This paper presents BIG-B13, a reduced structural model of observation-like updating and self-measurement within Boundary Information Geometry (BIG). In BIG-B13, a system is represented by a boundary field, internal resonance modes, and a hidden-depth memory state. Observation is modeled as a boundary-folding interaction between a system and an observer, where boundary proximity gates a coupling between hidden-depth states. The purpose of this paper is not to derive quantum measurement, the Born rule, consciousness, or AI perception. Rather, it defines a testable reduced mathematical layer for observation-like hidden-depth updating, binary basin selection, Born-like reference comparison, and self-model stability analysis. Numerical experiments are presented in two parts. First, a spin-like binary observation toy model is tested. The baseline model produces ordinary basin selection rather than Born-rule behavior. However, a low-barrier finite-noise regime reproducibly approaches a Born-like reference curve, with robustness checked across random seeds. Observer coupling is then shown to preserve, bias, or destroy this finite-noise Born-like regime depending on observer-axis symmetry and coupling strength. Second, a reduced self-observation model is analyzed through fixed points, Hessian spectra, discrete-time stability, and noise-induced basin switching. The model produces a multistable self-observation landscape with four locally stable fixed points. Noise tests show that local Hessian stability is distinct from global noise robustness: lower-energy self-model basins remain more persistent under high noise than higher-energy basins. BIG-B13 is therefore presented as a reduced, testable model of boundary-folding observation and self-model stability. It does not constitute a theory of quantum measurement or consciousness, but provides a mathematical scaffold for future extensions.
This paper presents BIG-B13, a reduced structural model of observation-like updating and self-measurement within Boundary Information Geometry (BIG). In BIG-B13, a system is represented by a boundary field, internal resonance modes, and a hidden-depth memory state. Observation is modeled as a boundary-folding interaction between a system and an observer, where boundary proximity gates a coupling between hidden-depth states. The purpose of this paper is not to derive quantum measurement, the Born rule, consciousness, or AI perception. Rather, it defines a testable reduced mathematical layer for observation-like hidden-depth updating, binary basin selection, Born-like reference comparison, and self-model stability analysis. Numerical experiments are presented in two parts. First, a spin-like binary observation toy model is tested. The baseline model produces ordinary basin selection rather than Born-rule behavior. However, a low-barrier finite-noise regime reproducibly approaches a Born-like reference curve, with robustness checked across random seeds. Observer coupling is then shown to preserve, bias, or destroy this finite-noise Born-like regime depending on observer-axis symmetry and coupling strength. Second, a reduced self-observation model is analyzed through fixed points, Hessian spectra, discrete-time stability, and noise-induced basin switching. The model produces a multistable self-observation landscape with four locally stable fixed points. Noise tests show that local Hessian stability is distinct from global noise robustness: lower-energy self-model basins remain more persistent under high noise than higher-energy basins. BIG-B13 is therefore presented as a reduced, testable model of boundary-folding observation and self-model stability. It does not constitute a theory of quantum measurement or consciousness, but provides a mathematical scaffold for future extensions.
This working paper presents BIG-B13.1, a reduced follow-up to BIG-B13 within Boundary Information Geometry (BIG). BIG-B13 introduced boundary folding as a reduced model of observation-like hidden-depth updating, binary basin selection, and self-observation stability. The present paper extends that analysis by testing rotated-basis basin selection, extracting an empirical selection kernel, analyzing its barrier-noise scaling, and studying sequential observation through post-selected hidden-depth clouds. The first result is that finite-noise basin selection is not restricted to a fixed original basis. Across several rotated basis angles, the selection probability remains moderately close to a basis-aware Born-like reference curve. However, shifted-coordinate collapse shows that the empirical response is not identical to cos². Instead, it is much better described by a logistic basin-boundary kernel, K_BIG(δ) = 1 / (1 + exp[-κ(cos 2δ - c)]). For the baseline finite-noise regime b = 1.0 and σ = 0.35, this kernel fits the empirical collapsed curve substantially better than the cos² reference. A scaling scan over barrier strength b and noise amplitude σ shows that the kernel sharpness approximately follows κ(b,σ) ≈ 0.532 b^0.569 σ^-1.659, with R² ≈ 0.934 in log-space. This supports the interpretation that the apparent Born-like behavior is governed by a noise-softened basin-boundary response rather than by a direct cos² probability law. The sequential observation tests reveal a further distinction. A point-like pointer reset remains close to the BIG logistic basin-boundary kernel. In contrast, actual noisy post-selected hidden-depth states produce a conditional response closer to cos². This is explained by a finite-width cloud-averaging mechanism: the sequential response is accurately modeled by a Gaussian convolution of the BIG logistic kernel, K_cloud(δ;w) = E[K_BIG(δ + ζ)], ζ ~ N(0,w²). For the actual post-selected sequential curve, the best-fit cloud width is approximately 12 degrees, with RMSE ≈ 0.0048. This fitted width agrees with independent estimates of the post-selected cloud width from centroid norms, supporting the interpretation that cos²-like sequential behavior emerges from cloud averaging of a noise-softened basin kernel. This paper does not derive quantum measurement, the Born rule, wave-function collapse, or consciousness. It provides a reduced, testable structural model showing how rotated-basis basin selection, logistic basin-boundary response, barrier-noise scaling, and finite-width post-selected clouds can generate Born-like or cos²-like reference behavior in a limited observation-like toy setting.
BIG-B15 presents a reduced operator representation for comparing observation traces and boundary history within Boundary Information Geometry (BIG). BIG-B13 and BIG-B13.1 studied observation-like updates in hidden-depth states, including basin selection, Born-like behavior, an empirical logistic selection kernel, cloud averaging, and sequential consistency. BIG-B14 introduced boundary history as a trace retained on the active boundary, with reduced operations such as writing, retention, fusion, division, and lineage-level persistence. The present paper asks whether these two forms of trace retention can be described within a common state-update language. The goal is not to derive quantum measurement, the Born rule, biological memory, heredity, or standard quantum channels. Rather, the paper provides a reduced representation framework in which hidden-depth clouds, boundary histories, kernel responses, and boundary-history update maps can be compared as history-dependent transformations of state. The paper first embeds hidden-depth states into a real two-dimensional Hilbert-like representation and shows that finite hidden-depth clouds can be represented by density-like mixtures. Increasing cloud width lowers purity and smooths the projective response curve. It then compares the empirical BIG logistic kernel with the Born cos² reference and affine qubit-like probability forms. The results show that the BIG logistic kernel is not identical to the Born cos² reference. An unconstrained affine fit improves the error slightly but violates probability bounds, while the constrained affine probability form returns to the Born cos² reference. Higher-feature representations approximate the BIG kernel much more accurately. The paper then lifts B14 boundary history profiles HΣ(s) into density-like mixture representations. This representation captures normalized trace geometry, localization, and spread, while absolute history strength must still be retained separately. Finally, representative BIG update maps are classified as linear, affine, nonlinear, or state-dependent. Forgetting/smoothing and fixed-partition division are linear under fixed parameters, writing with a fixed stimulus is affine, normalization into density-like weights is nonlinear, fusion addition is linear, and fusion compression is structurally overlap-dependent. B15 should therefore be read as a representation and comparison paper. It does not identify BIG with standard quantum theory. Instead, it clarifies which parts of BIG admit Hilbert-like, density-like, or operator-like representations, and which parts require nonlinear, normalized, state-dependent, coarse-grained, or higher-feature descriptions. In this sense, B15 provides a bridge between observation traces, boundary history, and a possible future boundary-history operator framework.
This manuscript formulates Boundary Information Geometry (BIG) as a layered family of models rather than a single universal equation. The framework connects boundary formation, local landing behavior, global stability, shape and topology transitions, sustained capture, state selection, and boundary history through the state S(t)=(φ, Σφ, R, h, HΣ). Numerical results from the existing B3–B15 modules are audited and separated from a newly proposed coupled model linking the field, internal modes, hidden-depth states, and a history field on a moving boundary. The manuscript does not claim a quantitative theory of quantum measurement, nuclear fission or fusion, or consciousness. It presents a technical model proposal with explicit limitations, falsification criteria, and a reproducibility package.
This is Paper 157 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops a finite structural account of quantum measurement based on boundary-supported record formation, decoherence, and global admissible-history closure. In pq-spacetime, measurement is not a fundamental collapse process. A system, apparatus, environment, and observer become incorporated into an enlarged β-boundary whose dynamics stabilize distinguishable record channels. Environmental coupling suppresses accessible interference between incompatible records, while each observer continuation inherits a definite local outcome through its embedding in one stable β-record ledger. The paper distinguishes this branch-local definiteness from the global structure of the theory. The finite amplitude law and No-Fence Principle do not permit admissible history sectors to be deleted by an additional measurement rule. The global metaobject therefore remains closed over the complete admissible-history structure even though individual observers occupy definite record continuations. Collapse is consequently recovered only as an effective conditional update of local access rather than as an ontological destruction of alternatives. The analysis connects measurement directly to the earlier pq-spacetime treatments of entanglement, decoherence, and black-hole information. In each case, reduced access to part of the admissible-history structure can produce locally mixed or effectively classical descriptions without implying destruction of the globally closed state. Born weights are inherited from the prior no-fence scalarization result rather than introduced as an independent measurement postulate. The Measurement Ledger Theorem summarizes the result: quantum measurement is the formation of stable branch-local β-record ledgers through boundary enlargement; consciousness is not a measurement mechanism; primitive collapse and hidden selectors are unnecessary; pointer states correspond to dynamically stable record channels; and the globally admissible history structure remains closed throughout the measurement process. The resulting picture identifies classicality with persistent, redundant record stability rather than with a separate classical ontology. Definite experimental outcomes and globally closed quantum evolution are therefore treated as complementary aspects of the same finite pq-spacetime dynamics.
Measurement-induced phase transitions in monitored many-body systems and collapse-like irreversibility in measurement contexts are often treated as distinct phenomena. We show that both arise as effective four-dimensional shadows of the same reduced-dynamical mechanism in Modal Triplet Theory: noninvertible projection onto an admissible coherent sector together with basin stabilization. We formulate a general projected evolution map for reduced states, define admissible basins via contractivity margins, and prove that both measurement-induced phase transitions and collapse thresholds correspond to the same loss of contractivity at basin boundaries. Using a local Ornstein–Uhlenbeck/Kramers reduction near basin boundaries, we derive a finite-strength, nonanalytic crossover (“knee”) and establish generic protocol dependence, including Zeno and anti-Zeno regimes. We further show that linear measurement-only or decoherence-only models cannot reproduce this combined structure without introducing state-dependent stabilization equivalent to basin dynamics. All results are slab-local and admissibility-conditioned, providing a unifying reduced-dynamical bridge between monitored-circuit transitions, continuous measurement, and irreversibility.
For stationary, stable linear systems, quantum form (symplectic form, complex structure, Hilbert inner product) is not a set of external postulates but geometric architecture already carried by the classical action and its flow; read through a local projection, for which decoherence is the explicit candidate, only one postulate remains: that states are sections. The Born measure is not derived here. This is a foundations paper. It adds no new dynamics, and most of its mathematics is known; its contribution is to arrange known results inside one ontology and to check what that ontology can and cannot carry. The ontology has a static space of consistent configurations, a single actualized history, a many-to-one projection Π and a selection operator A, and was proposed to account for quantum structure, the arrow of time and the Born rule. Taking each claim to the smallest system on which it can be computed, we find that the natural explicit candidate for Π, a projection onto locally accessible degrees of freedom, is standard decoherence theory; we do not show that Π must take this form. For linear systems with stable dynamics the quantum structure the ontology presupposes is reconstructed from the classical action: the symplectic form from the boundary term, the complex structure from the flow, and the Hilbert inner product from their combination. Stability of the linearized dynamics, rather than positivity of the action's Hessian, is the condition for quantum form to exist. Given the section postulate, the topological invariants of the ontology become the integrality conditions of geometric quantization; under the preservation assumption of Proposition 1, a dynamics that couples subsystems admits a single value of ħ; and if topology fixes dimensionless constants at all, it can do so only at a boundary, the measured values being images under renormalization-group flow (an illustrated pattern, not a derivation). The hidden sector splits into a gauge part and a temporally accessible part, which exposes a dilemma for structural accounts of irreversibility. A global selection cannot signal under three stated conditions, the central one being that its criterion reads only unconditioned reduced states in the causal past; we conjecture that selection is triggered by redundant environmental records. A hypothesis that the history begins at a regular horizon is tested against the Einstein equations in homogeneous models. Each result is labelled as known, synthesis or conjecture. The code-backed checks are listed in Tables 2–4 and reproduced by the attached scripts (run_all.py, with assertions on the quoted values; tested on Python 3.11/Linux and Python 3.14/Windows). Of the original framework's three predictions, one coincides with standard quantum mechanics in its safe form, one is withdrawn, and one lacks a mechanism and is bounded by binary-pulsar timing. How to check this paper: every result carries a status tag (known / synthesis / conjecture / withdrawn / scoped numerical evidence) collected in the ledger (Table 1). Code-backed claims are listed with their IDs in Tables 2–4. Unzip scripts_v15.zip and run python run_all.py (a few minutes; --rederive also redoes the symbolic Bianchi IX derivation); it asserts the numbers quoted in the text and exits non-zero on any mismatch. Corrections are welcome.