The Born Rule as the Unique Refinement-Stable Induced Weight on Robust Record Sectors

The Born R ule as the Unique Reﬁnemen t-Stable Induced W eigh t on Robust Record Sectors Mark o Lela OR CID: 0009-0008-0768-5184 Marc h 24, 2026 Abstract This pap er pro ves a conditional structural uniqueness theorem for induced w eight on robust record sectors within an admissible Hilb ert record lay er. Its theorem target and additiv e carrier diﬀer from those of the standard Born-rule routes: additivity is not placed on the full projector lattice, but on disjoin t admissible contin uation bundles through an extensiv e bundle v aluation, from whic h the sector-level additiv e law is inherited under admissible reﬁnemen t. Accordingly , the result is not a Gleason-t yp e represen tation theorem in diﬀeren t language, but a distinct uniqueness theorem ab out induced sector w eight inherited from bundle additivit y on admissible contin uation structure. Under t wo explicit structural conditions, in ternal equiv alence of admissible binary reﬁnemen t proﬁles and suﬃcien t admissible reﬁnement richness, the quadratic assignment is the only non-negative reﬁnement-stable induced weigh t on robust record sectors. In the main theorem, reﬁnement richness is secured by admissible binary saturation. A supplemen tary prop osition shows that dense admissible saturation already suﬃces if contin uity of the proﬁle function is added. Under normalization, the result reduces to the standard Born assignment. 1 In tro duction Wh y the quadratic quantum w eight is singled out remains a central question in the foundations of quantum theory . Most standard routes to the Born rule b egin b y ﬁxing a broad target and then proving that the quadratic assignmen t is the unique ob ject compatible with that target. In Gleason-t yp e approac hes, the target is a measure on the full lattice of pro jections or, more generally , on generalized observ ables [ 5 , 1 , 2 ]. In Ev erettian decision-theoretic approaches, the target is rational preference or rational credence in branc hing settings [ 3 , 14 , 15 ]. In env ariance-based approaches, the target is a probability assignment extracted from symmetry prop erties of entangled states [ 18 , 19 ]. Other routes app eal to self-lo cating uncertaint y , evidential coherence, op erational reconstruction, or normativ e Bay esian constraints [ 12 , 6 , 8 , 4 ]. The present pap er therefore starts from a diﬀerent theorem target, a diﬀeren t additive carrier, and hence a diﬀerent uniqueness problem. The present pap er asks a diﬀeren t question. It do es not b egin with a measure on all pro jectors, with a rationalit y p ostulate, or with a symmetry principle on entangled states. Instead, it asks whic h non-negative weigh t assignments are structurally admissible on systems that can function as robust in ternal records. The target is therefore narrow er but also sharp er: induced weigh ts on robust record sectors inside an admissible Hilb ert record lay er. 1 This shift in target c hanges the logical lo cation of the crucial additiv e step. Rather than p ostulating additivit y on the full pro jector lattice, the pap er places the additive primitive on disjoin t admissible con tinuation bundles through an extensive bundle v aluation. The sector-level additive law is then inherited from contin uation partition under admissible reﬁnement. The resulting theorem is therefore not a global measure-representation theorem on pro jectors, but a conditional structural uniqueness theorem for induced weigh ts on robust record sectors. The diﬀerence from standard routes is structural, not merely verbal: the ob ject b eing weigh ted is diﬀerent, the structure carrying additivit y is diﬀeren t, and the uniqueness problem is posed at a diﬀerent level. The second key mo ve is to imp ose an in ternal equiv alence condition at the level of admissible binary reﬁnemen t proﬁles. This do es not yet assume that induced w eight is a function of norm. It sa ys only that record situations with the same admissible binary reﬁnement proﬁle must carry the same induced weigh t. The reduction to a one-v ariable function of the norm of the projected record comp onent is obtained only after admissible binary saturation is added, b ecause saturation mak es the admissible binary reﬁnement proﬁle fully classiﬁable by that norm. Under this additional structural condition, the problem reduces to a function g : R ≥ 0 → R ≥ 0 of the norm of the pro jected comp onent. The remaining question is then whether the admissible reﬁnemen t structure is rich enough to force the functional equation g  q r 2 1 + r 2 2  = g ( r 1 ) + g ( r 2 ) for all r 1 , r 2 ∈ R ≥ 0 . In the main theorem, this reﬁnemen t richness is sec ured b y admissible binary saturation. A supplemen tary proposition sho ws that dense admissible saturation already suﬃces if con tinuit y of the proﬁle function is added. Non-negativit y then forces the additiv e function to b e linear, by lemma 2 . The quadratic assignment follows directly . The resulting theorem is conditional and should b e read as such, but its conditional form is part of its con tent rather than a defect. It neither derives Hilb ert structure nor assigns weigh ts to arbitrary pro jectors, and it do es not claim that ev ery orthogonal decomp osition is physically meaningful or that a global probability calculus has b een reconstructed from no assumptions. What it do es show is sharp er and more limited: it iden tiﬁes a precise structural threshold within an admissible Hilbert record lay er at which the quadratic assignment b ecomes the only non-negative reﬁnemen t-stable induced w eight on robust record sectors. This conditional form is a metho dological virtue. F oundational arguments often become diﬃcult to assess b ecause the assumptions doing the actual w ork are disp ersed across the pro of. The present route instead isolates them and turns them into an explicit threshold statement. One structural condition concerns internal equiv alence at the level of admissible binary reﬁnement proﬁles. The other concerns suﬃcient admissible reﬁnement richness to force the functional equation, realized in the main theorem by admissible binary saturation and weak ened in a supplementary prop osition to dense admissible saturation plus contin uity . Whether one accepts or rejects the conclusion can therefore b e tied to clearly iden tiﬁable commitmen ts rather than to hidden background assumptions. The main theorem may b e summarized informally as follo ws. Let Ψ b e a global represen tational state and let R b e a robust record sector in an admissible Hilb ert record lay er. If induced weigh t is ﬁxed by internally accessible admissible reﬁnement structure, and if the admissible reﬁnemen t structure is rich enough to force all norm-compatible binary proﬁles, then the quadratic assignment 2 is the only non-negative reﬁnement-stable induced w eight on robust record sectors. Equiv alen tly , there exists a constan t c ≥ 0 such that W Ψ ( R ) = c ∥ Π R Ψ ∥ 2 . Under normalization, this reduces to the standard Born assignment W Ψ ( R ) = ∥ Π R Ψ ∥ 2 . The no velt y of the presen t route therefore lies less in the ﬁnal mathematical form than in the theorem target, the additive carrier, and the structural lev el at which the uniqueness problem is p osed. This pap er do es not replace Gleason’s theorem. It prov es a diﬀerent uniqueness theorem ab out a diﬀeren t ob ject under a diﬀerent additive carrier. Gleason concerns global additiv e assignmen ts on the pro jector lattice. The present pap er concerns induced weigh ts on robust record sectors, with the additive primitiv e placed on disjoint contin uation bundles. Once one fo cuses on robust record sectors and on the con tinuation structure that makes them internally stable, the quadratic assignmen t is forced to be the only non-negative reﬁnemen t-stable induced w eigh t compatible with the stated conditions. The pap er is organized as follows. Section 2 in tro duces the admissible Hilb ert record lay er, robust record sectors, con tinuation bundles, and admissible orthogonal reﬁnements. Section 3 sho ws how the uniqueness problem reduces to a norm-based one-v ariable weigh t function. Section 4 states the t wo structural conditions and deriv es the cen tral functional equation. Section 5 prov es the quadratic uniqueness theorem. Section 6 clariﬁes the precise domain and limits of the result. Section 7 compares the presen t route with the ma jor existing families of Born-rule arguments, and Section 8 closes with the main conceptual takea w ay and directions for further work. 2 F ramew ork W e w ork conditionally within an admissible Hilb ert record lay er. The aim of this section is to ﬁx the objects on whic h the later weigh t assignmen t is deﬁned and to state the precise meaning of admissible reﬁnement. No attempt is made here to derive Hilb ert structure itself. That question b elongs to a separate foundational program. 2.1 A dmissible record lay er Let H b e a Hilb ert space serving as an admissible Hilb ert record lay er for a class of internally accessible record structures. The role of this lay er is represen tational rather than ontologic al: it pro vides a faithful co ordinatization of robust record alternatives and their admissible reﬁnements. The basic ob jects in the present pap er are not arbitrary pro jectors on H , but robust record sectors that corresp ond to internally stable record conten t. Deﬁnition 1 (Robust record sector) . A robust record sector is a close d subsp ac e R ⊆ H satisfying al l of the fol lowing c onditions: (i) In ternal discriminability : R r epr esents a r e c or d typ e that is internal ly r e adable and distin- guishable fr om inc omp atible admissible alternatives. (ii) Short-horizon p ersistence : the r e c or d c ontent r epr esente d by R is stable under admissible micr o-r e c o dings and under imme diate admissible c ontinuation, in the sense that its identifying 3 r e c or d c ontent do es not disapp e ar under arbitr arily smal l r epr esentational p erturb ations or one-step admissible evolution. (iii) A dmissible reﬁnement closure : R b elongs to an admissible exclusivity structur e in which me aningful next-step alternatives c an b e r epr esente d by admissible ortho gonal r eﬁnements of R . These conditions are functional rather than microscopic. They do not require a particular dynamical mec hanism in the deﬁnition itself. Their purpose is to exclude arbitrary closed subspaces from coun ting as record sectors merely by formal inclusion in the Hilb ert lay er. R emark 1 . P aradigmatic physical examples of robust record sectors are decoherence-stabilized p ointer sectors and closely related stable record subspaces. The p resen t pap er do es not build decoherence in to the deﬁnition itself, b ecause the theorem is meant to apply at the level of admissible record structure rather than at the level of any one sp eciﬁc stabilization mechanism. W e will only consider robust record sectors that b elong to the admissible lay er in the ab ov e sense. In particular, not every closed subspace of H is automatically treated as a ph ysically meaningful record sector. 2.2 State-relativ e contin uation bundles Let Ψ ∈ H denote a global represen tational state. Unless normalization is explicitly imp osed later, represen tational states in the present pap er are not assumed to hav e unit norm. The admissible Hilb ert record lay er is tak en to b e closed under p ositive scalar multiplication: if Ψ lies in the lay er and λ ≥ 0 , then λ Ψ lies in the same representational la yer. This closure is used only to iden tify the full realized norm domain b efore normalization is imp osed. F or each robust record sector R , we asso ciate a state-relative con tinuation bundle C Ψ ( R ) ⊆ A exp , where A exp denotes a structured family of experientially admissible contin uation bundles built from exclusiv e contin uation alternativ es within the admissible record framework used in this pap er. Informally , C Ψ ( R ) is the bu ndle of internally admissible contin uations of Ψ in which the record sector R is realized as the relev an t next record conten t. W e assume that this family of admissible contin uation bundles carries a non-negativ e set function µ . W e will refer to µ as an extensive bund le valuation . A t this stage, the only structural prop erty imp osed on µ is ﬁnite additivity on disjoint admissible bundles. Deﬁnition 2 (Induced record w eight) . F or a glob al state Ψ and a r obust r e c or d se ctor R , the induc e d weight of R r elative to Ψ is W Ψ ( R ) := µ ( C Ψ ( R )) . A t this stage, W Ψ ( R ) is only a state-relativ e weigh t on robust record sectors. It is not yet interpreted as a probabilit y measure on all pro jectors. 4 2.3 Extensiv e bundle v aluation and its logical role The additive primitive in this pap er is placed on disjoint admissible con tin uation bundles. This is a p ositiv e structural choice, not a mere reformulation of projectoral additivity . A contin uation bundle is an extensional object: it is a set of admissible con tinuations, individuated b y the record conten t they realize. Disjoint contin uation bundles are disjoin t sets of contin uations. Finite additivity on such bundles is motiv ated directly b y the exclusivit y of the con tinuations themselv es: if no contin uation b elongs to both B 1 and B 2 , then the com bined weigh t of the tw o bundles is the sum of their individual weigh ts. This is the same motiv ation that underlies classical probabilit y on mutually exclusiv e ev en ts. It requires no algebraic structure on the carrier. That exclusivit y is not disp ensable: if the relev an t record alternatives w ere allow ed to ov erlap already at the sector level, then the asso ciated con tin uation bundles would no longer form a canonical disjoint carrier, and additiv e induced weigh t would lose its clear domain of application. A pro jector, by con trast, is an algebraic ob ject in the op erator algebra of a Hilb ert space. Additivit y on orthogonal pro jectors presupposes that algebraic structure and is deﬁned at the level of the pro jector lattice. The presen t pap er does not b egin there. It places the additive primitive on a family of exclusive contin uation bundles, and the sector-lev el additive law is then inherited from admissible contin uation partition. The broader comparison with Gleason-t yp e routes is deferred to section 7 . R emark 2 (Why contin uation structure is not optional for induced weigh t in the present framew ork) . In the present framew ork, induced w eight is not introduced as a primitive assignment on record sectors, but as the derived quantit y W Ψ ( R ) = µ ( C Ψ ( R )) , with the additiv e primary carrier lo cated on state-relativ e con tinuation bundles. This is not a disp ensable represen tational c hoice, b ecause it ﬁxes the problem type: the question is not which weigh t one may directly p ostulate on sectors, but which sector weigh t is inherited from a deep er additiv e structure on admissible contin uations. Instan taneous collections at a single time slice do not carry the s ame seman tics, b ecause con tinuation bundles are not sync hronic co existence classes but diachronic compatibilit y classes of forward dev elopmen ts relativ e to a record sector. A ccordingly , a mo diﬁcation preserves the presen t problem type only if sector weigh t remains deriv ed from a deeper carrier and additivit y remains anchored on that carrier rather than b eing imp osed directly on sectors. What the framework therefore needs is not p ersistence in some maximal sense, but enough short-horizon stabilit y that under small time shifts or immediate admissible evolution the classiﬁcation of which forward developmen ts still realize the same record function do es not arbitrarily change. 2.4 Pro jected record comp onen ts F or eac h robust record sector R , let ϕ R := Π R Ψ denote the comp onent of Ψ in R . The later reduction step will show that, under explicit structural conditions, the induced weigh t W Ψ ( R ) dep ends only on structural information carried by ϕ R . In the present section, we only ﬁx the notation. 5 2.5 A dmissible orthogonal reﬁnements The cen tral structural op eration in the pap er is not an arbitrary decomp osition of a Hilb ert subspace, but an admissible reﬁnemen t of a robu st record sector in to mutually exclusive robust sub-sectors. Deﬁnition 3 (Admissible orthogonal reﬁnement) . L et Ψ ∈ H b e a ﬁxe d glob al r epr esentational state, and let R b e a r obust r e c or d se ctor. A n admissible orthogonal reﬁnement of R r elative to Ψ is a de c omp osition R = R 1 ⊕ R 2 such that: (i) R 1 and R 2 ar e themselves r obust r e c or d se ctors, (ii) R 1 and R 2 r epr esent mutual ly exclusive internal ly r e adable r e c or d alternatives, (iii) the r eﬁnement faithful ly r esolves the next-step observable exclusivity structur e c arrie d by R , in the sense that every admissible c ontinuation in C Ψ ( R ) fal ls under exactly one of R 1 or R 2 , and no admissible c ontinuation r emains outside this p artition. Condition (iii) is essen tial. It excludes purely formal orthogonal decomp ositions that do not corresp ond to a gen uine partition of exp erientially admissible contin uations relative to the ﬁxed state Ψ . Thus, admissibilit y is stronger than orthogonality alone and is state-relative in the sense relev an t for the induced weigh t construction. Whenev er R = R 1 ⊕ R 2 is an admissible orthogonal reﬁnemen t relativ e to Ψ , the projected comp onen ts satisfy ϕ R = ϕ R 1 + ϕ R 2 , ϕ R 1 ⊥ ϕ R 2 , ∥ ϕ R ∥ 2 = ∥ ϕ R 1 ∥ 2 + ∥ ϕ R 2 ∥ 2 . 2.6 Con tin uation partition lemma The next lemma records the precise bridge b et ween admissible orthogonal reﬁnement in the Hilb ert la yer and disjoint contin uation structure at the level of admissible contin uation bundles. Lemma 1 (Contin uation partition under admissible reﬁnement) . L et R b e a r obust r e c or d se ctor, and let R = R 1 ⊕ R 2 b e an admissible ortho gonal r eﬁnement r elative to Ψ . Then the asso ciate d c ontinuation bund les satisfy C Ψ ( R ) = C Ψ ( R 1 ) ˙ ∪ C Ψ ( R 2 ) . Pr o of. Since R 1 , R 2 ⊆ R , every contin uation that realizes R 1 or R 2 also realizes R . Hence C Ψ ( R 1 ) ∪ C Ψ ( R 2 ) ⊆ C Ψ ( R ) . Con versely , admissibility means that the reﬁnemen t R = R 1 ⊕ R 2 fully resolves the next-step observ able exclusivit y structure represented by R . This follows b ecause condition (iii) of deﬁnition 3 requires the reﬁnemen t to resolv e the full observ able exclusivit y structure of R at the next step, 6 lea ving no con tin uation in C Ψ ( R ) unaccoun ted for. Therefore ev ery contin uation in C Ψ ( R ) realizes exactly one of the reﬁned record sectors R 1 or R 2 . Hence C Ψ ( R ) ⊆ C Ψ ( R 1 ) ∪ C Ψ ( R 2 ) . Because R 1 and R 2 represen t mutually exclusiv e record alternativ es, no admissible con tinuation can realize b oth simultaneously . Th us C Ψ ( R 1 ) ∩ C Ψ ( R 2 ) = ∅ . Com bining the tw o inclusions with disjointness yields C Ψ ( R ) = C Ψ ( R 1 ) ˙ ∪ C Ψ ( R 2 ) . By ﬁnite additivit y of the extensive bundle v aluation µ on disjoint con tinuation bundles, ev ery admissible orthogonal reﬁnemen t automatically induces W Ψ ( R ) = W Ψ ( R 1 ) + W Ψ ( R 2 ) . Deﬁnition 4 (Reﬁnement-stabilit y) . A n induc e d weight W Ψ ( · ) on r obust r e c or d se ctors is c al le d reﬁnemen t-stable if for every r obust r e c or d se ctor R and every admissible ortho gonal r eﬁnement R = R 1 ⊕ R 2 r elative to Ψ , W Ψ ( R ) = W Ψ ( R 1 ) + W Ψ ( R 2 ) . F or induced w eights arising from an extensive bundle v aluation, this prop ert y is automatic under admissible orthogonal reﬁnement b y the con tinuation partition lemma and ﬁnite additivit y of µ . The term r eﬁnement-stable is retained only to name the prop ert y singled out by the uniqueness theorem. The subsequent sections determine whic h non-negative induced weigh ts can hav e this prop ert y on robust record sectors. 2.7 Wh y the additive primitiv e is placed on con tin uation bundles The additiv e primitive sits on contin uation bundles, not on the pro jector lattice. The sector-level additiv e law used throughout the rest of the argument is not introduced as a new p ostulate but is inherited from admissible contin uation partition, as established in the contin uation partition lemma ab o ve. The follo wing sections use exactly this inherited additivity for the proﬁle reduction and the deriv ation of the functional equation. 2.8 Scop e of the framew ork The framew ork ﬁxed ab ov e is delib erately narrow er than a global measure-theoretic treatment of all pro jectors on H . The argument dev elop ed in this paper concerns only robust record sectors and their admissible orthogonal reﬁnements. This restriction is essen tial: the later uniqueness result will b e a theorem ab out internally stable weigh t assignments on robust record sectors, not a reconstruction of a probabilit y calculus on the full pro jector lattice. 7 3 Proﬁle Reduction This section isolates the reduction step that turns sector-lev el weigh ts into a one-v ariable function of the pro jected record comp onent. The k ey p oint is that, within the admissible Hilb ert record lay er, the internally relev an t reﬁnement data of a pro jected comp onent is captured by its admissible binary reﬁnemen t proﬁle. A norm-based representation of the w eight is obtained only once the in ternal equiv alence principle is com bined with the norm classiﬁcation of proﬁles supplied by admissible binary saturation. 3.1 A dmissible binary reﬁnement proﬁles Let R b e a robust record sector and let ϕ R := Π R Ψ b e the pro jected comp onent of the global representational state Ψ in R . W e no w formalize the reﬁnement data carried by ϕ R . Deﬁnition 5 (A dmissible binary reﬁnement proﬁle space) . L et R b e a r obust r e c or d se ctor and let ϕ R = Π R Ψ . The admissible binary reﬁnement proﬁle space of ϕ R is the set D ( ϕ R ) of al l p airs ( r 1 , r 2 ) ∈ R 2 ≥ 0 for which either (i) ther e exists an admissible ortho gonal r eﬁnement R = R 1 ⊕ R 2 such that ∥ Π R 1 Ψ ∥ = r 1 , ∥ Π R 2 Ψ ∥ = r 2 , or (ii) ( r 1 , r 2 ) = ( ∥ ϕ R ∥ , 0) , which is include d by c onvention as the trivial self-pr oﬁle r epr esenting the absenc e of any pr op er binary r eﬁnement. Th us, D ( ϕ R ) records which binary norm proﬁles can o ccur under admissible reﬁnement of the record sector R relativ e to the state Ψ , together with the trivial self-proﬁle, within the admissible reﬁnemen t structure under consideration. 3.2 In ternal equiv alence at the proﬁle level The later uniqueness theorem will rely on a structural principle according to which internally indistinguishable record situations must carry the same weigh t. At the present stage, we only deﬁne the corresp onding equiv alence relation. 8 Deﬁnition 6 (Binary proﬁle equiv alence) . Two p airs (Ψ , R ) and (Ψ ′ , R ′ ) , with R and R ′ r obust r e c or d se ctors, ar e c al le d binary-proﬁle-equiv alen t if D (Π R Ψ) = D (Π R ′ Ψ ′ ) . The guiding idea is that binary-proﬁle-equiv alent pairs carry the same admissible binary reﬁnemen t structure and are therefore indistinguishable at the lev el relev ant for internal record stabilit y . The corresp onding w eight principle will b e stated explicitly in the next section. 3.3 Norm classiﬁcation of proﬁles The following prop osition is an immediate consequence of admissible binary saturation together with norm equality . Its role is simply to record that, within the admissible binary reﬁnemen t class used in this pap er, the admissible binary reﬁnement proﬁle is fully classiﬁed b y the norm of the pro jected comp onent. Prop osition 1 (Norm determines the admissible binary reﬁnemen t proﬁle) . L et (Ψ , R ) and (Ψ ′ , R ′ ) b e two p airs with R and R ′ r obust r e c or d se ctors inside the admissible Hilb ert r e c or d layer. A ssume that the admissible r eﬁnement structur e is binary-satur ate d in the sense that it r e alizes every admissible binary ortho gonal r eﬁnement c omp atible with the squar e d-norm de c omp osition of the c orr esp onding pr oje cte d c omp onents. Then D (Π R Ψ) = D (Π R ′ Ψ ′ ) ⇐ ⇒ ∥ Π R Ψ ∥ =   Π R ′ Ψ ′   . Pr o of. Set ϕ := Π R Ψ , ϕ ′ := Π R ′ Ψ ′ . First assume ∥ ϕ ∥ =   ϕ ′   = r . Under admissible binary saturation, b oth sectors realize exactly the admissible binary squared-norm decomp ositions of total size r 2 . Hence D ( ϕ ) = D ( ϕ ′ ) . Con versely , supp ose D ( ϕ ) = D ( ϕ ′ ) . The degenerate binary proﬁle ( ∥ ϕ ∥ , 0) b elongs to D ( ϕ ) by the trivial self-proﬁle conv ention in the deﬁnition of D ( ϕ ) . Hence it also b elongs to D ( ϕ ′ ) . Likewise, (   ϕ ′   , 0) b elongs to D ( ϕ ) . Since the total squared norm is ﬁxed across ev ery admissible binary proﬁle of a given pro jected component (as every pair in D ( ϕ ) satisﬁes r 2 1 + r 2 2 = ∥ ϕ ∥ 2 b y the Pythagorean relation for admissible orthogonal reﬁnements), equalit y of proﬁle spaces implies ∥ ϕ ∥ 2 =   ϕ ′   2 . 9 Because norms are non-negativ e, ∥ ϕ ∥ =   ϕ ′   . This pro ves the claim. R emark 3 . This prop osition records only a classiﬁcation consequence within the admissible binary reﬁnemen t class. It do es not say that arbitrary Hilb ert decomp ositions are ph ysically meaningful. It states only that, once admissible binary saturation is in place, the resulting proﬁle space is fully classiﬁed b y the norm of the pro jected comp onen t. 3.4 Reduction to a one-v ariable w eigh t function W e can now isolate the exact form of the later reduction. Corollary 1 (Existence of a norm-reduced weigh t function) . A ssume the internal e quivalenc e principle to gether with admissible binary satur ation. Then ther e exists a function g : R ≥ 0 → R ≥ 0 such that for every glob al state Ψ and every r obust r e c or d se ctor R , W Ψ ( R ) = g ( ∥ Π R Ψ ∥ ) . Pr o of. Let N := {∥ Π R Ψ ∥ | Ψ ∈ H , R a robust record sector } ⊆ R ≥ 0 b e the set of realized pro jected norms. By proposition 1 , which is a v ailable under admissible binary saturation, the binary proﬁle-equiv alence class of (Ψ , R ) is completely determined b y the single non-negativ e num b er ∥ Π R Ψ ∥ . If induced we ight is constan t on binary-proﬁle-equiv alence classes, then it dep ends only on that n umber. This deﬁnes a function g : N → R ≥ 0 , g ( r ) := W Ψ ( R ) , for an y pair (Ψ , R ) satisfying ∥ Π R Ψ ∥ = r . The deﬁnition is indep endent of the chosen representativ e pair b y proﬁle-equiv alence inv ariance. A t this stage represen tational states are not assumed normalized, and by the p ositive-scaling closure stated in section 2 , the admissible Hilb ert record lay er remains closed under p ositive scalar m ultiples. Hence, on any non-v acuous admissible record lay er, one has N = R ≥ 0 : if R 0 is an y non-zero robust record sector and 0  = ϕ ∈ R 0 , then for ev ery r ∈ R ≥ 0 the state Ψ r := r ∥ ϕ ∥ ϕ satisﬁes ∥ Π R 0 Ψ r ∥ = r . Th us g may b e regarded as a function g : R ≥ 0 → R ≥ 0 . Since induced w eights are non-negative, the co domain of g is R ≥ 0 . 10 3.5 Role in the main argumen t The presen t section do es not yet determine the form of g . It sho ws only how the reduction w orks once tw o ingredien ts are com bined: the in ternal equiv alence principle, which makes induced weigh t constan t on binary-proﬁle-equiv alence classes, and admissible binary saturation, which mak es those classes norm-classiﬁable. The next section states these tw o structural conditions explicitly: (i) the internal equiv alence principle that turns binary proﬁle-equiv alence in to weigh t-equiv alence, (ii) the admissible binary saturation condition that ensures enough binary reﬁnements b oth to classify proﬁles b y norm and to force the functional equation for g . The reduction dev elop ed in this section acquires gen uine force only if admissible reﬁnemen t structure is not trivial. If every record situation carried only its trivial self-proﬁle, then binary-proﬁle equiv alence would cease to induce a non-trivial in ternal classiﬁcation, and inv ariance at that level w ould impose no substan tive restriction on induced w eigh t. What is therefore still needed is t wofold: ﬁrst, an inv ariance principle that limits weigh t-relev ance to in ternally accessible reﬁnement structure rather than represen tational surplus; second, a ric hness condition ensuring that admissible reﬁnemen t structure is strong enough to support a constraining functional equation. The next section states these tw o requiremen ts explicitly in the form of the in ternal equiv alence principle and admissible binary saturation. In this wa y , the present section isolates the correct internal structural carrier, while the next section supplies the additional assumptions under which that carrier b ecomes theorematically decisiv e. Once these are in place, the contin uation partition lemma from the framework section yields the additiv e reﬁnement law, and the quadratic assignment follows. It is worth making the logical order explicit to forestall a p oten tial circularity concern. Condition 1 is stated at the lev el of binary reﬁnemen t proﬁles, b efore an y connection to norms has b een established. That binary-proﬁle-equiv alen t pairs are norm-classiﬁed is the conten t of proposition 1 , which is a later consequence of admissible binary saturation. The quadratic assignment then follows from the Cauc hy lemma, applied to the functional equation that admissible binary saturation makes a v ailable. Condition 1 does not assume norm-dependence of w eigh t; it is one input to the argumen t that ev entually forces it. 4 T w o Structural Conditions The framew ork and the proﬁle reduction established so far do not yet determine the form of the induced w eight. They identify the correct ob jects and show how the sector-lev el assignmen t reduces to a one-v ariable function of the norm of the pro jected comp onent only once internal equiv alence is com bined with admissible binary saturation. The remaining argument rests on tw o explicit structural conditions. They should not b e read as hidden bac kground assumptions. On the contrary , they mark the precise p oints at which the present theorem restricts the class of admissible record sectors under consideration. 11 4.1 In ternal equiv alence principle The proﬁle spaces introduced in section 3 provide the in ternal structural carrier on which record situations can b e compared. F or this comparison to yield a non-arbitrary induced w eight, the w eight must b e constrained to depend only on that in ternally accessible structure, not on outer redescription. The ﬁrst condition makes this requirement precise. The ﬁrst condition states that in ternally indistinguishable record situations m ust carry the same induced w eight. Condition 1 (In ternal equiv alence principle) . W e say that an induced weigh t satisﬁes the internal e quivalenc e principle if binary-proﬁle-equiv alen t pairs carry the same induced weigh t: D (Π R Ψ) = D (Π R ′ Ψ ′ ) = ⇒ W Ψ ( R ) = W Ψ ′  R ′  . R emark 4 (In ternal equiv alence as representation in v ariance) . The internal equiv alence principle is b est read as an in v ariance requiremen t on induced w eight. It sa ys that induced w eight m ust dep end only on internally accessible admissible reﬁnemen t structure, not on external co ding, am bient em b edding, or other representational surplus. Without such an internal-accessibilit y requirement, induced weigh t in the present framework ceases to b e ﬁxed by admissible reﬁnement structure alone and b ecomes sensitive to representational surplus. In the present framew ork, binary-proﬁle equiv alence is the relev an t notion of in ternal sameness. The p oin t is not merely that matc hing proﬁles are necessary for internal sameness, but that admissible binary reﬁnement structure is the full internal structural carrier that the present theorem allo ws induced w eight to see. Once external co ding, ambien t embedding, and other representational surplus are set aside, no further internally accessible w eight-relev ant datum remains within the theorem target b eyond the admissible binary reﬁnemen t proﬁle itself. If tw o record situations supp ort the same admissible binary reﬁnement proﬁles, then no diﬀerence b etw een them is av ailable at the structural lev el that the weigh t is mean t to trac k. A weigh t assignment that nev ertheless distinguished them w ould fail to b e intrinsic to the record structure itself. Rejecting the principle therefore comes at a deﬁnite price: one must allo w induced weigh t to dep end on some further feature not ﬁxed by the admissible in ternal reﬁnemen t proﬁle, and hence on represen tational surplus rather than on record structure alone. In this sense, binary-proﬁle equiv alence is the natural in ternal equiv alence for the presen t problem: it trac ks exactly the distinctions carried by the internally accessible admiss ible reﬁnement structure. Any ﬁner relation would rein tro duce representational surplus, whereas an y coarser relation would collapse distinctions already present in the in ternal proﬁle. The principle is therefore selective rather than measure-theoretic. It do es not postulate a global measure on pro jectors. It constrains which features of a record situation are allow ed to matter for induced w eight. R emark 5 (What the in ternal equiv alence principle do es and does not assume) . The in ternal equiv alence principle do es not assume that induced w eight is a function of norm. The norm classiﬁcation of proﬁles is a later consequence of admissible binary saturation, established in prop osition 1 . What the principle do es assume is proﬁle-suﬃciency: induced weigh t is ﬁxed by the admissible binary reﬁnement structure of the record situation and cannot dep end on representational features not enco ded in that proﬁle. 12 4.2 A dmissible binary saturation The second condition ensures that the admissible record sector is suﬃcien tly saturated to realize the full family of norm-compatible binary reﬁnements needed for the functional equation. Condition 2 (A dmissible binary saturation) . W e sa y that the admissible record lay er satisﬁes admissible binary satur ation if for every global state Ψ , every robust record sector R , and every pair of non-negativ e num b ers r 1 , r 2 ∈ R ≥ 0 satisfying r 2 1 + r 2 2 = ∥ Π R Ψ ∥ 2 , there exists an admissible orthogonal reﬁnement R = R 1 ⊕ R 2 suc h that ∥ Π R 1 Ψ ∥ = r 1 , ∥ Π R 2 Ψ ∥ = r 2 . This condition is stronger than mere Hilb ert-space decomposability . It is a statement ab out the admissible reﬁnemen t class, not ab out arbitrary orthogonal splittings of a subspace. Its role is to ensure that the additive reﬁnement law deriv ed from con tin uation partition can b e instan tiated for ev ery norm-compatible binary proﬁle. The term “binary saturation” is meant to emphasize that the admissible reﬁnemen t structure is saturated enough to realize every such binary proﬁle within the stated scop e. R emark 6 . Admissible binary saturation is not merely a scop e restriction. It identiﬁes the structural threshold at which the quadratic assignment b ecomes the only viable non-negative reﬁnement-stable induced w eight. The theorem’s con tribution is precisely to isolate this threshold explicitly . The condition remains a gen uine restriction on the domain of application. Whether physically relev an t systems in dimension greater than t wo satisfy admissible binary saturation is an open structural question that this pap er do es not resolv e. Decoherence-stabilized p ointer sectors are natural candidates f or the kind of record structure the theorem targets, but whether suc h sectors realize every norm-compatible binary reﬁnemen t as an admissible orthogonal decomp osition dep ends on ph ysical details that lie outside the presen t scop e. The theorem should therefore b e read as a threshold result: it identiﬁes the structural requirement that forces the quadratic assignmen t and leav es op en, as a separate empirical and structural question, whic h systems meet that requiremen t in dimension greater than tw o. The tw o-outcome spin example of section 6.10 illustrates the minimal binary setting targeted by the theorem; it does not sho w how large that domain is. R emark 7 (Wh y reﬁnement richness matters) . Binary saturation is not merely a tec hnical conv enience. T o see this, consider a to y reﬁnement class in whic h, for each total norm s ≥ 0 , the only admissible non-trivial binary reﬁnemen t is the equal split  s √ 2 , s √ 2  . Then the con tinuation-partition law yields only the relation g ( s ) = 2 g  s √ 2  for all s ∈ R ≥ 0 , 13 rather than the full quadratic functional equation. This restricted relation do es not force the quadratic form. F or an y ﬁxed 0 < ε < 1 , deﬁne g ε (0) := 0 , and for s > 0 , g ε ( s ) := s 2  1 + ε sin  4 π log 2 s  . Then g ε is con tinuous and non-negative on R ≥ 0 , and it satisﬁes g ε ( s ) = 2 g ε  s √ 2  for all s ∈ R ≥ 0 , since sin  4 π (log 2 s − 1 2 )  = sin  4 π log 2 s − 2 π  = sin  4 π log 2 s  . But g ε is not of the form c s 2 . Th us contin uity alone do es not restore uniqueness when the admissible reﬁnemen t class is to o sparse. What matters is suﬃcien t reﬁnemen t ric hness to force the full functional equation. 4.3 Reduction to the functional equation W e now collect the consequences of the previous section together with the tw o structural conditions stated ab o ve. By corollary 1 , once the internal equiv alence principle is com bined with admissible binary saturation, there exists a function g : R ≥ 0 → R ≥ 0 suc h that W Ψ ( R ) = g ( ∥ Π R Ψ ∥ ) for ev ery global state Ψ and every robust record sector R . Let R = R 1 ⊕ R 2 b e an admissible orthogonal reﬁnement. By the con tin uation partition lemma from the framework section, W Ψ ( R ) = W Ψ ( R 1 ) + W Ψ ( R 2 ) . Using the norm-reduced form, this b ecomes g ( ∥ Π R Ψ ∥ ) = g ( ∥ Π R 1 Ψ ∥ ) + g ( ∥ Π R 2 Ψ ∥ ) . Since admissible orthogonal reﬁnemen t satisﬁes the Pythagorean relation, ∥ Π R Ψ ∥ 2 = ∥ Π R 1 Ψ ∥ 2 + ∥ Π R 2 Ψ ∥ 2 , w e obtain g  q ∥ Π R 1 Ψ ∥ 2 + ∥ Π R 2 Ψ ∥ 2  = g ( ∥ Π R 1 Ψ ∥ ) + g ( ∥ Π R 2 Ψ ∥ ) . 14 A dmissible binary saturation no w allows this relation to b e realized for ev ery pair r 1 , r 2 ∈ R ≥ 0 with admissible total norm. Since representational states are not assumed normalized at this stage, and since the admissible Hilb ert record lay er is closed under p ositiv e scalar multiples as stated in section 2 , the realized norm domain is R ≥ 0 . Therefore the follo wing functional equation holds on all of R ≥ 0 : g  q r 2 1 + r 2 2  = g ( r 1 ) + g ( r 2 ) for all r 1 , r 2 ∈ R ≥ 0 . (4.1) Since induced weigh ts are non-negative b y deﬁnition, the function g satisﬁes g ( r ) ≥ 0 for all r ∈ R ≥ 0 . This regularit y condition will b e used in the next section to exclude pathological solutions of the functional equation. 4.4 In terpretiv e status of the conditions The argumen t has now b een reduced to a single functional equation, but this reduction is only as strong as the t wo structural conditions that supp ort it. The in ternal equiv alence principle is the deep er of the t w o. It expresses the claim that induced w eight m ust b e ﬁxed by in ternally accessible reﬁnemen t structure and not by represen tational surplus. Admissible binary saturation is more tec hnical, but it is not merely cosmetic: it deﬁnes the class of record sectors on which the uniqueness result will hold. Neither condition should b e hidden in the prose of the pro of. Their role is precisely to make explicit where the present route diﬀers from standard deriv ations. The main theorem will sho w that, under these conditions, the quadratic assignment is forced. It will not claim that the conditions themselves hold in ev ery conceiv able representation lay er. 4.5 T ransition to the main theorem A t this p oint the framew ork, the proﬁle reduction, and the tw o structural conditions hav e reduced the problem to a single functional equation. The next section uses this reduction to establish the uniqueness of non-negative reﬁnement-stable induced w eigh ts on robust record sectors. The explicit quadratic form is the direct expression of this uniqueness. 5 Main Theorem W e no w combine the results of the preceding sections to establish the uniqueness of non-negative reﬁnemen t-stable induced weigh ts on robust record sectors. The explicit quadratic form obtained b elo w is the direct expression of this uniqueness. Lemma 2 (Non-negative additive functions on R ≥ 0 are linear) . L et f : R ≥ 0 → R ≥ 0 satisfy f ( u + v ) = f ( u ) + f ( v ) for al l u, v ∈ R ≥ 0 . Then ther e exists a c onstant c ≥ 0 such that f ( x ) = cx for al l x ∈ R ≥ 0 . Pr o of. Set c := f (1) ≥ 0 . F or each p ositive integer n , additivity gives f ( n ) = nf (1) = cn . Since f (1) = f ( n · 1 n ) = nf ( 1 n ) , w e obtain f ( 1 n ) = c n . Therefore f ( q ) = cq for ev ery non-negative rational q . 15 Since f is non-negative and additiv e, it is monotone: for 0 ≤ x ≤ y w e hav e f ( y ) = f ( x ) + f ( y − x ) ≥ f ( x ) . F or arbitrary x ∈ R ≥ 0 , c ho ose rational sequences q − k ↑ x and q + k ↓ x . Monotonicity gives cq − k = f ( q − k ) ≤ f ( x ) ≤ f ( q + k ) = cq + k . P assing to the limit yields f ( x ) = cx . Theorem 3 (Structural uniqueness of reﬁnemen t-stable induced weigh t) . A ssume: (i) an extensive bund le valuation µ on the family of admissible c ontinuation bund les, satisfying ﬁnite additivity on disjoint admissible bund les, as sp e ciﬁe d in se ction 2 ; (ii) the internal e quivalenc e principle, c ondition 1 ; (iii) admissible binary satur ation, c ondition 2 . Then the quadr atic assignment is the only non-ne gative r eﬁnement-stable induc e d weight on r obust r e c or d se ctors. Equivalently, ther e exists a c onstant c ≥ 0 such that for every glob al state Ψ and every r obust r e c or d se ctor R , W Ψ ( R ) = c ∥ Π R Ψ ∥ 2 . Pr o of. By corollary 1 , there exists a function g : R ≥ 0 → R ≥ 0 suc h that W Ψ ( R ) = g ( ∥ Π R Ψ ∥ ) for ev ery pair (Ψ , R ) . By equation (4.1) , g satisﬁes g  q r 2 1 + r 2 2  = g ( r 1 ) + g ( r 2 ) for all r 1 , r 2 ∈ R ≥ 0 . Deﬁne f : R ≥ 0 → R ≥ 0 b y f ( x ) := g ( √ x ) . Then for all u, v ∈ R ≥ 0 , f ( u + v ) = g  √ u + v  = g  q ( √ u ) 2 + ( √ v ) 2  = g ( √ u ) + g ( √ v ) = f ( u ) + f ( v ) , so f is additive on R ≥ 0 . Since induced weigh ts arise from a non-negative extensiv e bundle v aluation, f ( x ) ≥ 0 for all x ≥ 0 . By lemma 2 , there exists c ≥ 0 such that f ( x ) = cx . Returning to g , g ( r ) = f ( r 2 ) = cr 2 for all r ∈ R ≥ 0 , and therefore W Ψ ( R ) = c ∥ Π R Ψ ∥ 2 . This is the quadratic form asserted. Since every non-negativ e reﬁnement-stable induced weigh t m ust satisfy this equation by the preceding reduction, the quadratic assignmen t is the only such w eight. In op erational settings, one ma y regard exact realization of every norm-compatible binary split as stronger than needed: it is enough that admissible reﬁnemen ts approximate such splits arbitrarily w ell, provided the induced proﬁle function v aries contin uously with the realized norm. 16 Prop osition 2 (Dense saturation under contin uity) . L et g : R ≥ 0 → R ≥ 0 b e c ontinuous. A ssume that for every s ≥ 0 ther e exists a dense subset A s ⊆ [0 , s ] such that g ( s ) = g ( t ) + g  p s 2 − t 2  for al l t ∈ A s . Then ther e exists c ≥ 0 such that g ( r ) = c r 2 for al l r ≥ 0 . Pr o of. Deﬁne f : R ≥ 0 → R ≥ 0 b y f ( x ) := g ( √ x ) . Since g is con tinuous, so is f . Fix s ≥ 0 and set B s := { t 2 : t ∈ A s } ⊆ [0 , s 2 ] . Because A s is dense in [ 0 , s ] , the set B s is dense in [ 0 , s 2 ] . F or ev ery x ∈ B s , writing x = t 2 with t ∈ A s , the assumed iden tity gives f ( s 2 ) = f ( x ) + f ( s 2 − x ) . The map x 7→ f ( x ) + f ( s 2 − x ) is con tinuous on [0 , s 2 ] . Since f ( s 2 ) = f ( x ) + f ( s 2 − x ) holds on the dense set B s , it extends to all x ∈ [0 , s 2 ] . Hence, for arbitrary u, v ≥ 0 , taking s = √ u + v and x = u , we obtain f ( u + v ) = f ( u ) + f ( v ) . Th us f is additiv e on R ≥ 0 . Since f is non-negativ e, it is monotone on R ≥ 0 , and b y lemma 2 there exists c ≥ 0 suc h that f ( x ) = c x for all x ≥ 0 . Returning to g , we obtain g ( r ) = f ( r 2 ) = c r 2 for all r ≥ 0 . This sho ws that full admissible binary saturation can b e w eakened to dense admissible saturation once con tinuit y of the weigh t function g is assumed. R emark 8 (Wh y contin uit y can b e plausible) . Con tinuit y of the proﬁle function g is not derived b y the present framew ork. In the dense-saturation route, it is an additional regularit y assumption. Its app eal is op erational rather than measure-theoretic: if admissible reﬁnemen ts approximate a norm-compatible binary split arbitrarily well, then it is natural to require that the induced weigh t should not undergo ﬁnite jumps under arbitrarily small changes in the realized norm. This makes con tinuit y p lausible when induced weigh t is mean t to track stable record structure rather than threshold-sensitiv e co ding artifacts. But the assumption should not b e treated as free. If on e allows genuinely discontin uous dep endence on norm, dense admissible saturation need not determine the weigh t uniquely . Moreov er, as remark 7 shows, contin uit y alone still does not suﬃce when the admissible reﬁnemen t class is to o sparse. 17 Corollary 2 (Born assignment as normalized realization) . A ssume in addition: (a) the induc e d weight is normalize d: the total admissible next-step c ontinuation bund le c arries total induc e d weight 1 ; (b) these se ctors form a c omplete ortho gonal de c omp osition of the r elevant next-step c ontent, so that X i ∥ Π R i Ψ ∥ 2 = ∥ Ψ ∥ 2 = 1 . Then W Ψ ( R ) = ∥ Π R Ψ ∥ 2 for every glob al state Ψ and every r obust r e c or d se ctor R in the de c omp osition. Pr o of. By theorem 3 , the induced w eight has the form W Ψ ( R ) = c ∥ Π R Ψ ∥ 2 with a constan t c ≥ 0 . By assumption (b), the squared norms of the pro jected comp onents sum to ∥ Ψ ∥ 2 = 1 . The total induced w eight of the admissible next-step contin uation bundle therefore equals X i W Ψ ( R i ) = c X i ∥ Π R i Ψ ∥ 2 = c. By assumption (a), this total equals 1 , so c = 1 . Therefore W Ψ ( R ) = ∥ Π R Ψ ∥ 2 . R emark 9 . The theorem do es not construct a probability measure on the full pro jector lattice of H . It establishes a uniqueness result only for induced weigh ts on robust record sectors inside an admissible Hilb ert record lay er. R emark 10 . The pro of uses no app eal to Gleason’s theorem, no decision-theoretic axiom, and no en v ariance argument. The quadratic assignment is forced directly by sector-level contin uation partition, norm-proﬁle reduction, and the tw o structural conditions isolated in section 4 . R emark 11 (Compression viewp oint) . The theorem can also b e read as a compression statemen t. Once admissible induced weigh ts are reduced to a norm-proﬁle function g , the en tire admissible w eight structure collapses to the one-parameter form g ( r ) = c r 2 . No further dep endence on the in ternal realization of a robust record sector surviv es b ey ond the norm of its pro jected comp onent. Under the normalization condition of corollary 2 , ev en the constan t c is ﬁxed, so the induced weigh t is determined uniquely b y squared norm alone. The next section clariﬁes the exact scope of this result, the sense in whic h it is conditional, and the limits of what has and has not b een shown. 18 6 Scop e and Limits The result established in theorem 3 is deliberately conditional and tightly focused. Its p oint is not to s olv e every surrounding foundational problem at once, but to isolate the precise structural threshold at which the quadratic assignment b ecomes unav oidable. In that sense, the narrowness of the theorem is part of its conten t: it identiﬁes exactly which assumptions do the forcing work and exactly where the uniqueness claim b egins. 6.1 What the theorem establishes The main theorem sho ws that, within an admissible Hilb ert record la y er, and under the tw o structural conditions stated in conditions 1 and 2 , the induced weigh t on robust record sectors is forced to tak e the quadratic form W Ψ ( R ) = c ∥ Π R Ψ ∥ 2 . With the additional normalization stated in corollary 2 , this b ecomes W Ψ ( R ) = ∥ Π R Ψ ∥ 2 . The conten t of the theorem is therefore not that one may cho ose the quadratic assignmen t, but that, once the stated structural threshold is met, no other non-negativ e reﬁnemen t-stable assignmen t remains compatible with the framew ork on the s tated class of record sectors. 6.2 What the theorem do es not establish It is equally imp ortant to state what has not b een shown. First, the theorem do es not derive a probability measure on the full projector lattice of H . The argumen t is restricted from the outset to robust record sectors and admissible orthogonal reﬁnements among them. This restriction is essential to the logic of the pap er. Second, the theorem do es not derive Hilb ert structure itself. The admissible representation lay er is assumed to hav e Hilb ert-sector form. The presen t argument then shows what w eigh t assignment is forced within that lay er. The deep er question of wh y admissible represen tation lay ers should be Hilb ertian is a separate foundational problem. This is not an ev asion but a delib erate scop e decision: the presen t argument is designed to sho w what is forced within a Hilb ert record lay er, indep endently of whic hever deep er justiﬁcation one might accept for that lay er. The conditional structure makes the result applicable across multiple foundational p ositions on the Hilb ert-structure question. Third, the theorem do es not claim that ev ery orthogonal splitting of a Hilb ert subspace is physically meaningful. Only admissible orthogonal reﬁnements enter the argument. A dmissibility is stronger than orthogonalit y: it requires faithful representation of the observ able next-step exclusivit y structure of the record sector b eing reﬁned. F ourth, the theorem do es not eliminate the need for substantiv e structural assumptions. On the con trary , it mak es them explicit. The pro of dep ends essen tially on the internal equiv alence principle and on admissible binary saturation. These are not cosmetic additions. They deﬁne the domain on whic h the uniqueness claim is v alid. 19 Fifth, the theorem do es not establish that physically relev ant systems in dimension greater than t wo fall within its domain of application. The tw o-outcome spin example of remark 14 illustrates only the minimal tw o-dimensional setting targeted b y the framew ork. Whether robust record sectors satisfying admissible binary saturation, or the w eaker dense saturation condition of prop osition 2 together with contin uit y , exist for physically relev ant systems in higher dimensions remains an op en question. The theorem iden tiﬁes the structural threshold that forces the quadratic assignment; it do es not determine which physical systems realize that threshold. This is the precise sense in whic h the result is conditional: not merely that it assumes its premises, but that the physical scop e of those premises remains to b e mapp ed. The conditional structure of the argument can b e summarized by the following failure map. If this is remo ved What remains What is lost Extensiv e bundle v aluation with ﬁnite additivit y on disjoint con tinuation bundles admissible sectors and reﬁnemen ts may still b e deﬁned no sector-level additiv e law, hence no route to the functional equation In ternal equiv alence reﬁnemen t additivity may still hold on admissible splits no reduction to a one-v ariable proﬁle function g , hence no scalar uniqueness problem Suﬃcien t admissible reﬁnemen t richness only partial additivit y relations on the realized reﬁnemen t class no uniqueness of the induced w eight in general; see remark 7 Normalization the quadratic form W Ψ ( R ) = c ∥ Π R Ψ ∥ 2 the constan t c remains undetermined, so the Born assignmen t is not yet ﬁxed The framework also carries four distinct structural loads. First, exclusivity of record alternativ es ensures that con tinuation bundles form a canonical disjoint carrier for µ ; without it, ﬁn ite additivity loses its unam biguous domain of application. Second, con tinuation structure with short-horizon stabilit y makes W Ψ ( R ) = µ ( C Ψ ( R )) gen uinely induced rather than directly p ostulated; without it, sector w eight b ecomes primitive and the problem type c hanges. Third, internal reﬁnement ric hness supplies the non-trivial proﬁle structure needed for the later functional equation; without suﬃcien t ric hness, proﬁle equiv alence b ecomes to o sparse to constrain induced weigh t. F ourth, internal accessibilit y of admissible reﬁnemen t structure is what preven ts induced weigh t from dep ending on represen tational surplus; without it, the internal equiv alence principle loses its ground. 6.3 Wh y the conditional form matters The v alue of the conditional form is that it mak es the threshold structure of the result explicit while keeping mathematical uniqueness separate from physical applicability . The theorem settles the former on its stated domain and leav es the latter to the structural and empirical question of whic h systems instantiate the required record structure. 20 6.4 The role of the in ternal equiv alence principle The in ternal equiv alence principle, condition 1 , is the deep er of the t wo structural conditions. It requires induced weigh t to b e ﬁxed b y internally accessible admissible reﬁnemen t structure rather than by representational surplus. Rejecting it means allo wing internally indistinguishable record situations to carry diﬀerent induced weigh ts, so that weigh t is no longer ﬁxed by admissible reﬁnemen t structure alone. 6.5 The role of admissible binary saturation A dmissible binary saturation, condition 2 , is the richness condition that turns con tinuation-partition additivit y in to a constraining functional equation for g . It should not be read as a demand that arbitrary orthogonal splittings b e physically realizable, but as the requiremen t that a robust record sector carry enough stable in ternal diﬀerentiation to realize norm-compatible binary reﬁnements, exactly or at least densely . Without such reﬁnemen t richness, the induced w eight need not b e uniquely determined. As prop osition 2 shows, full admissible binary saturation ma y b e weak ened to dense admissible saturation if contin uit y of the proﬁle function is added. 6.6 No on tological ranking of branc hes The theorem should also not be read as assigning diﬀeren t de gr e es of r e ality to diﬀeren t record sectors. What has been established is a uniqueness result for induced weigh ts under reﬁnemen t stabilit y conditions. That is a structural statemen t ab out admissible w eight assignments, not an on tological ranking theorem. F or that reason, the language of this pap er remains delib erately neutral. It sp eaks of induced weigh t on robust record sectors, not of branch reality , branch substance, or metaph ysical thickness. In that sense, the theorem is interpretation-neutral: it do es not privilege Everettian branching o ver collapse-based, relational, or pragmatist readings, provided they admit the relev an t record structure on the stated domain. 6.7 Wh y the restriction to record sectors is substantiv e One migh t w orry that restricting the theorem to robust record sectors makes the result to o narrow. W e take the opp osite view. The restriction is exactly what gives the theorem a distinct foundational target. The presen t question is not how to assign a measure to arbitrary formal subspaces, but which weigh t assignmen ts remain viable on structures that can function as stable, in ternally accessible records. That is a sharp er and more ph ysically motiv ated question. The resulting theorem is narrow er than a global measure theorem, but also closer to the problem of stable internal record structure. 6.8 A sc hematic decoherence-stabilized record mo del The follo wing sk etch illustrates ho w the framework of this pap er can b e read in a setting closer to physical practice. It do es not derive decoherence, do es not establish a p ointer basis, and do es 21 not pro ve that admissible binary saturation holds for an y sp eciﬁc physical system. The theorem itself does not dep end on any sp eciﬁc decoherence mo del; the present sketc h only exhibits one ph ysically familiar w a y of reading the formal notions introduced ab ov e. Its purp ose is only to mak e the abstract framew ork physic al ly le gible in a standard schematic setting. Setup. Consider a ﬁnite-dimensional system S coupled to an environmen t E , and supp ose that the in teraction selects a decoherence-stabilized record structure with macroscopically distinct alternativ es indexed by i [ 17 , 10 ]. Rather than identifying record sectors with one-dimensional p oin ter states, let R i denote coarse-grained subspaces asso ciated with stable record alternativ es of the comp osite S E -system, or of an eﬀective record-b earing subsystem. F or present purp oses, these sectors are treated as an idealized exactly orthogonal and short-horizon stable decomp osition, even though in realistic decoherence mo dels such structure is typically only approximate. In this setting, deﬁnition 1 receiv es the follo wing reading: internal discriminability corresponds to the record alternativ es b eing op erationally distinguishable, short-horizon p ersistenc e corresp onds to their stability o ver the timescale relev an t to the next step of record use, and admissible r eﬁnement closur e corresp onds to the p ossibilit y of resolving a coarse record alternative into ﬁner stable sub-records. A dmissible c ontinuations and r eﬁnements. The contin uation bundles C Ψ ( R i ) can then b e read as the class of p ost-in teraction contin uations of the comp osite dynamics in which the record conten t remains consisten t with the sector R i . An admissible orthogonal reﬁnement R i = R i, 1 ⊕ R i, 2 corresp onds schematically to a further stable discrimination within the same coarse record alternative, pro vided the reﬁned sectors are again m utually exclusiv e and robust enough to function as record b earers. What is structur al ly set, what is plausible, what r emains op en. What is set by this schematic is only the reading of robust record sectors as decoherence-stabilized record subspaces, of con tin uation bundles as record-consisten t contin uations, and of admissible reﬁnemen ts as further stable discriminations within a coarse record structure. What is plausible is that suc h a setting may plausibly supp ort a suﬃcien tly rich family of admissible reﬁnements, and that dense admissible saturation may therefore b e more realistic than full saturation when exact realization of ev ery norm-compatible split is op erationally to o strong. What remains op en is whether admiss ible binary saturation, or even the w eaker dense saturation condition of prop osition 2 , holds for any concrete physical system. The theorem iden tiﬁes the structural threshold, the sc hematic makes it physically legible, and the gap b et ween legibility and certiﬁcation remains a separate empirical and structural question. R emark 12 (A concrete candidate regime for dense saturation) . A plausible physical regime for the w eaker dense-saturation condition is a high-dimensional decoherence-stabilized record sector with redundan t environmen tal enco ding [ 9 , 20 ]. One should think here not of a one-dimensional p ointer state, but of a coarse macroscopic record, such as a detector outcome or p ointer reading, that is stably carried b y many microscopically distinct apparatus-environmen t conﬁgurations. In such a setting, an admissible reﬁnement need not arise from arbitrary one-dimensional splittings. It can instead arise from further stable discrimination within the same coarse record sector by grouping families of robust micro-record subspaces that preserv e the same coarse record conten t. When the num b er of such approximately indep endent record-b earing comp onen ts is large, and when the sector admits suﬃcien tly ﬁne robust sub-record decomp ositions whose binary groupings remain admissible, prop osition 3 pro vides a concrete route by which the realized admissible binary reﬁnemen t norms can become dense in the interv al allo wed by the total sector norm. As in the 22 sc hematic mo del of section 6.8 , exact orthogonalit y of the sub-records is an idealization; in practice the relev an t structure is only appro ximately orthogonal, and the dense-saturation condition is stated here as applying to the idealized version. This do es not establish dense admissible saturation for any concrete mo del, and it do es not show that ev ery norm-compatible split is physically realizable. Its role is narrow er: it iden tiﬁes a non-trivial ph ysical regime in which the dense v ersion of the saturation condition is at least credible. The natural candidate regime is that of redundan t macroscopic records supp orted b y many approximately orthogonal en vironmental subrecords in a large apparatus-environmen t sector. 6.9 A lo cal structural criterion for dense admissible s aturation The candidate regime just describ ed can be sharp ened in to a lo cal structural criterion. The p oin t is not to certify a concrete ph ysical mo del, but to isolate an explicit microstructural h yp othesis under whic h dense admissible saturation follows for a given pair (Ψ , R ) . Prop osition 3 (Lo cal criterion for dense admissible saturation) . Fix a glob al state Ψ and a r obust r e c or d se ctor R , and write s := ∥ Π R Ψ ∥ . A ssume that ther e exists a se quenc e of ﬁnite ortho gonal de c omp ositions R = N n M i =1 S ( n ) i ( n ∈ N ) such that e ach S ( n ) i is itself a r obust r e c or d se ctor, and that the fol lowing hold for every n : (i) for every subset A ⊆ { 1 , . . . , N n } , the gr oup e d se ctors R ( n ) A := M i ∈ A S ( n ) i , R ( n ) A c := M i / ∈ A S ( n ) i form an admissible ortho gonal r eﬁnement of R ; (ii) with w ( n ) i :=     Π S ( n ) i Ψ     2 , one has max 1 ≤ i ≤ N n w ( n ) i ≤ ε n ∥ Π R Ψ ∥ 2 for some se quenc e ε n → 0 . Then the set of r e alize d norm values D Ψ ,R :=      Π R ( n ) A Ψ         n ∈ N , A ⊆ { 1 , . . . , N n }  is dense in [0 , s ] . In p articular, if the same criterion holds for every p air (Ψ , R ) in the r elevant admissible domain, then the dense-r e alizability hyp othesis use d in pr op osition 2 is satisﬁe d on that domain. 23 Pr o of. If s = 0 , then D Ψ ,R = { 0 } , so there is nothing to prov e. Assume s > 0 , and set W := s 2 = ∥ Π R Ψ ∥ 2 . F or eac h n and each subset A ⊆ { 1 , . . . , N n } , orthogonalit y gives     Π R ( n ) A Ψ     2 = X i ∈ A w ( n ) i . Fix x ∈ [0 , W ] . If x = W , choose A = { 1 , . . . , N n } , so there is nothing to prov e. Assume x < W , and for eac h n choose among all subsets A satisfying X i ∈ A w ( n ) i ≤ x one for whic h the sum is maximal. W rite x ( n ) A := X i ∈ A w ( n ) i . W e claim that 0 ≤ x − x ( n ) A < max i w ( n ) i . Indeed, if x − x ( n ) A ≥ max i w ( n ) i , then A cannot b e the full index set, since x ( n ) A < W . Hence there exists j / ∈ A , and for ev ery such j one has w ( n ) j ≤ max i w ( n ) i ≤ x − x ( n ) A , so x ( n ) A + w ( n ) j ≤ x, con tradicting the maximality of x ( n ) A . Therefore 0 ≤ x − x ( n ) A < max i w ( n ) i ≤ ε n W . Since ε n → 0 , the realized quadratic sums are dense in [ 0 , W ] . Because the square-ro ot map is a homeomorphism from [0 , W ] to [0 , s ] , the realized norm v alues in D Ψ ,R are dense in [0 , s ] . R emark 13 (A work ed ﬁnite illustration of the lo cal criterion) . The mec hanism b ehind prop osition 3 can b e seen in a simple ﬁnite example. Let R be a robust record sector relative to Ψ with ∥ Π R Ψ ∥ 2 = 1 , and supp ose that R = S 1 ⊕ S 2 ⊕ S 3 ⊕ S 4 is a ﬁnite orthogonal decomp osition in whic h each S i is itself a robust record sector and ev ery binary grouping of these four sectors forms an admissible orthogonal reﬁnement of R . If w 1 = 0 . 40 , w 2 = 0 . 30 , w 3 = 0 . 20 , w 4 = 0 . 10 , then the realized quadratic shares are exactly the subset sums 0 , 0 . 10 , 0 . 20 , 0 . 30 , 0 . 40 , 0 . 50 , 0 . 60 , 0 . 70 , 0 . 80 , 0 . 90 , 1 . 24 This set is not dense in [0 , 1] , but it makes the com binatorial mechanism explicit: admissible binary groupings of ﬁner internal record components generate realized norm splits through subset sums of their pro jected wei ghts. If one passes instead to a sequence of ﬁner admissible decomp ositions for whic h the largest pro jected w eight tends to zero, then proposition 3 implies that the realized quadratic shares become dense in [0 , 1] , and therefore the realized norm v alues b ecome dense in [0 , ∥ Π R Ψ ∥ ] as w ell. This is only an illustration of the mechanism. It does not certify that any concrete physical apparatus realizes the required admissible reﬁnement structure. Its role is narrow er: it shows explicitly ho w dense admissible saturation can arise from suﬃcien tly ﬁne robust internal sub-record structure once admissible binary groupings are av ailable. 6.10 A to y structural example: tw o-outcome spin system The following example is purely structural. It does not derive a pointer basis from decoherence, and it do es not claim to settle any in terpretiv e question ab out quan tum measurement. Its sole purp ose is to verify that the deﬁnitions of section 2 are non-empt y and that the theorem delivers the exp ected result in the simplest p ossible case. R emark 14 (T oy structural example) . Let H = C 2 and let Ψ = α | ↑⟩ + β | ↓⟩ , | α | 2 + | β | 2 = 1 . Deﬁne t wo robust record sectors by R ↑ := span {| ↑⟩} , R ↓ := span {| ↓⟩} . V eriﬁc ation of deﬁnition 1 . Eac h of R ↑ and R ↓ is a one-dimensional closed subspace of H . They represen t m utually exclusive, in ternally readable record alternatives: no admissible con tinuation can realize b oth sim ultaneously . Eac h subspace is stable under small represen tational p erturbations in the sense that its one-dimensional span is not disrupted by admissible micro-reco dings. The pair { R ↑ , R ↓ } forms an admissible exclusivit y structure in whic h R ↑ ⊕ R ↓ = H . A dmissible ortho gonal r eﬁnement. The decomp osition H = R ↑ ⊕ R ↓ is an admissible orthogonal reﬁnemen t of the full next-step sector relative to Ψ : ev ery admissible contin uation falls under exactly one of R ↑ or R ↓ , and no con tin uation remains outside this partition. The pro jected components are ϕ R ↑ = Π R ↑ Ψ = α | ↑⟩ , ϕ R ↓ = Π R ↓ Ψ = β | ↓⟩ , with    ϕ R ↑    2 = | α | 2 and    ϕ R ↓    2 = | β | 2 . Binary satur ation. In this t wo-dimensional setting the decomp osition H = R ↑ ⊕ R ↓ is the only non-trivial admissible binary reﬁnemen t singled out b y the ﬁxed record basis. The example is therefore not mean t to establish admissible binary saturation in general. Its purp ose is only to illustrate the minimal binary setting in whic h the theorem applies once admissible binary saturation is assumed for the sector under consideration. Conclusion fr om the or em 3 and c or ol lary 2 . By theorem 3 , the only non-negative reﬁnemen t-stable induced w eight takes the form W Ψ ( R ) = c ∥ Π R Ψ ∥ 2 . Since R ↑ and R ↓ form a complete orthogonal decomp osition with | α | 2 + | β | 2 = 1 , corollary 2 giv es c = 1 . Therefore W Ψ ( R ↑ ) = | α | 2 , W Ψ ( R ↓ ) = | β | 2 . 25 This is the expected Born assignment. The example shows that the structural apparatus of the pap er is non-v acuous: the deﬁnitions are satisﬁable, the conditions are chec kable, and the theorem deliv ers the correct quadratic assignment in this minimal case. 6.11 T ransition to the literature comparison The scope of the theorem can now b e stated succinctly . Within an admissible Hilbert record la yer, and under explicit structural conditions tied to internal equiv alence and suﬃcient admissible reﬁnemen t richness, the quadratic assignment is the only non-negativ e reﬁnement-stable induced w eight on robust record sectors. The next section compares this route with several established families of Born-rule arguments and iden tiﬁes the precise logical p oint at which the presen t approac h diﬀers from them. 7 Relation to Existing Routes The present theorem b elongs to a familiar landscap e of attempts to accoun t for the quadratic quan tum weigh t, but its logical target and its carrier structure diﬀer from the standard routes. The comparison is therefore b est made not only at the level of conclusion, but at the level of what each route tak es as primary , where its additiv e or constraining structure is placed, and what each route is trying to pro ve. 7.1 Measure-theoretic uniqueness routes A natural comparison p oint is the family of Gleason-t yp e results. Gleason’s theorem shows that, on Hilb ert spaces of dimension at least three, suﬃcien tly well-behav ed additiv e measures on the lattice of pro jections are represented by density op erators [ 5 ]. Busch’s extension shifts the setting from pro jection-v alued assignments to generalized observ ables and thereb y remo v es the qubit restriction [ 1 ]. These results are mathematically p ow erful, but they b egin with a global measure-theoretic target. A dditivity is built in to the starting p oin t as a prop erty of the measure on projections or eﬀects. The presen t pap er does not replace that route, and it do es not attempt to subsume it. It prov es a diﬀeren t uniqueness theorem with a diﬀeren t theorem target and a diﬀerent additive carrier. Its target is narro wer, namely induced weigh ts on robust record sectors only . Its additiv e primitive is not a measure on the full pro jector lattice, but ﬁnite additivity on disjoin t con tinuation bundles through an extensive bundle v aluation. The sector-level additive law is then inherited from con tinuation partition under admissible reﬁnemen t. The diﬀerence therefore runs through the ob ject b eing w eighted, th e structure that carries additivity , and the level at which the uniqueness question is p osed. This diﬀerence in additiv e carrier was already built in to the framework level in section 2 and is here restated at the lev el of theorem comparison. A recen t critical analysis by Zhang argues, from a diﬀerent angle, that additivity is indisp ensable across several ma jor Born-rule deriv ations and cannot b e eliminated in fav our of weak er assumptions suc h as non-contextualit y and normalization alone [ 16 ]. That diagnosis is broadly compatible with 26 the presen t result, but the presen t pap er sharp ens the structural p oin t in a diﬀerent w ay: it do es not treat additivit y as a global measure-theoretic p ostulate on the pro jector lattice, but relo cates the additive primitive to disjoin t con tinuation bundles and then identiﬁes the further structural conditions under whic h quadratic weigh t is forced on robust record sectors. 7.2 Decision-theoretic Ev erettian routes A second ma jor route is the Everettian decision-theoretic program initiated b y Deutsc h and developed in more formal detail b y W allace [ 3 , 14 , 15 ]. In that framew ork, the Born rule is connected to constrain ts on rational preference or rational b etting b ehaviour in branc hing quantum situations. Critical discussion of that route has emphasized b oth its formal ingen uity and the non-trivial role pla yed by its rationality assumptions [ 7 ]. The presen t theorem do es not app eal to rational c hoice, utility , diachronic consistency , or b etting b eha viour. It is not a theorem ab out ho w agents should app ortion credence in a bran c hing setting. It is a theorem ab out which non-negative weigh t assignments are structurally admissible on robust record sectors under explicit reﬁnement-stabilit y conditions. The con trast is therefore not only b et ween normativ e and non-normative language, but b etw een t wo diﬀeren t theorem targets: rational credence constrain ts on the one hand, and reﬁnemen t-stable induced w eight on the other. 7.3 En v ariance and symmetry-based routes A third route is Zurek’s env ariance program, which seeks to derive the Born rule from symmetry prop erties of entangled states [ 19 ]. That approach has b een inﬂuential precisely b ecause it attempts to replace probabilistic p ostulates with a more primitiv e symmetry argument. At the same time, critical analyses hav e argued that substantial assumptions remain active in the deriv ation, esp ecially in the passage from symmetric situations to probability assignments [ 11 ]. The present route do es not rely on en tanglement symmetry , sw ap inv ariance, or environmen t-assisted in v ariance. The core functional equation arises instead from tw o diﬀerent ingredients: sector-level con tinuation partition under admissible reﬁnemen t, and the reduction of induced w eight to the norm proﬁle of p ro jected record comp onents. The active inv ariance principle here is therefore not a symmetry of en tangled states, but inv ariance of induced weigh t under in ternally indistinguishable admissible reﬁnemen t structure. 7.4 Self-lo cating uncertaint y and eviden tial routes Another line of work approac hes the Born rule through self-lo cating uncertaint y and closely related questions ab out conﬁrmation in Everettian settings [ 12 , 6 ]. These approaches ask how an observ er should reason when branching has o ccurred or is about to o ccur, and ho w observed frequencies can coun t as evidence in a theory with m ultiple successor observ ers. The present theorem is logically prior to suc h questions. It do es not b egin with p ost-branch credence, self-lo cation, or theory conﬁrmation. Its concern is earlier and more structural: which w eight assignmen ts are compatible with in ternally stable record reﬁnement in the ﬁrst place. On the present route, eviden tial and self-locating questions arise only after the admissible w eigh t form has already b een ﬁxed. 27 7.5 Op erational reconstruction routes A further comparison is with op erational reconstructions in which the measurement p ostulates are argued to b e ﬁxed b y the rest of quantum structure together with additional op erational principles. A notable recent example is the claim that the measurement p ostulates of quantum mechanics are op erationally redundan t [ 8 ]. That line diﬀers from the presen t one b oth in scop e and in target: it addresses the structure of measuremen ts and state up date more broadly , whereas the present pap er isolates only the uniqueness of induced weigh t on robust record sectors. The presen t result is therefore not a reconstruction of measurement theory , but a structural uniqueness theorem for record-sector w eight once the admissible framework is ﬁxed. The subsequent critical discussion of hidden assumptions in such reconstructions also underscores the imp ortance of making the active structural conditions fully explicit [ 13 ]. 7.6 Normativ e Bay esian routes Finally , there are approaches, most prominently in QBism, that do not treat the Born rule as an ob jectiv e branc h-weigh t law at all, but as a normativ e coherence condition gov erning an agen t’s probabilit y assignmen ts [ 4 ]. On that reading, the Born rule is not deriv ed from the structure of branc hing worlds or record sectors, but interpreted as an empirically constrained rule for consistent probabilistic judgemen t. The present theorem is diﬀeren t in b oth aim and ob ject. It do es not prop ose a normative rule for an agen t’s b etting commitments. It iden tiﬁes the only non-negative reﬁnement-stable induced weigh t on robust record sectors under explicitly stated conditions on admissible reﬁnement and internal indistinguishabilit y . 7.7 Logical diﬀerence of the presen t route The clearest w ay to summarize the diﬀerence is to ask where each route places its primary structure. Standard routes t ypically b egin with one of the following: (i) a global measure on pro jections or eﬀects, (ii) rational preference in branc hing settings, (iii) symmetry of en tangled states, (iv) self-lo cating credence after branching, (v) broad op erational p ostulates ab out measurement, (vi) normativ e coherence constraints on probability assignments. The presen t pap er b egins elsewhere. It starts from robust record sectors, admissible con tinuation bundles, an extensive bundle v aluation on disjoint bundles, and explicit structural conditions on admissible reﬁnemen t. Its theorem is correspondingly narrow er than a full reconstruction of quantum probabilit y , but sharp er in logical target. What is shown is not that ev ery pro jector carries Born w eight by abstract measure representation, nor that ev ery rational Everettian agent should b et in accordance with amplitude-squared weigh t, but that within an admissible Hilb ert record lay er the 28 quadratic assignment is the only non-negativ e reﬁnement-stable induced w eight on robust record sectors. The presen t route should therefore not b e describ ed as a replacemen t for Gleason, but as a distinct conditional uniqueness theorem at a diﬀerent structural level. 8 Conclusion This pap er has identiﬁed a conditional structural uniqueness theorem for quadratic weigh t that diﬀers in target, additive carrier, and logical structure from the standard deriv ations. The argument do es not b egin with a probabilit y measure on the full pro jector lattice, with rational b etting constrain ts, with en tanglement symmetries, or with self-locating credence. It b egins with robust record sectors, admissible con tinuation bundles, and explicit structural conditions on admissible reﬁnement. Within an admissible Hilb ert record lay er, the argument pro ceeds in three steps. First, admissible orthogonal reﬁnement induces a partition of con tinuation bu ndles and therefore an additiv e law for sector-lev el induced weigh t. Second, internal equiv alence together with the norm classiﬁcation supplied by admissible binary saturation reduces the sector-lev el assignmen t to a one-v ariable function of the norm of the pro jected record comp onent. Third, suﬃcien t admissible reﬁnemen t ric hness forces the resulting functional equation: in the main theorem this is secured by admissible binary saturation, while proposition 2 sho ws that dense admissible saturation already suﬃces if con tinuit y of the proﬁle function is added. Non-negativit y then forces the additive function to b e linear b y lemma 2 . The result is the quadratic assignment W Ψ ( R ) = c ∥ Π R Ψ ∥ 2 , with the normalized case giving the standard Born assignment W Ψ ( R ) = ∥ Π R Ψ ∥ 2 . The main conceptual p oint is therefore not merely that a quadratic expression reappears. It is that, on the stated domain, the quadratic assignmen t is the only non-negative reﬁnement-stable induced w eight. The theorem isolates a sp eciﬁc structural threshold: once the additiv e primitive is placed on con tinuation bundles, binary-proﬁle-equiv alen t pairs carry the same induced weigh t, and the admissible reﬁnement structure is ric h enough to force the functional equation, no other non-negativ e induced weigh t can b e reﬁnement-stable. This source of uniqueness sits at a diﬀerent structural lev el from the standard global measure-theoretic and decision-theoretic routes. The framew ork is not merely a con venien t domain restriction. It isolates a functional setting in which induced w eight on record-like alternatives is b oth non-arbitrary and structurally constrainable. Just as imp ortant is the restricted scop e of the result. The theorem neither deriv es Hilb ert structure nor assigns w eights to arbitrary pro jectors, and it do es not claim that ev ery orthogonal decomp osition is physically meaningful or that broader metaph ysical conclusions ab out branc hing structure follow b y themselves. What it do es show is narrow er and sharp er: once one w orks within an admissible Hilb ert record lay er and imp oses the stated structural conditions, the quadratic assignment is forced as the only non-negativ e reﬁnement-stable induced weigh t on robust record sectors. That conditional form should b e regarded as a strength. It makes the threshold structure of the argumen t transparen t and k eeps the uniqueness claim tied to explicitly stated premises. It also 29 separates tw o questions that are often blurred together: the mathematical uniqueness question and the physical applicabilit y question. F uture w ork can therefore pro ceed in a con trolled wa y . One direction is to inv estigate whether the in ternal equiv alence principle can b e justiﬁed from a still deep er c haracterization of internally accessible record structure. Another is to study whic h ph ysically realistic record sectors satisfy admissible binary saturation, or at least dense admissible saturation together with con tinuit y , and where these conditions break down. A third is to ask whether the present route can b e extended b eyond binary reﬁnements while preserving the same degree of explicitness. The presen t result should therefore b e read as a fo cused theorem ab out induced weigh t on robust record sectors. Its claim is not maximal generalit y , but explicit structural uniqueness at a clearly iden tiﬁed threshold. Under the stated conditions, the quadratic assignment is not one admissible c hoice among many . 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The Born Rule as the Unique Refinement-Stable Induced Weight on Robust Record Sectors

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