Constraint-Based Realization: A No-Internal-Alternative Theorem for Outcome Realization

Conditional Uniqueness of the Canonical Law-Form within a Burden-Bearing Class of Outcome-Realization Theories

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Abstract

Let 𝓣 denote the class of outcome-realization theories satisfying context specification, nonempty admissibility, pre-outcome fixation, non-circular comparison, operational uniqueness or pre-declared tie discipline, Born compatibility or explicitly bounded deviation, non-reduction to ordinary decoherence, parameter fixity, and explicit defeat conditions. The principal result of this paper is a No-Internal-Alternative Theorem: within 𝓣, every burden-satisfying outcome-realization theory is canonically equivalent to Constraint-Based Realization up to operational equivalence. Equivalently, once the law-candidate burdens are held fixed, no materially distinct internal alternative survives.

That result addresses a precise residual gap in the CBR program. The reconstruction paper establishes that a disciplined law of individual outcome realization naturally takes CBR-form: a physically specified context C, an admissible class 𝒜(C), a context-fixed burden structure ℛ_C, and a selected realization channel Φ∗_C defined up to operational equivalence. The canonical closure paper then states a mature law-candidate in closed form, with restricted uniqueness, local probability closure within canonical admissibility, operational accessibility, and a finite strong-null failure condition. What remained open was not whether one canonical CBR object could be written down, but whether another materially distinct internal law-form could satisfy the same full burden structure without reducing to that canonical object.

The argument proceeds in three steps. First, a burden-preserving representation lemma shows that any theory in 𝓣 admits comparison with canonical CBR at the level of context, admissible quotient structure, burden ordering, and selected operational verdict class. Second, an embedding dichotomy shows that any candidate either enters that comparison without burden loss, violates at least one class-defining burden, or exits 𝓣. Third, a noncanonical departure analysis shows that every materially noncanonical departure must appear as admissibility leakage, hidden weighting, unresolved operational multiplicity, decoherence reduction, or burden-structure deformation. The conclusion is therefore conditional rather than universal. It is not empirical confirmation of CBR, not closure over all conceivable interpretive or metaphysical frameworks, and not a replacement for platform-level empirical instantiation. Its claim is narrower and stronger for present purposes: within a declared burden-bearing theorem class, the canonical CBR law-form is conditionally unique.

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1. Introduction

1.1 The exact remaining gap

Constraint-Based Realization has already been developed beyond the stage of suggestive framework. The reconstruction paper identifies the minimal representational structure required of a disciplined law of individual outcome realization. On that account, a candidate realization law must be posed over a physically specified context C, a nonempty admissible class 𝒜(C), a context-fixed burden structure ℛ_C, an operational equivalence relation ≃_C, and a selected realization channel Φ∗_C or selected operational verdict class. The canonical closure paper then compresses the mature law-candidate into a single theorem-bearing object. It fixes a restricted admissibility architecture, states a canonical realization law, proves restricted uniqueness up to operational equivalence, establishes local probability closure within canonical admissibility, introduces an operational accessibility parameter, and places the instantiated model under a finite public failure burden.

Those achievements are substantial, but they do not yet settle one further question. A canonically stated and finitely exposed law-candidate may still leave open the possibility that some materially distinct internal alternative satisfies the same burden structure while differing in realization content. The present paper is written to address exactly that residual possibility. Its concern is not whether CBR can be formulated coherently, nor whether one mature canonical version has now been specified. Its concern is whether any materially distinct internal rival survives once the same class-defining burdens are held fixed.

The gap is therefore exact. The reconstruction paper shows why a burden-bearing realization law naturally takes CBR-form. The canonical closure paper shows how one mature CBR law-candidate can be canonically fixed, narrowed, and exposed. Neither paper, by itself, proves that no other materially distinct internal outcome-realization theory remains available within the same burden-bearing class. That is the target of the present work.

1.2 Why this gap matters

A law-candidate becomes stronger not merely by becoming coherent, not merely by being stated canonically, and not merely by incurring a finite empirical burden. It becomes stronger when the internal space of admissible alternatives contracts. So long as a materially distinct internal rival can plausibly claim to satisfy the same burdens, the canonical object remains one serious candidate among several internally surviving possibilities. Once that internal rival space is removed, the status of the canonical object changes. The central issue is no longer whether one can formulate a serious realization law. It becomes whether the only burden-satisfying internal law-form that remains is vindicated by the world.

This point is methodological before it is metaphysical. The CBR program already distinguishes evolution, registration, and realization. Evolution concerns ordinary quantum dynamics. Registration concerns the formation of record-bearing structure. Realization concerns the further law-governed selection of an outcome-channel once the relevant dynamical and record structure is in place. A program that has already imposed that distinction, restricted admissibility, required non-circularity, preserved probability discipline, refused reduction to ordinary decoherence, and accepted explicit defeat conditions cannot indefinitely leave its internal rival space unspecified. At that stage, either materially distinct internal alternatives remain, or they do not. A mature paper must say which.

1.3 Target of the present paper

The present paper asks one exact question:

Within a declared burden-bearing class of outcome-realization theories, does any materially distinct internal rival survive?

That question is narrower than a general comparison among interpretations of quantum mechanics, narrower than a universal closure claim across all conceivable realization-law architectures, and narrower than an empirical verdict on nature itself. It is also sharper than a generic request for uniqueness. The issue here is not whether CBR can be rewritten in another notation, nor whether operationally inert reformulations can be eliminated. The issue is whether a theory that remains materially distinct in realization content can still satisfy the same full burden structure while targeting the same problem of individual outcome realization.

The target is therefore conditional and internal. “Conditional” means that the result is claimed only within a declared theorem class. “Internal” means that the rival, to count as relevant, must accept the same target of explanation and the same class-defining burdens rather than escape by changing the subject. What is at stake is not total closure over every philosophical or metaphysical alternative, but the elimination, or survival, of internal burden-satisfying alternatives to the canonical law-form.

1.4 Main theorem

The central result is stated at the outset because the remainder of the paper is organized to support it and because nothing is gained by delaying the exact claim.

No-Internal-Alternative Theorem. Within the declared burden-bearing class of outcome-realization theories, every burden-satisfying candidate is canonically equivalent to CBR up to operational equivalence.

The force of this theorem is specific. It does not say that every conceivable realization-law proposal collapses to CBR. It does not say that CBR is empirically confirmed. It does not say that all non-CBR interpretations of quantum mechanics are defeated. It says something narrower and stronger for the present theorem class: once the class-defining burdens are held fixed, no materially distinct internal alternative survives.

The remainder of the paper is devoted to making that statement exact. It first fixes the theorem class and its imported objects. It then defines what counts as an internal rival, what counts as material distinctness, and what it would mean for a rival to survive. It next proves that any candidate theory in the class must either admit burden-preserving comparison with canonical CBR, fail one of the class-defining burdens, or leave the class. It then shows that every genuinely noncanonical departure pays a cost somewhere within the burden structure. The main theorem follows from those two steps together.

1.5 Main contributions

The paper makes four contributions, each subordinate to the same exact aim. First, it declares an explicit theorem class for internal comparison. Rather than speak loosely of “alternatives,” it identifies the class of outcome-realization theories to which the theorem applies and fixes the burdens required for membership in that class. Second, it develops a burden-preserving representation and embedding framework. This prevents a putative rival from evading comparison by appeal to mere differences of presentation, notation, or formal packaging. Third, it proves a no-internal-alternative theorem. The result is not that CBR is one well-behaved candidate among several internally surviving forms, but that no materially distinct internal law-form survives once the class-defining burdens are fixed. Fourth, it states exact defeat conditions for the strengthened claim. The paper therefore does not seek safety through vagueness. It specifies what would count against its own central result.

1.6 What this paper does not claim

The paper does not claim that CBR is experimentally confirmed. It does not claim universal closure over all realization-law alternatives. It does not claim final universal Born-neutrality closure across every admissibility geometry. It does not replace platform-level empirical instantiation, calibration, baseline validation, nuisance control, or detectability analysis. It does not refute every interpretation of quantum mechanics, and it does not settle every metaphysical dispute associated with measurement. Its claim is conditional, class-bounded, and internal to a declared burden-bearing comparison class.

These limits are not protective qualifications added after the fact. They are part of the method. A theorem of the present kind becomes stronger by stating exactly what it is entitled to establish and refusing everything beyond that scope. The reconstruction paper already proceeds by a conditional burden-to-structure argument rather than by a claim of empirical vindication. The canonical closure paper already distinguishes its restricted and local achievements from stronger universal claims it does not make. The present paper follows the same discipline.

1.7 Program placement

The reconstruction paper asks why a disciplined realization law naturally takes CBR-form. The canonical closure paper states the completed CBR law-candidate in canonically compressed form. The empirical exposure papers specify where context-sensitive versions of that law-candidate become vulnerable to test. The present paper has one narrower role: to determine whether, within the declared burden-bearing theorem class, any materially distinct internal rival remains once the law-candidate burdens are fixed.

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2. Fixed objects and declared theorem class

2.1 Fixed objects

The present paper does not redevelop the CBR architecture from the beginning. It imports the canonical comparison structure already fixed elsewhere and treats that structure as standing. The imported objects are: the physically specified measurement context C; the admissible class 𝒜(C) of realization-compatible candidates; the context-indexed burden functional ℛ_C; the selected realization channel Φ∗_C; and the operational equivalence relation ≃_C. These are not introduced here as optional notation. They are the fixed terms relative to which internal rival survival is assessed.

Their roles must remain exact. The context C denotes the physical situation relevant to individual outcome realization, not merely an observable label. The admissible class 𝒜(C) is the restricted class surviving the standing admissibility burdens, not the set of all formally writable candidates. The burden functional ℛ_C is the context-fixed comparison structure under which admissible candidates are evaluated, not a retrospective scoring device. The selected channel Φ∗_C is the selected realization channel or operational verdict class under the declared law-form, not a label attached after the fact to whatever occurred. The relation ≃_C removes physically irrelevant formal multiplicity and identifies the level at which uniqueness claims in this paper are to be read.

2.2 Inherited burden discipline

The present paper inherits, rather than reproves, the burden discipline already used to reconstruct and constrain the CBR law-form. The relevant burdens are: context specification, admissibility restriction, non-circular comparison, operational uniqueness, Born compatibility or formally declared bounded deviation, non-reduction to ordinary decoherence, parameter fixity, and explicit vulnerability to failure. These are not newly introduced safeguards. They are the standing conditions under which a realization-law proposal was already argued to count as disciplined rather than ad hoc.

Each burden performs a distinct anti-arbitrariness role. Context specification prevents the theory from floating above the physical setup it is supposed to address. Admissibility restriction prevents the candidate space from being widened until any verdict can be engineered. Non-circular comparison prevents the law from favoring the very outcome it is supposed to explain. Operational uniqueness prevents unresolved multiplicity from masquerading as selection. Born compatibility prevents probability from becoming a free internal design choice. Non-reduction to decoherence prevents the theory from collapsing into a non-selective map while still claiming independent realization content. Parameter fixity prevents post hoc tuning. Explicit vulnerability to failure prevents the law-candidate from insulating itself against defeat. The theorem to be proved concerns what survives when these burdens are held fixed.

2.3 Definition of the theorem class 𝓣

Let 𝓣 be the class of outcome-realization theories satisfying the following conditions.

A theory in 𝓣 must define a physically meaningful context rather than appeal to an unspecified or retrospectively redescribed measurement situation. It must define a nonempty admissible candidate class relative to that context. That class must be fixed prior to the outcome whose realization is at issue. The theory must supply a context-fixed comparison rule sufficient to distinguish among admissible candidates. It must either secure operational uniqueness or provide a pre-declared tie discipline compatible with the rest of the burden structure. It must preserve Born compatibility unless a bounded deviation is explicitly declared in advance and treated as part of the theory rather than as a rescue maneuver. It must not reduce entirely to ordinary non-selective decoherence. It must fix all parameters relevant to comparison before the selected verdict is known. It must state explicit defeat conditions.

The significance of 𝓣 is methodological. It is the comparison class the present paper is entitled to address while still speaking about disciplined outcome-realization laws rather than arbitrary formal constructions. The main theorem is not about every conceivable proposal one might imagine. It is about theories that continue to target the same problem while accepting the same minimum burdens required for a serious law-candidate.

2.4 Internal rival

A theory in 𝓣 counts as an internal rival only if it targets the same explanatory question as canonical CBR, namely a law-form for individual outcome realization, while claiming to satisfy the same basic burden structure. This excludes two easy escapes. A proposal does not count as an internal rival merely because it differs verbally from CBR while changing the target of explanation. Nor does it count as an internal rival merely because it rejects one or more class-defining burdens and therefore leaves 𝓣 altogether.

The point of “internal” is therefore exact. An internal rival is not simply any theory outside CBR. It is a theory that remains inside the burden-bearing comparison space while purporting to differ materially in realization content. The question of the paper is whether such a rival survives.

2.5 Material distinctness

A rival is materially distinct only if it differs in realization content, not merely in formal packaging. Difference of notation, parameter naming, quotient choice, symbolic normalization, or operationally inert order-preserving transformation does not suffice. Nor does a reformulation that leaves the admissible ordering, selected operational verdict class, and defeat structure unchanged. To count as materially distinct, a rival must alter law-content rather than presentation.

This definition is necessary because a theorem about internal alternatives cannot be allowed to proliferate trivial counterexamples. If every syntactic restatement counted as a different theory, no uniqueness result would be meaningful. The paper therefore adopts a law-content criterion of distinctness rather than a merely textual one. The operational level is decisive: a rival counts as materially distinct only if it differs in the realization role performed by the theory, not merely in how that role is described.

2.6 Burden-preserving embedding

A rival burden-preservingly embeds into canonical CBR only if it reproduces the same realization role without illicitly changing the burdens that define membership in 𝓣. More exactly, a burden-preserving embedding requires that admissibility discipline, uniqueness discipline, weighting discipline, parameter discipline, and defeat structure remain intact under comparison with the canonical law-form. A rival does not embed burden-preservingly if it preserves only the appearance of the selected verdict while silently widening admissibility, importing hidden weighting, relaxing uniqueness, loosening parameter fixity, or softening defeat conditions.

This notion is therefore stronger than superficial representability. It is not enough that a rival can be restated in symbols resembling the canonical structure. It must preserve the burden-bearing role of the law. The later embedding theorem depends on precisely that distinction.

2.7 Operational reading convention

All identity, uniqueness, and equivalence claims in this paper are read at the level of operational meaning rather than bare symbolic presentation. This convention does not trivialize the formal content of the paper. It prevents law-level claims from being held hostage by physically irrelevant differences in expression. When the main theorem states that a burden-satisfying theory in 𝓣 is canonically equivalent to CBR up to operational equivalence, the point is not that every symbol must coincide. The point is that no materially distinct law-content survives once operationally irrelevant formal multiplicity is removed.

Accordingly, “uniqueness” in the present paper does not mean typographic uniqueness under every conceivable reparameterization. It means conditional uniqueness of the law-form at the level at which physical verdicts, admissible ordering, and defeat structure are genuinely fixed.

2.8 Assumption Registry

The theorem architecture of the paper rests on the following standing assumptions.

A1. Dynamical compatibility. The realization law does not covertly replace ordinary quantum dynamics outside realization selection.

A2. Context specification. Each candidate theory defines the physical context to which its realization claim applies.

A3. Nonempty admissibility. For each admissible context, the candidate class is nonempty.

A4. Representational invariance. Physically irrelevant reformulations do not alter realization verdicts.

A5. Record-structural relevance. Realization depends only on physically relevant record-bearing structure.

A6. Pre-outcome fixation. Admissibility, comparison structure, and relevant parameters are fixed before the selected verdict is known.

A7. Operational uniqueness discipline. The theory secures operational uniqueness or a pre-declared tie discipline.

A8. Probability compatibility discipline. Born compatibility is preserved unless a bounded deviation is explicitly declared in advance.

A9. Non-reduction to decoherence. Independent realization content is not exhausted by an ordinary non-selective decoherence-compatible map.

A10. Parameter fixity. Adjustable parameters are not tuned after the relevant verdict is known.

A11. Admissibility stability under refinement. Admissibility is stable under physically legitimate refinement within the declared comparison class.

A12. Burden monotonicity. Burden comparisons behave monotonically under physically irrelevant expansion and inert reformulation.

A13. Quotient regularity. The admissible quotient under operational equivalence is sufficiently well-behaved for comparison and selection arguments.

A14. Explicit defeat structure. The theory states conditions under which its realization-law claim fails.

Each later theorem cites the assumptions it uses. The registry exists to keep every subsequent claim tied to declared scope and to prevent the strengthened result from floating above its actual burden base.

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3. What would it take for an internal rival to survive?

3.1 Survival criterion

An internal rival survives only if two conditions are simultaneously met. First, it must remain materially distinct from canonical CBR in realization content. Second, it must preserve the full burden-bearing law-role required for membership in 𝓣. Either condition alone is insufficient. Mere difference without burden preservation does not produce a serious internal alternative; it produces burden failure or class exit. Apparent burden preservation without material distinctness does not produce a genuine rival; it produces re-description.

The survival criterion is stringent by design. The present paper is not concerned with every way one might speak differently about measurement. It is concerned only with whether another law-form, still targeting individual outcome realization and still bearing the same disciplinary burdens, can remain genuinely distinct while doing the same explanatory work.

3.2 Why prior papers do not yet settle survival

The earlier anchor papers do not settle this question because their jobs are different. The reconstruction paper asks what representational structure naturally arises once a realization-law candidate accepts explicit burdens. Its answer is that CBR-form is the minimal structural answer to that demand. The canonical closure paper asks how one mature CBR law-candidate can be canonically stated, narrowed, and exposed to finite empirical failure. Its answer is a theorem-bearing canonical object with restricted uniqueness, local weighting closure, accessibility structure, and a strong-null failure criterion.

Neither paper, however, needs to prove that another materially distinct internal law-form cannot satisfy the same burdens. Reconstruction is about structural naturalness, not internal elimination. Canonical closure is about mature canonical specification, not exhaustion of internal alternatives. The present paper therefore occupies a genuinely new place in the architecture of the program. Its question is not why CBR can arise, nor how one CBR law-candidate is canonically fixed, but whether any other materially distinct internal law-form remains once the same burden structure is fixed.

3.3 Internal escape routes

If an internal rival is to survive, it must do so through one of a limited number of routes.

The first is admissibility inflation: the rival appears to preserve the law-role by widening the admissible class beyond the canonical one while claiming that the widening is harmless. The second is hidden weighting insertion: the rival appears to preserve verdicts by covertly importing weighting or preference structure not fixed independently of the outcome. The third is unresolved operational multiplicity: the rival leaves more than one operationally distinct candidate standing while still claiming to have specified a realization law. The fourth is decoherence-reductive pseudo-selection: the rival retains only the appearance of realization content while collapsing in substance to a non-selective decoherence-compatible description. The fifth is burden-structure deformation under verdict preservation: the rival preserves the same outcome labels or selected verdicts in chosen cases while altering the burden architecture that gave those verdicts their law-level meaning.

These five routes suffice for the present paper because they exhaust the ways a materially distinct internal rival could attempt to survive without openly rejecting the theorem class itself. If a rival escapes by changing the explanatory target, discarding the burden discipline, or leaving 𝓣, then it is no longer an internal rival in the relevant sense.

3.4 Why these are the right escape routes

Each route marks a different form of slippage between apparent and genuine survival. Admissibility inflation allows a rival to claim greater generality while concealing arbitrariness inside an enlarged candidate space. Hidden weighting allows it to preserve selected verdicts while importing undeclared preference structure. Unresolved multiplicity allows it to appear selective while declining to choose among operationally distinct candidates. Decoherence reduction allows it to retain the language of realization while surrendering independent realization content. Burden-structure deformation allows it to preserve visible outputs while quietly altering the conditions under which those outputs count as the result of a disciplined law.

The importance of identifying these routes in advance is methodological. A theorem about internal alternatives must not proceed by saying, after the fact, that every failed rival “must have violated something.” It must specify in advance where survival could plausibly be attempted and then show why those routes do not remain open within the declared class. The present section is therefore not a rhetorical catalog. It is the finite map of where internal survival could still try to occur.

3.5 Proposition of internal survival exhaustion

The preceding analysis can now be stated in theorem-facing form.

Proposition (Internal Survival Exhaustion). Within 𝓣, any putative materially distinct internal rival must survive, if it survives at all, through at least one of the following forms: admissibility inflation, hidden weighting insertion, unresolved operational multiplicity, decoherence reduction, or burden-structure deformation under verdict preservation.

The proposition does not yet prove that no such rival survives. It does something prior and necessary. It closes the possibility of indefinite rival drift by exhausting the internal routes through which survival could be attempted. The later argument therefore need not answer an open-ended question of the form “what else might a rival do?” It needs only to show that the finite survival routes identified here do not remain available once burden-preserving comparison with canonical CBR is imposed.

A serious realization-law candidate must eventually show either that no internal burden-satisfying alternative remains, or exactly where the residual internal rival space still lies. The proposition above fixes the second task in finite form so that the first can be attempted non-rhetorically.

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4. Burden-Preserving Representation and Embedding

4.1 Comparative burden

Let T ∈ 𝓣. The present section establishes that T cannot remain an internal rival by claiming that its law-content is formally incomparable to canonical CBR. Once the class-defining burdens are fixed, comparison is forced at the level relevant to the present paper: physically specified context, admissible candidate structure, burden ordering, and selected operational verdict class. If a putative rival survives, it must survive as a burden-preserving alternative inside that comparison. It cannot survive by withdrawing into difference of notation, packaging, or raw symbolic form.

For T, write 𝒜_T(C) for its admissible candidate class in context C, ≃^T_C for its operational equivalence relation, and [Φ^T_∗]_C for its selected operational verdict class. Canonical CBR supplies the comparison data 𝒜(C), ≃_C, ℛ_C, and [Φ∗_C]. The task of the section is to show that any T ∈ 𝓣 admits comparison with those canonical objects on the admissible quotient, and that failure of such comparison is not a harmless formal fact. It is either burden failure or class exit.

4.2 Lemma 1 — Burden-Preserving Representation Lemma

Lemma 1. Let T ∈ 𝓣. Then T admits canonical comparison with CBR at the level of context, admissible quotient structure, burden ordering, and selected operational verdict class.

Proof sketch. Membership in 𝓣 already fixes the relevant comparison data. By context specification, T determines the physical situation to which its realization claim applies. By nonempty admissibility and pre-outcome fixation, it determines an admissible candidate class independently of the verdict to be explained. By non-circular comparison and parameter fixity, it determines a context-fixed comparison structure over that class. By operational uniqueness discipline, it determines either a selected operational verdict class or a pre-declared tie resolution compatible with the rest of the burden structure. By representational invariance and quotient regularity, comparison is properly conducted on 𝒜_T(C)/≃^T_C rather than on raw representatives. These objects are sufficient for canonical comparison with 𝒜(C)/≃_C, the ordering induced by ℛ_C, and [Φ∗_C].

The lemma establishes comparability, not identity. It removes only one refuge: a putative internal rival may no longer appeal to incomparability of form.

4.3 Admissible quotient comparison

All comparison in this paper is conducted on admissible quotient classes. Raw syntactic comparison is insufficient for three reasons. First, it confuses representational multiplicity with law-content. Two candidates may differ formally while remaining operationally indistinguishable in every realization-relevant respect. Second, uniqueness in the present paper is operational rather than typographic. The issue is whether one operational verdict class survives, not whether every possible symbolic representative has been collapsed. Third, burden-preserving embedding itself is well-posed only on the quotient. A candidate differing from canonical CBR solely by equivalent encoding, inert reparameterization, or order-preserving formal restatement is not materially distinct and must not be counted as such.

Accordingly, when T is compared with canonical CBR, the relevant objects are 𝒜_T(C)/≃^T_C and 𝒜(C)/≃_C together with their induced order structures and selected verdict classes. Comparison below that level introduces distinctions the present theorem is not entitled to treat as law-level distinctness.

4.4 Lemma 2 — Embedding Failure Lemma

Lemma 2. Let T ∈ 𝓣. If T cannot be brought into burden-preserving comparison with canonical CBR without altering at least one class-defining burden, then T is not a surviving internal rival.

Proof sketch. By definition, an internal rival must remain both materially distinct and burden-preserving within 𝓣. If T can be compared with canonical CBR only by relaxing admissibility restriction, abandoning pre-outcome fixation, weakening non-circular comparison, loosening uniqueness discipline, modifying probability discipline, surrendering non-reduction to decoherence, altering parameter fixity, or softening defeat conditions, then T no longer preserves the law-bearing burdens that qualify it for internal consideration. In that case T is either burden-defective or no longer a member of 𝓣. Neither possibility yields a surviving internal rival.

Within the present theorem class, incomparability is therefore not neutral. It is diagnostic.

4.5 Theorem 1 — Burden-Preserving Embedding Dichotomy

Theorem 1. Let T ∈ 𝓣. Then one of the following obtains:

(i) T enters burden-preserving comparison with canonical CBR up to operational equivalence;

(ii) T violates at least one class-defining burden;

(iii) T exits the theorem class 𝓣.

Proof sketch. Lemma 1 supplies canonical comparison data for any T ∈ 𝓣. Lemma 2 shows that failure of burden-preserving comparison cannot remain internal to 𝓣. If comparison fails only at the price of altering a class-defining burden, then either that burden is violated or T no longer belongs to 𝓣. Hence any T ∈ 𝓣 either enters burden-preserving comparison with canonical CBR or ceases to count as a burden-satisfying member of the class.

The theorem does not yet imply collapse to CBR. It does imply that a putative internal rival survives, if at all, only inside burden-preserving comparison.

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5. Noncanonical Departure Analysis

5.1 Restricted departure space

After Theorem 1, the problem is no longer open-ended. Any surviving internal rival must now appear as a burden-preserving departure within canonical comparison. The present section shows that materially noncanonical departure is finite in form. Every such departure must alter the law-bearing role of the theory through one of a small number of mechanisms, and each mechanism carries a determinate cost.

The purpose of the section is not yet elimination. It is exhaustion. Once the departure space is finite, the main theorem no longer bears the burden of answering an indefinite question of the form “what else might a rival do?” It need answer only whether the finite departure modes identified here remain available within 𝓣.

5.2 Lemma 3 — Admissibility Leakage Lemma

Lemma 3. Any genuine widening of admissibility beyond the canonical admissible class either introduces burden-violating candidates or is operationally inert.

Proof sketch. Let a putative rival enlarge admissibility relative to canonical CBR. If the enlargement leaves the admissible quotient, burden ordering, and selected operational verdict class unchanged, then the surplus admissibility does no realization work and therefore introduces no material distinctness. If it changes any of those items, then law-relevant candidates not admitted by canonical admissibility have been introduced. That requires burden loss at the level of admissibility restriction, record-structural relevance, pre-outcome fixation, or related standing conditions. Hence admissibility widening either disappears on the quotient or leaks into burden failure.

5.3 Lemma 4 — Hidden Weighting Lemma

Lemma 4. Any rival preserving apparent verdicts only by covert weighting import violates non-circularity, probability discipline, or parameter fixity.

Proof sketch. Suppose a rival preserves selected verdicts by introducing undeclared weighting or preference structure not fixed independently of the verdict at issue. If that structure is introduced after the relevant verdict is known, non-circularity fails. If it alters ensemble discipline without a pre-declared bounded deviation, probability compatibility fails. If it remains adjustable at comparison time, parameter fixity fails. If none of these failures occurs, then the weighting structure was already part of the fixed burden-bearing law-role and therefore does not constitute a covert noncanonical departure.

5.4 Lemma 5 — Multiplicity Failure Lemma

Lemma 5. Any rival leaving operationally distinct minima unresolved fails uniqueness discipline unless it supplies a pre-outcome tie rule that itself preserves the burden structure.

Proof sketch. Let a rival leave several operationally distinct admissible candidates co-minimal. If no tie discipline is supplied, operational uniqueness fails. If a tie discipline is supplied only after the relevant verdict is known, non-circularity and parameter fixity fail. If a tie discipline is supplied in advance but alters admissibility, weighting, or defeat structure, burden preservation fails. Only a pre-outcome tie rule preserving the standing burdens avoids these failures, and such a rule remains subject to the same comparison logic as canonical CBR.

5.5 Lemma 6 — Decoherence Reduction Lemma

Lemma 6. Any rival exhausted by a non-selective decoherence-compatible map fails as an independent realization law.

Proof sketch. Membership in 𝓣 includes non-reduction to ordinary decoherence. If a candidate adds no realization content beyond a non-selective decoherence-compatible channel, then it does not supply an independent law of individual outcome realization. It redescribes evolution or registration structure while leaving realization unaccounted for. Such a candidate therefore fails a class-defining burden and cannot survive as an internal rival.

5.6 Proposition — Noncanonical Departure Proposition

Proposition. Every materially noncanonical departure from canonical CBR appears, within 𝓣, as at least one of the following: admissibility leakage, hidden weighting, unresolved operational multiplicity, decoherence reduction, or burden-structure deformation.

Proof sketch. By Theorem 1, any surviving internal rival must enter burden-preserving comparison with canonical CBR. Once comparison is imposed on the admissible quotient, material non-canonicity must alter some law-bearing component of the candidate’s realization role. If it alters admissibility, Lemma 3 applies. If it preserves verdicts by covert preference import, Lemma 4 applies. If it leaves operationally distinct co-minima unresolved, Lemma 5 applies. If it adds no realization content beyond decoherence-compatible structure, Lemma 6 applies. Any remaining departure that preserves visible verdicts while altering the law-bearing conditions under which those verdicts are produced is burden-structure deformation. No further internal departure type remains once membership in 𝓣 and quotient-level comparison are held fixed.

The proposition exhausts the internal departure space. It does not yet show that no departure survives. It shows that there is no sixth route hiding outside the argument.

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6. Main Theorem

6.1 Reduced question

The argument is now reduced to a single question. By Theorem 1, any surviving internal rival must enter burden-preserving comparison with canonical CBR. By the proposition of Section 5, any materially noncanonical departure within that comparison must occur through one of a finite set of departure modes. The only remaining issue is whether any such mode remains compatible with burden-satisfying membership in 𝓣.

6.2 Theorem 2 — No-Internal-Alternative Theorem

Theorem 2. Within the declared burden-bearing class 𝓣, every burden-satisfying outcome-realization theory is canonically equivalent to CBR up to operational equivalence.

Proof sketch. Let T be a burden-satisfying theory in 𝓣. By Theorem 1, T must enter burden-preserving comparison with canonical CBR, since the only alternatives are burden violation or class exit, both excluded by hypothesis. Assume that T is nevertheless materially distinct from canonical CBR. Then, by the proposition of Section 5, its noncanonical distinctness must appear as admissibility leakage, hidden weighting, unresolved operational multiplicity, decoherence reduction, or burden-structure deformation.

Lemmas 3–6 eliminate the first four possibilities as burden-preserving modes of internal survival. The fifth possibility, burden-structure deformation, is not an independent refuge. If the deformation is operationally inert, then T is not materially distinct. If it is operationally effective, then some class-defining burden has been altered and T is no longer a burden-satisfying member of 𝓣. Hence no materially distinct internal rival survives within 𝓣. Therefore every burden-satisfying theory in 𝓣 is canonically equivalent to CBR up to operational equivalence.

This theorem is conditional and class-bounded. It does not show that every conceivable account of measurement collapses to CBR. It shows that, within 𝓣, no materially distinct internal alternative remains once the law-candidate burdens are held fixed.

6.3 Immediate consequence

A materially distinct internal rival does not survive. This is the operational content of the theorem. The result is not that CBR is merely one disciplined member of the class. It is that burden-satisfying membership in 𝓣 collapses to the canonical law-form up to operational equivalence.

6.4 What the theorem establishes

The theorem establishes conditional uniqueness of the canonical law-form within 𝓣. More exactly, it establishes that the burden-bearing realization-law role targeted in this paper has only one surviving internal form once operationally irrelevant multiplicity is removed. Within the declared theorem class, canonical CBR no longer stands as one candidate among an indefinite family of internally surviving alternatives. It stands as the only burden-satisfying internal law-form remaining after comparison, exhaustion of departure modes, and elimination of noncanonical survival.

The result is structural rather than empirical. It concerns the space of internal alternatives under fixed burdens. It does not by itself decide whether nature obeys the surviving law-form.

6.5 What the theorem does not establish

The theorem does not establish empirical truth. It does not establish universal closure across all conceivable interpretations, ontologies, or realization-law proposals. It does not establish any claim beyond 𝓣. It does not replace platform-level empirical instantiation, calibration, baseline validation, nuisance control, or detectability analysis. It does not eliminate the need for further work on the scope and sharpness of the burden class. Its force is internal, conditional, and bounded by the standing assumptions.

6.6 Corollary — Conditional Uniqueness of the Canonical Law-Form

Corollary. Once the theorem class 𝓣 and its burdens are fixed, no materially distinct internal burden-satisfying alternative remains to canonical CBR.

This corollary states Theorem 2 directly at the level of law-form identity. It is the exact sense in which the canonical CBR law-form is conditionally unique.

6.7 Corollary — Residual rivals are external, not internal

Corollary. Any remaining rival must reject the declared burden framework, alter the target of explanation, or leave the theorem class 𝓣.

This corollary fixes the boundary of the result. The paper has not shown that no rival of any kind exists anywhere in quantum foundations. It has shown that, after the present comparison is completed, any remaining rival is no longer an internal burden-satisfying alternative to canonical CBR. Whatever remains must therefore live outside the declared class, not within it.

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7. Functional Rigidity

7.1 Residual issue

Theorem 2 eliminates materially distinct internal rivals at the level of burden-satisfying law-form within 𝓣. One residual question remains. Could verdict-level collapse coexist with law-level underdetermination? More precisely, could ℛ_C be materially replaced by a distinct burden functional while preserving the selected operational verdict class and thereby leaving the canonical law-form non-rigid at its comparison core? The present section addresses that narrower issue.

The claim made here is limited. It is not a universal theorem of functional uniqueness across all conceivable realization laws. It is a class-bounded result internal to 𝓣. Its purpose is to show that, once the burdens defining 𝓣 are held fixed, material replacement of ℛ_C is possible only by departing from those burdens or by changing nothing of law-content.

7.2 Operationally inert transformations

Let ℛ_C be the canonical burden functional on 𝒜(C). A transformation τ of ℛ_C is operationally inert in context C if and only if it preserves all law-relevant comparison content on 𝒜(C)/≃_C. For the purposes of the present paper, that requires three conditions.

First, τ ∘ ℛ_C must preserve admissible ordering on 𝒜(C)/≃_C. If [Φ₁]_C and [Φ₂]_C are admissible quotient classes, then ℛ_C([Φ₁]_C) ≤ ℛ_C([Φ₂]_C) if and only if τ ∘ ℛ_C assigns them the same weak order. Second, τ ∘ ℛ_C must preserve the selected operational verdict class. The minimizer structure on the admissible quotient, modulo the standing tie discipline where relevant, must remain unchanged. Third, τ ∘ ℛ_C must preserve the burden-bearing role of the functional under admissibility restriction, non-circular comparison, parameter discipline, and defeat structure. A transformation that preserves selected labels in some cases while altering the operative comparison geometry is not inert.

Operational inertness is therefore stronger than numerical monotonicity taken in abstraction. What matters is not whether ℛ_C can be rewritten, but whether any law-relevant feature of comparison, selection, or defeat changes once comparison is conducted on the admissible quotient within 𝓣.

7.3 Proposition 1 — Ordering Preservation Proposition

Proposition 1. A transformed burden functional counts as the same law-content, in the sense relevant to this paper, only if it preserves both admissible ordering on 𝒜(C)/≃_C and the selected operational verdict class.

Proof sketch. ℛ_C functions as a comparison law over admissible candidates. If a transformation changes admissible ordering, then it changes which candidates are less burdened, equally burdened, or more burdened than others and therefore changes law-content directly. If ordering is preserved but the selected operational verdict class changes, then law-content again changes directly. Preservation of both quotient-level ordering and selected verdict structure is therefore necessary for sameness of law-content. It is also sufficient for the present paper, because no further law-relevant distinction survives once operational equivalence and the standing burden structure are held fixed.

7.4 Proposition 2 — Anti-Plasticity Proposition

Proposition 2. Any deformation of ℛ_C that preserves selected outcome labels while altering burden geometry in a materially significant way is illegitimate within 𝓣.

Proof sketch. Let ℛ′_C preserve selected labels in a chosen family of cases while differing materially from ℛ_C. If the deformation is inert on the admissible quotient, then no material distinctness has been introduced. If it alters quotient-level burden geometry in a law-relevant way, then one of two things occurs. Either the selected operational verdict structure changes, in which case verdict-level sameness fails, or verdict-level sameness is maintained only by compensating changes in admissibility, weighting, tie discipline, parameter choice, or defeat structure. In the latter case the deformation is not burden-preserving and the candidate leaves 𝓣 as a materially distinct internal alternative. Hence material plasticity of ℛ_C is incompatible with continued burden-preserving membership in 𝓣.

7.5 Corollary — Functional Rigidity

Corollary. Once the burden structure defining 𝓣 is held fixed, ℛ_C is determined up to operationally inert transformations preserving admissible ordering and selected operational verdict structure.

The corollary is narrower than Theorem 2 and subordinate to it. Its function is to block the residual claim that canonical CBR might be unique only at the level of selected verdict while remaining non-unique at the level of the burden law that produces that verdict. Within 𝓣, that separation is unavailable. Verdict-level collapse and law-content rigidity separate only by inert transformation.

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8. Counterexample Screening

8.1 Function of explicit screening

Sections 4–7 establish the result abstractly. The present section tests that result against explicit rival templates. This is not a change of method but a stress condition on the one already adopted. A no-internal-alternative theorem is stronger when the principal internal escape routes are instantiated and screened directly rather than left at the level of general assurance.

The rival families below are schematic constructions internal to the argument. They are designed to realize, in direct form, the plausible modes of attempted survival isolated earlier: admissibility widening, covert weighting, unresolved multiplicity, decoherence reduction, and burden-functional deformation. The question in each case is the same: does the construction remain a materially distinct internal alternative while preserving membership in 𝓣?

8.2 Counterexample family A — Widened-admissibility rival

Let T_A agree with canonical CBR on target, context, and comparison form, but replace 𝒜(C) with a strictly larger admissible class 𝒜_A(C) ⊋ 𝒜(C).

If the added candidates are inert on 𝒜_A(C)/≃_C, then T_A is not materially distinct from canonical CBR. If the added candidates alter quotient-level ordering or selected verdict structure, then admissibility has been widened in a law-relevant way. That widening can be sustained only by weakening admissibility restriction, record-structural relevance, or pre-outcome fixation. In the former case the rival collapses; in the latter it no longer remains a burden-satisfying member of 𝓣. Hence T_A does not survive as a materially distinct internal alternative.

8.3 Counterexample family B — Hidden-weighting rival

Let T_B preserve the visible verdicts of canonical CBR by importing an additional weighting map not fixed independently of the verdict at issue.

If the weighting enters only after the relevant verdict is known, non-circularity fails. If it alters effective ensemble discipline without a pre-declared bounded deviation, probability compatibility fails. If it remains adjustable within the comparison procedure, parameter fixity fails. If none of these failures occurs, then the weighting is already part of the fixed comparison structure and does not supply a materially distinct rival. Hence T_B does not survive as a materially distinct internal alternative.

8.4 Counterexample family C — Multiplicity-tolerant rival

Let T_C preserve the comparison framework but allow multiple operationally distinct co-minimizers to remain unresolved.

If no tie rule is supplied, uniqueness discipline fails. If a tie rule is supplied only after the relevant verdict is known, non-circularity and parameter discipline fail. If a tie rule is supplied in advance but alters admissibility, weighting, or defeat structure, burden preservation fails. If a tie rule is supplied in advance and preserves the standing burdens, then T_C re-enters the same comparison logic already governed by Theorem 1 and Section 5. In none of these cases does a materially distinct internal rival survive. Hence T_C fails.

8.5 Counterexample family D — Decoherence-reductive rival

Let T_D purport to describe individual outcome realization while adding no realization content beyond a non-selective decoherence-compatible map.

Such a candidate does not supply an independent law of realization. It redescribes evolution or registration structure while leaving realization unaccounted for. The non-reduction burden is therefore violated. T_D is not a burden-satisfying member of 𝓣 and does not survive as a materially distinct internal alternative.

8.6 Counterexample family E — Burden-deformed functional rival

Let T_E preserve selected labels in chosen cases while replacing ℛ_C by a materially different functional ℛ′_C.

If ℛ′_C preserves quotient-level ordering and selected operational verdict structure, then by Section 7 the deformation is operationally inert and introduces no material distinctness. If ℛ′_C alters burden geometry in a law-relevant way, then verdict preservation can be maintained only by compensating changes in admissibility, weighting, tie discipline, parameter choice, or defeat structure. In that case T_E no longer preserves membership in 𝓣 as a materially distinct internal rival. Hence T_E fails.

8.7 Screening summary proposition

Proposition. No rival in the screened families A–E survives as a materially distinct internal alternative while preserving membership in 𝓣.

Proof sketch. Family A collapses by admissibility inertness or admissibility leakage. Family B collapses by non-circularity failure, probability-discipline failure, or parameter-fixity failure. Family C collapses by uniqueness failure or by re-entry into the standing comparison structure. Family D collapses by non-reduction failure. Family E collapses by operational inertness or burden-structure deformation incompatible with 𝓣. No screened rival remains both materially distinct and burden-preserving.

No screened rival remains a materially distinct internal alternative while preserving the burdens of 𝓣.

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9. Exact Defeat Conditions

9.1 Necessity of explicit defeat

The paper claims conditional uniqueness of the canonical law-form within 𝓣. That claim is stronger than the earlier reconstruction and canonicalization steps in the program and therefore incurs a correspondingly sharper defeat burden. The result does not become stronger by being insulated from counterexample. It becomes stronger by specifying exactly what would break it.

The present section fixes those failure modes within declared scope. The list below is not ceremonial. It identifies the exact points at which the central argument could collapse: by survival of a genuine internal counterexample, by failure of the comparison framework, by illicit narrowing of the theorem class, or by collapse of the law-content rigidity the paper claims to preserve.

9.2 Defeat conditions

F1. A materially distinct rival in 𝓣 survives Theorem 2.
That is, some candidate remains both burden-satisfying and internally distinct from canonical CBR after the embedding and departure arguments have been applied.

F2. The burden-preserving embedding dichotomy fails.
That is, some theory remains a burden-satisfying member of 𝓣 while neither entering canonical comparison nor incurring burden failure nor exiting the class.

F3. A noncanonical admissibility structure survives without burden cost.
That is, admissibility can be widened or altered in a law-relevant way without operational inertness and without violation of the standing admissibility burdens.

F4. A decoherence-reductive candidate reproduces all realization content relevant to the paper’s target while remaining a burden-satisfying member of 𝓣.
In that case the non-reduction burden has been mischaracterized or overstated for the theorem class actually under consideration.

F5. There exists a burden-bearing internal rival space excluded from 𝓣 without justification from the paper’s own target, definitions, and standing assumptions.
The issue here is not merely that 𝓣 is bounded. The issue would be that the paper’s own target requires confrontation with burden-bearing internal alternatives that 𝓣 omits without argument.

F6. Functional rigidity collapses at the level of law-content.
That is, ℛ_C proves materially replaceable in a law-relevant way without either operational inertness or loss of class-defining burden preservation.

These defeat conditions are exhaustive for the paper in the following sense: any internal failure of the no-internal-alternative claim must appear either as survival of a genuine counterexample, collapse of the embedding framework, illegitimate narrowing of the theorem class, or collapse of the law-content rigidity claimed in Section 7.

9.3 Defeat Proposition

Proposition. If any of F1–F6 is exhibited within declared scope, the no-internal-alternative claim of this paper fails.

Proof sketch. F1 defeats the conclusion directly. F2 removes the hinge transition from class membership to forced comparison. F3 defeats the admissibility branch of the departure analysis. F4 defeats the non-reduction branch. F5 defeats the theorem’s significance by showing that the declared comparison class omits burden-bearing internal rivals required by the paper’s own target. F6 defeats the claim that verdict-level uniqueness is accompanied by law-content rigidity. Any one of F1–F6 therefore breaks a necessary component of the central result.

9.4 Force of explicit defeat

A law-candidate does not become stronger by insulating itself from possible defeat. It becomes stronger by stating exactly what would count against it. The present paper therefore presents conditional uniqueness together with explicit failure modes rather than apart from them. That is part of the theorem’s force, not a concession against it.

The central claim stands only so long as no counterexample satisfies F1–F6 within declared scope.

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10. Consequence for the Empirical Program

10.1 What the present result changes

The present paper does not confirm CBR empirically. It changes the theoretical meaning of empirical testing. If the canonical law-form is conditionally unique within 𝓣, then a valid platform-level instantiation no longer tests one burden-satisfying internal candidate among several. It tests the only burden-satisfying internal law-form remaining within the declared class. Accordingly, if such an instantiation fails under its own locked conditions, that failure cannot be absorbed by appeal to another burden-satisfying internal alternative within 𝓣.

The point is limited and exact. The present result does not convert a protocol-specific null into a universal refutation of CBR. It establishes something narrower: once an instantiation genuinely represents the canonical law-form under the standing burdens, internal fallback within 𝓣 has already been removed. The empirical burden thereby becomes sharper in significance, not broader in scope.

10.2 What the present result does not replace

The present paper discharges none of the empirical burdens already incurred elsewhere in the program. It does not construct η. It does not validate a comparator. It does not separate nuisance structure. It does not set detectability budgets, sensitivity floors, or platform-level locking conditions. It does not transform a finite, protocol-specific empirical program into a universally platform-independent one.

Those tasks remain exactly where they belong: at the level of explicit instantiation. The present result changes what empirical failure would mean once such instantiation is achieved. It does not supply the instantiation itself.

10.3 Empirical sequel

The natural empirical sequel is a single locked platform-level instantiation in which context, η, comparator, nuisance envelope, sensitivity floor, and verdict rule are fixed in advance.

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11. Relation to Alternatives

11.1 Decoherence-only accounts

The present theorem does not address decoherence as an account of interference suppression, record stabilization, or effective classicality. Its target is narrower: a law of individual outcome realization. A decoherence-only position enters the present comparison only if it claims to satisfy that target while remaining within 𝓣. Otherwise it lies outside scope.

11.2 Collapse-style law proposals

The present paper is not a survey of modified dynamics. Its target is a burden-bearing law of outcome realization under a standing distinction among evolution, registration, and realization. A collapse-style proposal enters the present comparison only if it accepts that target and those burdens. If it changes the explanatory problem by rewriting the underlying dynamics, it is external to the theorem.

11.3 Branching or multiplicity ontologies

The present theorem is not an argument against multiplicity ontology as such. Its target is single-outcome law-form. A branching view that denies that target is not refuted here. A branching-style proposal becomes relevant only if it simultaneously claims to remain within the burden-bearing class of outcome-realization theories. Only that internal claim is excluded.

11.4 Hidden-variable or selection-rule approaches

The present comparison is not a total classification of hidden-variable or selection-rule programs. It is an internal burden-based result. Such a proposal enters the theorem only if it accepts the present target and the burdens defining 𝓣. If it does, it faces the same embedding, departure, and rigidity constraints. If it does not, it lies outside scope.

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12. Conclusion

12.1 Result

The paper has not widened CBR and has not claimed universal closure. It has done one narrower thing: it has removed the remaining internal space in which a materially distinct burden-satisfying alternative could survive within 𝓣.

12.2 Final claim

Within the declared burden-bearing class 𝓣, the canonical CBR law-form is conditionally unique up to operational equivalence.

12.3 Final sentence

This paper does not prove that nature obeys CBR. It proves that, within the declared theorem class, no materially distinct internal burden-satisfying alternative remains.

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Theorem Spine

Lemma 1 — Burden-Preserving Representation Lemma.
For every theory T ∈ 𝓣, the realization claim of T admits quotient-level comparison with canonical CBR at the level of admissibility, burden ordering, and selected operational verdict class.

Lemma 2 — Embedding Failure Lemma.
If a candidate theory cannot enter burden-preserving comparison with canonical CBR without altering a class-defining burden, then it does not survive as an internal rival.

Theorem 1 — Burden-Preserving Embedding Dichotomy.
For every theory T ∈ 𝓣, exactly one of the following obtains:
(i) T enters burden-preserving comparison with canonical CBR;
(ii) T violates at least one standing burden;
(iii) T exits 𝓣.

Proposition — Noncanonical Departure Proposition.
Within 𝓣, every materially noncanonical departure occurs through at least one of the following: admissibility leakage, hidden weighting, unresolved operational multiplicity, decoherence reduction, or burden-structure deformation.

Theorem 2 — No-Internal-Alternative Theorem.
Within 𝓣, every burden-satisfying outcome-realization theory is canonically equivalent to CBR up to operational equivalence.

Corollary — Conditional Uniqueness of the Canonical Law-Form.
Once the burdens defining 𝓣 are fixed, no materially distinct internal burden-satisfying rival remains.

Corollary — Functional Rigidity.
Once the burden structure defining 𝓣 is fixed, ℛ_C is determined up to operationally inert transformations preserving quotient-level ordering and selected operational verdict structure.

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Appendix A. Assumption Ledger

This appendix records the standing assumptions of the paper and the exact work each performs. For each assumption, three items are fixed: statement, first indispensable use, and failure if dropped. No assumption is included for convenience alone.

A1. Dynamical compatibility

Statement. The realization law does not covertly rewrite ordinary quantum evolution outside the realization task.
First indispensable use. Lemma 6.
Failure if dropped. The target of the paper shifts from a law of realization to an unrestricted modification of dynamics; the comparison class is no longer fixed.

A2. Context specification

Statement. Each candidate theory fixes a physically specified context C for its realization claim.
First indispensable use. Lemma 1.
Failure if dropped. Admissibility, ordering, and verdict selection lack determinate domain; quotient-level comparison cannot be stated.

A3. Nonempty admissibility

Statement. For each admissible context, the candidate class is nonempty.
First indispensable use. Lemma 1.
Failure if dropped. Comparison and minimization become vacuous; no realization law is under evaluation.

A4. Representational invariance

Statement. Physically irrelevant reformulations do not change law-relevant comparison content or realization verdict.
First indispensable use. Lemma 1 through quotient-level comparison.
Failure if dropped. Formal surplus can masquerade as law-content; operational equivalence loses force.

A5. Record-structural relevance

Statement. Only physically relevant record-bearing structure enters realization comparison.
First indispensable use. Lemma 3.
Failure if dropped. Admissibility may be widened by purely formal distinctions without immediate burden cost; realization collapses toward branch-labeling.

A6. Pre-outcome fixation

Statement. Admissibility, comparison structure, and relevant parameters are fixed independently of the verdict at issue.
First indispensable use. Lemma 1.
Failure if dropped. Non-circularity collapses; post hoc admissibility, weighting, and tie resolution become available.

A7. Operational uniqueness discipline

Statement. The theory selects a unique operational verdict class or supplies a pre-declared tie discipline consistent with the standing burdens.
First indispensable use. Lemma 1.
Failure if dropped. Multiplicity tolerance becomes compatible with apparent realization-law status; Lemma 5 and Theorem 2 fail.

A8. Probability compatibility discipline

Statement. Born compatibility is preserved unless a bounded deviation is declared in advance as part of the theory.
First indispensable use. Lemma 4.
Failure if dropped. Hidden weighting can survive by shifting burden into undeclared probability structure.

A9. Non-reduction to decoherence

Statement. A candidate realization law must add realization content not exhausted by a non-selective decoherence-compatible map.
First indispensable use. Lemma 6.
Failure if dropped. Decoherence-reductive candidates become admissible internal rivals; the theorem no longer targets realization law as such.

A10. Parameter fixity

Statement. Adjustable parameters relevant to comparison are not tuned after the verdict is known.
First indispensable use. Lemma 1, with decisive later use in Lemmas 4 and 5.
Failure if dropped. Apparent internal rivals can be engineered by retrospective adjustment.

A11. Admissibility stability under refinement

Statement. Admissibility remains stable under physically legitimate refinement within the declared class.
First indispensable use. Lemma 3.
Failure if dropped. Law-relevant admissibility widening can be reclassified as harmless refinement; departure exhaustion fails.

A12. Burden monotonicity

Statement. Burden comparisons remain stable under physically irrelevant expansion and inert reformulation.
First indispensable use. Lemma 3; later decisive use in Section 7.
Failure if dropped. Inert transformation and genuine law-content change cannot be distinguished reliably.

A13. Quotient regularity

Statement. The admissible quotient under operational equivalence is sufficiently well-behaved to support ordering, minimization, and comparison.
First indispensable use. Lemma 1.
Failure if dropped. Quotient-level comparison is not secure; uniqueness and canonical equivalence cannot be stated at the correct level.

A14. Explicit defeat structure

Statement. Each candidate theory states conditions under which its realization-law claim fails.
First indispensable use. Lemma 2.
Failure if dropped. Apparent internal membership can be preserved while defeat liability is softened; the comparison class loses evaluative discipline.

Dependency compression

The proof architecture decomposes into four burden clusters.

Comparison forcing: A2, A3, A4, A6, A7, A10, A13.
Departure exhaustion: A5, A8, A9, A11, A12.
Internal elimination: A1–A14 through Theorem 2.
Residual rigidity and defeat: A4, A12, A13, A14.

No assumption is idle. Several are stronger than the first result that uses them, but none is dispensable for the full theorem chain as stated.

Appendix B. Proof Skeletons

This appendix states only the hinge logic of each result.

B1. Lemma 1

Inputs: A2, A3, A4, A6, A7, A10, A13.
Hinge move: Membership in 𝓣 already fixes context, admissible quotient structure, comparison rule, and selected verdict class.
Output: Quotient-level canonical comparison is defined.

B2. Lemma 2

Inputs: Definition of internal rival; class-defining burdens.
Hinge move: If canonical comparison requires burden alteration, burden-preserving internal status is lost.
Output: Comparison failure implies burden failure or class exit.

B3. Theorem 1

Inputs: Lemmas 1 and 2.
Hinge move: Comparison is forced; burden-preserving incomparability is unavailable.
Output: Every T ∈ 𝓣 either enters comparison, violates a burden, or exits 𝓣.

B4. Lemma 3

Inputs: A5, A6, A11, A12.
Hinge move: Law-relevant admissibility widening either changes nothing on the quotient or weakens admissibility discipline.
Output: Admissibility widening is inert or burden-leaking.

B5. Lemma 4

Inputs: A6, A8, A10.
Hinge move: Covert weighting must be post hoc, probability-altering without declaration, or adjustable.
Output: Hidden weighting fails as a burden-preserving rival mechanism.

B6. Lemma 5

Inputs: A6, A7, A10.
Hinge move: Unresolved multiplicity requires either no tie rule, post hoc tie rule, or burden-altering tie rule.
Output: Multiplicity tolerance does not yield a surviving internal rival.

B7. Lemma 6

Inputs: A1, A9.
Hinge move: A candidate exhausted by decoherence-compatible structure adds no realization law-content.
Output: Decoherence reduction fails as internal survival.

B8. Noncanonical Departure Proposition

Inputs: Theorem 1; Lemmas 3–6.
Hinge move: Once comparison is forced, every material departure must alter a law-bearing component in one of the finite ways already classified.
Output: Departure space is exhausted.

B9. Theorem 2

Inputs: Theorem 1; noncanonical departure proposition.
Hinge move: Every materially distinct internal rival must depart through one of the finite modes; none survives burden-preservingly.
Output: No materially distinct internal rival remains within 𝓣.

B10. Conditional Uniqueness Corollary

Input: Theorem 2.
Hinge move: Internal rival space is empty once burdens are fixed.
Output: The canonical law-form is conditionally unique.

B11. Functional Rigidity Corollary

Inputs: Section 7 propositions; A4, A12, A13.
Hinge move: Law-relevant deformation either changes quotient-level law-content or requires burden alteration.
Output: ℛ_C is rigid up to inert transformation.

B12. Defeat Proposition

Inputs: F1–F6.
Hinge move: Each defeat condition breaks a necessary theorem component or the significance claim attached to it.
Output: Any one of F1–F6 defeats the central claim in the relevant sense.

Appendix C. Full Proofs

C.I. Comparison Forcing

C.I.1 Proof of Lemma 1

Let T ∈ 𝓣. By A2, T fixes a physical context C. By A3, it fixes a nonempty admissible class 𝒜_T(C). By the class definition together with A6 and A10, admissibility and comparison are fixed independently of the verdict at issue. By A7, T determines either a unique selected operational verdict class or a pre-declared tie discipline preserving the standing burdens. By A4 and A13, physically irrelevant multiplicity is quotiented out and quotient-level comparison is well-posed. Hence T admits canonical comparison with CBR at the level of admissibility, burden ordering, and selected operational verdict class.

C.I.2 Proof of Lemma 2

Suppose T cannot enter burden-preserving comparison with canonical CBR without altering a class-defining burden. An internal rival must remain both materially distinct and burden-preserving. If comparison requires altering admissibility, pre-outcome fixation, uniqueness discipline, probability discipline, non-reduction, parameter fixity, or defeat structure, then T no longer preserves the law-bearing burdens that qualify it for internal membership in 𝓣. Therefore T either violates a standing burden or exits the class. In neither case does it survive as an internal rival.

C.I.3 Proof of Theorem 1

Let T ∈ 𝓣. By Lemma 1, T admits canonical comparison. If comparison is burden-preserving, clause (i) holds. If it is not, Lemma 2 implies that T does not survive as an internal rival; since the altered conditions are class-defining, this occurs by burden violation or class exit. Hence exactly one of (i)–(iii) obtains.

C.II. Departure Exhaustion

C.II.1 Proof of Lemma 3

Let 𝒜_A(C) strictly enlarge canonical admissibility 𝒜(C). If the additional candidates induce no change in quotient-level ordering or selected verdict structure, the enlargement is inert. If they do induce such change, then noncanonical admissibility has become law-relevant. Because canonical admissibility is fixed under A5, A6, and A11, such change requires relaxation of admissibility discipline or an equivalent burden-bearing condition. Hence widened admissibility either disappears on the quotient or leaks burden failure.

C.II.2 Proof of Lemma 4

Let a candidate preserve apparent verdicts by an auxiliary weighting map not fixed independently of the verdict. If it is chosen after the verdict is known, A6 fails. If it alters probability discipline without declared bounded deviation, A8 fails. If it remains adjustable, A10 fails. If none of these occurs, then the weighting was already part of the fixed law-bearing structure and no covert rival has been produced.

C.II.3 Proof of Lemma 5

Let a candidate leave several operationally distinct quotient classes co-minimal. If no tie rule is supplied, A7 fails. If tie resolution is supplied post hoc, A6 and A10 fail. If tie resolution is supplied in advance but changes admissibility, weighting, or defeat structure, burden preservation fails. If tie resolution preserves the standing burdens, the candidate re-enters the same comparison logic governed by Theorem 1. Hence unresolved multiplicity does not generate a surviving materially distinct internal rival.

C.II.4 Proof of Lemma 6

Let a candidate be exhausted by a non-selective decoherence-compatible map. Then it adds no realization content beyond evolution or registration structure. Since A9 is constitutive of 𝓣, the candidate fails as an internal realization law and cannot survive as a materially distinct internal rival.

C.II.5 Proof of the Noncanonical Departure Proposition

Let T be materially noncanonical and assume it survives Theorem 1 as a burden-preserving comparison partner. Material non-canonicity must then alter a law-bearing component of the realization role. If it alters admissibility, Lemma 3 applies. If it preserves verdicts by covert weighting, Lemma 4 applies. If it leaves operational multiplicity unresolved, Lemma 5 applies. If it adds no realization content beyond decoherence-compatible structure, Lemma 6 applies. Any remaining case preserves visible verdicts while altering the law-bearing conditions under which they are produced; this is burden-structure deformation. No sixth internal departure mode remains once comparison is fixed on the admissible quotient.

C.III. Internal Elimination

C.III.1 Proof of Theorem 2

Let T ∈ 𝓣 and suppose T is burden-satisfying. By Theorem 1, T must enter burden-preserving comparison with canonical CBR. Assume for contradiction that T remains materially distinct. Then, by the noncanonical departure proposition, that distinctness must occur through admissibility leakage, hidden weighting, unresolved operational multiplicity, decoherence reduction, or burden-structure deformation. Lemmas 3–6 exclude the first four as burden-preserving survival routes. Consider burden-structure deformation. If it is quotient-inert, then T is not materially distinct. If it is law-relevant, then it changes admissibility, ordering, tie discipline, weighting, parameter discipline, or defeat structure in a way incompatible with burden-preserving membership in 𝓣. Both cases contradict the assumption that T is both burden-satisfying and materially distinct. Therefore no materially distinct internal rival survives within 𝓣. Hence every burden-satisfying outcome-realization theory in 𝓣 is canonically equivalent to CBR up to operational equivalence.

C.III.2 Proof of the Conditional Uniqueness Corollary

Immediate from Theorem 2.

C.IV. Residual Rigidity

C.IV.1 Proof of the Ordering Preservation Proposition

Let τ transform ℛ_C. If τ fails to preserve quotient-level ordering, then the comparative burden status of admissible candidates changes and law-content changes directly. If τ preserves ordering but changes the selected operational verdict class, law-content again changes directly. Conversely, within the scope of the paper, preservation of quotient-level ordering and selected verdict structure is sufficient for sameness of law-content because A4, A12, and A13 remove further operationally relevant distinctions.

C.IV.2 Proof of the Anti-Plasticity Proposition

Let ℛ′_C preserve selected labels while differing materially from ℛ_C. If the difference is inert on the admissible quotient, it produces no law-level distinctness. If it is law-relevant, then either selected verdict structure changes or verdict preservation is maintained only through alteration of admissibility, weighting, tie discipline, parameter structure, or defeat conditions. In the latter case burden preservation fails within 𝓣. Hence material plasticity of ℛ_C is illegitimate within the theorem class.

C.IV.3 Proof of the Functional Rigidity Corollary

Immediate from the two preceding propositions.

C.V. Defeat Structure

C.V.1 Proof of the Defeat Proposition

F1 defeats the conclusion directly. F2 removes the comparison-forcing step. F3 defeats the admissibility branch of departure exhaustion. F4 defeats the non-reduction branch. F5 defeats the significance claim by showing that 𝓣 omits burden-bearing internal rivals required by the paper’s own target and assumptions. F6 defeats the residual law-content rigidity claim. Since the central result depends on all of these components, any one of F1–F6 defeats the paper in the relevant sense.

Appendix D. Admissible Quotient Construction

Fix a context C and admissible class 𝒜(C). Define ≃_C on 𝒜(C) by

Φ₁ ≃_C Φ₂

iff Φ₁ and Φ₂ are operationally indistinguishable in every realization-relevant respect available within C. “Realization-relevant” here means relevant to admissibility, burden ordering, selected verdict structure, and defeat conditions. Under A4 and A13, ≃_C is treated as a well-behaved equivalence relation. The admissible quotient 𝒜(C)/≃_C is then the set of quotient classes [Φ]_C.

The quotient is not adopted by convenience. It is forced by the target of the paper.

Proposition (Quotient Adequacy). Comparison below 𝒜(C)/≃_C is too fine for the target of the paper, and comparison above it is too coarse.

Proof sketch. Comparison below the quotient is too fine because it treats operationally inert formal multiplicity as law-content. On that level, equivalent redescriptions can appear as materially distinct rivals. Comparison above the quotient is too coarse because it suppresses distinctions in admissibility, burden ordering, or selected verdict structure that remain law-relevant within 𝓣. The admissible quotient is therefore the unique level at which the paper’s target can be pursued without either overstating rivalry or understating law-content.

The main theorem is therefore stated up to operational equivalence by necessity, not by convenience. The paper is concerned with material internal alternatives in realization-law content, and the quotient is the exact level at which that concern becomes mathematically well-posed.

Appendix E. Counterexample Constructions

Each counterexample family is stated in a strict template: construction, survival claim, required burden preservation, failure point, verdict.

E.1 Family A — Widened-Admissibility Rival

Construction. Replace 𝒜(C) by 𝒜_A(C) with 𝒜(C) ⊊ 𝒜_A(C), leaving the remaining structure as close as possible to canonical CBR.
Survival claim. Material distinctness by enlarging admissible possibility space while preserving the appearance of the same law-form.
Required burden preservation. Compatibility with A5, A6, and A11.
Failure point. Either the enlargement is inert on the quotient or it becomes law-relevant and leaks admissibility burden failure.
Verdict. No survival.

E.2 Family B — Hidden-Weighting Rival

Construction. Add an auxiliary weighting map w_C not fixed independently of the verdict at issue.
Survival claim. Preserve visible verdicts while generating covert law-level distinctness.
Required burden preservation. Compatibility with A6, A8, and A10.
Failure point. Post hoc choice violates A6; undeclared probability alteration violates A8; adjustability violates A10; fixed use collapses into the standing comparison structure.
Verdict. No survival.

E.3 Family C — Multiplicity-Tolerant Rival

Construction. Permit multiple operationally distinct co-minimizers to remain selected.
Survival claim. Preserve comparison structure while denying uniqueness as a burden.
Required burden preservation. Compatibility with A7 and the rest of the class-defining burdens.
Failure point. No tie rule violates A7; post hoc tie rule violates A6 and A10; burden-altering tie rule leaves 𝓣; burden-preserving tie rule re-enters the standing comparison logic.
Verdict. No survival.

E.4 Family D — Decoherence-Reductive Rival

Construction. Identify realization content with a non-selective decoherence-compatible map Φ_mix.
Survival claim. Preserve realization language while avoiding distinct realization law-content.
Required burden preservation. Compatibility with A9.
Failure point. No realization content beyond decoherence-compatible structure is supplied.
Verdict. No survival.

E.5 Family E — Burden-Deformed Functional Rival

Construction. Replace ℛ_C by ℛ′_C while preserving selected labels in chosen cases.
Survival claim. Preserve visible verdicts while introducing a distinct law at the functional level.
Required burden preservation. Quotient-level law-content must remain fixed without burden alteration.
Failure point. If inert, no material distinctness results; if law-relevant, verdict preservation requires burden alteration or verdict change.
Verdict. No survival.

No counterexample family survives as a materially distinct internal burden-preserving alternative.

Appendix F. Defeat Conditions and Diagnostics

The defeat conditions divide into two kinds: theorem defeat and significance defeat.

F1. Surviving materially distinct internal rival

Type: Theorem defeat.
Diagnostic question: Does there exist T ∈ 𝓣 that remains burden-satisfying and materially distinct from canonical CBR after Theorem 2 is applied?
If yes: The conclusion fails directly.

F2. Failure of the embedding dichotomy

Type: Theorem defeat.
Diagnostic question: Does there exist T ∈ 𝓣 that neither enters burden-preserving comparison nor violates a standing burden nor exits 𝓣?
If yes: Theorem 1 fails.

F3. Cost-free noncanonical admissibility

Type: Theorem defeat.
Diagnostic question: Can admissibility be altered in a law-relevant way without quotient inertness and without burden violation?
If yes: Lemma 3 and the departure proposition fail.

F4. Cost-free decoherence reduction

Type: Theorem defeat.
Diagnostic question: Can a decoherence-reductive candidate reproduce all realization content relevant to the paper’s target while remaining a burden-satisfying member of 𝓣?
If yes: Lemma 6 and the use of A9 fail.

F5. Unjustified exclusion of burden-bearing internal rivals

Type: Significance defeat.
Diagnostic question: Does the paper’s own target, definitions, and standing assumptions require confrontation with burden-bearing internal rivals excluded from 𝓣 without sufficient argument?
If yes: The theorem may remain valid for 𝓣, but the significance claimed for Theorem 2 fails.

F6. Collapse of functional rigidity at the level of law-content

Type: Theorem defeat for the residual rigidity claim.
Diagnostic question: Is ℛ_C materially replaceable in a law-relevant way without inertness and without loss of burden preservation?
If yes: the functional rigidity corollary fails.

Defeat Proposition. If any of F1–F6 is exhibited within declared scope, the paper’s central claim fails either as theorem or as significance.

Proof sketch. F1–F4 and F6 each break a necessary theorem component. F5 breaks the relevance claim by showing that the theorem class omits internal rivals the paper’s own target requires it to address. Hence any one of F1–F6 defeats the paper in the appropriate sense.

Appendix G. Dependency Map to Prior CBR Papers

Imported from reconstruction

The burden-to-structure logic of disciplined outcome-realization law; the objects C, 𝒜(C), ℛ_C, ≃_C, and Φ∗_C; and the standing burdens of context specification, admissibility restriction, non-circular comparison, operational uniqueness, probability discipline, non-reduction to decoherence, parameter fixity, and explicit defeat conditions.

Imported from canonical closure

The mature canonical law-form; restricted admissibility; the canonical comparison perspective; restricted uniqueness up to operational equivalence; local probability-closure discipline within canonical admissibility; and the program’s explicit non-claim and failure-condition discipline.

Proved here

Forced comparison within 𝓣; exhaustion of materially noncanonical departure modes; the No-Internal-Alternative Theorem; the conditional uniqueness corollary; and the class-bounded rigidity consequence for ℛ_C.

Deferred to the empirical sequel

Platform-level empirical instantiation: explicit construction of η, comparator validation, nuisance separation, sensitivity analysis, and advance locking of verdict conditions. What this paper contributes to that sequel is narrower: once such instantiation is achieved, internal fallback within 𝓣 is no longer available.