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)
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)
🌀 Phase-Pi-Quantum-Prior Topological Superselection Z/6Z: Analytical Phase Derivation and Dissipative Stabilization for FTQC Cryptanalysis The analytic core of the Z/6Z topological superselection. The perfect geometric duality between the resonant channels (blue and red) demonstrates the Zero-Leakage adaptive strategy driven by the exact holonomic phase shift Δϕ=π.* 🎯 TL;DR – The Essentials 🔬 Theoretical Breakthroughs 📐 Analytical Phase Discovery: Proof that the optimal initialization phases (ϕ1,ϕ2) are not heuristic. ϕ2=π emerges from the unit group isomorphism (Z/6Z)×≅Z/2Z acting as a non-commutative Berry Holonomy. 🔄 Renormalization Group (RG) Origin: The thermodynamic phase ϕ1≈Rfund/10 is strictly derived as an infrared fixed point of the Callan-Symanzik beta function, coupled to the Euler-Kronecker invariant. ⚡ Polynomial Complexity: Exact state preparation via Matrix Product States (MPS) with constant topological bond dimension χ≤6, avoiding the exponential O(2n) overhead of arbitrary distributions. 🛡️ Parent Lindbladian & NEE Phase: Construction of a frustration-free dissipative superoperator with a non-collapsing Liouvillian gap Δ=Ω(1). Bounded by the Modified Logarithmic Sobolev Inequality (MLSI), this guarantees strict rapid mixing, proving the system enters a Non-Ergodic Extended (NEE) phase that defeats Eigenstate Thermalization (ETH). 📊 Computational Validation (N=60 Qubits) 📉 Dissipative Area Law: Under a depolarizing noise rate of p=0.015, the MPDO bipartite entropy S2 strictly saturates at ≈1.65 bits, defying the Ergodic Volume Law (30 bits). 🧪 Robustness Plateau: Sensitivity analysis confirms the NEE phase is structurally stable against gauge fluctuations in the range ϕ1∈[0,0.1] rad. 🚀 FTQC Resource Gain: Structural search space reduction (66.6% purge) enabling a massive reduction in the Toffoli gate pre-factor (~189 Billion T-gates saved for RSA-2048). This drastically compresses the space-time volume of Surface Code syndrome cycles (τ), relaxing magic state distillation thresholds. 🔍 Research Overview: Beyond Uniform Superposition Standard quantum algorithms (e.g., Shor, Grover) initialize registers in a state of maximum ignorance: the uniform superposition H⊗n|0⟩. While easy to prepare, this forces the oracle to process an exponential volume of arithmetically impossible trajectories, demanding impossible coherence times. This research introduces the Z/6Z Topological Prior, a structured Quantum State Preparation (QSP) protocol that injects arithmetic intelligence into the vacuum state. By aligning the quantum amplitude with the modular density of prime numbers, we transform integer factorization from a blind search into a topologically tuned resonance, optimized for Noisy Intermediate-Scale Quantum (NISQ) and early Fault-Tolerant (FTQC) hardware. 🧭 Conceptual Framework 📊 Experimental Results & Macroscopic Scaling The following tables summarize the thermodynamic and computational performance of the Z/6Z state under open-system dynamics (Lindblad Master Equation) using Matrix Product Density Operators (MPDO) up to N=60 qubits: 1. Thermodynamic Resilience (N=60, p=0.015) Metric Modular Prior (NEE Phase) Uniform Baseline (Ergodic ETH) Advantage Bipartite Entanglement (S2) 1.6495 bits 30.0 bits 94.5% Reduction Scaling Law Area Law (S≤log2χ) Volume Law (S∼N/2) Thermal Immunity Liouvillian Gap (Δ) Ω(1) (Rapid Mixing) →0 (Slow Mixing) Frustration-Free 2. FTQC Resource Projection (RSA-2048) Metric Classic Shor (H⊗n) MST Adaptive Strategy Impact Search Space Entropy 100% (All channels) 33.3% (Resonant only) Zero-Leakage Purge Estimated T-Count ∼2.83×1011 gates ∼9.45×1010 gates ~1.89e11 Gates Saved Hardware Implication Deep FTQC Required Shallow FTQC / Late NISQ Threshold Relaxation 🚀 Reproducibility and Computational Lab The empirical validation suite is divided into three comprehensive Jupyter Notebooks. Each notebook is meticulously designed to prove the theoretical claims of the manuscript through functional code, exact tensor contractions, and thermodynamic simulations. 📓 Notebook I: Arithmetic and Thermodynamic Foundations This notebook provides the experimental framework and empirical validation for the Z/6Z Topological Prior. It demonstrates how to break away from the traditional, maximum-entropy uniform superposition (used in algorithms like Shor's) by confining the quantum amplitude strictly to resonant arithmetic channels. Key Validations & Discoveries: Arithmetic Confinement (Zero-Leakage): Proves that 100% of prime numbers p>3 reside exclusively in the 1(mod6) and 5(mod6) congruence classes, completely nullifying amplitudes in sterile channels (0,2,3,4). Informational Impedance (Rfund): Derives and validates the thermodynamic constants governing the mapping of ternary arithmetic topology onto binary qubits. To isolate channel 1 with 99.98% fidelity, the system strictly requires the phase ϕ1≈Rfund/10. Dual Phase Optimum & Isomorphic Transfer: Demonstrates that absolute isolation for class 5 is achieved at ϕ2=π. Transitioning between channels requires no system recalibration, only the injection of this pure geometric phase, confirming the multiplicative inverse duality (5≡−1(mod6)). Discrete Symmetry and Modular Parity: Numerically validates the π-periodicity of the partition function Z(ϕ) and the exact duality P1(ϕ)=P5(ϕ+π). The preservation of the modular parity invariant ⟨P^⟩ ensures holonomic rotations are fundamentally stable and entropy-free. Conclusion: This framework enables quantum state preparation using Matrix Product States (MPS) with a bounded bond dimension (χ≤6). This strictly evades the O(2n) exponential depth limit and protects the system from decoherence via the Non-Ergodic Extended (NEE) Phase. 📓 Notebook II: The NISQ Hardware Challenge and MPS Compression In this notebook, we transition from abstract mathematics to Quantum Hardware Engineering. We demonstrate that the Z/6Z Topological Prior is not merely a theoretical construct, but a valid, unitary quantum state vector that can be encoded into the density matrix of a real-world quantum computer, overcoming the historical exponential circuit depth O(2n) typically required for sparse state preparation. Key Validations & Discoveries: Perfect Unitarity & Zero-Leakage (via Qiskit): Proves that the exact phases ϕ1=Rfund/10 and ϕ2=π generate a physically valid quantum state that perfectly nullifies amplitudes in sterile channels, strictly bounding entanglement complexity. The Path to MPS Compression: Demonstrates that the superselection rules act as a 6-state finite automaton. This allows the state to be compiled using Matrix Product States (MPS) with a maximum bond dimension χ≤6, guaranteeing an efficient polynomial circuit depth O(poly(n)). Steady-State Uniqueness (λ0=0): Through exact spectral analysis, we confirm that the dominant eigenvalue is exactly zero, proving the MPS state is the unique "dark state" of the Parent Lindbladian. Macroscopic Protection & Rapid Mixing (Δ=Ω(1)): Shows that the Liouvillian spectral gap does not collapse in the thermodynamic limit (N→∞). Instead, the topological penalty scales additively (Δ∝N), guaranteeing ultra-fast relaxation into the protected subspace and evading the Eigenstate Thermalization Hypothesis (ETH). Conclusion: The Z/6Z topological subspace acts as a strictly stable, frustration-free dissipative attractor. The Non-Ergodic Extended (NEE) phase is not a pre-thermal transient or a finite-size artifact, but an asymptotic macroscopic reality that mathematically protects the hardware from decoherence. 📓 Notebook III: Hybrid Cryptanalysis and Optimal Adaptive Strategy In this final notebook, we translate our thermodynamic and hardware-level discoveries into the domain of Applied Cryptanalysis. We simulate the orthogonal partitioning of the Hilbert space against target cryptographic moduli (e.g., RSA), rigorously quantifying the net reduction in oracle evaluations compared to the maximum-entropy initialization of standard Shor's algorithm. Key Validations & Discoveries: The Optimal Adaptive Strategy: Validates a fault-tolerant, two-round search protocol exploiting chiral inversion symmetry and the modular parity invariant (⟨P^⟩). Round 1 injects ϕ1≈Rfund/10 to collapse amplitude onto channel 1(mod6). If unsuccessful, Round 2 applies a holonomic rotation Δϕ=π to integrally transfer the probability mass to channel 5(mod6) with zero algorithmic leakage. FTQC Resource Compression: Proves that purging 66.66% of sterile search trajectories before the first oracle call directly reduces the polynomial pre-factor of the search algorithm by 2/3. Massive T-Count Reduction (RSA-2048): Projects the strategy onto a macroscopic RSA-2048 modulus. While standard Shor initialization demands ∼2.83×1011 T-gates, the Modular Search Strategy (MST) circumvents roughly 189 billion of these highly expensive operations. Coherence Threshold Relaxation: Demonstrates that the drastic reduction in logical depth significantly lowers the required gate fidelity and extends the effective coherence time needed for a successful factorization. Conclusion: The MST strategy transcends a mere mathematical speedup. It provides a thermodynamic and structural "shortcut" that depressurizes error correction thresholds, making Shor's algorithm feasible on Fault-Tolerant Quantum Computing (FTQC) hardware with significantly higher noise floors than previously estimated. Last Update: May, 2026 | Status: Under Review at Quantum Science and Technology ID: QST-105397 | Built with ⚛️ & 🐍
Two established results have never been composed. The fidelity susceptibility of a quantum state to a Hamiltonian parameter equals one quarter of its quantum Fisher information, F_Q = 4·chi_f; and an exact calculation in the Rosenzweig-Porter (RP) ensemble shows that eigenfunction fidelity susceptibility is enhanced in the fractal non-ergodic-extended phase (Skvortsov, Amini and Kravtsov, PRB 106, 054208, 2022). Composed, they make the eigenvector fractal dimension D2 — until now a diagnostic of localization physics — an operational metrological figure of merit: along the RP fractal phase, at fixed distance from criticality and for field-like (local) couplings to the sensed parameter, the finite-size enhancement exponent of the quantum Fisher information over the ergodic reference is 1 − D2. We confirm this in-house under preregistered criteria on the generalized-RP ensemble with D2 measured from inverse-participation-ratio scaling (not assumed): enhancement = 1 − D2 within 0.15 at 4/4 interior points of the phase (deviations 0.009–0.078; correlation 0.996). Two further preregistered experiments fix the criterion's boundaries: geometric fractality alone — a disorder-free Sierpinski-gasket lattice with certified eigenpairs and degenerate-multiplet treatment — shows no effect (enhancement −0.12; the RP formula does not extend), while critical power-law banded random matrices extend the law at all three tested bandwidths (deviations 0.016–0.064). The criterion is therefore: multifractal eigenstates of random ensembles, sensed through local couplings — randomness, not fractal geometry, is the missing ingredient, and an operator-structure scope law (diagonal inherits; short-range inherits once its range fits the fractal support; collective and dense couplings are D2-blind) gives the design rule. On the platform side we construct — because it cannot be cited — the minimal model that carries the physics onto cold atoms: an interacting tilted+disordered lattice in which the tilt is only the sensed parameter (the bare Wannier-Stark ladder is integrable, so sourcing RP statistics from gravity would be immediately refutable) and the dense RP-like spectrum comes from interactions and disorder. A preregistered coexistence test on the interacting t–V chain finds a textbook ergodic-to-localized crossover whose typical susceptibility peaks at the crossover's ergodic edge — not mid-crossover — with the predicted heavy x^−2 tail (Hill index 1.96–2.04) in the chaotic regime and estimator-dependent rare-event gains inside the crossover. The platform statement is scoped accordingly, and one instrument caveat is stated up front: this attaches to lattice/Bloch gravimetry, not to free-fall light-pulse interferometry. Tier-2 throughout: a shared random-matrix description with preregistered numerical evidence, not a hardware claim.
Abstract We identify the constant π as the optimal relative phase for stabilizing chiral quantum channels under Z/6Z superselection. Within a modular substrate governed by the quotient ring Z/6Z, we prove that the phase shift ϕ2=π maximizes the fidelity of the C5 chiral channel (residue class 5(mod6)) relative to the C1 channel. This result follows from the exact Z2 symmetry of the unit group (Z/6Z)×≅Z2 and the elementary trigonometric identity sin(θ+π)=−sin(θ). We further demonstrate, via high-precision numerical simulations, that chiral interference under this phase suppresses parity-symmetric Gaussian Unitary Ensemble (GUE) noise by 45.2%, yielding a net signal-to-noise ratio (SNR) gain of +6.07dB. Continuous open-system master equation dynamics (Lindblad bath with 85% collective amplitude damping and 15% local dephasing) confirm that the topologically shielded chiral singlet ∣S⟩ preserves long-term state fidelity (F=0.8113), cutting open-system decoherence by 63.4% (+4.48dB net gain). The algebraic core---including the optimal phase theorem and the parity transformation of symmetric noise operators---is formally certified in the Lean 4 proof assistant with zero omitted axioms (sorry-free). 📂 Repository Contents & File Guide 1. Manuscript Files π_as_the_Optimal_Phase_in_MQS.pdf: Final compiled PDF preprint. π_as_the_Optimal_Phase_in_MQS.tex: Complete LaTeX source code. 2. Interactive Notebooks & Executables LEAN_π_as_the_Optimal_Phase_in_MQS.ipynb: Interactive Google Colab notebook for the Lean 4 formal verification suite. Installs elan, pins toolchain v4.11.0, fetches pre-compiled Mathlib4 caches, and compiles all proofs. LEAN_π_as_the_Optimal_Phase_in_MQS.pdf: Static PDF printout of the Lean 4 notebook showing full build execution logs (0 sorrys). PYTHON_π_as_the_Optimal_Phase_in_MQS.ipynb: Interactive Google Colab notebook for numerical simulations (DSP M=6 polyphase filters, Qiskit gate-level DFS, Monte Carlo trajectories, and QuTiP open-system master equation). PYTHON_π_as_the_Optimal_Phase_in_MQS.pdf: Static PDF printout of the Python simulation notebook with embedded benchmark reports and plots. 3. Verification Source Code & Publication Figures mst_f1_verification_clean.zip: Clean standalone Lean 4 Lake project source code for local compilation and VS Code integration (lake build). DSP_Polyphase_Filter.pdf: Publication-ready vector graphic (Figure 1 in manuscript) demonstrating GUE noise collapse. lindblad_dynamics_paper_f1.pdf: Publication-ready vector graphic (Figure 2 in manuscript) showing continuous Lindblad fidelity and coherence protection. README.md: Complete instruction manual and build guide. 🛠️ Replication Instructions Cloud Execution (Google Colab - Recommended) Open Google Colab. Upload either LEAN_π_as_the_Optimal_Phase_in_MQS.ipynb or PYTHON_π_as_the_Optimal_Phase_in_MQS.ipynb. Select Runtime > Run all (Ctrl + F9). All dependencies install automatically and results reproduce in under 3 minutes. Local Execution (Lean 4 Command Line) Ensure elan is installed on your local machine. Unzip mst_f1_verification_clean.zip. Run the build commands in terminal: Bash cd mst_f1_verification lake exe cache get lake build 📜 License Code & Proofs: MIT License. Manuscript & Data: Creative Commons Attribution 4.0 International (CC-BY 4.0). 📌 Publication Status: Submitted to *Journal of Physics A: Mathematical and Theoretical* in July 2026 - Manuscript ID: JPhysA-125419.
This repository contains the full computational framework accompanying the article "Dissipative Protection of Modular Quantum States: A Parent Lindbladian for Z/6Z Superselection". The modular ring Z/6Z partitions a quantum register into two resonant channels (C₁, C₅) and four sterile ones (C₀, C₂, C₃, C₄). While the companion work establishes how to prepare a topological state confined to the resonant subspace, this article addresses the subsequent challenge: protecting that state against thermal decoherence on NISQ hardware. We construct a Parent Lindbladian—a frustration-free dissipative generator whose unique steady state is exactly the topologically superselected density matrix. The jump operators, derived from the modular deterministic finite automaton, continuously pump amplitude from the sterile channels back to the resonant ones, acting as a passive error-correction mechanism. Repository contents: Dissipative Protection of Modular Quantum States.pdf — The complete article (LaTeX source available in the accompanying GitHub repository). Dissipative_Protection_of_Modular_Quantum_States.ipynb — Reproducible Colab notebook implementing all numerical experiments: Exact diagonalisation of the Lindblad superoperator for spin chains up to N=6 qubits. Dissipative phase transition: strict mask (δ=0) enforces an Area Law; breaking the mask (δ>0) restores the Volume Law (ETH). Leakage parameter sweep (δ=0 to 0.5) showing a sharp crossover at δ_c ≈ 0.05. Liouvillian spectral gap computation confirming the topological state as a stable attractor. Macroscopic MPDO tensor network contraction (N=10 to 60 qubits) under depolarising noise, demonstrating S₂ entropy saturation at ~1.65 bits—far below the ergodic limit. Area_Law_Proof.png — Thermodynamic extrapolation of the bipartite Rényi entropy, proving the Area Law in the macroscopic limit. Transition_ETH_AreaLaw.png — Dissipative phase transition: von Neumann entropy vs. system size for strict and broken topological masks. Liouvillian_Gap.png — Evolution of the spectral gap as a function of the leakage parameter. Delta_Sweep.png — Entanglement entropy crossover at fixed N=4, identifying the critical noise threshold. All experiments are fully deterministic and reproducible on any standard Google Colab instance using QuTiP, NumPy, SciPy, and Matplotlib.
Five-Part Validation-Grade Suite on Universal Spectral Determinism and Non-Commutativity Residues: An Unconditional Resolution and Precision-Certified Certification of the \text{BQP} \not\subseteq \text{PH} Separation via Topological, Variational, and Information-Theoretic Invariants --- This 5-package resolution suite delivers an unconditional, non-relativizing, and non-algebrizing proof establishing the strict mathematical separation of Bounded-Error Quantum Polynomial-Time (\bm{\text{BQP}}) from the classical Polynomial Hierarchy (\bm{\text{PH}}) and Polynomial Time (\bm{\text{P}}). Rather than approaching the problem through symbolic logic or localized circuit lower bounds—which are structurally constrained by historic complexity barriers—this framework shifts the question into the continuous, precision-certified domains of Differential Geometry, Algebraic Topology, Category Theory, Information Thermodynamics, and Matrix Perturbation Theory. By analyzing the intrinsic geometric and numerical properties of complexity reductions under finite-precision constraints, the suite isolates a hard computational singularity. This singularity serves as an absolute structural boundary, proving that classical alternating configurations lack the algebraic and dimensional capacity to track non-local multi-particle quantum updates. Individual and Interlinking Functional Dynamics The 5-package suite operates as a mathematically closed, self-contained replication pipeline, where each package executes a specific analytical or numerical enforcement that feeds directly into the next. Package A: Analytic Foundations & Definitions • Individual Mechanics: This package translates the discrete states of algorithmic complexity into the continuous geometry of Riemannian manifolds. It formalizes the structural constraints of the Unitary Manifold (\bm{\mathcal{M}_{\text{BQP}}}) against the Classical Hypercube Manifold (\bm{\mathcal{M}_{\text{PH}}}). By utilizing Ricci flow to evaluate the state space reduction, it identifies a non-invertible Phase-Collapse Singularity (a "Neck-Pinch" obstruction), mathematically proving that the projection fails to preserve the underlying manifold structure. • Interlinking Pipeline: It establishes the primary mathematical universe, smooth structures, and geometric boundaries used by all subsequent packages to measure computational state deformations. Package B: Geometric Mapping & The Final Proof • Individual Mechanics: Building on the continuous manifold foundations, this package details the exact differential-geometric embedding of the computational manifolds utilizing the Fubini-Study metric (\bm{g_{\text{FS}}}). It formally characterizes the absolute dimensional mismatch between exponential quantum phase spaces and polynomial classical coordinate systems. By calculating the metric pullback field via the mapping Jacobian matrix (\bm{J_{\Psi}}), it demonstrates that the scalar curvature (\bm{R}) blows up and diverges as \bm{\mathcal{O}(2^n)}, establishing a geometric impossibility for polynomial reduction. • Interlinking Pipeline: It bridges the smooth geometric structures of Package A with the global topological invariants needed to escape localized oracle dependencies. Package C: Categorical Asymmetry • Individual Mechanics: This package elevates the separation proof into the abstract realm of Category Theory, defining the Quantum Monoidal Category (\bm{\mathbf{Quant}_n}) and the Classical Alternating Category (\bm{\mathbf{Alt}_n}). It frames the candidate complexity reduction as a natural transformation between computational functors. By applying the Yoneda Lemma, the text proves that such a natural transformation cannot exist because the quantum and classical domains possess fundamentally incompatible limit and colimit preservation profiles, ensuring they cannot be mapped without violating the Mac Lane Pentagon Coherence Invariant. • Interlinking Pipeline: By mapping the exact functorial boundaries, Package C transforms geometric and topological asymmetry into strict algebraic constraints, setting up the exact limits required to analyze information-theoretic capacities. Package D: The Entropic Seal • Individual Mechanics: This package models computation as a continuous thermodynamic channel, mapping quantum density operators within the Von Neumann entropy space against classical joint probability distributions within the Shannon alternating space. By invoking the Data Processing Inequality alongside Landauer's Principle, it argues that compressing exponential quantum entanglement capacity into polynomial classical bounds triggers an Entropic Seal Singularity. This singularity is characterized by an unphysical, exponential thermodynamic dissipation requirement (\bm{\Delta Q \sim \Omega(2^n)}), which physically precludes the containment of \bm{\text{BQP}} within \bm{\text{PH}}. • Interlinking Pipeline: This package creates the thermodynamic boundary conditions—demonstrating that a polynomial classical reduction violates physical laws. It defines the continuous informational flow equations that serve as the direct mathematical input for the final numerical validation engine. Package E: Error Analysis & Replication Framework • Individual Mechanics: Package E serves as the definitive numerical and empirical verification asset. It evaluates the complete reduction within a finite-precision digital environment using Moore-Rump Interval Enclosures to rigorously track non-linear error accumulation. Through Singular Value Decomposition (SVD) of the Morphic Coherence Matrix, it documents that the minimum singular value collapses exponentially (\bm{\sigma_{\min} \sim \mathcal{O}(2^{-n})}), driving an explosion of the condition number (\bm{\kappa \sim \Omega(2^n)}). When propagated, the target classical enclosures expand past unity (\bm{\omega \ge 1}), causing a total loss of tracking capability. • Interlinking Pipeline: Package E closes the entire suite. By tracking the relaxation kinetics of the non-commutativity residue, it isolates the power-law halting of the stable execution time-step (\bm{\Delta t \to 0}) under a critical break exponent \bm{\alpha = 1.0000 \pm 0.0001}. This bridges the theoretical abstractions of Packages A–D with an exact, error-bounded, replicable digital certificate (The Numerical Interlock Break). How the Suite Resolves, Validates, Seals, and Enables Replication 1. Resolves: The suite completely bypasses the Relativization, Algebrization, and Natural Proofs barriers. It does not construct localized classifiers over boolean functions; instead, it establishes a global functional space exclusion inside a Bounded-Variation Numerical Space \bm{\mathcal{V}_{\text{mach}}}. It proves that a polynomial classical reduction requires an impossible infinite-energy or zero-precision mapping. 2. Validates: Validation is handled dynamically by monitoring invariant spectral properties, metric pullbacks, and the convergence kinetics of the naturality residue. The system does not rely on heuristic approximations; it enforces absolute mathematical containment via strict interval-enclosure walls. 3. Seals: The resolution is sealed by the simultaneous convergence of independent physical and mathematical invariants: the phase-collapse neck-pinch, the scalar curvature divergence, the Mac Lane coherence failure, the Landauer dissipation wall, and the finite-time numerical halting. The complexity separation is locked because breaking the proof would require violating the fundamental laws of matrix calculus, differential geometry, and thermodynamics. 4. Enables Replication: Package E contains fully pre-wired, compilable LaTeX source code environments, exact citation keys anchored directly to foundational peer-reviewed literature (e.g., Golub & Van Loan, Page, Rump, Aaronson), and explicit numerical stability criteria. Independent replication teams can immediately deploy automated verification scripts to track the SVD spectrum, the scalar curvature metrics, and the \bm{\alpha = 1} critical break exponent, ensuring a completely transparent peer-to-peer review process. --- Note: The accompanying Agnostic Replication Kit (ARK) and Standard Academic Core (SAC) 17-package operational suite will be uploaded in the forthcoming version release to enable down-stream cross-institutional simulation and formal peer review.
ABSTRACT & MONOGRAPH OVERVIEW (PART IV) This preprint constitutes Part IV of the Arithmetic Modular Quantization (AMQ) framework, presenting a profound resolution to a century-old mathematical pathology in quantum mechanics: the inherent un-normalizability of canonical position and momentum eigenstates under the Stone-von Neumann (SvN) theorem. By enforcing a strict physical axiom—that physical irreducible operators must possess normalizable eigenstates—we demonstrate that the standard SvN theorem collapses, reducing to a mere linear degenerate limit of a broader non-linear W-algebra. We prove that the mixed non-commutative shear translation parameter (\gamma) acts as an intrinsic source of phase space Berry curvature, locking the Siegel cell boundaries into discrete topological invariants defined by integer Chern numbers \nu(\gamma). Consequently, the cell boundaries operate as a robust topological insulator. Furthermore, by mapping Max Born's Reciprocity Principle (X -> P, P -> -X) onto these non-commutative boundaries, we establish a strict chiral, uni-directional phase lock that algebraically forbids backscattering. Crucially, we demonstrate that the non-linear \gamma-extension commutes perfectly with the multidimensional Hecke discrete shifters, ensuring that the spectral rigidity and "multiplicity one" property of the fundamental benchmark systems (harmonic oscillator, particle in a box, hydrogen atom) remain entirely uncompromised. This shielding mechanism provides structural and topological protection to the emerging fractional anyonic statistics and their associated braid groups, proving that the quantum phase space-time naturally behaves as a self-regulating, error-correcting topological crystal. AUTHOR'S NOTE This paper marks the fourth installment of my independent theoretical physics monograph. Published as a recently retired Professor from the University of Zagreb, Faculty of Chemical Engineering and Technology, this work is placed in the open public domain under a Creative Commons Attribution 4.0 International License (CC BY 4.0) to preserve a permanent, open-access record of these non-linear, topological foundations of mathematical physics.