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 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 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.
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.
Quantum information, the fundamental building block of modern computation, possesses unique topological properties that profoundly impact its stability and robustness. This research delves into a novel mathematical framework, Quantum Information Topology, to better understand and quantify these topological aspects. We propose a new conceptualization of entanglement as a geometric entity, leveraging concepts from topology and geometry to derive rigorous predictions about the behavior of quantum systems. This framework seeks to provide a more comprehensive and predictive model for the evolution of quantum information, moving beyond traditional approaches that often fall short in capturing these intricate relationships. The core claim is that this framework will offer a deeper understanding of the topology of quantum information and its influence on system stability.
Quantum information geometry is a foundational concept in quantum information theory, offering a powerful framework for understanding and manipulating quantum states. This paper proposes a novel quantum information geometry Turing domain, characterized by its reliance on quantum entanglement and topological structures to define and analyze quantum information properties. The core mechanism involves constructing a quantum information coding framework based on topological structures, leveraging the unique properties of quantum entanglement to achieve robust and efficient information transfer and encoding. We argue that this approach significantly expands the scope of quantum information theory by extending the geometric interpretation of quantum states beyond traditional algorithmic applications.
Conscious Quantum Solver (CQS) v0.1 — Formal Specification is a technical whitepaper that defines an auditable framework for local optimization in structured instance families using fractional spectral regularization and Polyak–Łojasiewicz (PL) geometry. The specification includes explicit pullback conditions from relaxed variable space to a phase-only physical manifold, a scheduled-control (adiabatic tracking) condition, and an operational definition of computation based on certified readout before an inference horizon. Phase–Echo Theory (PET) is included only to formalize observer-access limits and to distinguish internal relaxation from observable computation. All guarantees are local, instance-restricted, and falsifiable; no claims are made regarding worst-case complexity, global optimization, or violations of known physical laws. This deposit provides the Conscious Quantum Solver (CQS) v0.1 — Formal Specification and associated diagrams. CQS is defined as a local optimization framework for restricted, structured instance families, grounded in spectral–topological geometry. The specification models a relaxed assignment space with energy F(x)=12x⊤Lαx+λ∑cϕc(x),F(x)=\tfrac12 x^\top L^\alpha x+\lambda\sum_c \phi_c(x),F(x)=21x⊤Lαx+λc∑ϕc(x), where LαL^\alphaLα is a fractional graph Laplacian operator and ϕc\phi_cϕc are smooth clause/constraint penalties. Under explicit assumptions (spectral expansion, penalty regularity, and a local smoothness basin), the energy satisfies a local Polyak–Łojasiewicz inequality, yielding exponential energy decay under gradient flow within a certified basin. The document further defines a phase-only physical state manifold θ∈Tm\theta\in\mathbb{T}^mθ∈Tm and an observation/decoder map x=D(θ)x=D(\theta)x=D(θ). Under explicit non-degeneracy (non-saturation) conditions, the pulled-back energy E(θ)=F(D(θ))E(\theta)=F(D(\theta))E(θ)=F(D(θ)) inherits a PL inequality, establishing a measurable bridge from relaxed optimization geometry to phase-space physical realizations. A scheduled-control condition is specified for adiabatic tracking under time-varying penalty weight λ(t)\lambda(t)λ(t). Operational computation is defined not by internal relaxation alone but by certified readout before an inference horizon TIT_ITI, determined by margin growth relative to noise floor. The certification rule is an auditable inequality using decoded margin m(t)m(t)m(t), residual proxies r(t)r(t)r(t), and noise σ(t)\sigma(t)σ(t). Phase–Echo Theory (PET) is included only to formalize observer-access limits (recoverability bounds and echo/variance tradeoffs) and to distinguish internal convergence from observable computation. The specification is intentionally conservative: all guarantees are local, instance-restricted, and falsifiable; no worst-case complexity claims, global optimization claims, or nonstandard-physics claims are made.