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.
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 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.
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 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.
Papers 1-2 of this series derived Born weights conditionally from the block’s Riemannian geometry: the canonical volume determines the Hilbert structure, the induced projective geometry determines the ray measure, and Gleason noncontextuality fixes the trace assignment. What remained was the confirmation problem: in a block ontology every compatible history and every compatible observer-instance exists, so probability must be a measure over realised record-relative viewpoints, not a chance of becoming actual. This paper builds the record-theoretic and statistical structure that closes that gap, up to two explicitly priced commitments. The central claim, calibrated. In a non-splitting block ontology, perfect record cells inherit the Born measure and its exact conditional calculus. Under a persistent-prefix distinguishability condition (Assumption PF, stated prominently), interference remains uniformly controlled through arbitrarily deep finite record refinements. The resulting static record filtration obeys martingale concentration without independence assumptions, and — conditional on the explicitly priced viewpoint-typicality principle (Commitment TY) — Born conditional probabilities are exponentially confirmed against persistently different rival policies, including adaptive history-dependent rivals. The supporting structure: an exact perfect-record projector calculus; a closed sequential interference bound whose index is refinement depth, not flowing time; an explicit block-compatible recorder family (translated Gaussian fragments, and intrinsically, normalised heat-kernel packets on the block’s own S3) for which the record-overlap exponents and minimum redundancy thresholds are derived in closed form — the recorder family itself is not yet derived from block dynamics; and a canonical Gram-polar construction that associates to any imperfect record family, at each fixed depth, an exact Boolean hierarchy of projectors within explicitly bounded distance of the raw records — a finite-depth canonical Booleanisation, with cross-depth projective consistency an open problem. The coherent event weight Q is used only comparatively: it is not an additive probability measure for imperfect records, and the paper states the resulting trade-off explicitly. Two commitments carry the non-mathematical load and are priced, not hidden. Commitment OL: observer continuity is represented by record-functional equivalence across compatible continuations. Commitment TY: rational expectation over the record-equivalent continuations follows the canonical conditional measure. Given TY, the block-derived Born measure is internally self-confirming on record-typical viewpoints; without it, the mathematics fixes the measure but not its first-person authority. No collapse, branch creation, privileged actuality, pointer-basis selection, or theory of consciousness is used. Finite-resolution results on the block’s S3 truncation ladder (exact spectral-leakage identities, a conditional Sobolev rate, the Gleason dimension threshold) are collected in Appendix B.
Part XI places the standard quantum probability and measurement package inside theVBRC representation layer. The paper does not claim that quantum mechanics is the uniquemicroscopic law of the unread side, nor does it identify the unread coordinate with a hiddenphysical sector. In accordance with Part I, unread content reaches retained physics onlythrough a licensed summaryFXI = ΣXI(IXII), ΣXI := ΣHXI,PXI ,with any first-order formulaFXI = DXIIIXIItreated as a representative realization rather than as a universal primitive.The starting point is an already generated R3 retained channel, inherited from the lawgeneration papers. Part XI asks how this retained channel becomes a quantum observationinstance. The core result is that, once a normalized retained protocol weight, possiblyrepresented by a Hilbert state or by a configuration density/current under an additional localconfiguration gate, and a readout channel are declared, outcome probabilities are obtainedas pushforwards and conditional laws of the declared observation protocol:νt = (Πδobs)#µt.Thus probability is not inserted as a new structural force; it is the protocol-level readout ofpartial observation under IIP.The usual Born, PVM/POVM, and measurement-update formulas are then recorded asthe standard Hilbert-compatible branch:pk = Tr(ρEk), ρ 7→ ρk,under the declared positivity, normalization, event-additivity, and update gates. Theseformulas are not presented as a derivation of a unique microscopic ontology, but as thecompatibility readout by which the retained R3 channel agrees with ordinary quantummechanics where the Hilbert/Born gates are valid.Finally, Part XI records the EPR/Bell compatibility condition. VBRC does not posita local hidden-variable factorization behind the retained readout. Bell-violating quantumcorrelations are therefore not excluded by the framework; they are handled by the standardbipartite Hilbert/Born readout, while local factorizing hidden-variable representations areclassified as incompatible gates for the quantum branch.
This is Paper 006 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops the emergence of effective operators and norm-preserving dynamics within the Anaxiomatic Mechanics / pq-Spacetime stack. Starting from finite binary-mechanical graph histories and their amplitude structure, it shows how boundary-resolution maps induce effective linear operators between distinguishable completion channels. Linearity is not assumed as a primitive quantum rule; it follows from the additivity of finite history amplitudes before probabilities are assigned. The paper then distinguishes closed boundary maps from open, conditioned, or coarse-grained maps. A closed boundary map carries admissible fine-history support forward without creation, deletion, duplication, selection, or loss. In the effective Hilbert representation, this closed support-preserving transfer appears as an isometry, and when the initial and final effective spaces have equal finite dimension, as a unitary map. Open or conditioned descriptions instead give nonunitary effective operations. The result provides the operator-theoretic bridge in the finite pq-spacetime construction: finite graph histories give amplitudes; amplitudes give Born-type weights; Hilbert representation gives state space; tensor-product structure gives composite records; and closed boundary maps give effective norm-preserving dynamics. Operators, projections, isometries, unitaries, and open-system maps are therefore not inserted as primitive assumptions, but arise as representation-level consequences of finite boundary-resolution structure.