

where K_stable counts the number of subcarrier configurations preserving above-threshold reconstruction.

B.6. Uncertainty propagation for η

Because η is built multiplicatively, its relative uncertainty is naturally handled in logarithmic form. Let the uncertainties in the primitive calibrated variables be δR, δP, δT, δD, and δS. Then to first order, assuming independent errors,

δη/η ≈ (1/5)√[(δR/R)² ＋ (δP/P)² ＋ (δT/T)² ＋ (δD/(1 − D))² ＋ (δS/S)²].

Equivalently,

δη ≈ (η/5)√[(δR/R)² ＋ (δP/P)² ＋ (δT/T)² ＋ (δD/(1 − D))² ＋ (δS/S)²].

If correlations among the primitive errors are non-negligible, covariance terms must be added:

(δη/η)² ≈ (1/25) ∑_{a,b} (∂ ln η/∂x_a)(∂ ln η/∂x_b) Cov(x_a, x_b),

where x_a ranges over R, P, T, 1 − D, and S in the calibrated representation. The main-text detectability theorem may use conservative envelopes instead of full covariance modeling, but the present appendix records the exact first-order propagation logic.

B.7. Accessibility-equivalence

The exact model requires a strict notion of when two platform realizations count as lying in the same accessibility regime.

Two realizations C₁ and C₂ are η-equivalent, written C₁ ≈_η C₂, if and only if:

they share the exact same signal–idler architecture and branch logic declared in Appendix A,
they are calibrated using the same primitive definitions and reduction map,
their accessibility estimates satisfy

|η̂(C₁) − η̂(C₂)| ≤ ε_η,

for a fixed declared tolerance ε_η,
4. any remaining differences are contained within the declared perturbative envelope and do not alter the realization-relevant observable structure except through that envelope.

This definition prevents arbitrary grouping of microscopically different experiments under a single η label while still allowing genuine implementation-equivalent realizations to be treated as the same accessibility regime.

B.8. Determination of the critical accessibility η_c

In the main text, η_c is the critical accessibility value at which the accessibility-sensitive realization burden becomes strong enough to alter the minimizing equivalence class of admissible channels. This appendix makes that determination explicit.

Let Φ_sub and Φ_sup denote the leading subcritical and supercritical admissible realization classes. Define the differential burden

Δℛ(η) ＝ ℛ_C(Φ_sup; η) − ℛ_C(Φ_sub; η).

Then η_c is defined by the balance condition

Δℛ(η_c) ＝ 0.

Using

ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ),

one has

Δℛ(η) ＝ αΔΞ(η) ＋ βΔΩ(η) ＋ γΔΛ(η),

where ΔΞ, ΔΩ, and ΔΛ are the burden differences between the two leading admissible classes.

In the exact main-text instantiation, ΔΞ and ΔΩ are taken to be effectively η-independent over the narrow transition window, while ΔΛ is linearized in η near the transition. Thus η_c is the unique solution of

αΔΞ ＋ βΔΩ ＋ γΔΛ(η_c) ＝ 0.

Under the main-text simplification ΔΛ(η) ∝ η − η_c, this becomes the exact point where the accessibility-sensitive correction changes sign and the selected realization regime switches.

B.9. Accessibility regimes

The exact construction of η supports the following regime partition:

low-accessibility regime: η close to 0, where at least one primitive accessibility ingredient strongly suppresses the effective record availability;
precritical regime: η < η_c but sufficiently near η_c that accessibility-sensitive effects are beginning to accumulate in the burden comparison;
critical regime: η in a neighborhood of η_c where the minimizing realization class is poised to change or has just changed;
postcritical regime: η > η_c, where the accessibility-sensitive correction is active in the exact CBR response;
asymptotic high-accessibility regime: η close to 1, where the record is maximally retrievable, stable, non-destructively available, and redundantly spread within the platform.

These regimes are not merely descriptive. They organize the main-text signature theorem and the detectability analysis.

B.10. Why this exact construction is sufficient

The accessibility construction defined here is sufficient for Volume V because it does everything the theory needs at the exact model level.

It maps a structured physical condition into a normalized scalar control variable.

It does so by using experimentally measurable quantities rather than abstract interpretive labels.

It supports uncertainty propagation and η-equivalence.

It ties directly into both the baseline response and the canonical CBR response.

It yields a critical accessibility value η_c that is mathematically meaningful inside the exact realization-law instantiation.

That is enough for the present volume. Broader generalization can wait. The task here is not to define the universal final ontology of accessibility across all conceivable experimental architectures. The task is to define one exact accessibility variable for one exact platform such that the resulting realization-law theory becomes mathematically explicit and experimentally vulnerable. This appendix completes that task.

Appendix C. Full Baseline Derivation

This appendix derives the exact standard-quantum baseline used throughout Volume V and fixes the corresponding baseline response class against which the instantiated CBR model is compared. Its purpose is not merely to summarize familiar interferometric facts, but to establish the ordinary comparator on the exact declared platform of Appendix A and with the exact accessibility construction of Appendix B. Every claim in the main text concerning non-equivalence, critical-regime structure, robustness, and invalidation depends on the baseline being explicit rather than schematic. The present appendix therefore derives the reduced signal response from the exact signal–idler state, incorporates the declared retrieval and erasure branch structure, ties the effective overlap to the calibrated accessibility variable η, and defines the exact baseline smooth-response class 𝒮_baseline used in the theorem spine of Volume V.

C.1. Baseline scope and assumptions

The baseline model is defined on the exact platform declared in Appendix A and makes only the following assumptions.

First, the total signal–idler system evolves according to ordinary quantum mechanics with no realization-law augmentation. Second, the initial state is the exact entangled preparation

|Ψ₀⟩ ＝ (1/√2)(|u⟩ ⊗ |r_u⟩ ＋ e^{iφ}|d⟩ ⊗ |r_d⟩).

Third, all visibility suppression and recovery effects are attributed to ordinary coherence loss, entanglement, basis change, and branch-conditioned reconstruction. Fourth, delayed-choice structure is treated in the standard way: later branch selection or record interrogation can alter conditional reconstruction logic but does not retrocausally alter the earlier marginal signal statistics. Fifth, the accessibility variable η influences the baseline only through its effect on the effective retrievable record overlap or distinguishability structure. No additional realization-sensitive law is introduced.

These assumptions are deliberately strong enough to give the standard comparator its best legitimate form. The baseline is therefore not a straw model. It is standard quantum interferometric reasoning stated on the same exact architecture later used for canonical CBR.

C.2. Joint density operator and reduced signal state

The exact initial density operator is

ρ₀ ＝ |Ψ₀⟩⟨Ψ₀|.

Expanding explicitly,

ρ₀ ＝ (1/2)∣u⟩⟨u∣⊗∣ru⟩⟨ru∣＋e−iφ∣u⟩⟨d∣⊗∣ru⟩⟨rd∣＋eiφ∣d⟩⟨u∣⊗∣rd⟩⟨ru∣＋∣d⟩⟨d∣⊗∣rd⟩⟨rd∣|u⟩⟨u| ⊗ |r_u⟩⟨r_u| ＋ e^{-iφ}|u⟩⟨d| ⊗ |r_u⟩⟨r_d| ＋ e^{iφ}|d⟩⟨u| ⊗ |r_d⟩⟨r_u| ＋ |d⟩⟨d| ⊗ |r_d⟩⟨r_d|∣u⟩⟨u∣⊗∣ru⟩⟨ru∣＋e−iφ∣u⟩⟨d∣⊗∣ru⟩⟨rd∣＋eiφ∣d⟩⟨u∣⊗∣rd⟩⟨ru∣＋∣d⟩⟨d∣⊗∣rd⟩⟨rd∣.

The reduced signal state is obtained by tracing over the idler subsystem:

ρ_s ＝ Tr_i(ρ₀).

Using linearity of the partial trace,

ρ_s ＝ (1/2)∣u⟩⟨u∣Tr(∣ru⟩⟨ru∣)＋e−iφ∣u⟩⟨d∣Tr(∣ru⟩⟨rd∣)＋eiφ∣d⟩⟨u∣Tr(∣rd⟩⟨ru∣)＋∣d⟩⟨d∣Tr(∣rd⟩⟨rd∣)|u⟩⟨u| Tr(|r_u⟩⟨r_u|) ＋ e^{-iφ}|u⟩⟨d| Tr(|r_u⟩⟨r_d|) ＋ e^{iφ}|d⟩⟨u| Tr(|r_d⟩⟨r_u|) ＋ |d⟩⟨d| Tr(|r_d⟩⟨r_d|)∣u⟩⟨u∣Tr(∣ru⟩⟨ru∣)＋e−iφ∣u⟩⟨d∣Tr(∣ru⟩⟨rd∣)＋eiφ∣d⟩⟨u∣Tr(∣rd⟩⟨ru∣)＋∣d⟩⟨d∣Tr(∣rd⟩⟨rd∣).

Since Tr(|r_u⟩⟨r_u|) ＝ Tr(|r_d⟩⟨r_d|) ＝ 1 and

Tr(|r_u⟩⟨r_d|) ＝ ⟨r_d|r_u⟩,
Tr(|r_d⟩⟨r_u|) ＝ ⟨r_u|r_d⟩,

one obtains

ρ_s ＝ (1/2)∣u⟩⟨u∣＋∣d⟩⟨d∣＋e−iφμ∣u⟩⟨d∣＋eiφμ★∣d⟩⟨u∣|u⟩⟨u| ＋ |d⟩⟨d| ＋ e^{-iφ}μ |u⟩⟨d| ＋ e^{iφ}μ★ |d⟩⟨u|∣u⟩⟨u∣＋∣d⟩⟨d∣＋e−iφμ∣u⟩⟨d∣＋eiφμ★∣d⟩⟨u∣,

where

μ ＝ ⟨r_d|r_u⟩

is the effective record overlap.

This is the standard reduced signal state for the exact declared platform before branch-conditioned refinements are imposed. It shows directly that the signal coherence is governed by the overlap structure of the record sector. All subsequent baseline visibility behavior follows from this fact.

C.3. Signal intensity in the interference basis

Let the signal reconstruction basis be

|χ(θ)⟩ ＝ (1/√2)(|u⟩ ＋ e^{iθ}|d⟩),

with θ the reconstruction phase. The corresponding signal intensity is

I(θ) ＝ ⟨χ(θ)|ρ_s|χ(θ)⟩.

Substituting the reduced state yields

I(θ) ＝ (1/2)⟨χ(θ)∣u⟩⟨u∣χ(θ)⟩＋⟨χ(θ)∣d⟩⟨d∣χ(θ)⟩＋e−iφμ⟨χ(θ)∣u⟩⟨d∣χ(θ)⟩＋eiφμ★⟨χ(θ)∣d⟩⟨u∣χ(θ)⟩⟨χ(θ)|u⟩⟨u|χ(θ)⟩ ＋ ⟨χ(θ)|d⟩⟨d|χ(θ)⟩ ＋ e^{-iφ}μ ⟨χ(θ)|u⟩⟨d|χ(θ)⟩ ＋ e^{iφ}μ★ ⟨χ(θ)|d⟩⟨u|χ(θ)⟩⟨χ(θ)∣u⟩⟨u∣χ(θ)⟩＋⟨χ(θ)∣d⟩⟨d∣χ(θ)⟩＋e−iφμ⟨χ(θ)∣u⟩⟨d∣χ(θ)⟩＋eiφμ★⟨χ(θ)∣d⟩⟨u∣χ(θ)⟩.

Using

⟨χ(θ)|u⟩ ＝ 1/√2,
⟨χ(θ)|d⟩ ＝ e^{-iθ}/√2,

one finds

I(θ) ＝ (1/2)1＋Re(μe−i(φ＋θ))1 ＋ Re(μ e^{-i(φ＋θ)})1＋Re(μe−i(φ＋θ)).

Equivalently, writing μ ＝ |μ|e^{iα},

I(θ) ＝ (1/2)1＋∣μ∣cos(φ＋θ−α)1 ＋ |μ| cos(φ ＋ θ − α)1＋∣μ∣cos(φ＋θ−α).

Thus the interference amplitude is |μ| and the baseline visibility is

V ＝ |μ|.

This result is exact for the declared initial state and signal-observable structure. It is the key bridge between record overlap and the experimentally reconstructed signal visibility.

C.4. Accessibility dependence of the effective overlap

The next step is to connect the effective overlap μ to the exact accessibility variable η constructed in Appendix B. In the baseline model, accessibility affects the visibility only by altering the effective distinguishability or retrievable separation of the record states. The simplest exact choice adopted in the main text is

μ(η) ＝ 1 − η,

with η ∈ [0,1].

This mapping satisfies the required endpoint behavior:

η ＝ 0 ⇒ μ ＝ 1,
η ＝ 1 ⇒ μ ＝ 0.

Thus at zero effective accessibility the record is operationally neutralized and the signal retains maximal visibility, while at unit accessibility the record is maximally available and the unconditioned signal visibility vanishes. This is the exact baseline law used throughout Volume V. It is not presented as the only conceivable overlap law in all experimental realizations, but as the exact declared overlap law for the exact declared main-text platform.

Substituting μ(η) ＝ 1 − η into the visibility formula gives

V_SQM(η) ＝ |1 − η|.

Since η ∈ [0,1], this reduces simply to

V_SQM(η) ＝ 1 − η.

This is the exact baseline visibility response used in the main text.

C.5. Justification of the exact baseline law

The relation μ(η) ＝ 1 − η is chosen because it is the minimal exact monotone law consistent with the platform’s interpretation of η as effective operational availability of the record. It is linear, normalized, endpoint-correct, and does not introduce any artificial critical structure absent realization-law augmentation. These are precisely the properties the baseline must have if it is to serve as a fair comparator in Volume V.

The linear law also separates cleanly what belongs to the baseline from what belongs to canonical CBR. Any localized critical-regime structure later appearing in V_CBR(η) cannot then be attributed to having already hidden such structure inside the standard comparator. This is one of the reasons the main-text theorem spine is so clean: the baseline is intentionally explicit, smooth, and non-thresholded.

Nothing in the baseline derivation requires accessibility to be interpreted as metaphysically primitive. It enters only through the effective overlap law, which is exactly where standard interferometric reasoning places path distinguishability effects in a signal–idler setup.

C.6. Retrieval branch and erasure branch in the baseline

The declared platform contains two operational branches: retrieval and erasure. In the baseline model, these branches alter conditional reconstruction logic but do not introduce a realization-law threshold.

C.6.1. Retrieval branch

In the retrieval branch, the idler subsystem is interrogated so that which-path information becomes operationally accessible. In the baseline model, this corresponds to maintaining or revealing the path-distinguishing record structure associated with η. The unconditioned signal visibility therefore follows the same overlap logic:

V_SQM^ret(η) ＝ 1 − η.

If retrieval is perfect and η approaches 1, the signal becomes fully distinguishable and the unconditioned visibility approaches 0.

C.6.2. Erasure branch

In the erasure branch, the idler record is measured or transformed in a basis that neutralizes path distinguishability at the retrieval level. In the baseline theory, this does not alter the earlier marginal signal statistics, but it does allow conditional reconstruction of interference in the appropriately sorted subensembles.

Let the erasure basis be

|e_±⟩ ＝ (1/√2)(|r_u⟩ ± |r_d⟩),

in the idealized symmetric case. Then conditioning on the erasure outcomes yields signal states with recovered phase coherence. In the ideal branch-resolved case, the conditional visibility approaches unity even when the unconditioned visibility is suppressed, provided the erasure basis is implemented exactly and residual distinguishability is absent.

Thus the baseline model distinguishes between:

unconditioned visibility, governed by η through effective overlap,
branch-conditioned recovery, governed by basis choice and conditional sorting.

This is standard and must be fully accounted for in the baseline before any realization-law effect is claimed.

C.7. Delayed-choice structure in the baseline

Because the retrieval-or-erasure choice may occur after signal detection in the declared platform, the baseline must also state clearly what ordinary quantum theory predicts under delayed choice.

The answer is exact and familiar in structure: delayed choice changes the interpretation and conditional reconstruction of the detected signal but not the already-recorded marginal signal statistics considered in isolation. The total state is treated as one entangled protocol, and later branch operations refine how the ensemble is partitioned, not what the earlier unconditioned counts were.

Thus the baseline contains no intrinsic critical accessibility phenomenon generated by delay alone. Delay affects conditioning logic, not the smooth functional structure of V_SQM(η). This is essential to the later theorem work, because it ensures that the critical-regime signature of canonical CBR is not being confused with ordinary delayed-choice reconstruction effects already present in the standard theory.

C.8. Exact baseline smooth-response class 𝒮_baseline

The exact representative baseline response is

V_SQM(η) ＝ 1 − η.

The baseline smooth-response class 𝒮_baseline is the class of functions on [0,1] satisfying the following conditions.

First, each member V(η) is globally continuous on [0,1].

Second, each member is piecewise C¹ and globally C¹ in any neighborhood not containing a declared apparatus discontinuity introduced independently of the theory test.

Third, each member is monotone nonincreasing in η.

Fourth, each member agrees with the exact endpoint conditions

V(0) ＝ 1,
V(1) ＝ 0,

up to declared perturbative tolerance.

Fifth, no member contains an intrinsic critical-regime derivative break, localized nonanalyticity, or threshold singularity near η_c unless that feature is independently imposed by declared apparatus design and modeled before comparison.

In the main-text exact model, the relevant representative is the single function

𝒮_baseline^0 ＝ {1 − η}.

Under perturbation analysis, one enlarges to a perturbed class 𝒮_baseline^pert consisting of smooth deformations of 1 − η that remain within the declared detector, erasure, environmental, and calibration tolerance envelope. What is excluded from 𝒮_baseline, by definition, is precisely the sort of localized critical-regime derivative break later predicted by the instantiated CBR model.

C.9. Baseline invariance under η-equivalent realizations

Let C₁ and C₂ be two η-equivalent realizations of the exact declared platform in the sense of Appendix B. Then they share:

the same signal–idler architecture,
the same primary observable,
the same calibrated accessibility value η within tolerance,
the same declared branch logic.

Since the baseline visibility depends only on η through

V_SQM(η) ＝ 1 − η,

it follows immediately that

C₁ ≈_η C₂ ⇒ V_SQM^{(C₁)}(η) ＝ V_SQM^{(C₂)}(η),

up to the declared perturbative tolerance envelope.

This invariance is not a minor technicality. It ensures that the baseline comparator is insensitive to superficial implementation detail once the accessibility regime has been fixed. Later, when canonical CBR is shown to generate a critical-regime structure under the same η-calibrated conditions, that difference cannot be dismissed as an artifact of using different engineering realizations of “the same” accessibility setting.

C.10. What the baseline does and does not establish

The exact baseline derivation establishes everything ordinary quantum interferometric reasoning is entitled to establish on the declared platform:

reduced signal coherence is controlled by effective record overlap,
unconditioned visibility degrades smoothly with effective accessibility,
conditional visibility can be recovered in erasure branches,
delayed choice alters conditional reconstruction logic without generating an intrinsic critical accessibility threshold.

What the baseline does not establish is any model-intrinsic reason for a critical-regime derivative break at η_c. Such structure is excluded by the exact baseline class unless it is imported through a different law or through undeclared apparatus effects. This is why the later non-equivalence and critical-regime theorems are meaningful. The baseline has now been fully derived, fixed, and bounded in scope.

Appendix D. Full Canonical CBR Derivation

This appendix develops the exact instantiation of canonical Constraint-Based Realization on the platform fixed in Appendix A, using the accessibility construction of Appendix B and the baseline comparator of Appendix C. Its purpose is to show, in complete form, how the canonized realization law

Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ),

with

ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ),

is converted into an exact model-level response on the accessibility-tunable delayed-choice quantum eraser declared in the main text. The appendix therefore performs five tasks. First, it defines the exact admissibility class 𝒜(C) for the platform. Second, it gives a concrete realization of the burden terms Ξ_C, Ω_C, and Λ_C. Third, it derives the accessibility-sensitive minimization structure and the critical accessibility value η_c. Fourth, it computes the induced response law V_CBR(η). Fifth, it places that response into direct comparison with the exact baseline V_SQM(η) derived in Appendix C.

The goal is not to introduce a second architecture distinct from the main text. It is to make explicit what the main text already used in compact form. No object defined here alters the principal law, the platform, the observable, or the accessibility construction. This appendix is therefore interpretive closure at the derivational level, not architectural expansion.

D.1. Context, admissibility, and realization channels

Let C denote the full measurement context associated with the exact platform of Appendix A. Thus C includes:

the signal Hilbert space ℋ_s,
the idler-record space ℋ_i,
the initial entangled preparation ρ₀,
the retrieval and erasure branch structure,
the delayed-choice timing relations,
the calibrated accessibility variable η,
the visibility observable V.

A realization channel Φ on this platform is a context-indexed effective map acting on 𝒟(ℋ_s ⊗ ℋ_i) whose physical role is not to modify the pre-realization quantum evolution, but to select a realization-compatible outcome structure consistent with the declared record architecture. In the present volume, one does not need the most general category-theoretic or operator-algebraic definition of such a channel. It is sufficient to treat Φ as an effective realization map satisfying the admissibility constraints stated below.

The admissible class 𝒜(C) is the set of realization channels Φ satisfying the following conditions.

First, dynamical compatibility: Φ does not alter the ordinary pre-realization state evolution already fixed by the declared signal–idler model. Any channel whose effect can be reinterpreted only as a hidden modification of the standard evolution law is excluded.

Second, representational invariance: Φ must assign the same realization verdict class under reparameterizations or relabelings of the signal–idler description that leave all realization-relevant observables unchanged. Any channel whose effect depends on arbitrary basis naming, record relabeling, or syntactic encoding is excluded.

Third, record-structural coherence: Φ must respond to the actual record architecture of the platform. A channel that either ignores stable path-correlated records when they are physically present or overweights formally definable but operationally inert record residues is excluded.

Fourth, accessibility consistency: Φ must treat η-equivalent realizations equivalently and must change realization behavior only through the declared accessibility structure rather than through undeclared implementation details.

Fifth, restricted Born-neutrality discipline: Φ may not insert the desired response weighting by brute definition. If weighting enters the instantiated law, it must do so through the declared burden structure and not by hiding the empirical answer in the channel class itself.

These constraints make 𝒜(C) a narrow class, not an unrestricted family of all formal maps on 𝒟(ℋ). For the exact main-text model, the admissible class can be reduced to two leading realization-equivalence classes and one critical boundary class:

Φ_sub, the subcritical realization class,
Φ_sup, the supercritical realization class,
Φ_crit, the critical limiting class at the transition point.

The task of the realization functional is then to determine, for each calibrated η, which of these classes minimizes the canonical burden.

D.2. Platform-specific burden decomposition

The realization functional is

ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ),

with α, β, γ ≥ 0 fixed at the level of the instantiated theory. Each burden term must now be tied concretely to the exact platform.

D.2.1. Representational invariance burden Ξ_C

The burden Ξ_C penalizes realization channels whose verdict depends on physically irrelevant description choices. In the exact delayed-choice quantum eraser, this includes path-basis relabelings, phase conventions that do not alter the measured visibility, and equivalent encodings of the idler record that preserve all declared observables and accessibility primitives.

Let 𝒢_C denote the group of physically irrelevant reparameterizations of the declared platform. For a candidate channel Φ, define the invariance defect

I_C(Φ) ＝ sup_{g ∈ 𝒢_C} d_op(Φ, g · Φ · g^{-1}),

where d_op is an operational-distance functional on realization verdict classes induced by the declared observable set. Then the representational burden is taken to be

Ξ_C(Φ) ＝ I_C(Φ).

For the leading admissible channel classes Φ_sub and Φ_sup in the main-text instantiation, Ξ_C is minimized by construction. That is, both classes are chosen to be representationally stable, while representation-sensitive pseudo-selection channels are excluded from 𝒜(C) altogether. Thus Ξ_C contributes no transition-driving term in the exact reduced model, but its explicit presence is essential because it justifies why the transition cannot be attributed to arbitrary redescriptions.

D.2.2. Record-structural coherence burden Ω_C

The burden Ω_C penalizes mismatch between the realization channel and the actual record organization of the idler subsystem. In the exact platform, the relevant structure includes:

the path-correlated states |r_u⟩ and |r_d⟩,
the retrieval and erasure branch logic,
the calibrated record-bearing properties summarized by R, P, T, D, and S.

Let 𝓡_phys(C) denote the declared physical record structure of the platform and 𝓡_Φ(C) the record structure effectively recognized by the candidate realization channel Φ. Let d_rec be a record-structural mismatch functional. Then define

Ω_C(Φ) ＝ d_rec(𝓡_Φ(C), 𝓡_phys(C)).

In the exact reduced model, Ω_C is treated as approximately constant within each of the two leading realization classes over the narrow transition region near η_c. Write

Ω_sub ＝ Ω_C(Φ_sub),
Ω_sup ＝ Ω_C(Φ_sup),

and define the record-structural differential

ΔΩ ＝ Ω_sup − Ω_sub.

In the main-text model, ΔΩ is taken as effectively η-independent over the critical window. This does not trivialize Ω_C. It means only that the transition is not driven by abrupt changes in record coherence itself, but by the accessibility-sensitive term competing against a fixed record-structural background burden.

D.2.3. Accessibility-consistency burden Λ_C

The burden Λ_C is the decisive term for Volume V. It penalizes realization channels whose verdict is inconsistent with the calibrated accessibility structure of the platform. To define it, recall that η is the exact scalar

η ＝ (R · P · T · (1 − D) · S)^(1/5),

and that η_c is the critical accessibility value at which the realization ordering changes.

The exact reduced instantiation assumes that the accessibility-sensitive differential between the two leading realization classes is locally linear in η near η_c. Thus define

ΔΛ(η) ＝ Λ_C(Φ_sup; η) − Λ_C(Φ_sub; η) ＝ η_c − η.

This sign convention is chosen so that for η < η_c, the supercritical class carries greater accessibility inconsistency burden than the subcritical class, whereas for η > η_c the opposite holds. In other words, the accessibility-sensitive ordering reverses at η_c, as required for a genuine regime change in the minimizing class.

Equivalently, one may write

Λ_C(Φ_sub; η) ＝ Λ₀ ＋ λ max{0, η − η_c},
Λ_C(Φ_sup; η) ＝ Λ₀ ＋ λ max{0, η_c − η},

up to an irrelevant additive constant Λ₀ and scaling λ > 0. The simpler differential form is enough for the main derivation; the explicit split form is useful for checking that the ordering reverses exactly once at η_c.

D.3. Reduced burden comparison and the critical point

The total burden difference between the supercritical and subcritical channel classes is

Δℛ(η) ＝ ℛ_C(Φ_sup; η) − ℛ_C(Φ_sub; η).

Substituting the burden decomposition gives

Δℛ(η) ＝ αΔΞ ＋ βΔΩ ＋ γΔΛ(η),

where

ΔΞ ＝ Ξ_C(Φ_sup) − Ξ_C(Φ_sub),
ΔΩ ＝ Ω_C(Φ_sup) − Ω_C(Φ_sub),
ΔΛ(η) ＝ Λ_C(Φ_sup; η) − Λ_C(Φ_sub; η).

By the exact reduced construction, the admissible leading classes are representationally matched, so

ΔΞ ＝ 0.

Thus

Δℛ(η) ＝ βΔΩ ＋ γ(η_c − η).

The critical accessibility value is defined by the balance condition

Δℛ(η_c) ＝ 0.

This yields

βΔΩ ＋ γ(η_c − η_c) ＝ 0,

so in the normalized reduced instantiation the transition point is encoded directly into the accessibility-sensitive term. More generally, if one had chosen a non-normalized linear form

ΔΛ(η) ＝ a − bη,

then η_c would be given by

η_c ＝ (a ＋ βΔΩ/γ)/b,

provided b > 0. The exact main-text model absorbs these constants into the definition of η_c and the scaling coefficient κ appearing later in the response law.

The minimizing class is therefore

Φ★_C(η) ＝ Φ_sub for Δℛ(η) > 0,
Φ★_C(η) ＝ Φ_crit for Δℛ(η) ＝ 0,
Φ★_C(η) ＝ Φ_sup for Δℛ(η) < 0.

Since Δℛ(η) is positive below η_c and negative above η_c in the exact reduced model, one obtains

Φ★_C(η) ＝ Φ_sub for η < η_c,
Φ★_C(η) ＝ Φ_crit at η ＝ η_c,
Φ★_C(η) ＝ Φ_sup for η > η_c.

This is the exact accessibility-sensitive minimization structure used in the main text.

D.4. Response functional induced by the selected realization class

The next step is to connect the selected realization class to the primary observable V. In the baseline model of Appendix C, visibility is

V_SQM(η) ＝ 1 − η.

Canonical CBR does not replace the entire baseline signal–idler structure. It modifies the response through the realization-law correction induced by the selected channel class. Thus the exact CBR response is written as

V_CBR(η) ＝ V_SQM(η) − ΔV_real(η),

where ΔV_real(η) is the realization-sensitive correction.

For the exact reduced model, ΔV_real(η) is zero below η_c and grows linearly with η − η_c above η_c. Therefore define

ΔV_real(η) ＝ κ max{0, η − η_c},

with κ > 0 a fixed theory-level coefficient representing the strength of the accessibility-sensitive realization correction in the supercritical regime.

It follows immediately that

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c}.

This is the exact main-text response law.

The physical meaning of κ is not that it is an ad hoc fit parameter inserted after the fact. It is the effective strength with which the accessibility-sensitive change in the minimizing realization class feeds into the primary observable on the exact platform. In a deeper microscopic derivation, κ would be expressed in terms of the platform-specific coupling between the selected realization class and the recovered signal contrast. For Volume V, it is fixed at the model level as part of the exact instantiated theory.

D.5. Piecewise form of the exact CBR response

The response law may be written piecewise as

V_CBR(η) ＝ 1 − η, for η ≤ η_c,

and

V_CBR(η) ＝ 1 − η − κ(η − η_c), for η > η_c.

Equivalently,

V_CBR(η) ＝ 1 − η, for η ≤ η_c,

V_CBR(η) ＝ 1 − (1 ＋ κ)η ＋ κη_c, for η > η_c.

This form makes three important facts explicit.

First, V_CBR is continuous at η_c. Indeed,

lim_{η→η_c^-} V_CBR(η) ＝ 1 − η_c,
lim_{η→η_c^+} V_CBR(η) ＝ 1 − η_c.

Second, the derivative changes at η_c. For η < η_c,

dV_CBR/dη ＝ −1,

while for η > η_c,

dV_CBR/dη ＝ −(1 ＋ κ).

Thus the exact response has a derivative break, or kink, at η_c.

Third, the low-accessibility regime is baseline-matched in the exact reduced model. The theory therefore does not demand visible departure where the accessibility-sensitive term is not yet realization-dominant. This is precisely why the critical regime becomes the focal point of the empirical burden.

D.6. Exact comparison with the baseline

The baseline and canonical responses now stand in exact comparison:

V_SQM(η) ＝ 1 − η,

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c}.

Subtracting gives the exact response difference

ΔV(η) ＝ V_CBR(η) − V_SQM(η) ＝ −κ max{0, η − η_c}.

Hence

ΔV(η) ＝ 0, for η ≤ η_c,
ΔV(η) ＝ −κ(η − η_c), for η > η_c.

This makes the later theorem spine almost immediate:

the two models are not globally identical if κ > 0,
their difference concentrates at the critical accessibility value η_c,
the morphology of that concentration is a derivative break in the strong form,
under smoothing or bounded perturbation, a non-baseline deviation band remains in the weak form.

Thus the full canonical derivation yields exactly the sort of critical-regime structure the main text claims, and it does so from the frozen law form, not from an arbitrary overlay on the baseline.

D.7. Strong-form and weak-form interpretations

The exact reduced response law gives the strong-form signature directly: a continuous function with a slope discontinuity at η_c. This is the sharpest manifestation of the realization-class transition under the chosen local linearization of the accessibility-sensitive burden.

However, the derivation also suggests the weak-form generalization. If one replaces the sharp term max{0, η − η_c} by a smoothed transition function g(η − η_c) satisfying

g(x) ≈ 0 for x ≪ 0,
g(x) ≈ x for x ≫ 0,

with a narrow smoothing width w > 0, then one obtains

V_CBR^smooth(η) ＝ 1 − η − κ g(η − η_c).

In that case the literal derivative discontinuity may be regularized, but the response still deviates from the baseline in a bounded neighborhood of η_c, provided κ is nonzero and the smoothing width remains finite. This is the mathematical basis of the weak-form signature theorem stated in the main text. It is not a different theory. It is the regularized version of the same exact accessibility-sensitive realization law.

D.8. Admissibility exclusions in the exact model

It is useful to record explicitly what kinds of channels are excluded by the exact instantiated derivation.

A channel is excluded if it:

reproduces the main-text response only by hard-coding κ or η_c into an otherwise accessibility-insensitive map,
changes the low-accessibility regime away from the exact baseline without justification from the declared burden structure,
introduces representational dependence into the transition location,
yields different critical points for η-equivalent realizations of the platform,
mimics the critical-regime structure only by hidden modification of the baseline signal–idler evolution rather than by realization-law selection.

These exclusions matter because they show that the exact CBR response is not the output of unrestricted model flexibility. It is the output of a narrow admissible class under the frozen canonical law.

D.9. Relationship between η_c and observability

The derivation makes clear that η_c is not an arbitrary marker on the horizontal axis. It is the accessibility value at which the realization burden ordering changes sign. This explains why the observable signature is concentrated near η_c rather than spread uniformly across all η.

Below η_c, the subcritical class remains burden-minimizing and the exact reduced model coincides with the baseline. Above η_c, the supercritical class becomes burden-minimizing and the realization-law correction turns on linearly with η − η_c. Thus η_c is simultaneously:

the burden-balance point,
the realization-regime boundary,
the onset of exact non-equivalence,
the location of the strong-form derivative break.

That convergence of roles is what gives the exact model its technical sharpness.

D.10. What this appendix establishes

This appendix completes the full canonical CBR derivation required by Volume V. It establishes:

the exact admissibility structure for the platform,
the explicit burden decomposition,
the accessibility-sensitive minimization rule,
the exact transition at η_c,
the induced primary response law V_CBR(η),
the direct difference from the declared baseline.

Nothing essential to the main-text theorem spine remains informal after this derivation. The exact instantiated theory is now mathematically explicit enough to support the proofs of non-equivalence, critical-regime signature, robustness, and binary invalidation that follow in the later appendices.

Appendix E. Proof of Non-Equivalence Theorem

This appendix gives the full proof of Theorem 4, the Baseline/CBR Non-Equivalence Theorem, for the exact instantiated platform of Volume V. The theorem states that once accessibility enters the canonical realization law nontrivially, the exact CBR response cannot remain globally identical to the exact standard baseline response across the admissible accessibility domain. In the main text this result was presented in compressed form, since its intuition follows quickly from the exact response laws already derived. The purpose of the present appendix is to state the result with full scope, fix the assumptions under which it holds, show precisely where the proof draws its force, and clarify what kinds of rival or degenerate cases do not count as counterexamples.

The proof is structurally important because it marks the point at which the instantiated CBR model ceases to be merely a sharpened framework and becomes a model with exact empirical burden. If global identity with the standard baseline remained possible even when accessibility entered the realization law nontrivially, then the entire critical-regime program of Volume V would collapse into formal decoration. The proof therefore does more than compare two functions. It shows that, for the exact declared platform and under the declared law form, accessibility-sensitive realization cannot remain observationally idle everywhere.

E.1. Statement of the theorem

For convenience, the theorem is restated here in full.

Theorem 4 (Baseline/CBR Non-Equivalence Theorem). For the exact declared accessibility-tunable delayed-choice quantum eraser of Volume V, suppose canonical CBR is instantiated by the law

Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ),

with

ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ),

where η ∈ [0,1] is the exact calibrated accessibility variable, η_c ∈ (0,1) is the critical accessibility value, and the instantiated primary response is

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

for some fixed κ > 0. Let the exact standard baseline response on the same platform be

V_SQM(η) ＝ 1 − η.

Then V_CBR(η) is not globally identical to V_SQM(η) on [0,1]. Equivalently,

∃η ∈ [0,1] such that V_CBR(η) ≠ V_SQM(η).

E.2. Assumptions and scope

The theorem rests on a specific chain of assumptions. These must be stated explicitly because the proof is exact only within this declared model and not as a universal statement about all conceivable realization-law frameworks.

First, the experimental platform is exactly the one fixed in Appendix A: a two-path signal subsystem, an idler-record subsystem, retrieval and erasure branch structure, delayed-choice timing, and visibility as the primary observable.

Second, accessibility is the exact scalar defined in Appendix B,

η ＝ (R · P · T · (1 − D) · S)^(1/5),

and is treated as the only main-text accessibility parameter.

Third, the exact baseline response is the one derived in Appendix C,

V_SQM(η) ＝ 1 − η,

and the baseline response class contains no undeclared realization-law threshold structure.

Fourth, the instantiated CBR response is the one derived in Appendix D,

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

with κ > 0.

Fifth, accessibility enters the realization law nontrivially. In the present exact model, this means that Λ_C contributes to the burden ordering in such a way that the minimizing realization class changes regime at η_c. A merely symbolic appearance of η inside the law without effect on the minimizing class is excluded from the theorem’s scope, because such a case would not qualify as nontrivial accessibility dependence in the first place.

Under these assumptions, the theorem is exact. Outside them, one may still draw related conclusions, but they would need separate proof.

E.3. Preliminary observation: equality below the critical regime

The exact instantiated CBR response is defined by

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c}.

For any η ≤ η_c, one has

max{0, η − η_c} ＝ 0.

Therefore, for all η ≤ η_c,

V_CBR(η) ＝ 1 − η.

Since the baseline is also

V_SQM(η) ＝ 1 − η,

it follows that

V_CBR(η) ＝ V_SQM(η) for all η ≤ η_c.

This preliminary fact is important. It shows that the theorem is not a claim of universal difference at every accessibility value. The exact reduced model permits baseline agreement throughout the entire subcritical regime. That is not a weakness of the theorem. It is part of its discipline. Non-equivalence is concentrated where the realization law itself predicts it should arise.

E.4. Strict inequality above the critical regime

Now consider any η > η_c. Then

η − η_c > 0,

max{0, η − η_c} ＝ η − η_c.

Substituting into the exact CBR response gives

V_CBR(η) ＝ 1 − η − κ(η − η_c).

Since κ > 0 and η − η_c > 0, the correction term κ(η − η_c) is strictly positive. Therefore

V_CBR(η) < 1 − η.

But the baseline response is exactly

V_SQM(η) ＝ 1 − η.

Hence

V_CBR(η) < V_SQM(η) for all η > η_c.

This proves immediately that the two responses cannot be globally identical on [0,1], because they differ strictly on every postcritical accessibility value.

E.5. Direct proof by contradiction

Although the preceding comparison already establishes the theorem, it is useful to write the proof in contradiction form because that clarifies the role of nontrivial accessibility dependence.

Assume, for contradiction, that

V_CBR(η) ＝ V_SQM(η) for all η ∈ [0,1].

Then by substitution,

1 − η − κ max{0, η − η_c} ＝ 1 − η for all η ∈ [0,1].

Subtracting 1 − η from both sides yields

−κ max{0, η − η_c} ＝ 0 for all η ∈ [0,1].

Equivalently,

κ max{0, η − η_c} ＝ 0 for all η ∈ [0,1].

Since κ > 0 by hypothesis, this implies

max{0, η − η_c} ＝ 0 for all η ∈ [0,1].

But this is false. Any η > η_c gives

max{0, η − η_c} ＝ η − η_c > 0.

Thus the assumption of global identity leads to contradiction. Therefore the two responses are not globally identical.

This contradiction proof isolates the real source of non-equivalence: once κ > 0 and η_c lies inside the admissible domain, the accessibility-sensitive correction cannot vanish everywhere.

E.6. Reformulation in terms of response difference

Define the exact response difference

ΔV(η) ＝ V_CBR(η) − V_SQM(η).

Substituting the derived responses gives

ΔV(η) ＝ −κ max{0, η − η_c}.

Thus

ΔV(η) ＝ 0 for η ≤ η_c,
ΔV(η) ＝ −κ(η − η_c) for η > η_c.

This form yields three immediate consequences.

First, ΔV is continuous on [0,1].

Second, ΔV vanishes on the entire subcritical regime and becomes strictly negative on the entire postcritical regime.

Third, the onset of non-equivalence occurs exactly at η_c.

Thus the theorem is stronger than a bare existence statement. It does not merely say that there exists some η for which the models differ. It gives the exact region of equality and the exact region of strict inequality. In the main-text theorem, only the existence claim was needed. The present appendix records the full structure.

E.7. Why the theorem is not a trivial artifact of parameterization

A natural objection is that the theorem appears almost too easy once the exact response functions are written down. But that ease is not evidence of triviality. It is evidence that the burden of the theory has already been pushed into the exact instantiation. The theorem is simple only because the hard work has already been done in earlier appendices:

Appendix A froze the platform.
Appendix B froze η.
Appendix C froze the standard baseline.
Appendix D froze the instantiated canonical law and derived the exact CBR response.

Once those objects are fixed, non-equivalence becomes a clean consequence rather than a vague interpretive claim.

A second objection would be that κ might simply have been inserted by hand to force the theorem. That objection fails inside the logic of Volume V, because κ is not introduced as a decorative discrepancy parameter. It is the effective coefficient induced by the accessibility-sensitive change in the minimizing realization class. In other words, κ > 0 is the exact model-level expression of the hypothesis that accessibility enters realization law nontrivially. If one sets κ ＝ 0, one does not preserve the same theory and merely simplify the algebra. One collapses the instantiated theory back into baseline equivalence by nullifying the accessibility-sensitive realization correction. That is a different case, not a counterexample.

E.8. Degenerate cases and excluded loopholes

The theorem excludes several apparent loopholes.

E.8.1. The case κ ＝ 0

If κ ＝ 0, then

V_CBR(η) ＝ 1 − η ＝ V_SQM(η)

everywhere. But this is not a counterexample to the theorem because the theorem explicitly assumes nontrivial accessibility dependence. Setting κ ＝ 0 removes that dependence at the response level and therefore exits the theorem’s scope.

E.8.2. The case η_c ∉ (0,1)

If η_c lies outside the admissible accessibility domain, the theory would either never transition or would already be entirely postcritical or subcritical throughout the domain. Such cases do not match the main-text instantiated model, which assumes η_c ∈ (0,1). They therefore do not count against the theorem as stated.

E.8.3. The case of hidden redefinition of η

One might try to rescue global equivalence by redefining η after the fact so that the postcritical correction is absorbed into a new baseline coordinate. That move is excluded because η is already fixed by the exact calibration rule of Appendix B. Reparameterizing η is therefore not a legitimate counterargument inside the exact platform. It is a change of theory or experiment, not a reinterpretation of the same one.

E.8.4. The case of baseline enrichment

One might also enlarge the standard comparator until it includes functions with the same critical-regime structure as V_CBR. That move is excluded by the definition of 𝒮_baseline in Appendix C, which permits only smooth perturbative deformations of the standard response and forbids importing realization-law threshold structure into the comparator by hand. The theorem therefore compares the exact CBR response to the exact baseline it was actually obliged to face.

E.9. Operational meaning of the theorem

The theorem may be read in two complementary ways.

At the mathematical level, it states that two exact functions on the same domain are not identical.

At the physical level, it states that once accessibility is allowed to alter realization ordering, the declared platform cannot remain observationally ordinary across the full accessibility range. There must exist a region in which the instantiated realization law leaves an observable trace not absorbed by the standard model.

This is the first point in Volume V at which the theory acquires exact operational burden. Before the theorem, canonical CBR could still be described as a sharpened law proposal instantiated on a platform. After the theorem, it is a platform-instantiated law proposal that must differ somewhere accessible if it is to remain the theory it claims to be.

E.10. Strengthened corollary

The main text states the existence corollary that canonical CBR has exact model-level empirical burden. The exact derivation here allows a stronger corollary.

Corollary E.1. Under the assumptions of Theorem 4, the exact instantiated CBR model and the exact baseline model coincide on the closed interval [0, η_c] and differ strictly on the open interval (η_c, 1].

This follows directly from the piecewise expression for ΔV(η). The corollary matters because it pinpoints where the empirical burden begins. It is not everywhere. It is exactly the postcritical regime, with onset at η_c.

E.11. Role of the theorem in the Volume V spine

The theorem is correctly placed as the first theorem in the Volume V spine because it does the foundational empirical work. Before one can discuss critical-regime morphology, robustness, or falsification, one must show that the instantiated law is not globally identical to the standard comparator. That is exactly what Theorem 4 proves.

Once Theorem 4 is established, the remaining theorems become more than speculative possibilities:

Theorem 5 identifies where the difference concentrates and what shape it takes.
Theorem 6 shows that the difference survives bounded perturbation.
Theorem 7 states what happens if the platform nevertheless exhibits only baseline behavior.

Thus Theorem 4 is the hinge between exact instantiation and exact empirical consequence.

E.12. Conclusion of the appendix

This appendix has completed the proof of the Baseline/CBR Non-Equivalence Theorem for the exact main-text model. Under the frozen platform, the frozen accessibility variable, the frozen baseline, and the frozen canonical CBR instantiation, global response identity is impossible once accessibility enters realization law nontrivially. The result is exact, not heuristic. The theorem therefore establishes the minimum empirical condition required for the rest of Volume V to matter: the theory must differ somewhere accessible if it is to remain itself.

Appendix F. Proof of Critical-Regime Signature Theorem

This appendix gives the full proof of Theorem 5, the Critical-Regime Signature Theorem, for the exact instantiated model of Volume V. The purpose of the theorem is not merely to restate the already-established fact that the instantiated canonical CBR response differs from the exact standard baseline somewhere on the admissible accessibility domain. That weaker conclusion was secured by Theorem 4 and proved in Appendix E. The present theorem does something more informative and more valuable: it identifies the morphology and localization of the difference. Specifically, it shows that under the exact main-text regularity assumptions, the non-equivalence concentrates at the critical accessibility value η_c as a derivative break or kink, and that if those stronger regularity assumptions are relaxed, a bounded non-baseline deviation class persists in a neighborhood of η_c.

The theorem is decisive for the empirical logic of Volume V because it explains why the theory does not need to predict large anomalies everywhere to be scientifically distinct. It needs only to predict one exact local regime in which the response law leaves the baseline smooth-response class for reasons internal to the realized channel selection. The proof below is therefore structured in two parts. The first establishes the strong form under the exact main-text piecewise response law. The second establishes the weak form under smoothed transition structure. Together they show that the theory’s distinctness is neither diffuse nor ill-defined. It is concentrated, mathematically characterizable, and directly tied to the accessibility-sensitive ordering change in the realization law.

F.1. Statement of the theorem

For convenience, Theorem 5 is restated here in exact form.

Theorem 5 (Critical-Regime Signature Theorem). Let the exact declared accessibility-tunable delayed-choice quantum eraser of Volume V be given, with exact standard baseline response

V_SQM(η) ＝ 1 − η

and exact instantiated canonical CBR response

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

where η ∈ [0,1], η_c ∈ (0,1), and κ > 0. Let 𝒮_baseline denote the declared baseline smooth-response class associated with the exact platform. Then, under the exact main-text regularity assumptions, V_CBR develops a critical-regime structure at η_c not contained in 𝒮_baseline. In particular, V_CBR is continuous at η_c but exhibits a derivative break there and therefore does not belong to 𝒮_baseline in any neighborhood of η_c. If the stronger regularity assumptions required for literal derivative discontinuity are relaxed while the accessibility-sensitive realization ordering remains nontrivial, then there exists a nonempty neighborhood U of η_c such that V_CBR lies in a bounded non-baseline deviation class on U not reducible to any member of 𝒮_baseline within the declared tolerance.

The theorem therefore has a strong form and a weak form. The strong form asserts a kink. The weak form asserts a bounded non-baseline deviation band.

F.2. Exact objects used in the proof

The proof uses only the exact objects already fixed in the earlier appendices.

The platform is the exact one declared in Appendix A.

The accessibility variable η is the exact calibrated scalar of Appendix B.

The standard baseline response is the exact law derived in Appendix C:

V_SQM(η) ＝ 1 − η.

The canonical CBR response is the exact law derived in Appendix D:

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c}.

The baseline smooth-response class 𝒮_baseline is the class of globally continuous, monotone nonincreasing, piecewise C¹ functions on [0,1] that are C¹ in any neighborhood free of declared apparatus discontinuity and that contain no intrinsic realization-law critical structure near η_c. In the exact main-text model, the representative baseline member is simply 1 − η, and the smooth-response class is generated by admissible smooth perturbative deformations of that representative.

Finally, the proof relies on the exact interpretation of η_c from Appendix D: η_c is the accessibility value at which the accessibility-sensitive burden differential reverses sign and the minimizing realization class changes.

F.3. Strong-form proof: continuity and derivative break

We begin with the exact main-text response

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c}.

This can be written piecewise as

V_CBR(η) ＝ 1 − η, for η ≤ η_c,

and

V_CBR(η) ＝ 1 − η − κ(η − η_c), for η > η_c.

Equivalently,

V_CBR(η) ＝ 1 − η, for η ≤ η_c,

V_CBR(η) ＝ 1 − (1 ＋ κ)η ＋ κη_c, for η > η_c.

F.3.1. Continuity at η_c

Evaluate the left and right limits at η_c.

From below,

lim_{η→η_c^-} V_CBR(η) ＝ 1 − η_c.

From above,

lim_{η→η_c^+} V_CBR(η) ＝ 1 − (1 ＋ κ)η_c ＋ κη_c ＝ 1 − η_c.

Thus

lim_{η→η_c^-} V_CBR(η) ＝ lim_{η→η_c^+} V_CBR(η) ＝ V_CBR(η_c),

so the function is continuous at η_c.

This matters because the predicted signature is not a jump discontinuity or an unphysical rupture in the observable itself. The theory predicts continuity of visibility across the critical regime, which is physically natural, but with altered local response law.

F.3.2. Left and right derivatives

For η < η_c,

dV_CBR/dη ＝ −1.

For η > η_c,

dV_CBR/dη ＝ −(1 ＋ κ).

Since κ > 0, these derivatives are not equal:

−1 ≠ −(1 ＋ κ).

Therefore the derivative fails to exist at η_c as a single two-sided value. Equivalently, V_CBR is continuous but not C¹ at η_c. It has a derivative break, or kink, there.

This establishes the strong-form morphology.

F.4. Exclusion from the baseline smooth-response class

The next step is to show that this critical-regime morphology is not contained in the declared baseline smooth-response class 𝒮_baseline.

By construction, 𝒮_baseline contains only functions that are globally continuous and locally C¹ in every neighborhood of η_c unless a discontinuity has been introduced independently by declared apparatus control. In the exact main-text model, there is no such apparatus discontinuity at η_c. The critical accessibility value is a theoretical transition point in the realization law, not a forced singular point of the declared baseline hardware. Therefore every admissible baseline response in the neighborhood of η_c is C¹.

But V_CBR is not C¹ at η_c. Hence

V_CBR ∉ 𝒮_baseline

in any neighborhood of η_c.

This is not merely a visual distinction. It is a category distinction. The baseline class and the CBR response differ in local regularity type. Therefore no member of the exact baseline smooth-response class can reproduce the CBR critical-regime morphology without ceasing to belong to that class.

This completes the proof of the strong form: the exact instantiated CBR response develops a critical-regime kink at η_c not contained in 𝒮_baseline.

F.5. Why the signal localizes at η_c

It is useful to state explicitly why the signature concentrates at η_c rather than appearing uniformly across all η. This is not an accident of parameterization. It follows from the structure of the realization law.

In Appendix D, the total burden difference between the leading supercritical and subcritical realization classes was

Δℛ(η) ＝ βΔΩ ＋ γΔΛ(η),

with ΔΛ(η) changing sign at η_c. Thus η_c is the point at which the minimizing realization class changes. Below η_c, the subcritical class remains selected and the exact reduced response coincides with the baseline. Above η_c, the supercritical class becomes selected and the accessibility-sensitive correction activates.

Therefore the strongest signal must occur at the onset of regime change. The theory does not predict a generic anomaly in every regime because the law does not change ordering in every regime. It changes ordering at η_c. The derivative break is simply the observable footprint of that ordering transition in the exact reduced model.

This localization is therefore a strength rather than a weakness. It shows that the theory’s signature is tied to an exact internal mechanism instead of being distributed arbitrarily across the entire domain.

F.6. Weak-form proof under smoothed transition

The strong-form proof relied on the exact piecewise correction κ max{0, η − η_c}. It is important to show that the theorem does not collapse if that sharp term is regularized by more refined platform modeling. The weak form addresses precisely that case.

Let g_w(x) be a one-parameter family of smooth transition functions with smoothing width w > 0 satisfying:

g_w(x) ≈ 0 for x ≪ −w,
g_w(x) ≈ x for x ≫ w,
g_w is continuous and differentiable for all x,
g_w converges pointwise to max{0, x} as w → 0.

Define the smoothed CBR response

V_CBR^w(η) ＝ 1 − η − κ g_w(η − η_c).

This response no longer has a literal derivative discontinuity if w > 0, since g_w is smooth. But its local behavior near η_c still differs from the baseline in a controlled way.

Define the response difference

ΔV_w(η) ＝ V_CBR^w(η) − V_SQM(η) ＝ −κ g_w(η − η_c).

Because g_w is not identically zero in any neighborhood crossing η_c whenever κ > 0, there exists a nonempty neighborhood U_w of η_c such that

|ΔV_w(η)| > 0

for some η ∈ U_w.

Now let 𝒮_baseline be the baseline smooth-response class as declared. Since all members of 𝒮_baseline are smooth deformations of the exact baseline without intrinsic critical-regime realization transition, there exists a tolerance envelope ε_base such that any baseline-compatible local deviation near η_c must remain within the declared smooth perturbative family. If

sup_{η ∈ U_w} |ΔV_w(η)| > ε_base,

then V_CBR^w cannot be absorbed into any member of 𝒮_baseline on U_w.

Thus, even when the strong-form kink is smoothed, there remains a bounded non-baseline deviation class in a neighborhood of η_c provided the accessibility-sensitive correction remains nontrivial. This proves the weak form.

F.7. Exact statement of the bounded deviation class

It is useful to make the weak form more explicit. Let U be a closed interval centered at η_c with width 2δ, where δ is small enough that the baseline remains locally well approximated by its declared smooth-response structure and large enough to contain the transition region of the smoothed response.

Then the bounded deviation class is the set of functions 𝒟_sig(U; ε_sig) satisfying

𝒟_sig(U; ε_sig) ＝ {f : ∃η ∈ U such that |f(η) − (1 − η)| ≥ ε_sig},

for some ε_sig > 0 fixed by the transition strength κ and smoothing width w.

The weak-form theorem says that, for nontrivial accessibility-sensitive realization and sufficiently small but finite smoothing width, there exists ε_sig > 0 such that

V_CBR^w ∈ 𝒟_sig(U; ε_sig)

while no member of the declared baseline class belongs to 𝒟_sig(U; ε_sig) unless the baseline class is improperly enlarged beyond its own definition.

This formalizes the fallback signature as a local amplitude-separation class rather than a regularity-class separation.

F.8. Exclusion of baseline mimicry

A skeptical objection would be that one could simply enlarge the baseline class until it contains the same local morphology or local deviation band. But that move is not allowed within the exact logic of Volume V.

The baseline class was fixed in Appendix C before the discriminating comparison. It consists of the standard response and its admissible smooth perturbative deformations on the declared platform. It does not include a realization-law critical point by fiat. Enlarging the class afterward to include the exact local structure the theory predicted would destroy the meaning of model comparison rather than defeat the theorem.

Similarly, one might try to absorb the signature into hidden apparatus discontinuity. That route is also excluded unless the discontinuity is independently declared and calibrated before the theory comparison. The theorem is about the exact declared platform, not about a baseline retrospectively rewritten to protect itself from predicted failure.

F.9. Relation between strong and weak forms

The strong and weak forms are not rival versions of the theory. They are two regularity levels of the same exact accessibility-sensitive realization law.

The strong form applies when the exact reduced main-text instantiation is taken literally. In that case the signal has a derivative break at η_c.

The weak form applies when one allows a more refined local smoothing of the realization-class transition while keeping the same exact platform, the same exact accessibility variable, and the same nontrivial dependence of the minimizer on η. In that case the derivative break may soften, but a bounded non-baseline deviation band remains.

Thus the weak form is not an escape hatch. It is the stable residual empirical content of the same theory under smoothing.

F.10. Corollary on local regularity separation

The strong-form proof implies a useful corollary.

Corollary F.1. Under the exact main-text regularity assumptions, there exists no open neighborhood U of η_c such that V_CBR restricted to U belongs to the same local differentiability class as the exact baseline response restricted to U.

This follows because V_SQM is C¹ on every neighborhood of η_c, while V_CBR is not C¹ at η_c. Therefore the local regularity classes differ.

This corollary sharpens the theorem by showing that the distinction is not merely numerical but structural.

F.11. Corollary on critical-regime concentration

A second corollary records the localization explicitly.

Corollary F.2. In the exact main-text model, the onset of non-equivalence between V_CBR and V_SQM occurs exactly at η_c and nowhere below it.

This follows from the piecewise response difference

ΔV(η) ＝ 0 for η ≤ η_c,
ΔV(η) ＝ −κ(η − η_c) for η > η_c.

The corollary matters because it shows that the theory is not predicting a hidden low-η anomaly that later becomes reinterpreted as a critical effect. The signal begins exactly where the accessibility-sensitive realization ordering changes.

F.12. Role of the theorem in the Volume V logic

Theorem 5 occupies the correct place in the theorem spine because it turns the abstract fact of non-equivalence into an experimentally recognizable form. Theorem 4 says that the theory differs somewhere. Theorem 5 says where and how. Only after that does it make sense to ask whether the difference survives realistic perturbations, which is the work of Theorem 6.

In this sense, Theorem 5 is the theorem that makes Volume V memorable. It does not simply assert that canonical CBR and the baseline are not the same. It says that if accessibility really alters realization, the exact platform should develop a local critical-regime signature whose morphology is calculable and whose absence under proper conditions would matter scientifically.

F.13. Conclusion of the appendix

This appendix has completed the proof of the Critical-Regime Signature Theorem in both strong and weak forms. Under the exact main-text regularity assumptions, the instantiated canonical CBR response is continuous but not differentiable at η_c and therefore leaves the baseline smooth-response class through a derivative break or kink. If the transition is smoothed while preserving nontrivial accessibility-sensitive realization ordering, the response still leaves the baseline class through a bounded non-baseline deviation band localized near η_c. The theory’s empirical distinctness is therefore neither diffuse nor rhetorical. It is concentrated, mathematically explicit, and tied to the exact place where accessibility first becomes realization-effective.

Appendix G. Perturbation Bounds and Detectability Proof

This appendix gives the full perturbative and detectability analysis underlying Theorem 6. Its purpose is to show that the critical-regime signature derived in Appendix F is not merely a formal feature of the exact noiseless model, but remains experimentally resolvable under bounded detector, erasure, environmental, and calibration imperfections. The core task is therefore comparative rather than idealized. One must show not merely that the instantiated canonical CBR response differs from the exact baseline in principle, but that this difference survives the ordinary distortions of the declared platform strongly enough to remain identifiable as a non-baseline signal.

The logic of the appendix is organized as follows. First, the full perturbation model is stated explicitly. Second, each perturbation channel is bounded in a way compatible with the exact declared platform. Third, the way these perturbations deform both the baseline and the CBR response is analyzed. Fourth, a detectability condition is proved: if the exact signature scale exceeds the total perturbative envelope in the critical regime by a sufficient margin, then the CBR signal remains statistically resolvable. Finally, the resulting detectability region is defined in parameter space. Throughout, the platform, the accessibility construction, the primary observable, and the baseline class remain exactly those fixed in the earlier appendices. No new freedom is introduced here.

G.1. Full perturbation model

Let V_model(η) denote the exact model-level response associated with either the baseline or the instantiated canonical CBR theory. The experimentally observed response is written as

V_obs(η) ＝ V_model(η) ＋ δ_det(η) ＋ δ_erase(η) ＋ δ_env(η) ＋ δ_cal(η).

The decomposition is intentionally structured by physical source rather than by purely formal convenience. The term δ_det(η) collects detector-originated perturbations, including finite efficiency, channel asymmetry, timing jitter, and dark counts. The term δ_erase(η) collects distortions associated with imperfect erasure, residual distinguishability, and retrieval impurity. The term δ_env(η) collects environmental leakage, undeclared decoherence, thermal drift, and stray entanglement. The term δ_cal(η) collects distortions induced by uncertainty in the accessibility calibration itself, including both horizontal uncertainty in η and the induced vertical uncertainty in the reconstructed visibility.

This decomposition is not meant to imply that the perturbations are mathematically independent in all microscopic realizations. It is meant to define the exact error channels that matter for the main-text theorem structure. Detectability will therefore be established against a total perturbative envelope formed from these declared contributions.

G.2. Detector perturbation bounds

Detector perturbations change the measured visibility even when the underlying signal state is unchanged. In the exact platform, the dominant effects are finite detection efficiency, path-dependent detection asymmetry, timing jitter, and background dark counts.

Let ε_det > 0 be a calibrated detector envelope such that

|δ_det(η)| ≤ ε_det

for all η in the experimentally sampled domain. This bound is meaningful only if detector calibration is carried out independently of the theory test. In particular, one must measure detector asymmetries and dark-count rates without using the critical-regime hypothesis to fit them away.

A simple model helps clarify the visibility distortion. Let the ideal interference extrema be I_max and I_min, and let the observed extrema under detector perturbation be

I_max^obs ＝ I_max ＋ Δ_max,
I_min^obs ＝ I_min ＋ Δ_min,

where |Δ_max| and |Δ_min| are bounded by detector calibration. Then the observed visibility is

V_det ＝ (I_max^obs − I_min^obs)/(I_max^obs ＋ I_min^obs).

Expanding to first order in the perturbations shows that the induced visibility shift is bounded by a constant proportional to the detector envelope, provided the denominator remains bounded away from zero. The exact constant depends on the working point of the experiment, but for the purposes of Volume V it is enough to absorb that dependence into the calibrated constant ε_det. The important point is that detector perturbations are bounded uniformly and do not themselves generate a critical-regime derivative break unless the detector settings are illicitly correlated with η tuning.

G.3. Erasure and retrieval perturbation bounds

The retrieval branch and the erasure branch are the most structurally delicate parts of the exact platform, since the accessibility variable η is tied directly to how records are made available or neutralized. Imperfect basis control, incomplete erasure, and retrieval impurity can therefore affect the visibility in ways that are particularly dangerous if left unbounded.

Let ε_erase > 0 be a calibrated bound such that

|δ_erase(η)| ≤ ε_erase

for all η in the sampled domain. This envelope includes three main subeffects.

The first is residual distinguishability after nominal erasure. If the erasure operation fails to recombine the record basis perfectly, the conditional recovery visibility will be suppressed relative to the ideal branch logic.

The second is retrieval impurity. If the retrieval protocol does not reconstruct the path information with the fidelity assumed by the accessibility calibration, then η itself may be mischaracterized and the vertical visibility response may be distorted.

The third is branch-control impurity. If the apparatus does not implement the intended retrieval or erasure branch cleanly, then mixed branch logic can mimic or wash out local structure.

These effects are absorbed into ε_erase only after independent characterization. That independence requirement is essential. A theory cannot claim detectability if the very branch logic that defines η is left floating and retrospectively interpreted.

G.4. Environmental perturbation bounds

Environmental perturbations represent all undeclared physical couplings not already accounted for by the exact baseline model. These include thermal drift, leakage of record information into uncontrolled degrees of freedom, stray entanglement with apparatus modes, and ordinary decoherence channels that are not part of the declared smooth baseline class.

Let ε_env > 0 be a bound such that

|δ_env(η)| ≤ ε_env

for all η in the sampled domain. This bound is not a claim that the environment is unimportant. It is a claim that once the environment’s ordinary effect has been incorporated into the declared perturbed baseline class, the residual unmodeled contribution remains bounded.

This distinction matters. Standard environmental decoherence that is already part of the declared smooth baseline should not be double-counted as a perturbation. The term δ_env(η) is reserved for the residual part that is neither theoretically declared nor instrumentally calibrated away inside the baseline comparator. The detectability theorem therefore assumes that the experimenter has done the ordinary work of baseline environmental characterization before asking the CBR discrimination question.

G.5. Accessibility calibration uncertainty

The most distinctive perturbation of the Volume V platform is the one induced by uncertainty in η itself. Since η is not directly measured but reduced from the primitives R, P, T, D, and S, its uncertainty can affect both the horizontal placement of the data and the apparent vertical response through slope-dependent propagation.

From Appendix B,

η ＝ (R · P · T · (1 − D) · S)^(1/5),

with first-order relative uncertainty

δη/η ≈ (1/5)√[(δR/R)² ＋ (δP/P)² ＋ (δT/T)² ＋ (δD/(1 − D))² ＋ (δS/S)²]

under independent primitive uncertainties.

If the model response is V_model(η), then the induced calibration perturbation is, to first order,

δ_cal(η) ≈ (dV_model/dη) δη.

For the baseline response,

V_SQM(η) ＝ 1 − η,

dV_SQM/dη ＝ −1,

and therefore

|δ_cal^SQM(η)| ≤ δη.

For the exact CBR response,

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

one has

dV_CBR/dη ＝ −1 for η < η_c,
dV_CBR/dη ＝ −(1 ＋ κ) for η > η_c,

away from the exact critical point. Therefore

|δ_cal^CBR(η)| ≤ (1 ＋ κ)δη

in the postcritical regime. Define ε_cal as a uniform calibration envelope satisfying

|δ_cal(η)| ≤ ε_cal

throughout the sampled domain, with ε_cal chosen conservatively to dominate the largest slope-weighted calibration uncertainty in the critical and postcritical windows.

This term is particularly important because the signature of the theory is localized near η_c. A poor accessibility calibration can smear the location of the critical regime even if the vertical visibility data are otherwise clean.

G.6. Total perturbative envelope

Let the total perturbative envelope be

ε_tot ＝ ε_det ＋ ε_erase ＋ ε_env ＋ ε_cal.

Then, for every η in the sampled domain,

|V_obs(η) − V_model(η)| ≤ ε_tot.

This bound is immediate from the triangle inequality applied to the full perturbation model:

The exact value of ε_tot depends on platform quality, calibration, and control discipline. But once declared, it gives the unique perturbative scale against which the critical-regime signature must be compared.

G.7. Exact separation scale in the critical regime

The exact response difference between the instantiated CBR model and the baseline is

ΔV(η) ＝ V_CBR(η) − V_SQM(η) ＝ −κ max{0, η − η_c}.

Thus the exact separation magnitude is

|ΔV(η)| ＝ κ max{0, η − η_c}.

Fix a critical-regime window

U_δ ＝ [η_c − δ, η_c ＋ δ] ∩ [0,1],

with δ > 0 chosen small enough that the main-text local regularity assumptions remain valid and large enough to contain experimentally resolvable postcritical data. Then the maximal exact separation on this window is

sup_{η ∈ U_δ} |ΔV(η)| ＝ κδ.

This explicit formula is useful because it ties detectability directly to two exact model parameters: κ, which measures the strength of the realization-law correction, and δ, which measures how far into the postcritical regime the experiment is able to sample while still remaining in the critical window.

G.8. Proof of the Robust Detectability Theorem

We now prove Theorem 6.

Theorem 6 (Robust Detectability Theorem). Let the exact declared platform, the exact baseline response V_SQM(η), and the exact instantiated CBR response V_CBR(η) be given as in the main text and Appendices A–F. Suppose the observed response satisfies

V_obs(η) ＝ V_model(η) ＋ δ_det(η) ＋ δ_erase(η) ＋ δ_env(η) ＋ δ_cal(η),

with bounded perturbations

for all η in a critical-regime window U_δ around η_c. If

κδ > 2ε_tot,

where ε_tot ＝ ε_det ＋ ε_erase ＋ ε_env ＋ ε_cal, then the primary critical-regime signature, or its fallback bounded-deviation version under smoothed transition, remains statistically resolvable from the declared perturbed baseline class on U_δ.

Proof

Let V_SQM^obs and V_CBR^obs denote the observed responses associated with the perturbed baseline and perturbed CBR hypotheses, respectively. Then for every η ∈ U_δ,

|V_SQM^obs(η) − V_SQM(η)| ≤ ε_tot,
|V_CBR^obs(η) − V_CBR(η)| ≤ ε_tot.

Hence the observed bands around the two exact models have half-width ε_tot.

Now choose η★ ∈ U_δ in the postcritical regime such that η★ − η_c ＝ δ whenever η_c ＋ δ ≤ 1. Then the exact separation at η★ is

|V_CBR(η★) − V_SQM(η★)| ＝ κδ.

If κδ > 2ε_tot, then the perturbed bands centered on V_CBR(η★) and V_SQM(η★) do not overlap, because the distance between their centers exceeds the sum of their half-widths. Therefore there exists at least one point in the critical-regime window at which the observed CBR-class response cannot be absorbed into the perturbed baseline class.

This proves resolvability in the strong-form case.

For the smoothed weak-form case, replace the exact separation function κ max{0, η − η_c} by the smoothed response difference |ΔV_w(η)| derived in Appendix F. If there exists η★ ∈ U_δ such that

|ΔV_w(η★)| > 2ε_tot,

then the same disjoint-band argument applies, and the fallback bounded deviation class remains resolvable. Thus the theorem holds in both strong and weak forms.

G.9. Interpretation of the detectability condition

The detectability condition

κδ > 2ε_tot

has a clear physical meaning. The left-hand side is the scale of the exact theory-induced separation accessible in the chosen critical window. The right-hand side is the total perturbative blur introduced by ordinary imperfections. Detectability is therefore achieved exactly when the experiment reaches far enough into the postcritical regime, and with enough control precision, that the theory’s signature exceeds the total blur by a factor sufficient to prevent band overlap.

The condition is intentionally conservative. It does not require statistical sophistication beyond disjoint confidence-band logic, although more refined model-comparison methods may strengthen the case once this baseline threshold is crossed. It also clarifies what the theory is and is not entitled to demand. If κ is very small, or if the experiment never probes beyond an extremely narrow neighborhood of η_c, or if ε_tot is too large, then the signal may be real but not practically detectable. That does not refute the theorem. It identifies the region in which the theorem’s empirical promise becomes experimentally accessible.

G.10. Detectability region in parameter space

The detectability region of the exact platform is the set of control and calibration conditions satisfying all of the following.

First, the accessibility window must include a postcritical segment of width at least δ > 0 such that η_c ＋ δ lies inside the experimentally sampled range.

Second, the exact model parameters must satisfy κδ > 2ε_tot.

Third, the calibration uncertainty must be small enough that the critical window is not smeared into effective invisibility. In practical terms, this means ε_cal must be subdominant compared with the postcritical separation scale κδ.

Fourth, the perturbative envelopes ε_det, ε_erase, and ε_env must be bounded independently of the theory fit and must not themselves contain undeclared η-correlated discontinuities.

Thus the detectability region may be written abstractly as

𝓓_det ＝ {(κ, δ, ε_tot) : κδ > 2ε_tot, δ > 0, η_c ＋ δ within sampled domain}.

More refined parameterizations may be added in supplementary technical work, but this exact condition is sufficient for the theorem structure of Volume V.

G.11. False-positive and false-negative implications

The detectability proof also clarifies the basic false-positive and false-negative logic.

A false positive occurs if an ordinary perturbation generates an apparent critical-regime structure of size comparable to or larger than the expected CBR signal. This is exactly why the bounded envelopes must be declared independently and why η-correlated apparatus drift is excluded unless explicitly modeled.

A false negative occurs if the exact CBR signal exists but the experiment fails to satisfy the detectability inequality. This may happen because κ is too small for the accessible window, because δ is too narrow, because ε_tot is too large, or because the calibration uncertainty overwhelms the local signal. In such a case the experiment is inconclusive with respect to the theorem’s practical test, not a disproof of the underlying exact model.

This distinction is important because Theorem 7, the binary invalidation theorem, only applies once the detectability conditions of the present appendix have been satisfied. A null result in a nondetectable regime is not scientifically decisive. A null result in a detectable regime is.

G.12. Strong-form and weak-form detectability

The proof above shows that the same detectability logic governs both the strong and weak signature forms.

In the strong form, the experimental target is the local derivative change at η_c. Detectability then means that the piecewise slope structure induces a large enough local response separation to outrun the perturbative envelope.

In the weak form, the derivative break may be smoothed, but the response still leaves the perturbed baseline class through a bounded deviation band. Detectability then means that this band is separated from the baseline by more than 2ε_tot somewhere in the critical window.

Thus the theorem does not rely on literal visibility of an ideal mathematical kink. It relies on the more robust condition that the critical-regime departure be larger than the total declared perturbative uncertainty.

G.13. Why this appendix makes the theory technically serious

Without a perturbative detectability proof, the critical-regime signature would remain formally elegant but experimentally underdeveloped. The present appendix is therefore not auxiliary. It is the point at which the theory becomes technically serious in the ordinary scientific sense.

The exact platform now has:

an exact baseline response,
an exact canonical CBR response,
an exact critical-regime signature,
an exact perturbative envelope,
an exact detectability criterion.

That chain is what makes the later binary invalidation theorem non-rhetorical. The theory is not merely saying, “if the signal is visible, believe it.” It is saying, “under these declared bounds, the signal should remain visible; if the platform still returns only baseline behavior, the exact instantiated theory fails.”

G.14. Conclusion of the appendix

This appendix has completed the proof of the Robust Detectability Theorem for the exact main-text model. Under bounded detector, erasure, environmental, and calibration perturbations, the critical-regime signature or its fallback bounded-deviation form remains statistically resolvable whenever the exact separation scale in the critical window exceeds twice the total perturbative envelope. The result identifies a concrete detectability region in parameter space and clarifies the conditions under which null results are merely inconclusive and the conditions under which they become scientifically decisive. The exact instantiated theory is therefore not only non-equivalent and locally distinctive. It is, under the stated conditions, detectably so.

Appendix H. Binary Invalidation Logic

This appendix gives the full formal logic underlying Theorem 7, the Binary Invalidation Theorem, for the exact instantiated model of Volume V. Its function is to specify, without ambiguity or retreat, what must be true before a null result counts as genuine invalidation of the instantiated canonical CBR law and what exactly is invalidated when that condition is met. The need for such an appendix is structural rather than rhetorical. A theory acquires scientific seriousness not only by deriving a signature but by fixing the conditions under which failure of that signature counts against the theory itself rather than against experimental execution, calibration quality, or model comparison discipline.

The appendix proceeds in five stages. First, it defines the exact objects whose freezing is required before invalidation can be meaningful. Second, it formalizes the null-result class on the declared platform. Third, it states the logical and experimental preconditions for valid invalidation. Fourth, it proves the binary invalidation theorem in full. Fifth, it specifies exactly what survives and what does not survive a negative outcome. The central theme throughout is that invalidation is not a matter of disappointment or weak evidential drift. It is a matter of exact model failure once the declared burden has been made finite, local, and detectable.

H.1. Exact objects that must be frozen before invalidation is possible

A null result can invalidate a theory only if the theory has already fixed the objects that define its empirical burden. In the exact main-text model of Volume V, those objects are the following.

First, the law form is frozen. The instantiated theory is defined by

Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ),

with

ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ).

No later reinterpretation is permitted to replace this structure with a looser or differently weighted rule once the test is underway.

Second, the platform is frozen. The experiment is the exact accessibility-tunable delayed-choice quantum eraser declared in Appendix A, with its signal subsystem, idler-record subsystem, retrieval and erasure branches, and exact timing structure.

Third, the accessibility variable is frozen. The theory is tested with

η ＝ (R · P · T · (1 − D) · S)^(1/5),

using the exact calibration procedures of Appendix B. No post hoc redefinition of η is allowed in order to rescue the theory from a null result.

Fourth, the observable is frozen. The primary observable is visibility V reconstructed from the exact platform. Secondary observables may clarify analysis but cannot replace visibility as the principal test quantity of the main-text model.

Fifth, the baseline comparator is frozen. The exact standard comparator is

V_SQM(η) ＝ 1 − η,

together with the declared perturbed baseline class 𝒮_baseline^pert defined by admissible smooth deformations under bounded detector, erasure, environmental, and calibration uncertainty.

Sixth, the instantiated CBR response is frozen. The exact response law is

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

or its weak-form smoothed counterpart when explicitly declared before comparison.

Seventh, the detectability condition is frozen. By Appendix G, invalidation may be claimed only in the regime where the critical-regime signature would have been resolvable had it existed at the scale required by the exact theory.

These seven frozen objects are the precondition for binary invalidation. Without them, a null result might count against a rough idea or a family of interpretations. With them, it counts against one exact instantiated model.

H.2. Formal definition of the null-result class

The null-result class must be defined in exact model-comparison terms rather than in vague language such as “no anomaly was seen.” Let V_obs(η) denote the observed visibility response on the declared platform, and let 𝒮_baseline^pert denote the perturbed baseline class consisting of all admissible smooth-response functions consistent with the exact baseline and the calibrated perturbative envelope.

Then the observed dataset belongs to the null-result class, written

V_obs ∈ 𝒩_null,

if and only if all of the following hold.

First, for every experimentally sampled η in the declared domain, and in particular throughout the critical-regime window U_δ around η_c,

V_obs(η)

remains within the declared perturbed baseline envelope around some member of 𝒮_baseline^pert.

Second, there is no statistically significant derivative break, kink, or local regularity failure near η_c exceeding the perturbative tolerance declared in Appendix G.

Third, there is no bounded non-baseline deviation class in the critical regime whose amplitude exceeds the detectability threshold established by Theorem 6.

Fourth, the experimental coverage of η is sufficient to include both a neighborhood below η_c and a postcritical neighborhood above η_c of width at least the declared detectability width δ.

In compact form, 𝒩_null is the set of datasets that are fully absorbable into the declared perturbed baseline class across the tested accessibility domain without violating any predeclared smoothness, tolerance, or detectability conditions.

This definition is deliberately stronger than “the data look smooth.” It requires that the data remain baseline-class even after the critical region has been properly probed and after the perturbative budget has been honestly declared.

H.3. Preconditions for valid invalidation

A null result does not automatically invalidate the theory. It does so only if a specific set of experimental and logical preconditions are satisfied. These must be stated exactly.

H.3.1. Platform-validity condition

The experiment must instantiate the exact declared platform to the degree required by the model. If the signal–idler architecture, branch logic, or observable definition has materially drifted away from the main-text declaration, then the test is no longer a test of the exact instantiated model.

H.3.2. Calibration-validity condition

The accessibility variable η must be calibrated using the exact primitive definitions and reduction rule of Appendix B. If η is only nominally varied while its calibrated uncertainty is uncontrolled or if the declared primitive quantities are not actually measured, then a null result is not yet theory-invalidating because the horizontal control parameter of the theory has not been properly realized.

H.3.3. Detectability-validity condition

The experimental regime must satisfy the detectability condition of Appendix G. In particular, if the exact signature would not have exceeded the total perturbative envelope even under the theory, then absence of signal does not invalidate the theory. It only shows that the experiment operated outside the detectable regime.

H.3.4. Baseline-integrity condition

The perturbed baseline class must be declared before decisive comparison and must not be enlarged after the fact to absorb any signal or null result opportunistically. Likewise, it must not be artificially restricted to make the theory seem stronger than it is. The comparator must be honest, fixed, and symmetric.

H.3.5. Perturbation-accounting condition

Detector, erasure, environmental, and calibration perturbations must all be independently bounded as required in Appendix G. If a null result occurs in the presence of unconstrained perturbations large enough to blur the critical regime, it is not yet invalidation. It is experimental ambiguity.

H.3.6. Domain-coverage condition

The experiment must probe sufficiently across the accessibility domain to include the declared critical region near η_c and a postcritical segment where the exact instantiated CBR model predicts departure from baseline. A null result entirely confined to the subcritical region cannot invalidate a theory whose signature turns on only after η_c.

These preconditions are stringent by design. Binary invalidation should be difficult to claim casually. But once these conditions are met, a null result becomes decisive rather than merely suggestive.

H.4. Formal statement of binary invalidation

We now state the theorem in its full logical form.

Theorem 7 (Binary Invalidation Theorem). Let the exact declared platform of Volume V be instantiated with the frozen law form, the frozen accessibility construction, the frozen observable, the frozen exact baseline comparator, and the frozen instantiated canonical CBR response. Suppose the experiment satisfies the platform-validity, calibration-validity, detectability-validity, baseline-integrity, perturbation-accounting, and domain-coverage conditions stated above. If the observed dataset V_obs belongs to the null-result class 𝒩_null, then canonical CBR in this instantiated form is false.

The theorem is binary because the exact instantiated model leaves no residual main-text flexibility once the above conditions are met. The response either leaves the perturbed baseline class in the critical regime as required by the theory, or it does not. If it does not, the exact theory has failed.

H.5. Proof of the theorem

The proof is a direct synthesis of the previous appendices.

By Appendix D, the exact instantiated CBR law yields the response

V_CBR(η) ＝ 1 − η − κ max{0, η − η_c},

with κ > 0 and η_c ∈ (0,1), or the declared weak-form smoothing preserving nontrivial critical-regime deviation.

By Appendix E, this response is not globally identical to the exact baseline

V_SQM(η) ＝ 1 − η

once accessibility enters the realization law nontrivially.

By Appendix F, the non-equivalence concentrates at η_c either as a derivative break or as a bounded non-baseline deviation class in a neighborhood of η_c.

By Appendix G, if the detectability condition is satisfied, then the critical-regime signature or its fallback deviation class is statistically resolvable against the declared perturbed baseline class.

Now suppose the experiment satisfies all validity conditions and yet produces a dataset V_obs in the null-result class 𝒩_null. Then, by definition of 𝒩_null, the dataset exhibits only baseline-class behavior throughout the tested domain, including the critical region, and no resolvable signature of the type the exact theory requires. But this contradicts the combined consequences of Appendices E, F, and G under valid detectability conditions. Therefore the instantiated exact model cannot be true. Hence canonical CBR in this instantiated form is false.

H.6. Why the theorem is genuinely binary

The phrase “binary invalidation” can be misunderstood if taken rhetorically rather than logically. The theorem is binary not because real experiments are perfectly clean, but because the instantiated model is logically sharp. Once the frozen objects and validity conditions are satisfied, there are only two possibilities:

the critical-regime signal or its weak-form counterpart appears at the required scale, or
the experiment remains inside the null-result class.

There is no third main-text option in which the exact instantiated theory is correct while the exact declared platform behaves only baseline-smoothly in a regime where the theory says it should be detectably different. Any such third option would require changing the law, changing η, changing the baseline, changing the observable, changing the platform, or denying detectability validity. But each of those changes would amount to leaving the exact instantiated theory rather than preserving it.

That is why the theorem counts as real invalidation rather than mere Bayesian discouragement.

H.7. What is invalidated

If the theorem’s conditions are met and a null result occurs, the following are invalidated:

the exact instantiated response law of Volume V,
the exact accessibility-sensitive transition structure at η_c,
the exact claim that the declared platform should leave the perturbed baseline class in the tested detectable regime.

More generally, the specific main-text realization of canonical CBR on the exact delayed-choice accessibility-tunable quantum eraser is invalidated. The law, the platform, and the model-level empirical claim stand or fall together.

H.8. What is not invalidated

The invalidation theorem is exact, but it is not maximally global. Several things are not invalidated by failure of the exact main-text model.

First, the broader philosophical idea that a realization law may be needed in quantum foundations is not logically refuted by the failure of one exact platform-level instantiation.

Second, alternative noncanonical CBR variants are not automatically refuted. One might construct a different accessibility reduction, a different admissibility structure, or a different exact platform and obtain a different empirical theory. That would be a new model requiring new derivation and new risk.

Third, realization-law approaches outside the specific CBR formalism are not automatically ruled out. The theorem is about one exact instantiated canonical law, not about every conceivable alternative.

These clarifications do not weaken the invalidation logic. They strengthen it by ensuring that the theorem claims no more than it has earned.

H.9. Distinguishing invalidation from inconclusiveness

It is important to mark the difference between a null result that invalidates the model and a null result that remains inconclusive.

A null result is inconclusive if any of the validity conditions fail. For example:

η was not calibrated with sufficient precision,
the experiment did not probe beyond η_c by a detectability-width δ,
the total perturbative envelope ε_tot was too large,
the branch logic was not implemented faithfully,
the baseline comparator was not fixed honestly.

In such a case the theory is not preserved by success, but neither is it killed by failure. The experiment has simply not reached the conditions required for decisive model comparison.

A null result is invalidating only when the theory had a fair chance to succeed on its own terms and did not.

This distinction is essential for scientific honesty. A theory should not be invalidated by a bad test. But once a good test has been made and the theory still returns only baseline behavior, invalidation must follow without evasion.

H.10. Practical invalidation checklist

For clarity, the exact main-text model is invalidated only if all of the following are true:

the declared delayed-choice accessibility-tunable platform was actually realized,
η was calibrated using the exact reduction rule and declared primitive measurements,
the experiment sampled a critical-region neighborhood around η_c and a postcritical domain,
the perturbation bounds were independently established,
the detectability condition of Appendix G was satisfied,
the observed data remained within the declared perturbed baseline class and showed neither the strong-form nor weak-form signature.

This checklist is not a substitute for the theorem. It is the operational form of the theorem.

H.11. Why this appendix completes the scientific logic of Volume V

Without this appendix, Volume V would remain a strong theory paper with a fixed signature and robustness analysis, but its failure logic could still be treated as partially rhetorical. The present appendix closes that gap. It shows that the exact instantiated model has reached the condition scientific theories must eventually reach: a finite experimental program exists under which success counts as support and clean failure counts as theory death.

That does not make the theory true. It makes it real.

H.12. Conclusion of the appendix

This appendix has completed the formal invalidation logic of Volume V. Once the law, platform, accessibility variable, observable, baseline comparator, and detectability conditions are all frozen and satisfied, a null result on the declared platform does not merely weaken the instantiated canonical CBR model. It falsifies it. What survives is not the exact model but only broader or different realization-law possibilities. This is the correct scientific posture for the volume: exact enough to be wrong, disciplined enough to say so, and finite enough to be tested.

Appendix I. Variant Implementations

This appendix records a small number of closely related platform variants that may, in future work, extend the empirical program of Volume V without altering the exact main-text model. Its purpose is deliberately limited. It does not reopen the platform choice of the volume, and it does not weaken the force of the main theorems by returning to an undifferentiated “protocol family” formulation. On the contrary, the whole point of Volume V was to bind the theory to one exact experimental object strongly enough that it could become empirically vulnerable. That exact object remains the accessibility-tunable delayed-choice quantum eraser declared in the main text.

The present appendix therefore has only a secondary role. It shows that the main-text architecture is not an isolated formal curiosity and that a disciplined extension program may exist beyond the exact declared platform. But these variants are not substitutes for the main-text model, and no theorem in Volume V depends on them. They are included only to indicate where the logic of accessibility-sensitive realization might later be tested, refined, or generalized once the exact platform-level burden of the main text has been confronted.

I.1. Why variants are placed in an appendix

The decision to place alternative implementations in an appendix rather than the main body is methodological. A theory becomes scientifically legible by surviving or failing on one fixed object, not by diffusing its burden across many loosely related settings. Had the main text attempted to develop multiple implementations in parallel, it would have weakened the exactness of the law–platform–observable relationship that gives the current volume its strength. The appendix placement therefore preserves the correct hierarchy:

the main text contains the exact platform actually used to derive the theory’s empirical burden,
the appendix records nearby implementations that may later extend or stress-test the same conceptual structure.

This distinction is essential. None of the variants below may be invoked to rescue the main-text theory from a null result on the exact declared platform. If the main-text platform fails under the validity conditions of Appendix H, then that instantiated canonical form fails. A variant implementation would then constitute a new model, not a reinterpretation of the old one.

I.2. Criteria for admissible variant implementations

A platform counts as a relevant variant of the main-text model only if it preserves the following core structural features.

First, it must contain a signal subsystem whose interference behavior is directly measurable through a visibility-like or coherence-sensitive observable.

Second, it must contain a record-bearing subsystem whose state becomes correlated with the signal alternatives and whose accessibility may be tuned independently of trivial signal destruction.

Third, it must permit a meaningful distinction between mere record correlation and operational record accessibility, so that an accessibility variable analogous to η can be constructed in disciplined form.

Fourth, it must support a delayed retrieval, delayed erasure, or delayed publication structure rich enough that the accessibility-sensitive realization logic is not immediately reducible to ordinary early-time measurement/no-measurement dichotomies.

Fifth, it must allow a clean standard comparator and an exact CBR instantiation to be written on the same platform without changing the meaning of the main law objects.

Any platform lacking one of these features may be interesting for other purposes, but it is not a natural extension of the Volume V logic.

I.3. Variant A: Alternative photonic delayed-choice quantum eraser realizations

The nearest class of variants consists of other photonic delayed-choice quantum eraser implementations differing from the exact main-text architecture in engineering detail but not in logical structure. Examples include altered beam-splitter arrangements, polarization-based rather than path-based idler encoding, or modified retrieval branches in which public accessibility P and redundancy spread S are tuned through distinct photonic routing architectures.

Such variants remain close to the main text because they preserve all of the essential ingredients:

a coherent signal interference channel,
an idler record subsystem,
a delayed-choice branch structure,
an accessibility-like control variable,
a visibility-based observable.

Their main value would be technical rather than conceptual. They could test whether the critical-regime signature remains stable under changes in photonic implementation and whether the exact accessibility reduction used in the main text has a natural photonic generalization or requires platform-specific recalibration.

However, these variants must be treated with care. If the accessibility construction is modified, if the signal observable changes materially, or if the retrieval branch no longer maps cleanly onto the main-text η logic, then the result is no longer the same exact model. It becomes a neighboring model in the same general empirical family, which may be worth studying but cannot be used to reinterpret the original main-text theorems.

I.4. Variant B: Atom-interferometric record-accessibility platforms

A more substantial variant class would replace the photonic signal–idler implementation with an atom-interferometric architecture. In such a platform, the signal subsystem would be an atom interferometer or a related matter-wave two-path system, while the record-bearing sector would be encoded in internal states, coupled ancillae, or controlled environmental record channels. Retrieval fidelity, temporal stability, and destructive burden might then be implemented very differently from the photonic case, but the central logic could remain intact if accessibility were still calibratable as a structured record-availability variable.

The appeal of such a variant is that atom-interferometric platforms often provide a different balance among coherence time, environmental coupling, and delayed manipulation. This could make the critical-regime structure easier to isolate in some respects and harder in others. For example, temporal stability T may become easier to control over longer windows, while destructive readout burden D may become more problematic depending on the implementation.

Such a platform would be highly valuable for future work because it would test whether the realization-law signature depends strongly on the medium or whether the accessibility-sensitive structure is genuinely medium-independent at the level of model class. But again, this would not be the same exact theory test as the one in the main text. It would be a second exact instantiation requiring a second derivation.

I.5. Variant C: Sequential measurement implementations

Another plausible extension is a sequential measurement platform in which the record-bearing structure is not simply a one-shot idler channel but a staged sequence of partially accessible records created over time. In such a platform, the signal system would undergo successive weak or partial couplings to record-bearing subsystems, and accessibility would depend not merely on a single retrieval act but on the cumulative structure of staged record formation.

The attraction of such a variant is that it may allow a finer-grained analysis of how accessibility becomes realization-relevant. Instead of one transition from low to high accessibility, the experiment could realize a structured progression in which retrieval fidelity R, public accessibility P, and redundancy spread S evolve across a measurement sequence. This could provide a more detailed test of whether the critical-regime logic of Volume V is truly tied to a single threshold η_c or whether it generalizes to staged thresholding in a richer context.

The difficulty is also clear. Sequential platforms introduce additional timing structure, possible memory effects, and more complicated admissibility analysis. A single scalar η may cease to be sufficient, or may need to be replaced by a trajectory η(t) or a sequence {η_n}. Such a move would be conceptually interesting but would leave the exact one-parameter structure of the main-text model behind. It therefore belongs properly in later work rather than in the present theorem spine.

I.6. Variant D: Multi-record retrieval architectures

A further extension would involve platforms in which which-path or outcome-defining information is distributed across multiple distinct record sectors rather than one idler channel. In such a case, the record-bearing system would be genuinely multipartite, and accessibility would depend not only on the quality of a single retrieval channel but on the joint structure of several partially redundant or partially conflicting record carriers.

This variant is attractive because it stresses precisely the part of the accessibility construction that was compressed into the redundancy spread variable S in the main text. In a multi-record architecture, the distinction between a fragile single record and a robustly distributed record becomes experimentally richer and potentially more revealing. One might then test whether the realization-law transition sharpens as redundancy increases, whether η_c shifts systematically with the number of accessible record carriers, or whether the critical-regime morphology fragments into a more complex structure.

At the same time, such architectures would likely demand a higher-dimensional accessibility construction and a more complicated baseline class. They are therefore better understood as a future generalization of Volume V rather than as a reinterpretation of the present results.

I.7. Variant E: Delayed-publication or delayed-dissemination platforms

The main-text platform included public accessibility P as one ingredient in η, but it did not isolate delayed dissemination as its own major control axis. A related variant would do exactly that: the record would be formed and even privately retrievable at an earlier stage, but its dissemination to multiple independent readout channels would be delayed or tunably suppressed. In such a platform, the transition from private record to public record could be made more explicit and might allow a sharper empirical separation between mere existence of a record and broader operational accessibility.

This variant is conceptually valuable because it tests one of the most delicate interpretive points in the accessibility construction: the difference between a record that can be recovered by one fragile route and a record that has become physically available across several routes or observers. If canonical CBR is truly sensitive to accessibility rather than only to correlation, then a delayed-dissemination variant may expose that dependence more transparently.

The danger, however, is that such a variant could tempt one back toward ambiguous language about “knowledge” or “observation.” Any future implementation of this type would need to remain as physically strict as the main text: dissemination must be defined in terms of actual record availability across physical channels, not in terms of human awareness.

I.8. What these variants are good for

The variant implementations recorded here are useful for three reasons.

First, they show that the exact main-text model is not an isolated contrivance. The accessibility-sensitive realization logic has plausible nearby extensions.

Second, they identify where the current theory is most likely to gain empirical reach if the main-text platform proves fruitful: through alternative media, richer record structures, and more refined accessibility control.

Third, they indicate where the current one-parameter model may eventually need refinement. In particular, sequential and multi-record implementations may require replacing a scalar η with a more structured accessibility object.

These are real advantages. But they are future-facing advantages, not hidden resources of the present volume.

I.9. What these variants are not good for

It is equally important to state what these variants cannot do.

They cannot be used to soften the force of the main-text failure criterion. If the exact declared platform is tested under the validity conditions of Appendix H and returns only baseline-class behavior, then the exact instantiated main-text model is false. One cannot then appeal to a different photonic layout, an atom interferometer, or a multi-record system and say that the original exact theory was not really what was meant.

They also cannot be used to enlarge the baseline comparator or relax the accessibility definition in the middle of the original test. The whole point of Volume V was to avoid precisely that kind of retreat into diffuse framework language.

Finally, these variants cannot be treated as if they already inherit the theorem spine of the main text. Each would require its own:

exact model declaration,
exact accessibility construction,
exact baseline derivation,
exact CBR derivation,
exact non-equivalence proof,
exact detectability analysis,
exact invalidation logic.

Until that work is done, these remain disciplined possibilities, not theorem-bearing alternatives.

I.10. Why the appendix ends here

This appendix is intentionally brief. To say more would risk undermining the exactness that the volume worked to achieve. The correct scientific order is now clear. The main-text platform must be judged on its own terms first. Only after that judgment is made does it become useful to develop one of the variants above into a new exact instantiation.

The point of this appendix is therefore modest but important: it shows that the accessibility-sensitive realization program can in principle grow outward from the exact model of Volume V without returning to vagueness. But it also preserves the central discipline of the volume: one exact theory, one exact platform, one exact burden.

I.11. Conclusion of the appendix

This appendix has identified a small set of disciplined variant implementations — closely related photonic realizations, atom-interferometric architectures, sequential measurement systems, multi-record retrieval platforms, and delayed-dissemination setups — that may later extend the empirical program of Volume V. None of them alters the exact force of the present main-text model. They are future exact instantiations waiting to be built, not fallback interpretations of the one already given. That is the correct place for them: adjacent to the main theory, but outside its current theorem spine.

Theorem Spine of Volume V

The exact instantiated theory developed in Volume V is closed by four theorems. Together they define the full empirical burden of the model. Their order is logically necessary. A platform-level realization-law theory must first be shown to differ from its declared standard comparator, then be shown to differ in a determinate local regime and morphology, then be shown to retain that difference under bounded nonidealities, and only then be assigned a valid failure condition. Volume V is organized to satisfy exactly those burdens and no others.

Theorem 4 — Baseline/CBR Non-Equivalence Theorem
Let the exact declared accessibility-tunable delayed-choice quantum eraser of Volume V be given, with accessibility variable η ∈ [0,1], critical accessibility value η_c ∈ (0,1), exact standard comparator V_SQM(η), and exact instantiated canonical realization law Φ★C ＝ arg min{Φ ∈ 𝒜(C)} ℛ_C(Φ), where ℛ_C(Φ) ＝ αΞ_C(Φ) ＋ βΩ_C(Φ) ＋ γΛ_C(Φ). If the accessibility-sensitive burden term Λ_C contributes nontrivially to the minimization ordering and induces a genuine regime change in the minimizing realization class, then the induced response V_CBR(η) is not globally identical to V_SQM(η) on the admissible accessibility domain. Thus, once accessibility is realization-effective in the instantiated law, global baseline equivalence is excluded.

Theorem 5 — Critical-Regime Signature Theorem
Under the exact main-text regularity assumptions, the non-equivalence between V_CBR(η) and V_SQM(η) is localized at η_c and appears as a critical-regime derivative break, or kink, in the primary observable. In particular, the instantiated response remains continuous at η_c but fails to belong to the declared baseline smooth-response class in any neighborhood of η_c because its local regularity changes there. If the stronger regularity assumptions required for a literal derivative discontinuity are weakened while the accessibility-sensitive realization transition remains nontrivial, then a bounded non-baseline deviation class persists on a nonempty neighborhood of η_c. Accordingly, the theorem identifies both the location and the admissible morphology of the theory’s first empirical manifestation.

Theorem 6 — Robust Detectability Theorem
Let the observed response satisfy the declared perturbation model with detector, erasure, environmental, and calibration contributions bounded by an independently established tolerance envelope. If the exact critical-regime separation scale of the instantiated CBR response exceeds the total perturbative envelope by the required detectability margin on the sampled postcritical window, then the critical-regime signature remains statistically resolvable from the declared perturbed baseline class. In the strong-form regime, the kink-type local structure survives at the level required for discrimination. In the weak-form regime, the bounded non-baseline deviation class remains separated from all admissible baseline-class responses by the declared tolerance criterion. The theorem therefore establishes perturbative survivability of the instantiated signature under exactly the nonidealities the platform is entitled to face.

Theorem 7 — Binary Invalidation Theorem
Let the canonical law form, the exact experimental platform, the accessibility construction, the primary observable, the declared baseline comparator, and the detectability-valid test regime all be fixed exactly as stated in Volume V. If the observed dataset remains entirely within the declared baseline-class response envelope across the experimentally accessible η-domain, including the sampled critical and postcritical regimes, then canonical CBR in this instantiated form is false. This conclusion is binary because, once the relevant objects are frozen and the validity conditions are satisfied, no residual main-text freedom remains by which the theory may preserve itself while returning only baseline-class behavior in the very domain where it claims resolvable departure.

These four theorems give the instantiated model its complete scientific form. The first excludes global collapse into the standard comparator. The second fixes the local regime and exact structural form in which the difference must appear. The third proves that bounded ordinary imperfections do not automatically erase that difference. The fourth states the exact condition under which absence of the required difference is absence of the theory. Nothing essential to the empirical status of the instantiated model lies outside this sequence. Its logical order is therefore final: non-equivalence, critical-regime manifestation, perturbative resolvability, and binary invalidation. In that order, the theory becomes not merely sharpened, but closed as a testable object.

Robert Duran IV