A Toy Rug: Three Sectors, Two Themes, and a Multifractal Error Measure on a Spin Chain Computing rug holonomy against a single interpolated chart and an averaged Lindblad baseline
Volume I of this pair introduced a Persian-rug ontology of quantum substrates: a sheaf of themed sectors with occupancy ledger Λ(α, θ) = (lX, lT, dH, dS) and a rug holonomy interrupted by wall operators Wi (Mamun, 2026a). The present volume constructs the minimal object that can put those definitions to a number. A six-site XXZ chain with a next-nearest perturbation is assigned three Hamiltonian sectors and two themes (even parity as a code theme; full-space dynamics as a charge theme). An even-parity initial state is transported around a triangular control loop that visits the three sectors. Three evolutions are compared: a piecewise sector path with optional knots; a smoothly interpolated single-chart Hamiltonian on the same triangle; and an averaged generator with local amplitude damping and dephasing. At vanishing knot strength the piecewise path already parts from the interpolated chart by a fidelity residual ΔF = 1 − F = 0.020. The residual rises monotonically with knot angle ε, reaching 0.080, 0.249 and 0.706 at ε = 0.10, 0.20 and 0.40. A detector-like wall that fails the theme grammar leaks even-parity weight from 1.000 to 0.882 and opens a further residual. An averaged Lindblad generator on the barycentric Hamiltonian does not track either unitary path (purity 0.450). Finite-N dimension estimators only weakly separate the three sectors; gap, echo envelope and correlator profile separate them cleanly. Also available on Academia.edu: https://www.academia.edu/attachments/134080798/download_file?s=portfolio
Volume I of this pair introduced a Persian-rug ontology of quantum substrates: a sheaf of themed sectors with occupancy ledger Λ(α, θ) = (lX, lT, dH, dS) and a rug holonomy interrupted by wall operators Wi (Mamun, 2026a). The present volume constructs the minimal object that can put those definitions to a number. A six-site XXZ chain with a next-nearest perturbation is assigned three Hamiltonian sectors and two themes (even parity as a code theme; full-space dynamics as a charge theme). An even-parity initial state is transported around a triangular control loop that visits the three sectors. Three evolutions are compared: a piecewise sector path with optional knots; a smoothly interpolated single-chart Hamiltonian on the same triangle; and an averaged generator with local amplitude damping and dephasing. At vanishing knot strength the piecewise path already parts from the interpolated chart by a fidelity residual ΔF = 1 − F = 0.020. The residual rises monotonically with knot angle ε, reaching 0.080, 0.249 and 0.706 at ε = 0.10, 0.20 and 0.40. A detector-like wall that fails the theme grammar leaks even-parity weight from 1.000 to 0.882 and opens a further residual. An averaged Lindblad generator on the barycentric Hamiltonian does not track either unitary path (purity 0.450). Finite-N dimension estimators only weakly separate the three sectors; gap, echo envelope and correlator profile separate them cleanly. Also available on Academia.edu: https://www.academia.edu/attachments/134080798/download_file?s=portfolio
Preprint. Third paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), Paper D (the observer-context category) and a synthesis note. The slogan "contextual holonomy encodes quantum contextuality" is attractive and, as usually stated, false. This paper gives it a sharp form. On the context stack of maximal abelian subalgebras the canonical degree-2 class has order exactly n, and its cocycle holonomy along closed context loops detects EXACTLY the all-versus-nothing sector of contextuality for closed-triple Weyl families at even local dimension. Both containments are strict, each by an explicit certificate: at d=3 the class is nontrivial of order 3 while the achievable state-independent AvN value is 0, and KCBS exhibits state-dependent contextuality where the class shadow vanishes. The Wigner-friend loop generated by H and S splits by level: its projective (Pancharatnam) holonomy is +1 while its determinant-section holonomy is exactly -i, so any interpretational claim about "the" holonomy of that loop must state which one it means. ATTRIBUTION. The classification of state-independent values for qudit Weyl families, the even/odd dichotomy and the fixed value d/2, on which the state-independent side of the equivalence rests, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024). Attainment at general even d is the Raussendorf-Okay-Zurel-Feldmann construction. This paper uses both and rediscovers neither. A RETRACTION IS CARRIED IN THIS PAPER, NOT BURIED. An earlier draft presented the tier-two facets of the d=4 census on a (1/(2*sqrt2))Z grid with a single coefficient magnitude 3/(2*sqrt2), and read the resulting unit-coefficient bounds 4*sqrt2 and 14*sqrt2/3 as placing Tsirelson's sqrt2 inside the tier-two geometry. The whole presentation was a normalization artifact and is withdrawn (Retraction 1): the stored lift has coefficients in {0,+-1} with integer bounds 6 and 7, every facet functional in the census is rational, and Q(sqrt2) is not among the eighteen canonical arithmetic fields. Certificate: d4_arithmetic_profile.py. WHERE THE EVIDENCE IS THIN, STATED IN THE PAPER. Two sweeps quoted here drew no positive instances at all: the d=4 half of the 2,100-family equivalence test, and the 6,500-trial false-certificate sweep, which draws no contextual family at any of d=2,4,8 and therefore establishes the absence of false certificates only vacuously. Both are disclosed where they appear. Theorem 2 is labelled machine-validated rather than proved: the closed form is established end to end for the deletion families treated, while the general closed-triple d=2 statement in compressed coordinates remains open and is recorded as open in the verification ledger. HARDWARE CAVEAT, STATED UP FRONT. The quoted significances, about 56 sigma for the ray-level Wigner-friend holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented witness model and conditional on compatibility and nondisturbance assumptions. They are not systematics-aware and neither measurement is loophole-free; a roughly 3.5 sigma preparation offset on one strand is itself evidence of unmodelled systematics. We also record that the pinned value S = 4.6125 cannot be recomputed from the stored counts (a naive recomputation gives 4.5662, a gap of about 2.7 standard errors); the archived value 4.558 +- 0.018 is the one an independent reader can reproduce from the deposited data. Read these as consistency checks on real hardware, not as precision measurements or certifications. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. It was held back from the first publication batch and released only after three adversarial review passes; what each pass found, and how each item was closed, is recorded in verification/INDEX.md rather than quietly fixed. One defect found there is NOT closed and belongs to the companion Paper B, which cites a false-certificate artifact that is not in fact in the repository; it is recorded rather than left for a reader to discover. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Note: https://doi.org/10.5281/zenodo.21630739 . Code archive: https://doi.org/10.5281/zenodo.21498498 AI-ASSISTANCE DISCLOSURE. Large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance by the scripts named in the text, and the author takes full responsibility for the results, including any errors. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Volumes I and II defined a rug substrate and computed a holonomy residual on a six-site chain (Mamun, 2026a; 2026b). The present volume asks a different question: if that space is real, what shapes can an observer record? Shape is not a picture of a manifold. It is the image of a finite list of maps from laboratory paths into numbers, curves and flags. Five observable shapes are classified. (S1) The occupancy constellation: the image of Λ in a four-dimensional ledger. (S2) The wall complex: the arrangement of regime crossings in control space. (S3) The holonomy curve: ΔF as a function of knot angle and winding. (S4) The theme flag: an invariant that stays put under legal walls and drops under detectors. (S5) The measure and echo shapes: correlator profiles, revival envelopes, and a finite-N singularity spectrum. On the Volume II toy these shapes are already distinct. The constellation separates in the (gap, lT) plane and crowds in the (IPR, d2) plane. The holonomy curve is monotone in ε, with an identity-wall intercept of 0.020. The theme flag is binary at legal knots and leaks to we = 0.882 at a detector. Echo envelopes and connected ZZ profiles are sector-specific; d2 is not. The laboratory translation is a short menu: report the shape, not only the fidelity to a target computational state. A single exponential T2, a single Berry phase on one Bloch sphere, and a single Lindblad rate are the shapes this ontology predicts one will not see once walls are present. C S Also available on Academia.edu: https://www.academia.edu/attachments/134081814/download_file?s=portfolio
J. Harlow's essay The Coming Substrate Crisis in Quantum Computing (29 August 2026) argues that every major hardware family — superconducting circuits, trapped ions, neutral atoms, and photonics — is being scaled on a false geometric premise: a single-sector continuum manifold with one curvature, one connection, and one holonomy group. The present KSCCN note accepts that diagnosis as a research programme, not as a completed theorem. It performs an anatomical review of the claim, contrasts it with the leading substrate theories of contemporary quantum geometry (loop quantum gravity and spin networks; causal sets; noncommutative geometry; algebraic quantum field theory and Type III factors; information geometry; AdS/CFT and holographic codes; scale-relativity and fractal spacetime; and post-quantum cryptography's discrete algebraic substrates), and records both strengths and liabilities. The constructive contribution is a Persian-rug ontology of substrates. Quantum substrates are not a single rolled continuum, nor a discrete lattice of identical cells. They are unfurled, like a hand-knotted rug, as a family of sectors and themes, each occupying a characteristic length and breadth in space and in time, each carrying its own Hausdorff–Besicovitch spectrum, and none identically repeating another. The relevant geometry is therefore neither Abelian nor Euclidean in the first instance. It is a stratified, multifractal, sector-indexed holonomy with theme-dependent dimension. Also available on Academia.edu: https://www.academia.edu/attachments/134053150/download_file?s=portfolio
TL;DR — The Core Finding Contrary to the initial expectation that arithmetic structures (e.g., the coprime-to-6 mask or the Möbius mask) generate distinct number-theoretic spectral signatures, this work provides numerical and formally certified evidence that the Non-Ergodic Extended (NEE) phase is entirely governed by the preservation of first-neighbor connectivity (d=1). Suppressing the d=1 hopping channel induces a topological collapse into strong spatial localization (D2≈0.288→0.045, a factor of 6.4), representing a single-particle spatial analogue of Hilbert Space Fragmentation (HSF). Furthermore, the deep arithmetic logic of the Möbius function acts as topological white noise, in compliance with the Möbius Randomness Principle (Sarnak conjecture): since 86.05% of the interaction variance is concentrated in short-range bonds (d≤5), deeper arithmetic channels are swamped by the finite-size GUE noise floor (σGUE∼1/N). Theoretical and Physical Framework Deterministic-stochastic hybrid Hamiltonian. Combines an exact Riemann–von Mangoldt diagonal potential (inverted deterministically via the Lambert WW function) with Power-Law Random Banded Matrix (PRBM) off-diagonal disorder. Sparse hopping constraints. Hopping is filtered by deterministic arithmetic sieves (coprime-to-6 mask Ξ6, square-free Möbius mask Ξμ, Liouville mask Ξλ) and controlled random Bernoulli ensembles. Single-particle Hilbert space fragmentation. Without d=1, adjacent spatial transitions (i→i+1) require higher-order virtual paths (i→i+5→i+1) that suffer severe energy detuning from the fluctuating diagonal potential. The one-dimensional metric graph shatters into decoupled spatial clusters. Coupling variance dominance. The power-law decay (d−1.5 in variance) concentrates 86.05% of the interaction weight in d∈{1,2,3,5}. Deeper arithmetic channels (d≥6) remain buried beneath the finite-size GUE noise floor. Parameter-free operating point. ν=0.75 (exact center of the critical window 0.5<ν<1); local resonance coupling analytically derived as ϵc=π2; Maslov phase 7/8. Mechanized Formal Verification in Lean 4 The algebraic, topological, and connectivity constraints governing the model are certified 100% sorry-free by the Lean 4 kernel (v4.34.1 with Mathlib4), across 13 Level N1 modules (23 theorems, 11 definitions, 34 total verified statements): WheelChannelStructure.lean — Confinement to active chiral sublattices (d≡1,5(mod6)). UnitCellMeasure.lean — Unit cell totient measure (∑d=16Ξ(d)=2 ⟹ φ(6)/6=1/3). ChiralPhaseAbsorption.lean — Invertibility of coprime residues modulo 6. KatoRellichRegime.lean — Self-adjointness domain window (1/2<ν<1). LocalResonanceCoupling.lean — Hybridization resonance condition (ϵc=π2). ChannelDensityContraction.lean — Strict density hierarchy (4/15<1/3<1/2<1). NonErgodicPhaseBound.lean — Non-triviality of the bipartite ceiling (1/4<1). PolyphaseFilterZ6.lean — Complete stopband annihilation for non-coprime channels. FirstNeighborPreservation.lean — Guaranteed base coordination degree (w1=2). DihedralQuantization.lean — Reflection involution and percolation threshold bounds. BipartiteInterferenceBound.lean — Certified upper bound D2≤1/4 under chiral equiprobability. ExpectedDegreeBound.lean — Mean degree lower bound (⟨deg⟩local≥2). FractalDimensionUpperBound.lean — Generalized Jensen inequality for the IPR and universal bound D2≤φ(m)/m. Empirical validation of Lean 4 axioms. Extensive GPU audits up to N=12 000 confirm that the chiral equiprobability axiom (c1=c5=1/2) holds strictly at the ensemble level (⟨W1−W5⟩=−0.000048≈0), with local mass fluctuations decaying monotonically as N→∞. Comprehensive Experimental Validation Matrix Observable Empirical Result Benchmark Physical significance Macroscopic alignment (R2) >0.999999 R2→1 Autonomous Weyl law tracking; zero empirical rescaling Bulk spacing ratio ⟨r⟩ 0.5891±0.0024 GUE 0.5996 / Poisson 0.3863 Intermediate non-ergodic extended statistics SFF ramp exponent γ 0.6086±0.0103 GUE γ=1 Sub-diffusive spectral ramp characteristic of NEE FSS thermodynamic limit D2∞ 0.113±0.003 Anderson D2=0 39σ rejection of Anderson localization Chalker scaling β=2D2 β=0.5549±0.0020 2D2=0.5562±0.0054 0.2σ agreement between transport and fractal support Multifractal curvature ΔD D0−D2=0.730 Ergodic ΔD=0 Strict convexity of Dq for q∈[−3,4]; width Δα=3.726 First-neighbor removal D2=0.0448±0.0009 Preserved: 0.2882 Sixfold collapse into strong localization (HSF) Short-range rescue (d=1) D2=0.2849±0.0007 Unforced: 0.0556 Forcing d=1 restores the NEE phase in random masks Topological pinning (ε=εc) D2no/D2with≈0.17 Healed ratio →1→1 Collapse cannot be overcome at the critical resonance Riemann zeros control (SFF) γzeros=0.94±0.12 GUE γ=1γ=1 Real zeros are ergodic; the NEE ramp is intrinsic to the mask Repository Structure and Reproducibility The codebase is fully vectorized, GPU-accelerated (PyTorch cuSOLVER in complex128), and audited via an automated SHA-256 manifest. The notebook regenerates every figure and table of the manuscript from a fixed random seed (SEED = 42). PRBM_Modular_Lean/ — Standalone Lean 4 (v4.34.1) project containing all 13 verified .lean files and lakefile.lean. Modules 00a–00c — System setup, forensic audit of the 100,000-zero Riemann dataset, and vectorized Hamiltonian engine (complex128). Modules 01–03 — Exact diagonalization at N=10 000; local level statistics P(s) and ⟨r⟩; spectral form factor ensemble averaging (M=100, N=15 000). Modules 04–06 — Channel-density scaling and finite-size scaling; FSS collapse of P(d2,N) across N∈{2000,4000,8000,12000}; AIC/BIC model selection. Modules 07–09 — Anderson transition sweep (ν∈[0.5,3.0]); systematic mask comparison suite; Möbius equivalence and density sweep. Modules 10–12 — Full multifractal spectrum Dq,f(α); microscopic GPU diagonalization at N=12 000; quantum dynamics (MSD, survival probability, Shannon and entanglement entropies). Modules 13–15 — Riemann zeros control SFF; coupling sweep over ε/εc∈[0.2,4.0]; chiral equiprobability audit ⟨Wr⟩ across residue classes. Module 16 — Automated high-resolution vector export (PDF/PNG 300 DPI) and cryptographic manifest generation (MANIFEST_SHA256.txt). Open-Access Dual Licensing Source code (Lean 4, Python, PyTorch): released under the MIT License. Free for academic, educational, and commercial replication. Manuscript, figures, manifests, and data: released under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Declarations Author contributions. Conceptualization, methodology, software, formal analysis, formal verification, investigation, data curation, validation, writing (original draft and revision), visualization, project administration. Competing interests. The author declares no competing interests, financial or otherwise. Funding. No external funding was received for this research. Ethics. Ethical approval was not required, as this research is theoretical and numerical and involves no human participants or animal subjects. Generative AI disclosure. Google Gemini and DeepSeek were used exclusively as auxiliary tools for language editing, adversarial logical review ("red-teaming"), and code debugging. They were not used to produce any original scientific content: all model design, numerical experiments, theoretical derivations, and interpretations are the author's own work. Every AI-assisted output has been reviewed and validated by the author. Scientific Epilogue "In the beginner's mind there are many possibilities, but in the expert's there are few." — Shunyu Suzuki What began as an inquiry into whether number-theoretic structures dictate quantum chaos revealed a simpler, deeper, and more universal physical law: geometry and local connectivity precede arithmetic complexity. The code, formal proofs, and data deposited here are provided to ensure complete auditability, verifiable down to the individual logical inferences of the Lean 4 kernel. Archival snapshot: October 2026 · Certified Level N1 · Built with Lean 4, PyTorch, NumPy, SciPy, and Google Colab. Version: v4 ( submission EPJB-D-26-00793)
Süß (2026) presents a concise yet technically ambitious five-page preprint that claims to close a longstanding axiomatic gap in the information-theoretic reconstruction of bipartite quantum correlations. By introducing an explicitly constructed N-bit Random Access Code (RAC) protocol—the "Eigenvector Protocol"—the author proves analytically and verifies numerically that the principle of Information Causality (IC) alone enforces the positive semi-definiteness (PSD) of all Fourier-harmonic components of the bipartite correlation kernel for every integer k ≥ 1. This result completes the derivation chain No-Signaling + Rotational Covariance + IC ⇒ E(Δθ) = −cos(Δθ) that had previously relied on partial or numerical evidence. The work is self-contained, device-independent in spirit, and directly addresses the open Mamun, SM (2026)……………………….1 Reviewing Süß (2026), "Information Causality Implies Kernel Positivity: A Complete Proof via the Eigenvector Protocol" question left by Pawłowski et al. (2009) and subsequent numerical scans (Süß, 2025, 2026a, 2026b). This paper outlines a detailed summary, methodological critique, comparison with prior literature, assessment of strengths and limitations, and discussion of broader implications. Mamun, SM (2026)……………………….2 Reviewing Süß (2026), "Information Causality Implies Kernel Positivity: A Complete Proof via the Eigenvector Protocol" Also available on Academia.edu: https://www.academia.edu/attachments/132327893/download_file?s=portfolio
Two walls stand at the edge of the two-ring instrument, and this paper is the map of both. Many qubits first. One setting reads two amplitudes and N qubits carry 2^N, and that wall is the only thing between the instrument and many-body quantum mechanics. Part I is the map of the attempt to scale through it, chains of certified rings in lockstep, each qubit a setting with its digit certificate, entanglement between neighbours as the shared-radius coupling of the companion paper, local turn laws carrying the gates, a geometric form of the tensor networks that are classical physics' own escape from the same wall. What carries over is stated exactly and verified. Across every cut of a chain the two sides share one spectrum, so the cut's radius is its certificate. An eight-link chain built by one layer of levers holds at most one bit per cut and is held by a chain of bond two to a fidelity of 1.000, and a random eight-qubit state has rank 16 at the middle cut and is held to 0.162, the wall exhibited. The move past it is the nesting of the companion repair paper, a ring inside every cell, done here as a setting inside every share. A setting whose two shares are themselves settings, applied down to single qubits, holds any pure state of N qubits exactly, 2^N − 1 settings for 2^N amplitudes, the standard's own count, verified on random states at every size to eight qubits and on the same random state the chain held to 0.162, with every setting on the ladder. At a node the entanglement of that qubit with the rest is sin θ √(1 − |⟨ψ₀|ψ₁⟩|²), the Wootters concurrence for a pair, largest at equal shares over mirror branches. A Z turn on qubit k turns the azimuths at depth k and nothing else, and a measurement reads the root's shares and leaves its branch. On ibm_kingston a random three-qubit state was prepared from its seven settings as the machine's own gate angles and its populations read within 0.048 of the tower's shares. The contraction that turns the chain into a reading is classical arithmetic, and the ladder guard of the companion paper stands on it, 112 fixed-point words carrying their base-60 digit pairs, 200 random corruptions of one to six bits, every one caught from the shadow alone and every word recomputed, while the unguarded contraction returned a wrong energy in 155 of the 200. The five problems of the chain program are each settled where they stand: the tower is the canonical form, a gate on any qubit needs no route through its neighbours, the contraction's cost is measured, the classes the tower compresses are counted, and the width a chain needs to follow the dynamics is the entanglement across its cuts, measured. Three levels second. A pure qutrit carries four real parameters and one setting reads two lengths. The populations are read by two levers, one inside the other, the tower of Part I at its smallest: the first splits level 0 from the rest and the second splits the rest between levels 1 and 2, so every split is reachable, a state fully in one level included. Two certificates ride them, the sum to one and the product law, the coherence between two levels at most the product of their two amplitudes and equal to it for a pure state, and on a thousand random qutrits every one-percent misread of the populations is caught by one of the two, 1,000 of 1,000, 992 of them by the sum alone. The threefold turn a qutrit wants is not a single-digit multiplier, and that is a theorem, not bad luck. The Pisano period over the rank of apparition is 1, 2 or 4 (Vinson 1963), confirmed by enumeration to modulus 250. But the sixty-ring carries the threefold turn as a two-term law, F(n + 20) ≡ 5F(n + 1) + F(n) (mod 10) at all sixty positions, and 178 moduli below 250 carry the same law at a third of their period. The two relative phases are read as well, by three two-level settings, one for each pair of levels, whose azimuths obey the addition law exactly, the third certified by the other two, so that three populations and two certified phases rebuild every pure qutrit, verified on a thousand. Every number is verified by the script in the appendix, and the document is not built unless every check prints True.