We study the information geometry of helical cylindrical lattices with N protofilaments and helical offset h. The microtubule geometry (N=13, h=3) serves as a structural template motivating the lattice topology; we do not model microtubule biophysics. Starting from the exact Bessel expansion of the partition function per node, we derive the entropic deficit C as a local geometric invariant independent of lattice size N, and compute the Fisher–Rao metric and its scalar curvature R on the (J₁, J₂, J₃) parameter space. All metric quantities are expressed as closed-form Bessel sums with no perturbative truncation. Main results: (1) R = +2.6245 at the reference point (β=0.5, J₁=J₂=1, J₃=0.8, h=3), stable to six significant digits; (2) R grows rapidly near the BKT transition with effective scaling R ~ β^4.79 (first computation of Fisher scalar curvature at a BKT transition); (3) helical topology inverts the curvature sign (R > 0 for helical vs. R < 0 for square lattices); (4) UV–IR Fisher manifolds share curvature sign only for N=13; (5) commensurability resonances governed by gcd(N,h). New results in version 3: (6) mutual information I(A:B) exhibits exact area-law scaling I = N · f(β, J₁, J₃, h) (analytical proof via Bloch decomposition); helical coupling increases boundary information density by 53% relative to the square lattice; (7) linear relationship R ≈ 5.9 · (I/N) − 0.088 (Pearson correlation +0.99) connecting Fisher scalar curvature to boundary mutual information.
We study the information geometry of helical cylindrical lattices with N protofilaments and helical offset h. The microtubule geometry (N=13, h=3) serves as a structural template motivating the lattice topology; we do not model microtubule biophysics. Starting from the exact Bessel expansion of the partition function per node, we derive the entropic deficit C as a local geometric invariant independent of lattice size N, and compute the Fisher–Rao metric and its scalar curvature R on the (J₁, J₂, J₃) parameter space. All metric quantities are expressed as closed-form Bessel sums with no perturbative truncation. Main results: (1) R = +2.6245 at the reference point (β=0.5, J₁=J₂=1, J₃=0.8, h=3), stable to six significant digits; (2) R grows rapidly near the BKT transition with effective scaling R ~ β^4.79 (first computation of Fisher scalar curvature at a BKT transition); (3) helical topology inverts the curvature sign (R > 0 for helical vs. R < 0 for square lattices); (4) UV–IR Fisher manifolds share curvature sign only for N=13; (5) commensurability resonances governed by gcd(N,h). New results in version 3: (6) mutual information I(A:B) exhibits exact area-law scaling I = N · f(β, J₁, J₃, h) (analytical proof via Bloch decomposition); helical coupling increases boundary information density by 53% relative to the square lattice; (7) linear relationship R ≈ 5.9 · (I/N) − 0.088 (Pearson correlation +0.99) connecting Fisher scalar curvature to boundary mutual information.
We study the scalar curvature of the Fisher information metric on the microscopic coupling-parameter manifold of lattice spin models at criticality. For a d-dimensional lattice with periodic boundary conditions and n=Ld sites, the Fisher manifold has m=d⋅n dimensions (one per bond), and we find ∣R(Jc)∣∼ndR with dR=(dν+2η)/(dν+η), where ν and η are the correlation-length and anomalous-dimension critical exponents. For 2D Ising (ν=1, η=1/4), this predicts dR=10/9, confirmed by exact transfer-matrix computations (L=6--9: dR=1.1115±0.0002) and multi-seed MCMC through L=24. For 3D Ising (ν=0.630, η=0.0363), the prediction dR=1.019 is consistent with MCMC on L3 tori up to L=10 (power-law fit: dR=1.040). For 2D Potts q=3 (predicted 33/29≈1.138), FFT-MCMC through L=40 shows deff oscillating non-monotonically around ∼1.20, consistent with O(1/(lnL)2) logarithmic corrections. For q=4 (predicted 22/19), effective exponents oscillate with strong logarithmic corrections. The Ricci decomposition identity R3=−R1/2, R4=−R2/2 holds to 5--6 digits for all models. This exponent is distinct from Ruppeiner thermodynamic curvature and reflects the collective geometry of the growing Fisher manifold. We provide falsification criteria and predictions for additional universality classes.
This paper develops a reciprocity‑based account of observation that treats interfaces as primary objects and links informational and geometric structure via a dimensionless reciprocity constant Ξ. On the informational side, a measuring instrument is modeled by a Fisher or Bures / quantum Fisher metric on a statistical manifold, from which an intrinsic scalar curvature ℛ[g_I] is constructed. On the geometric side, the Raychaudhuri equation is packaged into a focusing bracket ℬ, normalized to a dimensionless quantity ℬ̃. The central principle, Observation as Mutual Curvature (OMC), states that for a fixed instrument class operated in a stationary regime, a linear combination of informational curvature and focusing curvature is approximately invariant under reciprocity‑equivalent implementations of the same instrument. This motivates an operational reciprocity constant Ξ, obtained by regressing changes in ℛ[g_I] against changes in ℬ̃ across calibrated configurations. The framework is “interface‑first”: a Noether‑type balance theorem for moving boundaries yields a cut‑independent boundary ledger that tracks fluxes of energy, helicity, and information at photonic interfaces. Combining these elements, the paper defines Ξ‑light, a reciprocal dark–luminous architecture that employs quasi–bound states in the continuum, coherent‑perfect absorption, anapole modes, and time‑modulated media, together with a trap–meter–release protocol for estimating Ξ. A cosmological roadmap based on weak gravitational lensing is outlined, and the interpretive implications for objectivity and the role of information in physics are discussed. No new microphysics is assumed; all novel structures appear as balance principles, definitions, or falsifiable conjectures built on standard general relativity, quantum theory, and electromagnetism. Keywords: observation; information geometry; Fisher information; quantum Fisher information; Raychaudhuri equation; Noether charges; interfaces; metasurfaces; bound states in the continuum; anapole modes; coherent perfect absorption; photonic time crystals; weak gravitational lensing; reciprocity constant; Ξ‑light.
This article develops the generalized HKR theory as a unified geometric and information-theoretic model of the transition from context-dependent representations to path-independent objective facts. The construction combines a fact-state bundle over a Riemannian context manifold with the Hochschild-Kostant-Rosenberg filtration, Hopkins-Kuhn-Ravenel character-theoretic depth, Fisher-Rao information geometry, quantum Darwinism, and renormalization-group dynamics. In the reported simulations, the context-space scalar curvature has a hyper-stable minimum of approximately −2.25, while consensus concentrates the normalized Wilson-loop invariant at approximately 1.000. Fisher-Rao natural-gradient revision reaches tolerance in 15 iterations, compared with 179 ordinary-gradient iterations, giving an acceleration factor of 11.9. The quantum-Darwinist partial-information curve saturates at a system entropy of 1.000 bits for an environment fraction below 0.05 and yields redundancy 117.6. Finite-size scaling for lattice sizes from 64 to 1024 gives ν≈1.398 , γ/ν ≈1.731 , and β/ν ≈0.109 , establishing the reported second-order transition in the two-dimensional site-percolation universality class near the critical threshold 0.593. Active-inference, social-channel, biomimetic, and recursive tests further produce a self-referential score of 0.637 and a 16-claim composite score of 0.881. Together these results establish a scale-invariant, topologically coherent mechanism for emergent objectivity.
The preservation of quantum coherence in macroscopic biological environments at 300 K remains a primary challenge in open quantum systems, traditionally prohibited by the rapid thermalization dictates of the Lindblad master equation. We propose that the specific irrational scaling of the 13-protofilament biological microtubule circumvents this limit through a purely geometric isolation mechanism. By modeling the traversing quasiparticle (a Fröhlich polaron) within a ϕ-scaled chiral cylindrical lattice, we derive a rigorously constrained momentum-space topology governed by Valley-Orbit coupling. The structural Fibonacci sequence {5, 8, 13} acts as an effective non-Abelian gauge field, forcing the valley isospin to accumulate a quantized topological Berry phase of π/5. Utilizing a Jacobi-Anger expansion, we demonstrate that this C5 chiral holonomy acts as a spatial Bessel-function filter against the Super-Ohmic biological thermal bath, effectively zeroing the dissipator term (D[ρ] = 0) for low-frequency acoustic modes. Furthermore, the spatial parity-breaking of the 25-nanometer lattice induces a Harrison curvature-driven topological mass gap (Mtopo = Δβ(a/RMT )2 cos(π/5)σz), which structurally shields the system from high-frequency optical phonons. Utilizing Hierarchical Equations of Motion (HEOM) cavity dynamics, we show this geometry forces coherent non-Markovian information backflow, bounding the local Von Neumann entropy to an oscillating plateau (⟨dS/dt⟩ ≈ 0). Finally, we propose a strictly falsifiable test: substituting the local hydration layer with Deuterium Oxide (D2O) will spike the Fröhlich coupling constant, inducing polaron self-trapping. This mass-driven localization breaks the 2kF Fermi surface nesting, triggering a spontaneous Peierls transition that shatters the π/5 invariant and induces instantaneous, predictable catastrophic decoherence.
This paper establishes a local structural bridge between two recently developed information-geometric frameworks for observer-dependent probability representations on finite state spaces. Background. On the parametric side, the partition-adapted framework of Khomyakov (KLEO v3.24.1) defines observer entropy as the Kullback–Leibler divergence between a parametric distribution and the uniform lift of its coarse-graining, and shows that it scales quadratically with the resolution threshold, with a coefficient determined by the Fisher information matrix. On the empirical side, the Interference-First Reality (IFR) framework of Bianchi (2026) identifies multi-observer objectivity with the Fréchet barycenter of observer-induced distributions in the Jensen–Shannon metric, with almost-sure convergence guaranteed by metric barycenter theory. Principal results. The paper derives four main results: 1. Local Metric Bridge Lemma (Section 3): On the open probability simplex, the Jensen–Shannon divergence satisfies the local quadratic expansion D_JS(p ‖ p + εδ) = (ε²/8) I_p(δ) + O(ε³), where the explicit coefficient 1/8 arises from the midpoint structure of the symmetric divergence. 2. Multi-Observer Quadratic Aggregation Theorem (Section 4): For a partition-adapted parametric family satisfying conditions (A1)–(A3) of Khomyakov (KLEO v3.24.1), the empirical Jensen–Shannon Fréchet barycenter of an N-observer ensemble, centered on the within-class neutral locus, exists uniquely for small ε and coincides to leading order with the weighted Fisher–Rao barycenter of the within-class perturbation directions. The minimal Jensen–Shannon dispersion satisfies a quadratic scaling law governed by the within-class block I_ww(θ₀*) of the Fisher information matrix. 3. Leading-order coincidence of estimators (Corollary 6.1): The variational Fréchet minimizer p_F and the closed-form geometric overlap estimator p_G of Bianchi (2026) agree to order O(ε²) in Jensen–Shannon distance within the local regime. 4. Multi-observer Landauer corollary (Corollary 6.6): Under a per-channel irreversibility assumption and an additivity postulate for independent observer channels, the aggregate energetic cost of barycenter formation is bounded below by k_B T times the Jensen–Shannon dispersion. Scope. The bridge is local in ε (the within-class perturbation amplitude) and finite in N (the number of observers). It does not subsume Bianchi's almost-sure N → ∞ consistency theorem nor Khomyakov's ε → 0⁺ quadratic scaling result; it establishes their structural compatibility in the overlapping regime where both apply. The results are specialized to balanced exponential families (softmax), with explicit worked-example coefficients for four- and five-point state spaces. Domain-specific applicability is assessed against the cross-domain distinguishability hierarchy of Bianchi (2026), Table 14, identifying the low-dispersion regime (d_JS ≲ 10⁻² nats) as the primary domain of validity. Keywords: Jensen–Shannon divergence, Kullback–Leibler divergence, Fisher information, Fisher–Rao metric, multi-observer aggregation, Fréchet barycenter, probability simplex, partition-adapted parameterization, observer entropy, information geometry. MSC 2020: 62B10, 94A17, 53B12, 62F12.
Quantum information theory is traditionally formulated on projective state spaces, where the qubit is a point in ℂP¹. We introduce a new computational primitive—"the flubit"—defined as a point in the flag manifold Fl₁,₂(ℂ³), representing a line ℓ inside a plane P in ℂ³. This extends the hierarchy: bit → qubit → qutrit → flubit{0,1} → ℂP¹ → ℂP² → Fl₁,₂(ℂ³) Unlike projective spaces, the flag manifold carries intrinsic hierarchy: coarse (plane), fine (line), and their coupling via connection curvature Ω. We construct the Fisher information geometry of Fl₁,₂(ℂ³) and show it decomposes into base, fiber, and cross-term contributions. The cross-term vanishes for separable states and is nonzero precisely when the Weyl relation x² + xy + y² = 0 is activated. This yields a reinterpretation: entanglement is not a relation between subsystems, but the internal coherence of a single flubit. Standard two-qubit entanglement arises as the residual correlation left after projecting away the flag structure. We define the flubit–projective gap Δ(ψ), the excess Fisher information accessible through hierarchical measurements beyond independent projective ones. Δ(ψ) quantifies entanglement and provides a candidate measure of quantum computational advantage. On the product manifold M = Fl₁,₂(ℂ³) × ℂP¹, three structures coincide: Kähler geometry = quantized phase space (Kay) = information geometry (Fisher/Braunstein–Caves) Finally, we compute the Ricci curvature of the flag Fisher metric and show its eigenvalue structure constrains the gauge coupling hierarchy, with the Einstein constant λ = 2 emerging as the Ricci normalization. The flubit is the first non-projective computational primitive, and the flag manifold provides the minimal geometry where hierarchy, entanglement, and information are unified.