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IMM Paper 7: A Falsifiable Deviation from the Born Rule — Closed Form, Six Forcing Conditions, and a 27-Second Test

Travis Bergen

Published Jun 14, 2026• 0 citations
Publisher

Abstract

Six conditions uniquely force δP(i)=ε(P_QM(i)-1/n)F(x). Sign reversal at P_QM=1/n is exact and forced by normalization. Positivity bound -1≤εF≤1/(n-1) proved. Testable at ε=10⁻³ in 27 seconds on a transmon qubit (T₂=200μs). No decoherence model produces a zero fixed at 1/n under state sweep.
DOI
10.5281/zenodo.20365733

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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.
The Born rule sits at the heart of quantum mechanics as an unexamined assumption: probabilities are fundamental. For a statistician, that is uncomfortable. Probabilities are tools for describing ignorance — so what, precisely, are we ignorant of? That question leads to a deeper one: what is measurement, really? If measurement is a physical event, something in the world must change when it happens — something records it. WF-S formalises this with two physical inputs and one Lagrangian. Postulate 1 (WF-S kernel): a complex amplitude field φ(x) couples to a real substrate field s(x) via L = Lφ + Ls − g|φ|²s. Postulate 2 (record condition): macroscopic outcomes are weighted by cumulative entropy export σi = ΔSenv,i/kB during record formation. The substrate carries the irreversible imprint of that record. The Schrödinger equation and the Born rule are not independent postulates: they are the same Lagrangian in two regimes, separated by the physical threshold of record formation. The Born rule emerges as the entropy-uniform limit of a more general selection rule — derivable, not assumed. The entropy increments σi are thermodynamic quantities, defined through physical irreversibility independently of any probability assignment. The arrow runs: physical event → thermodynamic σi → outcome weighting → Born rule. The framework makes falsifiable predictions. For two outcome channels, the log-odds shift is ln(P1/P2) = ln(|a1|²/|a2|²) + Δσeff, where Δσeff ≈ εη Δσeng, Δσeng is an independently calibrated entropy contrast and εη is an experimentally bounded suppression factor. Swap-symmetry of the tilt (F1), linear response in engineered entropy contrast (F2), universality of the coarse-graining parameter across platforms (F3), and exact Born recovery in the entropy-uniform limit (F4-F5) are testable on existing platforms independently of the numerical value of εη.This paper is the first in a five-paper series; the substrate framework developed here is extended to forces, gravity, cosmology, and a full information-geometric synthesis in Papers II–V."
Theorem (One-Loop Path-Integral Form) + Falsifiable Prediction. Under the IMM action structure, with probability normalization as the only constraint, the deviation from standard Born statistics is forced to be P_IMM(i) = ||^2 + epsilon*(||^2 - 1/n)*F(x) with F(x) = |grad Delta|^2 / (|grad Delta|^2 + Delta_0^2). Three discriminating features: vanishes at uniform Delta (recovers QM), changes sign at P_QM=1/n (no decoherence model produces this), spatially structured by spectral gradient. Circuit-QED experiment proposed: 27-second measurement at epsilon=10^-3 for 3-sigma detection; 3-day falsification at epsilon >= 10^-5. The framework's principal experimental kill shot. Inherits from Paper 0 (axioms A1, A4, A5, A6); builds on Papers 4 and 5.
This paper develops a fully operator-theoretic account of quantum measurement and the Born rule within a constrained dynamical framework given by a coupled Dirac–Λ system. In this Tier-1 setting, a reversible Dirac carrier is coupled to an independent irreversible (record) channel through a scale-by-scale capacity inequality enforced via a Karush–Kuhn–Tucker (KKT) structure and a fixed ultraviolet anchor. All spectral schemes and determinant prescriptions are globally fixed and non-tunable. The main results are: • A canonical record (pointer) algebra defined intrinsically as the fixed-point algebra of the modular automorphism group of the saturated stationary state. No external measurement basis is inserted. • A quantitative convexity gap showing that persistent record-coherence carries strictly positive Dirac-side capacity cost. At any active (saturated) scale, complementary slackness forces record-diagonality. Collapse is therefore enforced as a feasibility condition, not postulated. • A rigidity theorem showing that modular implementability of the linearized saturation dynamics forces Hilbert (GNS/L²) geometry, excluding non-Hilbert Schatten p-structures and fixing the quadratic Born exponent. • A derivation of the Born rule as the unique positive, normalized, unitarily covariant bounded linear functional on pointer projections once L² geometry is enforced. • A quantitative irreversible activation cost at measurement, including ultraviolet and infrared asymptotics and a capacity-limited decoherence rate bound with 1/T suppression in the infrared regime. The framework is falsifiable: violations of the capacity inequality, failure of the ultraviolet anchor, instability of the saturated configuration, deviation from quadratic Born weights, or absence of the predicted infrared suppression regime would disconfirm the model. This work does not rely on interpretational postulates and does not invoke collapse as an axiom. Measurement and Born probabilities emerge structurally from saturation geometry within a capacity-limited reversible/irreversible coupling.