One mixer suffices: a minimal symmetry premise for the Born exponent in coordinate-additive probability rules
Paper 19A · Pødenphant Lund, T. (2026) · Technical note · Live on Zenodo
Derivations of the Born rule differ most decisively in how much structure they consume before the square appears. This note asks the accounting question in reverse: rather than how fast the exponent follows from the full unitary group, what is the minimal mixing premise that still forces it? The answer is exact for one sharply delimited class. For coordinate-additive rules, invariance under a single genuine two-route mixer already forces the exponent to be two whenever there are at least three routes, and polarisation then reconstructs the entire Hermitian inner product from the invariant functional alone. Within the class, demanding one mixer is equivalent to demanding the whole quadratic structure. It is a characterisation, not a derivation: the mixer premise is the unitarity assumption and receives no independent justification here.
| DOI (concept) | 10.5281/zenodo.21471576 |
| Type | Technical note (quantum foundations; exact results with numerical certificates) |
| Target venue | Foundations of Physics |
| Class | Coordinate-additive rules: one shared measurable g, distinguished basis, no cross terms |
| Author | Tomas Pødenphant Lund [ORCID] |
TL;DR
Work with unit vectors in an n-dimensional complex space in a distinguished basis whose indices are called routes. A coordinate-additive rule assigns route i the weight g(|ψi|) for one shared Lebesgue-measurable g with g(0) = 0 and g(1) > 0, normalised by the sum. The invariance functional is Φ(ψ) = ∑i g(|ψi|), and a unitary S is an admissible symmetry when Φ(Sψ) = Φ(ψ) for all unit states. Standard finite-dimensional complex Hilbert kinematics is in place throughout; the rule itself consumes only coordinate moduli, and the premise touches inner-product structure at exactly one point, namely that the mixer is unitary.
The result is a trichotomy with an empty middle: generalised permutations constrain g not at all; one genuine two-route mixer forces g(x) = g(1)x2; and no linear symmetry sits strictly between them. Polarisation then upgrades the forced square to the full inner product.
The results
- Theorem 1 (one genuine mixer suffices). For n ≥ 3, if Φ is invariant under a single genuine two-route mixer for all unit states, then g(x) = g(1)x2 on all of [0,1]. A two-route unitary acts on two fixed routes and leaves the rest alone; unitarity forces its block moduli into the pattern √c, √s with c + s = 1, and it is genuine when cs ≠ 0. The quantifier over all states supplies the phase freedom the proof uses, so no separate phase symmetry is postulated.
- Theorem 2 (freedom without mixing). If the admissible symmetries are all generalised permutations (a permutation matrix times a diagonal phase matrix), every measurable g passes: the invariance requirement is satisfied term by term and fixes nothing.
- Theorem 3 (no middle class, infinitesimal form). For power functionals Φp with p ≥ 1, the isometry group's Lie algebra has real dimension n for p ≠ 2 (diagonal phase generators only) and is all of u(n), dimension n2, at p = 2. At group level the classification of ℓp isometries as generalised permutations for p ≠ 2 is the Banach–Lamperti theorem; the note's contribution in this leg is the Lie-algebra-level verification.
- Proposition 4 (n = 2 is not enough). With two routes the requirement fixes only the symmetrisation, and an infinite-dimensional family of non-quadratic measurable g passes. The failure is structural, mirroring Gleason's dimension threshold, where the qubit case is likewise rescued only by enlarging the measurement class.
- Corollary 5 (polarisation). Under Theorem 1, Φ = g(1)‖ψ‖2 on the whole closed unit ball, so the scaled parallelogram law holds and the Jordan–von Neumann polarisation sum recovers g(1)⟨φ,ψ⟩. The mixer requirement and the quadratic structure are therefore equivalent within the class.
Why one mixer forces the square
The phase sweep. Pass to squared variables and consider states where two routes carry masses v and w while a third spectator route carries the remainder. The spectator is what n ≥ 3 buys: it decouples the pair's total mass from the normalisation, so every pair total is reachable. When the mixer acts, the redistributed mass depends on the relative phase between the two coordinates, and because invariance is required at every unit state, that mass sweeps across a whole interval. A single unitary therefore delivers a continuum of equations rather than one.
The centre move. Choosing the phase so the cosine term vanishes replaces the pair by one with the same sum and a smaller deviation from the midpoint, the deviation being multiplied by c − s. Genuineness gives |c − s| < 1, so iteration contracts any pair geometrically toward the balanced pair until the midpoint falls inside the sweep's reach.
Jensen's midpoint equation. What survives is the statement that the value at the midpoint equals the average of the endpoint values, for every admissible pair. The classical result is that measurability alone suffices to conclude affinity, with no continuity assumed. Because the domain is a triangle rather than a square, the gap is closed in two steps: affinity is established on the lower half, and the instance with a vanishing second coordinate supplies a doubling equation that extends the linear form to the whole interval. With G(0) = 0 this gives the square. The balanced Hadamard case is the one-step special case where the contraction chain is unnecessary.
The characterisation reading
Theorem 1 pins g on all of [0,1], not merely on the moduli patterns of unit vectors, so the quadratic form is derived on the closed unit ball rather than assumed, and no value of Φ outside the ball is used. The polarisation arguments all lie inside the ball, where the quadratic form is established. The invariance functional that the symmetry requirement singles out therefore is the Hermitian inner product up to scale.
The consequence for axiomatics: any axiom system on coordinate-additive rules that includes invariance under at least one genuine two-route mixer already entails the square, whatever its remaining axioms contribute; and a system whose symmetries are all generalised permutations cannot force uniqueness at all. Within the class, the puzzle "why is the exponent two" is precisely the puzzle "why does any genuine two-route mixing symmetry hold at all", which is the unitarity question the reconstruction programmes themselves leave open. Relocated, not reduced.
Relation to the literature
Against the Gleason line, Gleason consumes the whole orthogonality structure with the inner product as background; the nearest methodological relative is the reduction of frame-function requirements to a Cauchy functional equation, and this note also lands on a classical functional equation, but extracts it from one fixed unitary inside an additive ansatz, with the inner product entering the premise only through that single element's unitarity.
Against the operational reconstructions the relation is complementarity rather than competition: their non-quadratic alternatives carry cross terms and are not coordinate-additive, while the non-quadratic additive rules here are excluded from their classification by assumption rather than derived away. The single-mixer question cannot even be posed in machinery that requires a state's stabiliser to act transitively. Against symmetry derivations at fixed function class, those show how quickly the exponent follows once the full unitary structure is granted; this trichotomy shows that any genuine two-route blend already entails the inner product inside the additive class.
Operationally, coordinate-additivity is the rule-level analogue of the first classical step in Sorkin's interference hierarchy, namely pairwise additivity of route weights in the distinguished basis. Rules breaking the class assumption are exactly those with genuine pairwise cross terms, a two-slit quantity, which is the frame in which the class boundary becomes experimentally meaningful in principle.
Scope and open ends
Two-route mixers only. A closure argument through the group generated by the mixer and the phases fails at this regularity level, since g is merely measurable, so Φ need not be continuous and density in the unitary group cannot be invoked. The discrete-Fourier tritter and all k-route symmetries with k ≥ 3 are certified only numerically, and the corollary is therefore stated as "at least one genuine two-route mixer" rather than "any genuine route blending". Proving the k-route case directly is the natural next step.
The class assumption is load-bearing. The note characterises the symmetry price of the Born exponent within the coordinate-additive class and claims nothing outside it; rules with cross terms belong to the operational classifications. The restriction (one shared g, coordinate additivity, a distinguished basis) is a strong noncontextuality-type assumption, and the class may contain no physical theory besides quantum mechanics itself.
Numerical certificates, not proofs. Two deterministic scripts accompany the note, each certifying a leg independently: the balanced mixer at n = 3 gives a one-dimensional constraint kernel matching the square to about 10−15; the n = 2 control reproduces Proposition 4 on the grid; generalised permutations give an identically zero constraint; the polarisation formula recovers the inner product to about 10−15; and the isometry-algebra dimensions come out as predicted. The tritter case is covered numerically only and lies outside Theorem 1's proof.
Connections to other papers in the series
- Paper 19B (Beyond the arcsine law) — the sibling note. 19A treats the static probability rule, asking why the exponent is two; 19B treats the dynamics, deriving the exact time-in-lead statistics of a monitored qubit whose collapse recovers the same Born probabilities as mixture weights.
- Paper 10 (Race architecture) — the substrate-general race vocabulary. The coordinate-additive class arises naturally in commit-race models of sequential decision, where each route carries an independent commit weight computed from its own amplitude modulus. Nothing in the mathematics depends on that reading, and the note is written to stand entirely on its own as a quantum-foundations result.
Read the paper
The full note is on Zenodo (concept DOI 10.5281/zenodo.21471576):