AUTHOR: Evan T. Kotler
The quadratic form of probability in quantum theory is commonly introduced as a postulate or justified through interpretive, operational, or decision-theoretic arguments. This paper adopts a different perspective. Building on a framework in which probability emerges structurally as a compression-invariant frequency in finite relational systems, we treat representation as an explicitly licensed extension rather than a primitive assumption. We show that admissible composition in such systems is generically non-functorial, obstructing straightforward numerical or linear representations of probability. When this obstruction is respected rather than idealized away, any linear representation capable of encoding structural probability must be projective and phase-tolerant. Introducing a minimal public coherence requirement—additivity on mutually exclusive contexts—we prove, via a Gleason-class result, that probability assignments compatible with admissible composition necessarily factor through a positive quadratic form. The familiar Born rule thus appears not as a postulate of physics, but as the minimal consistent resolution of a representational obstruction. No claim is made that amplitudes or Hilbert spaces are fundamental; the result is conditional and concerns representation, not ontology.
Probability plays a central role in quantum theory, yet its mathematical form is strikingly rigid. Probabilities are not merely additive; they are quadratic in complex amplitudes. This feature is usually taken as a fundamental axiom or justified by appeal to measurement postulates, decision-theoretic rationality, or interpretive commitments.
This paper asks a different question:
If probability is not fundamental, but emerges structurally, what form must its representation take—if representation is attempted at all?
Previous work has established that probability can arise as a compression-invariant frequency in finite relational systems subject to admissible extension and non-collapse. In that setting, probability exists prior to any numerical or algebraic representation. Nothing in the structural derivation of probability requires amplitudes, Hilbert spaces, or even numbers.
Representation is therefore optional. But if one attempts it, the attempt is constrained.
The central claim of this paper is that the quadratic form of probability is not a physical postulate but a representational necessity. It arises as the minimal way to encode structural probability while respecting a deep obstruction: the non-functoriality of admissible composition.
We assume the results of previous work:
probability emerges as a compression-invariant frequency associated with survivor classes in a finite relational system;
probability is public, structural, and non-epistemic;
no numerical representation is assumed or required.
This paper introduces representation as an additional, explicitly licensed move.
A representation attempt is a mapping [ F : \mathcal W \rightarrow \mathcal D, ] where:
(\mathcal W) is the domain of admissible contexts and extensions,
(\mathcal D) is an algebraic or numerical domain intended to encode probability.
No structural properties of (F) are assumed a priori.
A natural expectation is that representation should respect closure under 𝔈-composition.
A representation attempt (F) is functorial if, for all admissible extensions (a,b), [ F(b\circ a) = F(b)\circ F(a). ]
This expectation fails.
For any representation attempt that respects admissibility and compression, there exist admissible extensions (a,b) such that [ F(b\circ a) \neq F(b)\circ F(a). ]
Admissible composition is contextual. Compression irreversibly removes distinctions, and admissible extensions may introduce new relational neighborhoods. No admissible operation preserves all information required to enforce strict compositional equality. Any representation enforcing functoriality implicitly introduces additional invariant structure, violating stability under admissible extension. ∎
Non-functoriality is not a technical flaw. It is:
invariant under admissible renaming,
stable under admissible extension,
independent of the chosen target domain.
Any viable representation must therefore absorb this obstruction rather than deny it.
To proceed, we introduce the only representational assumption of the paper.
There exists a representation attempt with the following properties:
Linear carrier: representational elements lie in finite-dimensional complex vector spaces.
Projective identification: physically meaningful distinctions correspond to rays (vectors modulo nonzero scalar multiplication).
Phase-tolerant composition: admissible extensions are represented by linear maps defined up to scalar phase.
Probability functional: probabilities are functions defined on rays.
This assumption does not assert:
that the world is linear,
that Hilbert space is fundamental,
or that quantum mechanics is derived.
It merely licenses a specific class of representations to be analyzed.
Any representational quantity invariant under admissible extension must be invariant under nonzero scalar multiplication.
Non-functoriality manifests as phase-like defects in composition. Scalar-sensitive quantities vary under admissible extension and therefore fail stability. Quotienting by nonzero scalars is the minimal operation restoring invariance. ∎
Projective structure is thus forced, not assumed.
Representation alone does not fix probability assignments. An additional coherence requirement is needed.
A context is a finite set of mutually exclusive rays representing alternatives that cannot co-occur within a single admissible continuation.
A probability assignment is publicly coherent if it assigns probabilities to rays independently of the context in which they appear, subject to normalization on each context.
For any two contexts (\mathcal C) and (\mathcal C') related by admissible extension or relabeling, [ \sum_{[v]\in\mathcal C} P([v]) = \sum_{[v']\in\mathcal C'} P([v']). ]
This axiom encodes only public consistency. It assumes no observers, measurements, or rational agents.
Let (P) be a publicly coherent probability functional on rays in a finite-dimensional complex vector space of dimension at least three. Then (P) must factor through a positive quadratic form: [ P([v]) = \frac{\langle v, \rho v\rangle}{\langle v, v\rangle}, ] for some positive semidefinite operator (\rho).
Under the representational bridge and projective noncontextual additivity, the hypotheses of a Gleason-class theorem are satisfied. Such theorems establish that any additive frame function in dimension (\ge 3) is necessarily quadratic. ∎
In dimension two, Gleason’s theorem does not apply. Additional admissibility or coherence constraints are required. This limitation is structural, not a defect of the present framework.
The result above does not claim that quantum mechanics is derived. It establishes that:
if probability is represented linearly,
if admissible composition is non-functorial,
and if public coherence is required,
then a quadratic rule is unavoidable.
The Born rule appears here as the minimal consistent resolution of a representational obstruction, not as a physical axiom.
This paper:
does not assume amplitudes as fundamental,
does not introduce measurement postulates,
does not claim ontological primacy of Hilbert space.
It addresses representation only.
Probability can exist without amplitudes, universality can exist without constants, and information can exist without probability. Representation, however, is unforgiving. Once one attempts to encode structural probability linearly while respecting admissible composition, the freedom disappears. The quadratic form of probability is not chosen; it is forced.
This appendix provides an explicit example demonstrating the failure of functoriality for admissible composition. The purpose is not generality, but existence: to exhibit a concrete pair of admissible extensions whose composition order matters once compression is taken into account.
Let (\mathcal S) be a finite relational system with configuration set [ \Omega = {a,b,c}. ]
Assume the initial relational structure satisfies:
(a) and (b) are distinguishable,
(b) and (c) are distinguishable,
(a) and (c) are behaviorally congruent under all admissible extensions currently available.
Thus, prior to any extension, the behavioral congruence classes are: [ [a] = {a,c}, \qquad [b] = {b}. ]
No interpretation is attached to these configurations; the structure is purely relational.
We now define two admissible extensions (E_1) and (E_2), each satisfying the admissibility criteria of Paper I.
(E_1) introduces additional relational context that attempts to distinguish (a) and (c), but does so in a way that is unstable under further admissible extension.
Concretely:
(E_1) introduces a relation (R) such that (R(a)) and (R(c)) differ,
however, the distinction depends on relational structure not preserved under arbitrary admissible extension.
As a result, under admissibility: [ a \sim c \quad \text{still holds after } E_1, ] and compression identifies them: [ [a]_{E_1} = {a,c}. ]
Thus, (E_1) is admissible but does not split the survivor class.
(E_2) introduces additional relational structure that merges (b) with the class ({a,c}) by eliminating the distinctions that previously separated (b) from them.
After (E_2), the behavioral congruence classes are: [ [a]_{E_2} = {a,b,c}. ]
This is an admissible coarse-graining: no new distinctions are introduced, and stability under extension is preserved.
We now examine the two compositions.
After (E_1): classes are ({a,c}), ({b}).
Applying (E_2): all distinctions are eliminated.
Final survivor structure: [ \Sigma_{E_2 \circ E_1} = {{a,b,c}}. ]
After (E_2): single class ({a,b,c}).
Applying (E_1): no refinement is possible, because compression has already eliminated the relevant distinctions.
Final survivor structure: [ \Sigma_{E_1 \circ E_2} = {{a,b,c}}. ]
At the structural level, both compositions yield the same survivor set. However, the relational history differs: in one case, an unstable refinement attempt is made before compression; in the other, it is blocked outright.
Let (F) be any representation attempt assigning linear operators to admissible extensions.
(F(E_1)) must encode a refinement attempt,
(F(E_2)) must encode coarse-graining,
but compression irreversibly erases the intermediate structure.
Thus: [ F(E_2 \circ E_1) \neq F(E_2) \circ F(E_1), ] because the latter composition presupposes preservation of information that admissibility explicitly forbids.
The obstruction is not algebraic; it is structural. The relational data required to enforce functoriality does not survive admissible extension.
This example demonstrates that:
admissible extensions are closed under composition,
but composition is not functorial with respect to any representation preserving admissibility and compression,
and the failure arises from irreversible structural compression, not from representational deficiency.
This establishes non-functoriality as a generic feature of admissible structure, not a pathological case.
