How much unitary symmetry does the Born exponent actually need? Within coordinate-additive probability rules—each outcome weighted by a function $g$ of its amplitude modulus—invariance under a single genuine two-coordinate mixer already forces the quadratic $g(x)=cx^2$ for every dimension $n\ge 3$, where the standard route (Gleason 1957; Hossenfelder 2021; Gogioso 2023) assumes the full unitary group. A single beam-splitter-like unitary on two coordinates, with phase freedom, reduces the invariance condition to a measurable Jensen (midpoint) functional equation whose only solutions are quadratic; the forced weight functional obeys the parallelogram law, so Jordan–von Neumann polarization reconstructs the inner product. The result is a characterization and a no-go, not a derivation of the Born rule: any axiomatization in this class demanding even one genuine mixing symmetry has already presupposed the inner product, so ‘why the exponent is two’ coincides with ‘why coordinate mixing is unitary’. Generalized permutations alone leave the exponent free; the forcing fails at $n=2$, mirroring Gleason’s dimension threshold; and the intermediate regime is empty for the two symmetry classes treated. Exact-arithmetic and numerical certificates are included. Target venue: Foundations of Physics. v2 (August 2026) — premise-level correction. The symmetry premise is restated at the level at which it holds, the weight functional: invariance of the normalised probability rule does not follow from it, and an explicit counterexample is given. A global probability-level invariance premise is shown to be vacuous for any genuine mixer with non-vanishing quadratic weight, and domain-of-definition conditions are made explicit. A prior-art and novelty assessment accompanies this version. Earlier versions remain in the version history.
How much unitary symmetry does the Born exponent actually need? Within coordinate-additive probability rules—each outcome weighted by a function g of its amplitude modulus—invariance under a single genuine two-coordinate mixer already forces the quadratic g(x)=cx2 for every dimension n≥3, where the standard route (Gleason 1957; Hossenfelder 2021; Gogioso 2023) assumes the full unitary group. A single beam-splitter-like unitary on two coordinates, with phase freedom, reduces the invariance condition to a measurable Jensen (midpoint) functional equation whose only solutions are quadratic; the forced weight functional obeys the parallelogram law, so Jordan–von Neumann polarization reconstructs the inner product. The result is a characterization and a no-go, not a derivation of the Born rule: any axiomatization in this class demanding even one genuine mixing symmetry has already presupposed the inner product, so ‘why the exponent is two’ coincides with ‘why coordinate mixing is unitary’. Generalized permutations alone leave the exponent free; the forcing fails at n=2, mirroring Gleason’s dimension threshold; and the intermediate regime is empty for the two symmetry classes treated. Exact-arithmetic and numerical certificates are included. Target venue: Foundations of Physics. v2 (August 2026) — premise-level correction. The symmetry premise is restated at the level at which it holds, the weight functional: invariance of the normalised probability rule does not follow from it, and an explicit counterexample is given. A global probability-level invariance premise is shown to be vacuous for any genuine mixer with non-vanishing quadratic weight, and domain-of-definition conditions are made explicit. A prior-art and novelty assessment accompanies this version. Earlier versions remain in the version history.
Quantum mechanics says that when you measure a system, the probability of an outcome is the squared modulus of that outcome's amplitude — not absolute value, not cubed, not some other power. The exponent has been two since Born wrote it down in 1926, and for a century it has been a postulate. Copenhagen accepts it as a brute fact; Many-Worlds tries to derive it from decision theory; QBism reframes it as Bayesian credence; Zurek derives it from environmental symmetries; Hardy derives it from operational axioms. Each program leaves at least part of the question unanswered: why squared, and not some other exponent? Gleason's 1957 theorem went part of the way: given a few structural conditions on quantum probability assignments, the squared form is mathematically forced. But Gleason leans on one substantive assumption — non-contextuality — that standard QM postulates rather than derives. This walkthrough shows how the Event Density framework derives the Born rule, including the squared exponent *and* the non-contextuality assumption underneath it, from a smaller set of structural commitments about what reality is made of at the substrate level. The chain has three load-bearing steps: (i) the participation measure is forced to the form √b · e^{iπ} via the Cauchy functional equation on bandwidth additivity plus Frobenius's theorem on real division algebras; (ii) the sesquilinear inner product on the participation-measure space is forced, delivering the Hilbert-space structure Gleason operates on; (iii) non-contextuality is forced by the channel-as-primitive ontology. With non-contextuality derived rather than postulated, Gleason–Busch applies directly. The squared exponent is a consequence of the framework, not a separate postulate.
Born-like squared magnitude is forced by stable composition, not assumed from physics. This paper derives a Born-like squared-magnitude rule as a representational necessity in a pre-physical program. Building on prior results that force (1) plural incompatible stabilization criteria, (2) route dependence with cancellation-capable summaries (phase-like structure), and (3) the need for a stable nonnegative "yield" map, we formalize a yield function rho from summaries to nonnegative real numbers under minimal requirements: phase invariance, additivity over exclusive outcomes, multiplicativity under independent composition, and symmetry under balanced regrouping (no preferred exclusive coordinates). Under mild regularity, multiplicativity restricts rho to a power-law family, and regrouping invariance uniquely selects the quadratic case. The result is a squared-magnitude extraction rule obtained without assuming Hilbert spaces, wavefunctions, inner products, dynamics, hbar, or measurement postulates. The paper concludes by identifying the next forced pressures toward complex realization and norm-preserving (unitary-like) evolution.
We study how much of the Hilbert-space and Born-rule structure of quantum theory can be obtained from operational primitives that involve records but not probabilities. The setting is a regime in which coherent alternatives recombine without leaving a stable record. We obtain three results of different kinds. Geometry. From composable contributions, record-inert reversible controls and amplitude scalings, under named conditions, we obtain a real inner-product event geometry. The existence of the inner product is equivalent to bounded mixing and is assumed in that form. In a worked interferometer with one mode per channel, the metric is not an input: metric-free port swaps and phase flips fix the form up to scale, and boundedness only forces the other controls to preserve it. Its uniqueness up to scale, the orthogonality of distinct outcomes, and the representation of events by orthogonal projections are derived. Obstruction. This geometry, even with full symmetry about the preparation and exact dyadic standards, does not select the quadratic weight: the preparation mixed with the maximally mixed state survives. The structural operations exhibited here certify only values on which the quadratic weight equals the counting weight. On restricted event families where Gleason’s theorem is unavailable, a calibration order oriented by the quadratic magnitude selects it: given exact standards, a state is the quadratic one exactly when it respects all such brackets. Whether such an orientation can be earned is the open problem. Given such brackets and refinement only to a finite depth K, every admissible state lies within 2^-K of the quadratic weight, without the Archimedean step. On the full projection lattice, Gleason’s theorem with null support, or sphere-transitive symmetry with null support, gives other conditional routes. Bridge. A predictive probability kernel is introduced separately. Its identification with the real quadratic weight ||P_A ψ||² / ||ψ||² is conditional on kernel-admission hypotheses, one of which is equivalent to the Born value; the complex form needs phase invariance in addition. The hypotheses select a class of contexts, and none is shown to be forced. For each we state the physical feature it expresses and a sufficient condition where one is known, and we work out one apparatus in which the geometry-stage hypotheses hold together. For all but one we give a context in which it fails (for the founding conditions, one that leaves the regime); for the remaining one, TCC-1, we give an illustration. This paper is the first in a series built on a common framework, Structural Ledger Consistency (in preparation).
We give a finite-dimensional geometric account of the Born weights that separates three logically distinct questions: selection of the projective measure, geometric realisation of the outcome weights, and their interpretation as empirical frequencies. Pure states are rays [ψ]∈CPN−1. Unitary symmetry fixes a unique invariant Borel probability measure on this homogeneous space, the Fubini–Study measure μFS. This uniqueness result is now machine-checked in Lean 4 at general finite N. The Born weights then follow by a separate geometric step. Under the toric moment map, m([ψ])=(∣⟨e0,ψ⟩∣2,…,∣⟨eN−1,ψ⟩∣2), the normalised Fubini–Study measure induces the uniform Dirichlet(1,…,1) measure on the probability simplex. For a preparation b=m([ψ]), the state-dependent barycentric region associated with outcome i has relative Fubini–Study volume μFS(Ri(ψ))=∣⟨ei,ψ⟩∣2. The formal result holds for all finite dimensions and includes vanishing Born weights. Symmetry therefore fixes the measure, not the measurement regions themselves. The latter are explicit geometric realisations whose physical implementation belongs to the measurement layer of CSD. A second, independent route based on Gleason-type operational consistency is retained as corroborating support rather than as the load-bearing derivation. Finally, the deterministic volume-typicality result of Paper A connects these weights to repeated-trial frequencies under its stated preparation assumptions. The resulting account derives the finite-dimensional Born weights without postulating them as a fundamental stochastic law.
This record contains a technical note on a conditional derivation of the quadratic Born density from a minimal two-cell structural mixing condition. The article starts from a Hilbert-space representation of states and assumes that event weights admit a local gauge-invariant pointwise modulus density. It then adds one structural invariance condition: the total weight of two equal cells is unchanged under a real rotation mixing their amplitudes. Under these assumptions, the local density is forced to be quadratic, f(r) = cr², yielding the usual Born weight after normalization. The result is not presented as a replacement for Gleason-type or decision-theoretic derivations of the Born rule. Its purpose is narrower: to isolate a simple local route by which the square law follows once the density form and a minimal two-cell basis-independence principle are granted. The functional-equation core and the two-cell reduction have been formalized in Lean 4 using Mathlib. The accompanying Lean development is archived separately at Zenodo: https://doi.org/10.5281/zenodo.20528721. The reported axiom footprint for every formalized theorem in the development, including the measure-theoretic born_density and born_rule_weight, is the standard classical Lean base, with no admitted steps (sorryAx) in any of the formalized theorems. The manuscript and the accompanying Lean development were prepared with extensive AI assistance. The author reviewed, selected, corrected, and revised the outputs and remains responsible for the final text, the mathematical claims, the Lean formalization, and the interpretive framing.
From a single axiom — that outcomes form one non-separable whole — two consequences fix the quantum state space: a possibility can differ from another in exactly two ways, by magnitude (how much) and by phase (from where). A quantity carrying both is a complex number. On this space, non-separability forces completion by the conjugate, the squared-magnitude weight, linearity, normalization, and the interference cross-term, yielding the Born rule p_i = |ψ_i|² with Σp_i = 1 and interference. The algebraic steps are machine-verified (script included); the structural steps are stated as named claims with every open seam labeled honestly. Hermiticity of observables (A = A†) is argued and machine-verified for the real-weighted case rather than assumed. The derivation presupposes no Hilbert space, inner product, probability measure, or operational structure, and reaches the same targets that existing reconstructions obtain from five or six axioms. Submitted for refutation. The theorem of Section 7B is unchanged, and its proof does not depend on check A10 — it runs on Lemmas 2 and 4 and A1. Check A10 in verify_all.py was defective. It tested whether the product ψ·X is real, rather than whether X completes ψ. Consequently it accepted the sign flip, a partner that A1 explicitly excludes as degree-negating. The check could not fail. A10 is replaced with an exact test of A1’s criterion — that a completer removes vantage while preserving degree — which has the unique solution X = conj(ψ), and which rejects both the sign flip and any neighbouring pole. Two sentences describing A10 (Sections 6 and 7B) stated that a pole multiplied by a neighbour “still carries vantage.” This is false: the product m²e^(−iδ) is vantage-free but does not complete. Both sentences are corrected. This version also revises Lemma 3 to the operator-level criterion, removes the section on Hardy’s Simplicity axiom as superseded by Masanes–Müller (2011), addresses the direct-sum counterexample to Seam 7, and adds eleven references.