We show the Born rule P = |ψ|² is the unique probability rule consistent with thermodynamic constraints on record formation, conditional on the regime where Landauer-type bounds apply. The derivation proceeds from five operational postulates: normalization, phase information loss in record formation, interference consistency, tensor product factorization, and continuity. The key physical insight is that creating a classical measurement record requires that phase information is not retained in the record accessible to observers, a not-reversible operation with entropy flow to the bath and Landauer cost k_B T ln 2 per bit. The squared modulus emerges as the unique probability rule that (i) eliminates phase to produce positive probabilities, (ii) preserves interference effects before measurement, and (iii) satisfies standard probability axioms. This thermodynamically motivated derivation complements Gleason's theorem: where Gleason proves the rule is mathematically necessary (dimension ≥ 3), we show it is the unique rule realizable through record formation under these constraints (all dimensions including d=2). The framework provides a concrete answer to "why squared?": the irreversible formation of a classical record, on a Hilbert space whose norm is preserved by unitary evolution, admits no other consistent probability rule.
We show the Born rule P = |ψ|² is the unique probability rule consistent with thermodynamic constraints on record formation, conditional on the regime where Landauer-type bounds apply. The derivation proceeds from five operational postulates: normalization, phase information loss in record formation, interference consistency, tensor product factorization, and continuity. The key physical insight is that creating a classical measurement record requires that phase information is not retained in the record accessible to observers, a not-reversible operation with entropy flow to the bath and Landauer cost k_B T ln 2 per bit. The squared modulus emerges as the unique probability rule that (i) eliminates phase to produce positive probabilities, (ii) preserves interference effects before measurement, and (iii) satisfies standard probability axioms. This thermodynamically motivated derivation complements Gleason's theorem: where Gleason proves the rule is mathematically necessary (dimension ≥ 3), we show it is the unique rule realizable through record formation under these constraints (all dimensions including d=2). The framework provides a concrete answer to "why squared?": the irreversible formation of a classical record, on a Hilbert space whose norm is preserved by unitary evolution, admits no other consistent probability rule.
We show the Born rule P = |ψ|² is the unique probability rule consistent with thermodynamic constraints on record formation, conditional on the regime where Landauer-type bounds apply. The derivation proceeds from five operational postulates: normalization, phase information loss in record formation, interference consistency, tensor product factorization, and continuity. The key physical insight is that creating a classical measurement record requires that phase information is not retained in the record accessible to observers, a not-reversible operation with entropy flow to the bath and Landauer cost k_B T ln 2 per bit. The squared modulus emerges as the unique probability rule that (i) eliminates phase to produce positive probabilities, (ii) preserves interference effects before measurement, and (iii) satisfies standard probability axioms. This thermodynamic derivation complements Gleason's theorem: where Gleason proves the rule is mathematically necessary, we show it is physically realizable through record formation at finite temperature. The framework provides a concrete answer to "why squared?" because squaring is the minimal phase-eliminating operation consistent with Landauer-cost record formation under the stated conditions.
Given the standard Hilbert space structure of quantum mechanics (including L^2 normalization from unitary dynamics), we show that the Born rule P = |psi|^2 is the unique probability rule compatible with phase-independent record formation. The derivation uses one thermodynamic axiom (classical records cannot retain quantum phase, an irreversible operation with Landauer cost k_B T ln 2 per bit) together with three structural axioms (normalization, tensor product factorization, continuity) and an operational interference consistency condition. The squared modulus is the unique function satisfying all five constraints; alternative powers |alpha|^n with n != 2 fail because the L^2 unit sphere is not an L^n unit sphere. This result complements Gleason's theorem: where Gleason proves the Born rule is mathematically necessary given non-contextuality, we identify the physical mechanism (phase information loss during record formation at finite temperature) that makes it realizable. The L^2 structure itself is not derived here; it is inherited from the requirement that unitary evolution preserve the inner product. The contribution is to show that, conditional on that structure, thermodynamics selects the squared modulus as the unique admissible probability map.
We prove a conditional uniqueness theorem for projective measurements on pure states: given the L2 Hilbert space structure of quantum mechanics, the Born rule P=∣ψ∣2 is the only outcome-local probability assignment compatible with five operational postulates (outcome-locality, normalization, phase independence of the classical record, tensor product factorization, and continuity), with interference consistency used only as a post-hoc check. Physically, the theorem addresses a boundary problem: amplitudes support interference and cancellation before measurement, whereas a stabilized classical record cannot retain the full relative-phase distinction structure in accessible form. Irreversible record formation motivates this phase-insensitivity requirement, although no thermodynamic quantity enters the formal proof. Mathematically, phase independence reduces the map to a function of modulus, factorization and continuity force a power law, and consistency with L2 normalization fixes the exponent to 2. Within this framework, the squared modulus is the unique classicalization map from phase-sensitive amplitudes to accessible record weights. The result is complementary in physical motivation to Gleason-type derivations but narrower in scope: it is confined to the pure-state projective setting and does not derive the general trace rule for mixed states and POVMs.
We show that the quadratic measure associated with the Born rule need not be introduced as an independent probabilistic postulate, but follows, under the stated structural and regularity assumptions, from the compatibility between reversible linear evolution and the formation of stable records. Physical processes alternate between reversible dynamics and episodes of record formation. Quantities that remain well defined throughout such processes must therefore admit a representation that remains consistent across both regimes. Prior to record formation, alternatives combine additively at the level of complex amplitudes, whereas successive record-forming distinctions admit a multiplicative representation through refinement. Requiring compatibility between these compositional structures constrains the admissible weight assignment to the family of weights proportional to the pth power of the amplitude magnitude. The finite-dimensional specialization of Lamperti's isometry theorem then singles out p = 2 as the unique value compatible with continuous reversible linear mixing while preserving total weight. The Born rule therefore emerges as the unique weight assignment compatible with reversible linear composition, multiplicative record refinement, phase invariance, and continuous reversible invariance, without invoking probabilistic assumptions or a specific interpretation of quantum measurement.
The Born rule — that the probability of obtaining outcome i in a measurement of a quantum system in state |ψ⟩ is given by P(i) = |⟨i|ψ⟩|² — is conventionally introduced as an axiom of quantum mechanics. We present a derivation of the Born rule based on a structural distinction within the substrate-field framework between F1, the unbounded equilibrium configuration of a fundamental scalar field that carries the unitary wave-function dynamics, and F2, the bounded observable configuration that records definite measurement outcomes. F1 → F2 sampling occurs at each Planck-scale time quantum and produces stochastic localization with a probability distribution that, by Gleason's theorem and the non-contextuality of the sampling mechanism, must be of the form Tr(ρ P_i). The squared-modulus form follows. We complement the Gleason argument with Zurek's environment-assisted invariance to address the equal-amplitude case explicitly and discuss the relationship to existing derivations including the decision-theoretic approach of Deutsch and Wallace, the QBist program of Caves, Fuchs, and Schack, and the Bohmian treatment of Dürr, Goldstein, and Zanghì. The contribution of the present argument is to provide a physical mechanism — F1/F2 sampling — that generates probabilities, eliminating the need to postulate them as primitive while retaining the mathematical structure already secured by Gleason's theorem.
The Born rule — the statement that the probability of obtaining outcome |ϕi⟩ from quantum state |ψ⟩ equals |⟨ϕi|ψ⟩|² — is normally taken as a fundamental postulate of quantum mechanics. This paper derives the Born rule from the bridge law connecting the the Fubini–Study geometry of quantum state space to physical time through irreversible contraction of the accessible future state space Ω. The bridge law, d(ln Ω)/dτ = −κ Γ(τ), specifies that each recording event depletes generative capacity at a rate set by the structural efficiency constant κ = π — the Fubini–Study volume of ℂP1. This identifies the Fubini–Study metric as the natural measure of state-space contraction. Three physical requirements — unitary invariance, projective consistency, and smooth locality — uniquely select this metric, leaving no freedom in the probability assignment. The probability of each outcome is the fraction of the local Fubini–Study contraction associated with that outcome, yielding pi = |⟨ϕi|ψ⟩|². The framework further predicts finite-time corrections to the Born rule. Because records form through irreversible thermodynamic processes with characteristic timescale τdet, the effective probability of registering independent events at short delay τ is modulated by the record-completion function R(τ) = 1 − e−|τ|/τdet. This produces a distinctive suppression of photon coincidences in the second-order correlation function g(2)(τ) at picosecond timescales in Hanbury Brown–Twiss measurements using superconducting nanowire single-photon detectors. The effect depends on detector material properties through the detector transition time τdet and is experimentally testable with existing instrumentation. The result therefore provides both a geometric derivation of the Born rule from the bridge law of irreversible record formation and a concrete experimental prediction linking quantum probability to the thermodynamics of measurement. This work forms part of the author's research programme Emergence as Regime Formation, investigating the origin of time, probability, and physical constants through irreversible record formation.