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From Settlement Choice to the Born Rule

琼金 黄

Published Jul 11, 2026• 0 citations
Publisher

Abstract

The quantum measurement problem—why measurement yields a single definite outcome, and why that outcome follows the Born rule P(i)=|c_i|²—has for nearly a century been treated as a postulate rather than a derivable consequence. This paper proposes that measurement is not an extraneous process tacked onto quantum mechanics; it is the natural execution of the settlement operation Ŝ(Θ). The core settlement cost functional is defined as C[ρ]=D_KL(ρ∥ρ_target)+β·Tr(ρH)−η·N(ρ). Three core results are obtained: (1) The Born rule P(i)=|c_i|² is strictly derived as the statistical limit of KL divergence minimization under projective measurement, requiring no additional postulates; (2) Wavefunction collapse is reinterpreted as irreversible settlement commitment. The stochasticity and nonlinearity of collapse originate from discrete Kraus branch selection of settlement operations; (3) A finite non-zero settlement delay τ_settle is predicted, which varies with consciousness parameters C, Q, A, I. High-coherence observers (C>0.8) possess far shorter settlement delay than ordinary humans or electronic detectors. This paper gives the explicit formula τ_setle=1/[η₀·C·Q·(1−A)·I] and designs a delayed-choice Bell experiment based on SPDC entangled photons for experimental verification. Predictions P1 and P3 from the series provide direct experimental channels to test this settlement framework. quantum measurement problem, Born rule, wavefunction collapse, settlement choice, settlement delay, consciousness parameters, two-phase settlement, Ξ-field

DOI
10.5281/zenodo.21310646

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The quantum measurement problem—why measurement yields a single definite outcome, and why that outcome follows the Born rule P(i)=|c_i|²—has for nearly a century been treated as a postulate rather than a derivable consequence. This paper proposes that measurement is not an extraneous process tacked onto quantum mechanics; it is the natural execution of the settlement operation Ŝ(Θ). The core settlement cost functional is defined as C[ρ]=D_KL(ρ∥ρ_target)+β·Tr(ρH)−η·N(ρ). Three core results are obtained: (1) The Born rule P(i)=|c_i|² is strictly derived as the statistical limit of KL divergence minimization under projective measurement, requiring no additional postulates; (2) Wavefunction collapse is reinterpreted as irreversible settlement commitment. The stochasticity and nonlinearity of collapse originate from discrete Kraus branch selection of settlement operations; (3) A finite non-zero settlement delay τ_settle is predicted, which varies with consciousness parameters C, Q, A, I. High-coherence observers (C>0.8) possess far shorter settlement delay than ordinary humans or electronic detectors. This paper gives the explicit formula τ_setle=1/[η₀·C·Q·(1−A)·I] and designs a delayed-choice Bell experiment based on SPDC entangled photons for experimental verification. Predictions P1 and P3 from the series provide direct experimental channels to test this settlement framework. quantum measurement problem, Born rule, wavefunction collapse, settlement choice, settlement delay, consciousness parameters, two-phase settlement, Ξ-field
Abstract:The quantum measurement problemthe fundamental conict between the deterministic, unitary evolution described by the Schrödinger equation and the stochastic, irreversible outcomes observed in experimentshas remained unresolved since the inception of quantum mechanics nearly a century ago. Here we demonstrate that the Chrono-Singularity Unication (CSU) framework provides a complete, rst-principles resolution of all ve components of the measurement problem with zero adjustable parameters. From a minimal informational substrate ΨI characterized by three internally-calibrated constants (α = ln 2, β = ln 2, w = −1), we derive: (i) the Born rule P = |ψ| 2 as a mathematical necessity via the GNS construction and spectral theorem; (ii) the emergence of the preferred (position) basis through Dobrushin concentration bounds on redundant environmental records; (iii) single denite outcomes from the sampled-history ontology without dynamical collapse; (iv) full compatibility with special relativity through topological linking that preserves no-signaling while permitting Bell-inequality violation; and (v) unique, falsi- able experimental predictions including a mass-dependent gravitational decoherence oor testable in near-future molecular interferometry experiments. Significance:This framework provides the exact geometric scaling required to resolve the Hubble tension and derives the Diósi-Penrose gravitational decoherence floor (τ=ℏ/ΔEG) strictly as a low-energy macroscopic limit. This paper serves as the theoretical foundation for a fully deterministic, parameter-free unification of quantum mechanics and general relativity.
The Born rule — the axiom that measurement probability equals |ψ|² — has resisted derivation from deeper principles since 1926. This paper proposes that it is not an axiom but a consequence of the coherence condition. A measurement outcome is realized if and only if it preserves the coherence of the combined system-observer state: K(S+O) > K_crit. The Born weight |c_i|² is identified as the coherence preservation measure — the fraction of the joint Hilbert space norm available for coherence maintenance in branch i. Outcomes with low |c_i|² are not improbable; they are incoherent — the resulting system-observer state lacks the temporal structure required for observation. This resolves the measurement problem without collapse, without branching, and without hidden variables: only coherent branches have observers, and only observers have outcomes. Falsifiable prediction: Born rule deviations at experimentally degraded observer coherence, testable with existing quantum optics apparatus. Part of the Spektre research corpus.
Abstract: The quantum measurement problem has been the deepest conceptual difficulty since the birth of quantum mechanics. Standard quantum mechanics accepts wavefunction collapse as an independent postulate but never provides a physical explanation. The decoherence approach attributes collapse to the interaction between the system and its environment, but faces two fundamental limitations: the “infinite regress” problem and the “insufficiency of decoherence” problem. Within the framework of the Self-Construction Theory (SCT), this paper provides a complete resolution of the measurement problem. The core thesis is that quantum measurement is not a fundamental process independent of the laws of physical evolution, but rather a local manifestation of constructive renormalization—the scale evolution of the spacetime complex field KKK from microscopic degrees of freedom to macroscopic effective descriptions. The origins of wavefunction collapse, decoherence, and the Born rule receive a unified first‑principles interpretation in this framework. SCT’s spontaneous decoherence mechanism is the key to the resolution. Constructive renormalization inherently involves information loss—the integrated‑out microscopic degrees of freedom no longer exist in the macroscopic effective description. During a measurement, the measuring apparatus possesses a huge number (∼1023\sim10^{23}∼1023) of microscopic degrees of freedom. When the apparatus interacts with the measured system, constructive renormalization triggers information compression of these degrees of freedom—the apparatus can no longer distinguish which branch of the superposition it is in, and quantum coherence is thus spontaneously lost. No “external environment” is needed—the constructive renormalization dynamics of spacetime itself is the ultimate source of decoherence, thereby completely solving the “infinite regress” problem. Among the probability distribution given by the Born rule, the weights of non‑maximal probability branches are further suppressed by information compression under the huge number of microscopic degrees of freedom, so that macroscopic observers can perceive only a definite outcome in the effective description. The information loss caused by constructive renormalization is ontological—it does not rely on any approximation or truncation, thus fundamentally solving the “insufficiency of decoherence” problem. The Born rule—that measurement outcomes occur with probabilities given by the squared amplitudes—is directly derived from the constructive integral structure in SCT. In the constructive renormalization limit, interference terms between different configurations are naturally suppressed by off‑diagonal phase randomization, and the system automatically switches to a classical probabilistic description. In the diagonal density matrix representation, the squared amplitude is the only natural weight that is simultaneously real and non‑negative. Hence the Born rule is not an additional postulate of quantum mechanics but a necessary and unique manifestation of the relative frequencies of the constructive integral in the constructive renormalization limit. This derivation is rigorous within the mathematical framework of constructive renormalization—it does not rely on any approximations or truncations. SCT’s quantum measurement theory is deeply unified with the black hole information paradox. The “collapse” in quantum measurement and the “information loss” in black hole evaporation are different manifestations of the same core mechanism—the redistribution of information due to constructive renormalization—in two extreme physical situations. In quantum measurement, information is redistributed into the degrees of freedom of the macroscopic apparatus; in black hole evaporation, information is redistributed into the time‑point configurations on the horizon. This unification is the strongest proof of the universality of constructive renormalization as an information‑processing mechanism. Keywords: quantum measurement; spontaneous decoherence; constructive renormalization; Born rule; wavefunction collapse; information redistribution; Self-Construction Theory
The standard measurement postulate of quantum mechanics stipulates an instantaneous, non-local update of the global wavefunction upon measurement of an entangled subsystem. This postulate has been a persistent source of foundational tension, both with the locality structure of relativistic physics and with the dynamical character of the decoherence program. We separate the measurement postulate into three logically independent claims: definite outcome realization, Born-rule probability assignment, and instantaneous projection for the global wavefunction. We argue that the third claim can be eliminated at the axiomatic level and propose a modified measurement postulate, Postulate M, that replaces it. Under Postulate M, a single pointer outcome is realized locally when environmental decoherence has einselected a pointer basis at the measurement site; the corresponding selection acts on the global wavefunction as a conditional restriction to a branch rather than as a physical operation on spacelike-separated regions. We show that Postulate M recovers the standard measurement postulate as a limiting idealization for vanishing decoherence time, derives Lüders’ rule for sequential measurements, and yields no-signaling as an automatic structural consequence — not only at the level of measurement statistics but at the level of physical states themselves. Two explicit illustrations in the EPR scenario — with one and with both particles coupled to local spin environments — demonstrate the framework concretely and exhibit its frame-independent, manifestly local character. The framework is built on standard decoherence theory, compatible with multiple candidate mechanisms for single-outcome realization including gravitationally-induced collapse proposals, and naturally situated within a ψ-ontic reading of the quantum state. It represents, we argue, a reformulation of quantum mechanics as a local, state-realist theory — preserving Einstein’s aspiration for an objective physical world in the only form consistent with the empirical content of quantum mechanics, namely realism about the wavefunction rather than about pre-measurement observable values.
We present a complete, mathematically rigorous, and observer-free resolution of the quantum measurement problem. Building entirely on the axiomatic framework established in the preceding work, which derives the Born rule exclusively from unitary dynamics, uniqueness of classical records, unitary invariance, and ergodic stability, we show that the entire measurement process is governed by a single deterministic law: unitary evolution. The apparent wavefunction collapse is not a physical process but an effective description of the emergence of a unique dynamically stable classical record under decoherence and unique ergodicity. No auxiliary postulates, collapse mechanisms, or observer-dependent inputs are required. We explain the origin of definite measurement outcomes, the emergence of objective probability, the elimination of quantum-classical duality, and provide a fully consistent resolution of the Wigner’s friend paradox. The framework reproduces all experimental predictions of quantum mechanics without contradictions and establishes a unified, self-contained foundation for quantum measurement theory. This work constitutes the first complete, minimal, and mathematically closed solution to the quantum measurement problem.