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 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 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 record hosts the preprint “Boundary-Condition Quantum Mechanics IV_b” (BCQM IV_b), a companion paper to BCQM IV in the Boundary-Condition Quantum Mechanics series. The BCQM programme develops a discrete, event-based extension of quantum mechanics with a finite coherence horizon W_coh, from which spacetime structure, inertial response, and an intrinsic inertial-noise floor are intended to emerge. BCQM IV_b focuses on the numerical side of this programme. It develops and tests a minimal inertial-noise pipeline built from a simple, W_coh-blind control kernel (“bcqm_toy_3”) defined at the level of a single event thread. The paper describes how acceleration time series are generated, how one-sided power spectra are estimated, and how a noise amplitude A and characteristic frequency omega_c are extracted and fit to log–log scaling laws. A first set of results demonstrates that, for a deliberately W_coh-independent kernel, the fitted exponent β in the scaling law A(W_coh) ∝ W_coh^{−β} is statistically compatible with zero, as required for a clean control model. The paper then extends the analysis to simple configuration-space cluster toys. An independent-probe cluster model (“bcqm_cluster_toy”) is used to study how the centre-of-mass (COM) inertial-noise amplitude scales with the number of probes N. The measured behaviour A_COM(N) ∝ N^{−1/2} and an almost N-independent spectral centroid confirm the expected COM suppression for independent probes and provide a baseline against which more realistic, correlated cluster kernels can be compared in future work. The dimensionless results are finally mapped into SI units for representative mechanical and atomic benchmark platforms, yielding order-of-magnitude estimates for the BCQM inertial-noise floor and associated thermal crossover temperatures. Taken together, these results provide a numerically validated control model, a documented pipeline from event-level dynamics to inertial-noise spectra, and simple few-body benchmarks that will be used in the gravitational and emergent-curvature analysis developed in BCQM V. The companion simulation and analysis code for this paper is archived separately on Zenodo under DOI 10.5281/zenodo.17815304 and is referenced in the text.
v0.3.0 of the multi-base post-processing analysis of Shor's order finding. The original five theorems (v0.2.1, DOI 10.5281/zenodo.20681847) are retained unchanged: (1) noise-invariant determinism, (2) logarithmic coverage time, (3) exact noise scaling 1/g_M(η), (4) conditional compatibility with Regev's 2023 multi-base framework, and (5) a hybrid (C)+Regev b-trick factoring algorithm with closed-form bound matching N=437 empirical mean within 4%. New in v0.3.0: §3.6 of v0.2.1 is *self-corrected*. The previously claimed 'boundary-flip + universal direction stochasticity + plateau/overload' framework is shown to be the finite-trial expression of a single closed form p(σ) = ρ + (p_0 - ρ)·exp(-σ²) derived from the standard phase-noise dephasing of the FFT. The closed form is verified across five algorithm classes — Grover (R²=0.88), Shor pure (0.95), QPE isolated (0.96), Simon (0.99), Hybrid (C)+b-trick (paper §3.6 setup, 0.91). Positioning: analytical complement to Yang-Markidis (arXiv:2605.16074, ICS Workshops '26), whose empirical recoverability model (1-ε)·P_s + ε·distractors has the same structural form; we provide the analytical foundation ε = 1 - exp(-σ²). Implication: SR-based factoring acceleration is precluded by a closed-form bound on |ΔK|. v0.3.0 retracts the 'boundary flip' lexicon as a distinct mechanism; all raw measurement data and theorems 1-5 are retained.
v0.3.0 of the multi-base post-processing analysis of Shor's order finding. The original five theorems (v0.2.1, DOI 10.5281/zenodo.20681847) are retained unchanged: (1) noise-invariant determinism, (2) logarithmic coverage time, (3) exact noise scaling 1/g_M(η), (4) conditional compatibility with Regev's 2023 multi-base framework, and (5) a hybrid (C)+Regev b-trick factoring algorithm with closed-form bound matching N=437 empirical mean within 4%. New in v0.3.0: §3.6 of v0.2.1 is *self-corrected*. The previously claimed 'boundary-flip + universal direction stochasticity + plateau/overload' framework is shown to be the finite-trial expression of a single closed form p(σ) = ρ + (p_0 - ρ)·exp(-σ²) derived from the standard phase-noise dephasing of the FFT. The closed form is verified across five algorithm classes — Grover (R²=0.88), Shor pure (0.95), QPE isolated (0.96), Simon (0.99), Hybrid (C)+b-trick (paper §3.6 setup, 0.91). Positioning: analytical complement to Yang-Markidis (arXiv:2605.16074, ICS Workshops '26), whose empirical recoverability model (1-ε)·P_s + ε·distractors has the same structural form; we provide the analytical foundation ε = 1 - exp(-σ²). Implication: SR-based factoring acceleration is precluded by a closed-form bound on |ΔK|. v0.3.0 retracts the 'boundary flip' lexicon as a distinct mechanism; all raw measurement data and theorems 1-5 are retained.
We propose a minimal ontological framework in which selection — theactualization of an element from a possibility space — is the soleprimitive. From this postulate, a recursively grown selection graph anda local coherence measure kappa induce temporal partial ordering,identity as basin stabilization, and the subject-object distinction asan internal basin partition. Description is selection; first-personstructure emerges as recursive self-indexed closure within stabilizedbasins; agency is self-indexed coherence elevation under indeterminacy. The framework introduces no new empirical predictions but constrainsadmissible ontological interpretations by eliminating categoryseparation between structure and experience. The central result is adissolution — rather than a solution — of the hard problem ofconsciousness: with selection as primitive, no additional primitive isavailable to ground a structure/experience distinction, and thedistinction cannot be coherently formulated within the framework. Thecoherence measure kappa is grounded as the zero-order approximation ofaverage pairwise mutual information, fixing the meaning of theneighbourhood scale k as a mutual-information horizon. Temporal order isderived by label-covariant sequential accretion, connecting the accountto causal set theory (Bombelli et al. 1987; Sorkin 2003), relationalquantum mechanics (Rovelli 1996), QBism (Fuchs, Mermin & Schack 2014),integrated information theory (Tononi 2004), and process ontology(Whitehead 1929). Open problems — Born-rule recovery, the numericalvalue of k, formal consistency verification, the combination problem,and the zero-depth limit — are stated without minimization. This preprint is part of an ongoing research program. A companionpaper, Selection Dynamics, Trace Ordering, and the Emergence ofTemporal Experience (doi:10.6084/m9.figshare.31225114), develops thetemporal-emergence claim at the phenomenological level, with aheart-rate-variability protocol for empirical test. Further companions:CDIL Unified Framework (doi:10.6084/m9.figshare.31566811); CoherenceBounds on Consciousness (doi:10.6084/m9.figshare.31724512); Kaluza-Kleinhidden-symmetry paper (doi:10.6084/m9.figshare.32916482). REFERENCES / RELATED MATERIALS (DOI field — cross-populates DataCite)---------------------------------------------------------------------10.6084/m9.figshare.3122511410.6084/m9.figshare.3156681110.6084/m9.figshare.3172451210.6084/m9.figshare.32916482