BCT Phase 2 — Supplementary Note S2: Born Rule (Revised) This Supplementary Note provides a revised and improved formulation of the Born rule within the Phase 2 framework of Bit-Collapse Theory (BCT). It supersedes the Born-rule note included in the current Phase 2 package. Key points: The Born rule is derived as the unique probability measure compatible with record-defined outcomes, accessibility constraints, and invariance requirements, without introducing new dynamics. Measurement outcomes are defined via stable record subspaces, avoiding observer-dependent or semantic assumptions. The revised formulation clarifies the role of standard assumptions (e.g. dimensionality, embedding of qubit cases) and their scope within Phase 2. The result is intended as a structural consistency result, not as a modification of quantum mechanics. This record is Supplementary Note S2 for the Phase 2 main package and is intended to be integrated into the next update of the main DOI. Related main record (Concept DOI): https://doi.org/10.5281/zenodo.18496128 Keywords: Bit-Collapse Theory, Born rule, quantum probability, records, accessibility, quantum foundations.
BCT Phase 2 — Supplementary Note S2: Born Rule (Revised) This Supplementary Note provides a revised and improved formulation of the Born rule within the Phase 2 framework of Bit-Collapse Theory (BCT). It supersedes the Born-rule note included in the current Phase 2 package. Key points: The Born rule is derived as the unique probability measure compatible with record-defined outcomes, accessibility constraints, and invariance requirements, without introducing new dynamics. Measurement outcomes are defined via stable record subspaces, avoiding observer-dependent or semantic assumptions. The revised formulation clarifies the role of standard assumptions (e.g. dimensionality, embedding of qubit cases) and their scope within Phase 2. The result is intended as a structural consistency result, not as a modification of quantum mechanics. This record is Supplementary Note S2 for the Phase 2 main package and is intended to be integrated into the next update of the main DOI. Related main record (Concept DOI): https://doi.org/10.5281/zenodo.18496128 Keywords: Bit-Collapse Theory, Born rule, quantum probability, records, accessibility, quantum foundations.
Bit-Collapse Theory (BCT) is an ontological interpretation of quantum mechanics in which the primitive is not a physical collapse, but informational identity. A (conscious) configuration is treated as a discrete informational macro-state: multiple microphysical realizations may instantiate the same macro-identity, so experiential “superposition” corresponds to physical multiplicity under one informational equivalence class. “Collapse” is defined as equivalence-breaking: when a distinguishing bit becomes macro-relevant and persistently recorded, a shared macro-description can no longer cover the alternatives, yielding distinct continuations. This release is designed for public dissemination and includes:(1) a technical note deriving Born weights as the unique identity-measure compatible with fine-graining invariance and symmetry, and linking the experiential arrow of time to the monotonic accumulation of persistent records, with an explicit bridge to subsystem entropy via entanglement and redundancy;(2) a short popular booklet introducing the framework without mysticism;(3) an objections & responses document;(4) a compact glossary. BCT introduces no modification to Schrödinger dynamics and is presented as an interpretive framework: it reorganizes definiteness, probability, and temporal directionality in terms of informational equivalence classes and persistent records. Note: This DOI currently contains an exploratory technical note, a popular booklet, a glossary, and a short objections-and-responses document. The package develops BCT as an exploratory interpretation centered on informational identity, Born measure, and the arrow of time from persistent records. One especially promising line within the package is the bridge from persistent records to entanglement, subsystem entropy growth, and the arrow of time. A fuller technical version would require additional work on the selection/relevance criterion for the descriptive map, a more explicit positioning relative to Quantum Darwinism, and a stronger treatment of the relation between the present Born-measure argument and standard fine-graining approaches. For now, this record is kept as a structured exploratory package rather than as a finalized journal-ready paper. AI assistance disclosure: Large language models (LLMs) were used during the development of this work. The author is solely responsible for the content and any errors. If you’d like to collaborate (technical critique, examples, simulations, or experimental test ideas), email me at batoritranslation@gmail.com — I’m happy to discuss and will credit substantial contributions in future releases. Keywords: quantum foundations; interpretation of quantum mechanics; measurement problem; Born rule; decoherence; quantum Darwinism; informational identity; equivalence classes; persistent records; arrow of time; entropy; macro-definiteness; information theory
Companion paper to the current BCT Phase-2 program. This note introduces the Record Threshold Principle (RTP) as a single operational spine for three connected sectors: observer-definiteness, irreversibility, and record-support assignment. RTP states that event-like distinctions and time-directed classifications are organized by threshold crossing in record channels: a distinction counts only when the relevant record is accessible, redundant/robust, and persistent beyond a calibrated threshold. No modified Schrödinger dynamics is introduced; the global state may remain unitary while “collapse” is treated as an agent-relative update of an equivalence-class label induced by stable record updates. The paper develops three sectors under one channel-threshold rule: a retention-organized measurement wedge with preregistered defeat conditions, a Loschmidt-style irreversibility criterion with a critical redundancy threshold, and a Gleason-style uniqueness route for a record-support measure within a frozen coarse-grained regime. The aim is compression: one generator relation applied to different physical channels, rather than three unrelated mechanisms. This record should be read as a companion note within the broader BCT corpus, alongside the main BCT Phase-2/Axiom 0 materials and the project website. Main BCT DOI:https://doi.org/10.5281/zenodo.18496128 Project website:Bit-Collapse Theory (BCT) — Phase 2 AI disclosure:AI tools were used as drafting and editorial assistants during the development of this manuscript. All conceptual responsibility, selection of claims, verification of scope, and final scientific judgment remain with the author.
Papers I and II derived gravitational geometry and second-order dynamics from finite accessibility without invoking measurement, observation protocols, or collapse as primitive notions. This paper extends the operational program to the measurement sector, deriving the structure of record formation and probabilistic weights from the same finite accessibility and compositional constraints. Finite accessibility restricts admissible operations on a subsystem to a proper operator subalgebra. Because bounded certification excludes operators requiring nonlocal coordination or arbitrarily fine phase resolution, the accessible algebra is strictly smaller than the full operator algebra. When a system becomes correlated with orthogonal environment sectors, off-diagonal operators between those sectors fall outside the accessible algebra. The reduced state is therefore indistinguishable, under all admissible operations, from its diagonal ensemble. Definiteness emerges as an algebraic consequence of sector restriction, not as a collapse postulate. The weights of these sectors are fixed uniquely by compositional consistency. Phase invariance, multiplicativity under independent refinement, and stability under record coarse-graining constrain any admissible assignment of sector weights to satisfy a functional equation whose only continuous, non-contextual solution is quadratic in amplitude. The Born rule therefore arises as the unique invariant measure compatible with finite accessibility. Irreversibility follows from action scaling: reversal requires operators outside the admissible tensor algebra and incurs action cost proportional to the number of correlated subsystems. Since admissible operations are bounded by the action gap, macroscopic record reversal is excluded. Measurement introduces no new physical primitives beyond finite accessibility, locality, and compositional stability. Record formation, classical definiteness, and Born probabilities are structural consequences of algebraic restriction under finite accessibility. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** Born rule derivation, measurement problem, quantum statistics, finite accessibility, operator subalgebra, record formation, definiteness emergence, wave function collapse, compositional consistency, accessible state spaces, action gap, irreversibility, sector restriction, quantum foundations, algebraic quantum mechanics, decoherence
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.
STATUS (18 October 2026). The physics programme that motivated this work was concluded in October 2026 and its claim of a unified derivation withdrawn. The results in this paper do not depend on that programme and stand as stated. NOTE ON SCOPE (18 October 2026). Where this paper says that the Standard Model's matter content, charges or gauge structure are derived from, forced by or emerge from the eight-bit error-correcting cell, read encoded in or assigned to. The finite results themselves stand: the code, its constraints and the anomaly arithmetic are exact. But two things are premises of the framework, not consequences: the identification of codewords with particles, and the choice of the eight-bit [8,4,4] cell. In particular, the step from read/write balance to a single CSS code is not derived [R22]. The author's October 2026 review of the programme concluded that the framework's Standard-Model content is assigned rather than derived. No observation currently distinguishes the framework's account from the Standard Model itself. This revised note argues that, within the accepted complex quantum-error-correcting record substrate, the quadratic Born form arises from linear record writing, orthogonality of record sectors, and isometric normalization before any probability measure is introduced. Conditional uniqueness is treated separately through Gleason-type and refinement-stable record-sector routes. The 30 August 2026 additivity-audit revision cites Jiaxuan Zhang's analysis of five Born-rule derivations (arXiv:2603.06211) and makes the premise accounting explicit: a Gleason frame function includes orthogonal additivity, while Marko Lela's route begins with a non-negative extensive valuation on disjoint continuation bundles. The closed-record-pair route assumes neither measure-level object, but establishes the quadratic form rather than a standalone uniqueness theorem. The resulting open frontier is to obtain both form and uniqueness without importing measure-level additivity. The paper also records the minimal [8,4,4] record cell and the bare monitored service rate alpha_0 = 1/137, while retaining the deeper reconstruction question as open.