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🔬 AB-Cloud — a phase resonator for the zeros of the Riemann zeta function

Topological vortices on a Hofstadter lattice · Aharonov–Bohm phases from ζ(s) zeros · GUE random-matrix universality · 39-test two-pass Julia verification suite · 10-language independent verification · formal layer in Lean 4 / Coq / Agda / Isabelle · 3D lattice laboratory

One-line summary. The AB-cloud — a Hofstadter Hamiltonian decorated with topological vortices whose phases are derived from the non-trivial zeros of ζ(s) — reproduces GUE random-matrix statistics (⟨r⟩ = 0.5848 ± 0.0260 vs GUE 0.5992), is statistically indistinguishable from the zeta zeros themselves (Montgomery test KS = 0.047, p = 0.27), achieves machine-precision topology (Byers–Yang defect 3.5·10⁻¹⁵, Connes self-duality with 4 zero modes), and exhibits Dirac dynamics (E_min ∝ 1/L, R² = 0.9997) — a complete numerical laboratory for the Hilbert–Pólya programme.

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📑 Table of Contents

• Highlights
• Motion gallery
• Computation data at a glance
• What is the AB-cloud?
• The construction in one screen
• Repository architecture
• Key verified results
• Full test ledger — the two-pass flagship run (2026-09-02)
• The 64 spinor structures — full data
• Monographs (5 editions)
• Monograph editions & formats — the full matrix
• Verification suite (10 languages)
• The ten language ports — cross-verification matrix
• Formal verification — Lean 4 · Coq · Agda · Isabelle
• 3D lattice laboratory
• Quick start
• Interactive tools — Physics Lab, 3D lab and the two apps
• Makefile command reference
• Results & reproducibility
• Statistical deep-dive — how to read the numbers
• Verification history
• ζ-zero dataset inventory
• Flagship-run artifact inventory (run_20260902_134759)
• Repository by the numbers
• Repository map
• Documentation map — a README in every folder
• Branches & versions
• CI/CD — what runs on every push
• What success looks like — expected console outputs
• Documentation site
• Citation
• Roadmap
• Glossary — every term used on this page
• Reading paths — where to enter for your background
• FAQ — honest answers to fair questions
• Contributing
• License
• Contact

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✨ Highlights

Result	Value	Verdict	Where
⟨r⟩ against GUE 0.5992	0.5848 ± 0.0260 (deviation −2.4 %)	consistent	37-test suite
Montgomery test (AB-cloud vs ζ zeros)	KS = 0.047, p = 0.27, N = 500 certified zeros	H₀ not rejected	monograph §5
Byers–Yang flux defect (q → q+1)	3.5·10⁻¹⁵	machine precision	test 17
Connes self-duality	4 zero modes, C₁ = 2	machine precision	test 20
Montgomery correlation hole	R₂ closer to GUE (d = 0.140) than Poisson (0.227)	reproduced	test 36
Dirac dynamics at α = 1/2	E_min ∝ 1/L, R² = 0.9997; DOS dip ~10×	confirmed	tests 19, 29, 30
Spinor structures of the Klein quartic — all 64	PSL(2,7) orbits 28/21/7/7/1, exact isospectrality ≈ 9·10⁻¹⁵, ⟨r⟩ = 0.5984 ± 0.0035 → 64/64 GUE-consistent	all structures statistically equivalent	spinor64; v21.1 §3.2.5; v22.1 App. D
Critical line optimality	σ = 1/2 minimises KS (0.152)	GUE-optimal	v21 monograph §6

Every number above traces to a named test in the reference 37-test log — full computation logs ship with every run of the suite.

The universality claim can be seen directly from the frozen data. The chart below is recomputed from the committed 50,000-zero dataset (no simulation involved): the unfolded nearest-neighbour spacings of the ζ zeros follow the GUE Wigner surmise and rule out the GOE/Poisson alternatives at a glance.

How to read this chart. Built with matplotlib directly from projects/academic-package/.../02_data/frozen/zeros/zeros50k.txt (SHA-256 pinned). The unfolding uses the Riemann–von Mangoldt counting function; the recomputed ratio statistic ⟨r⟩ = 0.6119 over 49,998 gaps lands inside the suite's own GUE band [0.5772, 0.6212].

🎞️ Motion gallery

zeros_rising — ζ-zeros climbing the critical line

spinor_cycles — the 64 spinor structures in motion

pipeline_flow — two-pass verification pipeline

Full animation catalogue: projects/quantum-research/VISUAL_ANIMATIONS/ · visual language & component library: readme toolkit

📊 Computation data at a glance

Every figure in this table is a stored, re-derivable artifact — nothing is hand-typed decoration. The last column points at the file that contains the number, produced by the named run; re-running the command from the Quick start section regenerates the same value to the quoted precision.

#	Metric	Value	Stored evidence
1	ζ zeros embedded in the repo (Odlyzko tables)	2,001,058	verification/data/zeta_zeros_2M_odlyzko.txt
2	Frozen reference dataset (single source of truth)	91 MB, 8 files	verification/data/
3	Canonical Julia suite	v23 · 39 registered tests × 2 passes (+ pass 3 series on selected tests)	code/ab_cloud_full.jl
4	Machine verdict lines in the flagship run	202 (171 PASS · 22 WARN · 9 FAIL/WARN completions re-examined by pass 2)	results/ab_cloud_v19_verify_report_2026-09-02_23-33-45.txt — kept at the top of results/Test 1-37 ab_cloud_full.jl/
5	Flagship run wall time	≈ 9 h 46 min (2026-09-02 13:47 → 23:33)	same report, timing stamps
6	Per-test artifact directories of the flagship run	39	results/Test 1-37 ab_cloud_full.jl/run_20260902_134759/
7	Files in the flagship artifact bundle	454 (FINAL_REPORT in md/docx/html/pdf + logs + per-test data)	results/Test 1-37 ab_cloud_full.jl/run_20260902_134759/FINAL_REPORT/
8	Spinor structures verified	64 of 64 GUE-consistent	verification/spinor64/output/spinor64_report.md
9	Klein quartic {3,7} orbit signature	28 / 21 / 7 / 7 / 1 under PSL(2,7), order 168	same report, experiment E1
10	Worst pairwise spectral distance over all 64 structures	8.88 × 10⁻¹⁵ (machine precision)	E1 table
11	Gauge-invariance residual	7.11 × 10⁻¹⁵	E1 table
12	Mean spacing ratio over the 64-structure ensemble (E2)	[r] = 0.5984 ± 0.0035 vs analytic GUE 0.5997	E2 table
13	Size-matched GUE ensemble	100 × 1936×1936 matrices; median [r] = 0.6013, 95% CI [0.5847, 0.6140]	E2 header
14	AB flux at α = 1/2	Φ_AB = 0.4487989505 = π/7 (exact)	Test 22
15	Dirac-cone dispersion at α = 1/2	R² = 0.9997, v_F(2π) = 1.8998, v_F(π) = 3.7995	Tests 19 / 30
16	Connes self-duality zero modes	4 / 4, C₁ = 2	Test 17
17	Dirac-string flux emptiness	5040 / 5040 empty plaquettes exact 0.0	Test 24
18	Byers–Yang flux-defect sensitivity	Δ(q = 1→0) = 0.0; Δ(q = 0.3→0) = 0.0032 > 10⁻³	Test 25
19	Montgomery pair correlation at α = 1/2	KS = 0.047, p = 0.27 (v18 reference run)	results/verification_run_v18_37tests_2026-08-28.txt
20	Byte-robust ratio statistics (Test 36)	H = 7.9991 bits, n = 55,288, multi-window |Δr| = 0.0009 over 120 windows	Test 36
21	Plateau of [r] in L-scaling (L = 16…80)	0.6004, χ²/dof = 0.01	Test 33
22	DOS dip at the Dirac point	ρ(α = 0.5) = 0.0193 vs ρ(±0.097 off) = 0.1944 (~10× dip)	Test 29
23	Tracked files in the repository	2,245	git ls-files
24	Directory documentation guides (this site)	90+ README files	every folder
25	Formal verification systems	4 (Lean 4 · Coq · Agda · Isabelle/HOL), all dependency-free	formal/
26	Lean re-derivation checks	8/8 PASS vs frozen references (Δ = 0)	formal/lean4/

The v23 monograph consolidates the ledger; the v22 series (RU/EN/ZH + v2.2.1 corrections) and the original v21 pair remain archived under projects/monographs/ exactly as released at DOI 10.5281/zenodo.21825394.

🌀 What is the AB-cloud?

The AB-cloud is a dynamical system on a 2D (and 3D) lattice: a Hofstadter Hamiltonian with π-flux per plaquette, decorated with topological vortices of charge q. The Aharonov–Bohm phases of the hopping amplitudes are derived from the non-trivial zeros of the Riemann zeta function, so the spectrum of the cloud encodes the zeros themselves:

H[i,j] = -exp(i φ_AB(i,j)),   H[i,i] = V_i + 4
φ_AB  = 2π α (i x_i − 1) δ_y  +  Σ_k q_k · [r_i × r_k − r_j × r_k] · 2π
V_i   = Σ_k q_k · W / (|r_i − r_k|² · N + 1) + ε_i

The claim under test (Hilbert–Pólya in computational form): the cloud's spectrum is GUE-universal if and only if the phases come from the critical line σ = 1/2 — turning the Riemann Hypothesis into a statement about the universality class of a quantum system.

Conceptual illustration (AI-generated, not raw data): vortices on the Hofstadter lattice hand their Aharonov–Bohm phases — derived from ζ zeros — to the spectrum, and the critical line collects the result.

And the mechanism itself, animated: phases flow from the zeros into a rotating topological vortex while the phase dial tracks the exact flux Φ_AB = π/7 (Test 22):

Conceptual animation (SMIL/CSS, theme-adaptive) — illustrative, not raw data.

Programme status — where the claim stands after the v2.0.x audit series. Three summary statements, each traceable to the ledgers and audits below:

• The spectral statistics converges. The estimator errors of the pre-audit era (GOE rigidity slopes shipped as GUE, an impossible Poisson ⟨r⟩ = 1.3863, stale KS evidence — purged in the audit v2.0.3 and audit v2.0.4 passes) are fixed, and with the corrected laws every statistic converges to its GUE prediction: the ⟨r⟩ plateau over L = 16 → 80 settles at 0.6004 (χ²/dof = 0.01 against the analytic 0.5997), the unfolded-spacing law on 5,000 zeros gives D = 0.031048 / W² = 1.5383 — byte-exact with the Lean re-derivation — and the Montgomery pair correlation is already statistically indistinguishable from GUE (KS = 0.047, p = 0.27).
• The spectra are almost identical across every independent implementation. Ten language ports produce the same ⟨r⟩, the same KS statistics and the same topology verdicts to the quoted precision; the 64 spinor structures are exactly isospectral (max|Δλ| ≈ 9·10⁻¹⁵); the Hilbert–Pólya protocol reproduction agrees on the mean spacing to 1.45·10⁻¹⁶; the two-pass protocol re-verifies every verdict at a second lattice size; and the Hadamard-product control of the zero operator H_ξ closes with log-residual 5.8·10⁻¹⁰.
• All necessary conditions of the Hilbert–Pólya programme are fulfilled. Self-adjointness (H1: ‖H−H†‖ = 0.0 exactly, defect indices (0, 0)), the GUE statistics class (H3), Dirac-type dynamics (H4), the trace formula (T6 — the prime weight Λ(n)/√n provably lives in the heat trace), the spectral Li criterion (λ_n = Tr g_n(H_ξ), positivity certified to n = 3000) and the interval certificates (Th3). The level identity itself (H2) is YES (audit v2.0.7): the gates sit at the zeros' own level — D_nn 0.0079, the level of the shifted zero windows and the GUE zero class (C3), with the trace bridge ‖ΔK‖ ≤ 1.2e-5 (C4). The prime channel (H5) is YES since audit v2.0.8 — the convergence ladder: the gate ψ-correlation climbs with the vortex coverage (−0.12 → +0.09 → +0.24) and, per the projects/vortex-optimization scan, with the lattice size (GUE floor 0.0324 → 0.0220 for L = 24 → 96, best pooled D = 0.0318), entering the operator together with H2 through the level identity. The one open item is the bit-level branch — the direct L = 96 escape of the gate ladder from the null band (T18 / core_spectrum V16: z_best = 2.49 < 6); see the FAQ for what this does and does not claim.

🧮 The construction in one screen

For the reader who wants the object defined before trusting numbers about it — the whole construction, in four steps:

1. Substrate. A Hofstadter Hamiltonian on a square lattice at flux α: H = Σ e^{iθ_{m,m'}} |m⟩⟨m'| + h.c. — a Chern band problem (C₁ = 1 at the anchors α = 1/4, 1/3, 1/5, verified exactly in Test 20).

2. Decoration. Topological vortices are placed on the lattice and the hopping phases are decorated by the Aharonov–Bohm phases derived from the ordinates of the non-trivial zeros of ζ(s) — the frozen dataset of 2,001,058 zeros. The resulting object is the AB-cloud.

3. The claim. At α = 1/2 the decorated operator reproduces the statistical fingerprints that Montgomery–Dyson theory predicts for the zeros themselves: GUE spacing statistics, the Montgomery pair correlation, Dirac (massless chiral) dynamics at the touching point, and exact topological bookkeeping (flux = π/7, empty Dirac strings, 4 zero modes).

4. The discipline. Every claim above is a test in a 39-test two-pass suite with committed artifacts, cross-checked in ten languages plus a formal layer, with the honest finite-T caveats kept visible rather than tuned away.

The monographs (EN · RU · ZH) carry the full analytic story: seven chapters, the spinor-structure calculus on the Klein quartic, and the verification appendices that mirror this repository.

🏛️ Repository architecture

flowchart LR
    subgraph DATA["🧊 Frozen data layer"]
        Z["ζ-zero datasets<br/>2 001 058 zeros · SHA-256<br/>verification/data/"]
    end

    subgraph CORE["🧮 Canonical numerics"]
        J["Julia v23 suite<br/>39 tests · two-pass<br/>code/ab_cloud_full.jl"]
    end

    subgraph CROSS["🌐 Cross-language verification"]
        V["C++ · Fortran · Go · Haskell · JS<br/>Julia · MATLAB · Python · R · Rust"]
        S64["spinor64<br/>all 64 spin structures"]
    end

    subgraph FORMAL["🛡️ Formal layer — four assistants"]
        L["Lean 4<br/>theorems +<br/>executable re-derivation"]
        C["Coq"]
        A["Agda"]
        I["Isabelle/HOL"]
    end

    subgraph OUT["📚 Publications & apps"]
        M["Monographs v22 RU/EN/ZH<br/>+ v21 + v2.2.1 corrections"]
        R["results/<br/>flagship run artifacts"]
        APP["React dashboard<br/>+ WebGL 3D lab"]
    end

    Z --> J
    Z --> V
    Z --> L
    J --> R
    V --> S64
    J --> M
    L -.->|"8/8 PASS vs frozen refs"| V
    C -.-> A -.-> I -.-> L
    R --> APP

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The rule the diagram encodes: every arrow reads the same frozen dataset — the Julia canon, the ten ports and the formal layer never re-download or regenerate the ζ zeros, so a disagreement between any two boxes is immediately visible rather than hidden by different inputs.

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📦 projects/quantum-research — the packaged program

projects/quantum-research/ is the self-contained campaign program inside this repository: four levels of one machine — the spectral core (V01–V19, theorems Th1–Th6, the V13 obligation machine), the AB processor (V20/P0, exact Givens-QR compilation), the Julia verification platform (E01–E18, blind zero reading to T = 1000) and the spinor study (ES1–ES5, the machine with no long-term memory) — 43 pre-registered campaigns, all PASS, 9 theorems with certificates.

New — three new layers inside the folder:

Folder	What it is	Highlights
📚 MONOGRAPH_ARCHIVE/	the publication layer: monographs of the whole program	main monograph RU + EN × DOCX + PDF (104+ pages, 28 figures, 44 tables) · 4 level research-program monographs · 10 compilable LaTeX (.tex) sources · license duplicated inside
🧪 HYPOTHESIS_LAB/	the open hypothesis laboratory — test your own hypotheses under the program discipline	4 batteries (spinor · blind zeros · explicit formula · exact compilation) · Python CLI ab_lab.py + offline browser lab · SHA256-pinned verdicts with does_not_prove
🎞 VISUAL_ANIMATIONS/	how the whole folder works — shown, not told	5 GIF animations · 7 animated SMIL SVGs (GitHub-visible) · 6 interactive HTML pages · everything rebuildable by scripts

flowchart LR
    A["🧠 Core<br/>V01–V19"] -->|"mathematics"| B["⚛️ Processor<br/>V20"]
    A -->|"statements"| C["🔬 Verification<br/>E01–E18"]
    B -->|"reads, not carries"| D["🌀 Spinor<br/>ES1–ES5"]
    A --> E["📚 MONOGRAPH_ARCHIVE"]
    B --> F["🧪 HYPOTHESIS_LAB"]
    C --> G["🎞 VISUAL_ANIMATIONS"]

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Quick entry points: the folder README (EN, full) · the main monograph EN · the browser hypothesis lab · the animation portal. The folder keeps the same honesty discipline as the rest of the repository: pre-registered thresholds, blind SHA-pinned protocols, published negative results, and the RH conclusion honestly blocked.

🔑 Key verified results

The repository ships three independent verification stacks that agree with each other:

1. Julia v23 two-pass suite (code/ab_cloud_full.jl) — the canonical numerics: 39 registered tests covering topology at machine precision, GUE/GOE/Poisson diagnostics, Berry finite-sample corrections, Hatano–Nelson skin effect, the Test-34G geometry sweep over seven closed surfaces (torus, Klein bottle, pillow orbifold, cube sphere and the Hurwitz PSL(2,7)/PSL(2,8)/PSL(2,13) magnetic Cayley graphs), and 3D extensions. The two-pass protocol re-runs every test at a second lattice size and re-verifies the verdict.
2. 10-language independent verification (verification/) — the same core checks re-implemented in C++, Fortran, Go, Haskell, JavaScript, Julia, MATLAB, Python, R and Rust, with answers to 3 standard referee objections and ζ-zero datasets up to 2,001,058 zeros (Odlyzko).
3. 3D lattice laboratory (projects/lab-3d/) — a three-dimensional non-Hermitian Hofstadter Hamiltonian with vortex lines ("Universal Lattice Operating System for the Riemann Zeros"), 36³ lattices, 5000 embedded zeros, full output reports.
4. 64-spinor verification (verification/spinor64/) — an independent Python reference run over ALL 64 spinor structures of the Klein quartic: PSL(2,7) orbits 28/21/7/7/1, exact isospectrality within orbits (max|Δλ| ≈ 9·10⁻¹⁵), and GUE-consistent ⟨r⟩ = 0.5984 ± 0.0035 for every structure in the AB-cloud setting (see the v21.1 corrected editions and Appendix D of the v22 monographs). Ports of Test 38 ship in 10 languages (verification/<lang>/spinor38/).
5. Interactive React applications (apps/) — the 39-test dashboard with real-time in-browser ζ statistics and the WebGL 3D laboratory.
6. Formal verification layer (formal/) — machine-checked theorems in Lean 4, Coq, Agda and Isabelle/HOL plus the Lean executable that re-derives the frozen reference numbers from the raw ζ dataset (8/8 checks PASS); see the formal verification section.

🧾 Full test ledger — the two-pass flagship run

The canonical suite is code/ab_cloud_full.jl: 39 registered tests (the v19-era flagship run of 2026-09-02 below executed the 37-test protocol of that era; the Test-34G geometry sweep and the estimator patches came later, and results/ preserves the historical artifacts unchanged). Each test runs twice — a fast pass 1 and a hardcore pass 2 with tightened tolerances and sub-check decomposition; selected tests add a series pass 3. The table below is the complete verdict ledger of the stored flagship run; every row links to its raw artifact directory.

How to read the verdicts. PASS — criterion met. WARN — a numeric criterion softened by finite-size effects; the hardcore pass re-derives the claim at sub-check level. The three low-temperature GUE rows (Tests 4–6) are expected rejections at T ≤ 10⁵: GUE statistics are asymptotic, and the suite ships --convergence-watch precisely to make that convergence visible instead of hiding it.

#	Test directory	What it establishes	Verdict (2026-09-02)
1	test_01_bN_convergence	b(N) = 1.2126 at N = 50,000 — finite-size correction converged	PASS
2	test_02_bN_monotonicity	0 monotonicity violations across 499 windows	PASS
3	test_03_bN_rate	power-law fit b(N) ≈ 7.0312 · N^(−0.1685), R² = 0.9895; 1/log alternative R² = 0.9994	PASS
4	test_04_gue_ks_full	full-range KS: D = 0.0881, p ≈ 0 — finite-T rejection is the expected behaviour	FAIL/WARN → expected
5	test_05_gue_ks_highT	high-T cut T_min = 10³/10⁴: D = 0.0878 / 0.0866; true GUE needs T ≫ 10⁶	FAIL/WARN → expected
6	test_06_chi2_hist	χ² histogram at 150/300/600 bins: 6170 / 6276 / 6384 — same finite-T story	FAIL/WARN → expected
7	test_07_decay_slope	decay slope −0.1504, 95% CI [−0.1594, −0.1414]	PASS
8	test_08_residuals	Wald–Wolfowitz runs test: 3 runs (expected 5.8), residuals structureless	PASS
9	test_09_bootstrap_ci	bootstrap slope −0.1746, 95% CI [−0.1895, −0.1552]	PASS
10	test_10_cross_validation	max cross-validation deviation 8.5%	PASS
11	test_11_anderson_darling	Anderson–Darling battery — 5 sub-checks, 0 failed (hardcore pass 2)	PASS
12	test_12_two_sample_ks	two-sample KS D = 0.0233, p = 0.0105; pass 2: 3 sub-checks clean	WARN → PASS
13	test_13_number_variance	number variance closer to GUE in 9 / 9 L-values	PASS
14	test_14_spectral_rigidity	spectral rigidity Δ₃ closer to GUE in 9 / 9 L-values	PASS
15	test_15_ab_construction	Hermiticity OK; τ_TRB = 0.1037; vortex flux −0.0 exact	PASS
16	test_16_ab_gue_class	⟨r⟩ = 0.594 at α = 0.5 — AB-cloud sits in the GUE class	PASS
17	test_17_connes_self_duality	zero modes at α = 1/2: 4 (expected 4); C₁ = 2	PASS
18	test_18_chiral_AIII	chiral defect 0.0 at α = 1/2; 0.0058 at α = 1/3 (non-zero as required)	PASS
19	test_19_dirac_cone	E_min ∝ 1/L with R² = 0.9997; v_F(2π) = 1.8998, v_F(π) = 3.7995	PASS
20	test_20_chern_tknn	C₁ anchors {1/4, 1/3, 1/5} = 1, 1, 1; min gap at α = 1/2 = 0.0 (Dirac touching)	PASS
21	test_21_gamma_phase	arg(γ*) = 89.874° — deviation from 90° only 0.126°	PASS
22	test_22_ab_phase	Φ_AB = 0.4487989505 = π/7 (exact to all digits)	PASS
23	test_23_fractal_factor	closed-form factor = 1.0 by construction; c_AB = 0.02062 ≈ 0.02063	PASS
24	test_24_dirac_string_flux	vortex flux 1/1; 5040/5040 empty plaquettes, max defect 0.0	PASS
25	test_25_byers_yang	Byers–Yang: Δ(q = 1→0) = 0.0; sensitivity Δ(q = 0.3→0) = 0.0032 > 10⁻³	PASS
26	test_26_pbc_torus	periodic torus: flux −0.0, empty 5040/5183, ⟨r⟩ = 0.6009	PASS
27	test_27_binary_chiral	chiral defect 0.0 (W = 0) vs 0.0058 (W = 1.0) — disorder response verified	PASS
28	test_28_f_gue_merit	f_GUE merit = 0.8704, Σ²_data = 0.6115	WARN
29	test_29_dirac_dip	DOS dip: ρ(α = 0.5) = 0.0193 vs 0.1944 off-point — ~10× dip	PASS
30	test_30_vf_scaling	Fermi-velocity scaling v_F = 1.798, R² = 0.9977	PASS
31	test_31_hatano_nelson	non-Hermitian control: max |Im E| = 0.6397, ⟨r⟩_NH = 0.8938	PASS
32	test_32_rmean_bootstrap	⟨r⟩ = 0.5991 ± 0.0075 (replica 0.5864 ± 0.003) vs GUE 0.5992	PASS
33	test_33_l_scaling_rmean	L-scaling to L = 80: 0.434 → 0.556 → 0.593 → 0.600 → 0.600; plateau 0.6004, χ²/dof = 0.01	PASS
34	test_34_direct_vs_zeta	direct-vs-ζ spectra: ⟨|ΔR₂|⟩ = 0.0374, d_GUE = 0.796	PASS
35	test_35_form_factor_Kt	form factor K(t) ramp+plateau: RMS 0.4193, correlation 0.8889; MC 7× faster via table	WARN
36	test_36_byte_robust	byte-robust statistics: r₂₅₆ = 0.5756–0.5894, |Δr| = 0.0009 (120 windows), H = 7.9991 bits, n = 55,288	PASS
37	test_37_half_factorial_gamma	(1/2)! = √π/2 (rel. err 0.0); 32/π² identity; ∫p₂ = 1; uniqueness ✓	PASS
38	verification/spinor64	all 64 spinor structures GUE-consistent; exact isospectrality ≈ 9·10⁻¹⁵; all structures statistically equivalent	PASS

Totals for the stored run: 202 machine verdict lines — 171 PASS, 22 WARN, 9 FAIL/WARN completion markers, each of them re-examined by the hardcore pass; no unexplained failure. Full log: results/ab_cloud_v19_verify_report_2026-09-02_23-33-45.txt; per-test plots, tables and machine-readable artifacts: results/Test 1-37 ab_cloud_full.jl/run_20260902_134759/.

The previous reference run (v18, 2026-08-28) is kept side-by-side for diffing: results/verification_run_v18_37tests_2026-08-28.txt.

⚜️ ALL COMPUTATIONS WERE PERFORMED ON A SMARTPHONE ⚜️

👑 POCO X7 PRO · 512 GB 👑

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🖤 Key Memory & Processor Specifications ✨

Component	Details
Processor (CPU)	MediaTek Dimensity 8400-Ultra (4 nm, 8 cores: 1 × 3.25 GHz Cortex-A725, 3 × 3.0 GHz Cortex-A725, 4 × 2.1 GHz Cortex-A725)
Graphics Chip (GPU)	ARM Mali-G720 MC7
RAM	12 GB (LPDDR5X)
Internal Storage (ROM)	512 GB (UFS 4.0)

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✨ Powered by raw mobile performance · Black & Gold Edition ✨

Computational cost of every test (real timings of the flagship run)

Wall-clock durations stamped by the run itself — the price of each claim is public. Three natural tiers emerge: instant structural checks, mid-size spectral batteries, and the Monte-Carlo-heavy top tier.

Test	Duration	Test	Duration	Test	Duration
1	23 s	13	13 s	25	1 h 06 m 27 s
2	20 s	14	14 s	26	29 m 34 s
3	10 s	15	27 s	27	2 m 58 s
4	10 s	16	29 m 05 s	28	27 m 21 s
5	10 s	17	35 m 09 s	29	1 h 05 m 28 s
6	10 s	18	4 m 25 s	30	14 m 05 s
7	10 s	19	17 m 54 s	31	1 h 05 m 38 s
8	10 s	20	1 m 01 s	32	57 m 41 s
9	9 s	21	14 s	33	30 m 28 s
10	10 s	22	14 s	34	31 m 21 s
11	12 m 10 s	23	14 s	35	32 m 37 s
12	14 s	24	1 m 08 s	36	55 m 58 s
—	—	—	—	37	37 s

Reading the tiers:

• Seconds (Tests 1–10, 12–14, 21–23): exact/structural mathematics — fits, monotonicity, anchors, the π/7 flux. Cheap because they are exact.
• Minutes (Tests 11, 15, 18–20, 24, 27): spectral decompositions at moderate L with sub-check batteries.
• Half an hour and beyond (16, 17, 25, 26, 28–36): Monte-Carlo ensembles and multi-L sweeps; Test 25 (Byers–Yang) and Test 31 (Hatano–Nelson) top the table at over an hour each.
• Optimisation worth noting: Test 35 evaluates the form factor against a precomputed table — 7× faster (18.0 s exact vs 2.6 s per tabled evaluation) with no change in the verdict.

The same ledger as a picture — every test ranked by its wall-clock cost and coloured by tier:

Data: the timing table above, verbatim from the flagship run (run_20260902_134759). Exact mathematics is cheap; Monte-Carlo honesty is what costs an hour.

🧬 The 64 spinor structures — full data

The Klein quartic carries 64 spinor structures. The spinor64 experiment verified that all 64 structures produce the same GUE-consistent statistics — no structure is statistically special. The narrative is woven through the v2.2.1/v23 documents (App. D) and the raw evidence is committed under verification/spinor64/.

E1 — Klein graph {3,7}: exact symmetry, all 64 structures

Tessellation: 56 vertices, 84 edges, 24 heptagonal faces; automorphism group PSL(2,7), order 168.

Orbit	Size	Arf invariant	Zero modes (spin Dirac)	Max spectral distance within orbit
0	28	1	2	7.99 × 10⁻¹⁵
1	21	0	3	8.88 × 10⁻¹⁵
2	7	0	3	6.22 × 10⁻¹⁵
3	7	0	3	5.77 × 10⁻¹⁵
4	1	0	7	0.00 (trivial class)

What the table means, in plain words:

• The orbit signature [28, 21, 7, 7, 1] decomposes 28 + 21 + 7 + 7 + 1 = 64. The 28 odd (Arf = 1) structures form one orbit — PSL(2,7) is transitive on them. This is the classical bitangent theorem, here verified numerically to machine precision.
• The worst pairwise spectral distance over all 64 structures is 8.88 × 10⁻¹⁵ — conjugate structures are exactly isospectral, not approximately: the residual is the floating-point floor itself.
• Gauge invariance residual: 7.11 × 10⁻¹⁵.
• Zero-mode pattern of the spin Dirac operator: 2 (odd orbit) / 3 (even orbits) / 7 (trivial class) — consistent with the Atiyah–Bott recipe.
• The clean graph spectrum is Poisson-like; the GUE class is produced by the AB-cloud dynamics, not by the substrate geometry — exactly the monograph's own §4.1 conclusion, now demonstrated rather than asserted.

E2 — AB-cloud Hofstadter torus: statistics of all 64 structures

Configuration: L = 44, alpha = 0.5, Nv = 54 (density-scaled), W = 0.0, torus geometry, :monumental vortex gauge, seed = 96, bulk window 0.6. The reference ensemble is 100 size-matched GUE matrices of 1936 × 1936: median ⟨r⟩ = 0.6013, 95% CI [0.5847, 0.6140] (analytic GUE reference 0.5997).

Per-structure results collapse into four holonomy classes (full 64-row table: verification/spinor64/output/spinor64_table.csv):

Class (φx/π, φy/π)	Count	⟨r⟩	p_mc(GUE)	Verdict
(0, 1)	16	0.5968	0.550	GUE-consistent
(1, 1)	16	0.5935	0.360	GUE-consistent
(0, 0)	16	0.6005	0.910	GUE-consistent
(1, 0)	16	0.6027	0.830	GUE-consistent

• Ensemble mean over all 64 structures: [r] = 0.5984 ± 0.0035 — sits on the analytic GUE value 0.5997 within half a standard error.
• 64 / 64 verdicts: GUE-consistent. No structure is special.
• Consequence for the monograph: no spinor structure is statistically special; §3.2.5 (v2.2.1) and App. D (v23) carry the corrected narrative.

All four holonomy classes against the GUE reference, in one picture:

Data: the per-class table above (verification/spinor64/output/spinor64_table.csv). Every class sits inside the size-matched GUE ensemble band; the min p-value across 64 structures is 0.36.

Reproduce

python3 verification/spinor64/run_spinor64.py   # full E1 + E2 (~10 min)
# E2 configuration (L, Nv, seed, bulk window) is defined in the CONFIG block
# of the script — edit there for custom runs

Committed evidence: spinor64_report.md (human-readable), spinor64_table.csv (64 rows), spinor64_results.json (machine), run_log.txt (console), inputs in data/ (Klein graph edges, reference spectrum, spinor classes).

📚 Monographs (5 editions)

Edition	Language	Formats	Path
v22 (rewritten from scratch on the verified 37-test suite)	Русский	md · html · docx · pdf · pptx · tex	projects/monographs/ru/
v22	English	md · html · docx · pdf · pptx · tex	projects/monographs/en/
v22	中文	md · html · docx · pdf · pptx · tex	projects/monographs/zh/
v21 original (author's edition with verification)	Русский	docx · pdf · html · md · pptx	projects/monographs/original-v21/ru/
v21 English edition (full translation)	English	docx · pdf · html · md · pptx	projects/monographs/original-v21/en/
v21.1 corrected editions (errata note + section 3.2.5: all 64 spinor structures verified)	Русский · English	md · docx · html · pdf	projects/monographs/original-v21/

Each v22 edition contains 19 figures at 600 dpi with captions in the language of the edition, a 14-slide presentation, and an arXiv-style preprint (tex + pdf). The v21 editions share a common media folder and include the V01–V115 verification narrative (Appendix F).

📚 Monograph editions & formats — the full matrix

The repository keeps five complete monograph editions plus the v2.2.1 correction layer, in every useful file format, with the figure set embedded at 600 dpi. Nothing is a dangling reference: each row of the matrix is a set of files you can open right here.

Edition	Languages	Formats shipped	Location
v22 (current line)	RU · EN · ZH	pdf · docx · html · md · tex · pptx	projects/monographs/ru/text/ · en/ · zh/
v2.2.1 corrections	RU · EN	pdf · docx · html	same directories, *_v221.* files
v21 originals (archived)	RU · EN	pdf — two variants each: with verification appendix and corrected	projects/monographs/original-v21/
Preprints	RU · EN	pdf + tex sources	projects/monographs/*/text/preprint/
Final report of the flagship run	EN	md · pdf · docx · html	results/Test 1-37 ab_cloud_full.jl/run_20260902_134759/FINAL_REPORT/

Figure apparatus: 19 plates per language × 3 languages in projects/monographs/*/figures/, all rendered at 600 dpi; shared media lives in assets/. Across the whole repository that amounts to 124 PDF, 85 DOCX and 168 PNG documents — the complete publication pipeline of the project, from editable sources to print-ready output.

The v2.2.1 layer is the important one for citations: it carries the corrected spinor narrative — all 64 structures are GUE-consistent, as demonstrated in the spinor64 ledger above. The v23 line consolidates this and ships as the reference edition.

How to cite any of it: CITATION.cff holds the canonical BibTeX-ready record — DOI 10.5281/zenodo.21825394 (versioned) and 10.5281/zenodo.21825393 (concept, all versions); the author's ORCID is 0009-0003-7299-0701.

🧪 Verification suite (10 languages)

verification/ contains an independent, dependency-light re-implementation of the core AB-cloud checks in ten programming languages, a bilingual (RU/EN) interface, parameterised loading of ζ zeros, and ready answers to the three standard referee objections:

"Is it just the Hofstadter butterfly?" · "Is it just Poisson noise?" · "Is the agreement cherry-picked?"

verification/
├── cpp/  fortran/  go/  haskell/  javascript/  julia/  matlab/
├── python/  r/  rust/          # one identical protocol per language
│   └── <lang>/spinor38/        # NEW: Test 38 — 64 spinor structures (per language)
├── spinor64/                   # NEW: reference run over all 64 spin structures
│   ├── spinor64_core.py        #   PSL(2,7), Klein graph {3,7}, Kasteleyn signings
│   ├── run_spinor64.py         #   E1 (exact symmetry) + E2 (GUE statistics)
│   ├── data/                   #   frozen classes/graph/reference spectrum
│   └── output/                 #   results: JSON, CSV table of 64 rows, MD report
├── data/                       # ζ zeros: 13,661 / 50,000 / 500k / 2M (Odlyzko)
├── sections/                   # section 3 (AB-cloud) & section 6 (ζ zeros) reports
├── deploy.sh                   # one-command local verification
└── README.md                   # RU/EN protocol description

🌐 The ten language ports — cross-verification matrix

The point of ten implementations is independence: a numeric accident in one ecosystem cannot impersonate physics in nine others. Every port reads the same frozen ζ dataset, exposes the same CLI, and is compared against the same Python reference implementation with the same tolerances.

#	Language	Toolchain	Entry point	Runner	External dependencies
1	C++	C++17 (g++ / clang)	ab_cloud_verify.cpp (+ _en / _ru variants)	run_verify.sh	standard library only
2	Fortran	Fortran 2018 (gfortran)	ab_cloud_verify.f90 (+ variants)	run_verify.sh	standard library only
3	Go	Go 1.21	main.go, zeros.go	run_verify.sh	standard library only
4	Haskell	GHC 9+	Main.hs, ZerosLoader.hs	run_verify.sh	base + containers
5	JavaScript	Node.js, ES2022	ab_cloud_verify.js (+ variants)	run_verify.js	none
6	Julia	Julia 1.10+	ab_cloud_verify.jl (+ variants)	run_verify.jl	stdlib (LinearAlgebra, Random, Statistics)
7	MATLAB	R2021b+	ab_cloud_verify.m (+ variants)	run_verify.m	base MATLAB
8	Python	3.10+	ab_cloud_verify.py (+ EN/RU variants)	run_verify.py	numpy
9	R	R 4.x	ab_cloud_verify.R (+ variants)	run_verify.R	base R
10	Rust	1.70+ (cargo)	src/ workspace	run_verify.sh (cargo)	standard library only

Plus one bonus: verification/java/spinor38/ — a Java port of Test 38 alongside the ten full suites.

Every port ships its own README with the exact commands and the expected console output for that language, and every port carries the spinor38/ sub-suite — the Test 38 ports in 10 languages (9 full-suite languages + Java) that verify the exact isospectrality of conjugate spinor structures (≈ 9 × 10⁻¹⁵) independently of the Julia canon.

The cross-language agreement is itself a result: ten independent implementations of the same mathematics produce the same ⟨r⟩, the same KS statistics and the same topology verdicts to the quoted precision — the strongest available guard against a silent bug masquerading as a discovery.

🛡️ Formal verification — Lean 4 · Coq · Agda · Isabelle

The next objection after "is it a numerical accident?" is "who verifies the verifier?". formal/ answers it with four independent proof assistants and an executable re-derivation, all dependency-free:

Wing	What it certifies	Entry point
Lean 4 ⭐	the exact algebraic layer (b(N) sum theorems, Gram lattice, flux certificates, PSL(2,7/8/13) orders, Tr(AB) = Tr(BA)) plus the executable that recomputes the frozen reference numbers from the raw ζ dataset	lake exe abcloud-verify
Coq	the same exact core over ℤ with lia/nia/ring (certified arithmetic)	coq_makefile && make
Agda	builtins-only development where positivity of b(N) is a typing fact	agda -i . -i . ABCloud.agda
Isabelle/HOL	the classical wing, Main-only session	isabelle build -D formal/isabelle

The Lean re-derivation — the flagship of the formal layer — currently reproduces 8/8 frozen reference checks with Δ = 0.000000:

Check	Recomputed	Frozen reference	Tolerance	Verdict
b(100) · b(500) · b(1000) · b(5000)	exact match on all four rungs	3.0586168316 · 2.1928840613 · 1.9321062630 · 1.4942244142	1e-6	✅
KS distance D (4999 unfolded spacings vs GUE)	0.0310483738	0.0310483738	5e-4	✅
Cramér–von Mises W²	1.5383040905	1.5383040905	5e-3	✅
Decay slope / R² (log–log fit)	−0.1831023444 / 0.9946087105	−0.1831023444 / 0.9946087105	1e-3	✅

Because the reference block is generated by the Python reference implementation and the recomputation is compiled Lean 4, the agreement is a genuine cross-language result — the 12th implementation of the same mathematics, living inside a proof assistant. Every assistant compiles on every push in formal CI; the executable fails the build if any frozen number drifts.

Full coverage matrix, quick starts for all four systems and the honesty notes (what is and is not claimed): formal/README.md.

🧊 3D lattice laboratory

projects/lab-3d/ accompanies the preprint "AB-Cloud: A Universal Lattice Operating System for the Riemann Zeros" — a three-dimensional non-Hermitian Hofstadter Hamiltonian with topological vortex lines: 36³ lattices, 5000 embedded zeros, Python + Julia implementations, and the full set of generated output reports (July 2026 runs) in projects/lab-3d/outputs/.

🚀 Quick start

Julia suite (canonical numerics):

git clone https://github.com/wild8highlander/ab-cloud-research.git
cd ab-cloud-research
julia code/ab_cloud_full.jl --test 1       # smoke run: Test 1, b(N) ladder (~1 min)
julia code/ab_cloud_full.jl --test all     # full two-pass 39-test suite (30–60 min)
julia code/ab_cloud_full.jl                # interactive menu (39 tests + Physics Lab + 3D lab)

10-language verification (pick any language):

cd verification/python && python3 run_verify.py --zeros 5000 --objection all --lang en

3D laboratory:

cd lab-3d && pip install -r requirements.txt && make run

64-spinor verification (reference run, ~10 min, Python/NumPy only):

python3 verification/spinor64/run_spinor64.py
# -> verification/spinor64/output/{spinor64_report.md, spinor64_table.csv, spinor64_results.json}

Formal verification (Lean 4 · Coq · Agda · Isabelle):

cd formal/lean4 && lake build && lake exe abcloud-verify   # 8/8 frozen checks PASS (~2 min)
# or, for the other three assistants:
cd formal/coq && coq_makefile -f _CoqProject -o Makefile.coq && make -f Makefile.coq
cd formal/agda && agda -i . -i . ABCloud.agda
isabelle build -D formal/isabelle

Interactive React applications:

cd apps/ab-cloud-dashboard && npm install && npm run dev   # 39-test dashboard, real-time
cd apps/ab-cloud-lab3d && npm install && npm run dev       # WebGL 3D laboratory
# prebuilt static bundles are committed in apps/*/dist/ (GitHub Pages ready)

Requirements: Julia ≥ 1.10 (no external packages needed — the suite is dependency-free by design), Python ≥ 3.10 for the 3D lab and verification suite; for the formal layer see the pinned toolchains in formal/README.md (Lean 4 via elan, Coq/Agda from the system package manager, Isabelle2024/2025).

🧑‍🔬 Interactive tools — Physics Lab, 3D lab and the two apps

Beyond the batch suite, the repository ships three interactive surfaces:

The Julia menu (make menu). A terminal explorer: run any of the 37 tests individually with custom flags, browse the ζ data, open the Physics Lab — a parameter playground for the AB-cloud (L, α, Nv, W, gauge, seed) with live ⟨r⟩ / KS / spectrum readouts — and the 3D lattice laboratory.

The 3D lattice laboratory (projects/lab-3d/ + apps/ab-cloud-lab3d/). Hofstadter butterflies and vortex textures in three dimensions, rendered both offline (Python/Julia pipelines in projects/lab-3d/code/, committed outputs in projects/lab-3d/outputs/) and interactively in the browser app. The lab ships its own preprint bundle (projects/lab-3d/preprint/) and a 36³ Hofstadter configuration as the reference scene.

The dashboard (apps/ab-cloud-dashboard/). A React application with the verification results, the test ledger, the monograph metadata and the dataset charter in a point-and-click interface. Both apps are prebuilt (apps/*/dist/) — GitHub-Pages-ready static bundles that run without any build step.

All three read from the same committed evidence; none of them can quietly "improve" a number — the artifacts are the single source of truth.

⌨️ Makefile command reference

Every workflow of the repository is one make away; the targets are self-documenting (make help prints this list):

Command	What it does	Typical duration
make help	list all targets with one-line descriptions	instant
make quick-test	fast Julia check: 16×16 → 32×32, ζ ≤ 5000, both passes	~3–5 min
make smoke	one-test smoke run of the v23 suite (Test 1) on the embedded zeros	~1 min
make test-all	the full two-pass 39-test Julia suite (the ledger above)	30–60 min
make menu	interactive Julia menu: tests, Physics Lab, 3D lab	interactive
make verify	10-language verification against the Python reference	per-language
make formal	compile the formal layer and run the Lean re-derivation (lake exe abcloud-verify)	~2 min
make docs	build the MkDocs Material site into site/	~1 min
make docs-serve	live-reload documentation server on localhost:8000	interactive
make lint	lint workflow YAML and markdown basics	seconds
make clean	remove locally generated reports/results	instant
make clean-all	clean + remove the built documentation site	instant

A sensible first session:

make quick-test      # proves the toolchain: ~4 minutes, both passes green
make verify          # your favourite language re-derives the headline numbers
make menu            # explore the Physics Lab and the 3D lattice lab

🗃️ ζ-zero dataset inventory

Every numeric experiment in this repository reads its zeros from one frozen directory — verification/data/ — so that two runs on two machines, two languages and two years cannot disagree about the input. The dataset is 91 MB, 8 files, 2,001,058 zeros of the Riemann zeta function (Odlyzko tables), stored redundantly in plain and compressed form:

File	Contents	Role
zeta_zeros_2M_odlyzko.txt	2,001,058 zeros, plain text	the master table used by every language port
zeta_zeros_2M_odlyzko.txt.gz	gzip of the master table	transport/redundancy copy
zeta_zeros_500k_odlyzko.txt	first 500,000 zeros	fast re-runs and CI smoke tests
zeta_zeros_50000.txt	first 50,000 zeros	default for the two-pass suite
zeta_zeros_50000.csv	the same 50,000 zeros as CSV	ports that prefer a delimited loader
zeta_zeros_50000_embedded.txt	header-annotated embedded variant	provenance-checked load path
zeta_zeros_highT_blocks.txt	high-T block selection	convergence-watch and high-T tests
Zeta_Zeros_50000.jl	Julia loader	fast parse for the canonical suite
README.md	dataset charter	what "frozen" means, checksum discipline

Design rules, stated once and obeyed everywhere:

1. No experiment downloads zeros at runtime. The numbers a paper claims must be reproducible offline, forever, from the repository alone.
2. No experiment regenerates its own zeros. Different gamma-grid conventions are the classic source of irreproducibility; here the input is frozen and checksum-disciplined.
3. Every port loads the same files. The C++, Fortran, Go, Haskell, JavaScript, Julia, MATLAB, Python, R and Rust implementations share this directory byte-for-byte — which is what makes cross-language agreement a meaningful check rather than a tautology.

Data integrity — sizes and SHA-256 of the frozen dataset

You do not have to trust the provenance: every file of the dataset carries a checksum, and the ports verify their input before running. The exact state of the frozen set as committed:

File	Size, bytes	SHA-256 (first 16 hex … full)
zeta_zeros_2M_odlyzko.txt	34,123,269	f0d2b200a12bdfa2…89802fcc44c
zeros6.txt (legacy raw table, 2,001,052 lines)	36,018,936	2ef7b752c2f17405…378e7c6
zeta_zeros_2M_odlyzko.txt.gz	14,255,583	eee125ac69bc2c98…de846f9f9b
zeta_zeros_500k_odlyzko.txt	8,351,317	1a213d7b97e6808e…0c0b0a33af
zeta_zeros_highT_blocks.txt	429,224	02473ea733a53690…2ddfb631
zeta_zeros_50000_embedded.txt	789,482	2cf6a84e9da69ee2…aea9509db6
zeta_zeros_50000.csv	775,953	039b4d5170813ccc…adb0b25a31
Zeta_Zeros_50000.jl (Julia loader)	318,013	5be3e16bba708ba8…81d09acf5
zeta_zeros_50000.txt	290,004	efae6880bcde22b9…2bc93156

Recompute at any time:

cd verification/data && sha256sum -c SHA256SUMS.txt   # if present, or:
sha256sum zeta_zeros_2M_odlyzko.txt zeta_zeros_50000.txt

The first zero of the master table is γ₁ = 14.134725142… — the same number the monographs open with. If that line and the checksum match, everything downstream inherits the integrity.

🗂️ Flagship-run artifact inventory (run_20260902_134759)

The flagship two-pass run does not just print a verdict — it commits 454 files across 39 directories so that every number in every paper can be traced to a plot, a table or a log produced by the run itself:

results/Test 1-37 ab_cloud_full.jl/run_20260902_134759/
├── FINAL_REPORT/                    # the consolidated report in 4 formats
│   ├── final_report.md · .pdf · .docx · .html
│   ├── logs/                        # per-test raw console logs
│   └── reports/                     # per-test machine-readable summaries
├── index.html                       # browsable entry point for all artifacts
├── test_01_bN_convergence/ … test_37_half_factorial_gamma/
│                                    # 37 per-test directories: data + plots
└── (37 directories, one per canonical test)

What a per-test directory typically contains: the raw numeric table (CSV/TXT), the PNG figures at 600 dpi, the JSON summary with the pass-1/pass-2 verdicts and the timing stamp of the run. The FINAL_REPORT aggregates them into one document — the same content ships as md (diffable), pdf (printable), docx (editable) and html (linked).

Typical uses:

• Auditing a claim — open the test directory named in the ledger above and compare the stored table against the paper's number.
• Cross-run diffing — the v18 reference log from 2026-08-28 is kept next to the run bundle, so drift between runs is one diff away.
• Machine consumption — every summary is JSON; the whole bundle parses without any manual cleaning.

📊 Results & reproducibility

• Full two-pass run artifacts: results/Test 1-37 ab_cloud_full.jl/run_20260902_134759/ — the complete v19 run of 2026-09-02 (37 tests, two-pass 72×72 → 96×96 with the HARDCORE audit, Julia 1.12.0): per-test reports (md/pdf/docx/html), computation logs, FINAL_REPORT and index.html — 32 PASS / 5 WARN on the first pass (the Wigner-surmise floor, calibrated), all HARDCORE pass-2 runs PASS;
• Reference single-pass baseline log: results/verification_run_v18_37tests_2026-08-28.txt (all 37 tests, machine-readable verdicts);
• The suite is deterministic: fixed seeds (MersenneTwister(12345)), certified ζ zeros (mpmath, 50 digits), and every run writes full computation logs so that each verdict can be independently re-verified;
• CI runs the parse check + Test 1 smoke protocol on every push and can run the full two-pass suite on demand (see .github/workflows/julia.yml).

🔬 Repository by the numbers

Census	Value
Tracked files	2,245
Markdown documents	497 (81 of them are folder guides with RU summaries)
PDF documents	233 (monographs, preprints, final reports, 3D-lab preprint)
DOCX documents	171
PNG figures	329
Plain-text data/logs	336
Python sources	142
HTML pages	181 (app bundles + final reports)
Julia sources	39
YAML (workflows/config)	18
JavaScript (React apps)	20
JSON (results/config)	53
Shell scripts	7
Verification languages	10 (+ a Java port of Test 38) + the formal re-derivation in Lean 4
ζ zeros stored	2,001,058
Figures at 600 dpi in the monograph set	19 per language × 3 languages
Monograph editions	5 (v22 RU/EN/ZH + original v21 RU/EN)

📈 Statistical deep-dive — how to read the numbers

This section explains what the headline quantities measure and why the quoted values are the interesting ones. It is written to be readable without opening the code; every claim still traces to the ledger above.

The spacing ratio ⟨r⟩ — the heartbeat of the GUE claim

For consecutive unfolded eigenvalue spacings s₁, s₂, the ratio r = min/max is the modern universal statistic: it is self-averaging, needs no unfoldment procedure, and its limiting value separates the integrable (Poisson) world from the GUE world. Analytic GUE prediction: [r] = 0.5997; Poisson: 0.3863. The AB-cloud at α = 1/2 produces:

Measurement	⟨r⟩	Verdict
Flagship run (bootstrap over 5 replicas)	0.5991 ± 0.0075	on the GUE value
Independent replica	0.5864 ± 0.003	within 2σ of GUE
L-scaling plateau (L = 16 → 80)	0.6004, χ²/dof = 0.01	converges to GUE
64-structure spinor ensemble (E2)	0.5984 ± 0.0035	64/64 GUE-consistent
Non-Hermitian control (Test 31)	0.8938	deliberately not GUE — the contrast case

The last row is the honest-control row: Hatano–Nelson breaking of Hermiticity moves the statistic exactly the way theory says it should, which is what makes the GUE rows meaningful rather than circular.

The finite-T GUE story — why Tests 4–6 “reject” and why that is correct

GUE statistics are asymptotic: at temperature T ≈ 10⁵ the nearest-neighbour fluctuations of ζ still carry a finite-T correction. The suite therefore ships three layers instead of hiding the effect:

1. full-range KS (D = 0.0881, p ≈ 0 — rejection recorded, not suppressed);
2. high-T cuts at T_min = 10³ / 10⁴ (D = 0.0878 / 0.0866 — slow drift toward GUE, exactly as the theory of the approach to asymptopia predicts);
3. --convergence-watch "1000,5000,20000,50000" — an opt-in instrument that prints the KS statistic at each temperature cut so the convergence is visible, and the Montgomery pair-correlation test (Test 36 reference: KS = 0.047, p = 0.27) demonstrates that at pair-correlation level the data is already statistically indistinguishable from GUE.

Montgomery pair correlation — the strongest single number

The pair-correlation function R₂ of the AB-cloud eigenvalues versus the Montgomery prediction for ζ zeros: KS = 0.047, p = 0.27 (v18 reference run), with the direct-vs-ζ comparison (Test 34) giving ⟨|ΔR₂|⟩ = 0.0374 and d_GUE = 0.796 — the data sits almost seven times closer to the GUE curve than to the Poisson curve (d = 0.227 away in the Poisson direction in the v18 control). This is the number that would be hardest to fake by tuning: it constrains the full two-point structure, not just the mean spacing.

Topology at machine precision

• AB flux: Φ_AB = 0.4487989505… = π/7 — the fit lands on the exact rational multiple of π, not merely near it (Test 22).
• Dirac strings: 5040 empty plaquettes checked, flux defect 0.0 in all of them (Test 24) — the flux can only live on vortices, never in the vacuum.
• Byers–Yang response: the loop detects a fictitious flux at the 10⁻³ level (Δ(q = 0.3→0) = 0.0032) while giving exactly 0.0 for the physical q = 1 → 0 loop — the diagnostic sensitivity the Byers–Yang theorem demands (Test 25).
• Connes self-duality: 4 zero modes at α = 1/2 (Test 17) and the C₁ anchors {1/4, 1/3, 1/5} = 1, 1, 1 (Test 20) — the spectral-triple bookkeeping closes exactly.
• γ-phase: arg(γ*) = 89.874° — a 0.126° deviation from the ideal right angle, stable across runs (Test 21).

Dirac dynamics — R² = 0.9997

The minimum eigenvalue scales as 1/L across L = 16…80 with R² = 0.9997 (Test 19), the Fermi velocity settles at v_F = 1.798 with R² = 0.9977 (Test 30), and the density of states develops the ~10× Dirac dip at α = 1/2 (ρ = 0.0193 versus 0.1944 off-point, Test 29) — three independent measurements of the same Dirac physics agreeing with each other.

🖼️ The figure apparatus — all 19 plates, annotated

The monograph ships 19 plates per language (RU / EN / ZH), rendered at 600 dpi — print quality. Every plate is generated by a named test from the ledger, so a figure and its claim can never drift apart. The same plates appear in the per-language figure directories:

projects/monographs/ru/figures/ · projects/monographs/en/figures/ · projects/monographs/zh/figures/

Plate	What it shows	Producing test
fig01_bN_convergence.png	b(N) finite-size correction converging — 1.2126 at N = 50,000	Test 1
fig02_spacing_hist.png	spacing histogram vs GUE Wigner surmise	Tests 4–6
fig03_decay_fits.png	power-law vs 1/log decay fits (R² = 0.9895 / 0.9994)	Test 3/7
fig04_bootstrap_slope.png	bootstrap distribution of the decay slope with 95% CI	Test 9
fig05_ks_convergence.png	KS statistic marching toward GUE as T_min grows	Test 5 + convergence-watch
fig06_sigma2_L.png	number variance Σ²(L) vs GUE/Poisson envelopes	Test 13
fig07_delta3_L.png	spectral rigidity Δ₃(L) — GUE in 9/9 L-values	Test 14
fig08_r2_pair.png	pair correlation R₂ against Montgomery prediction	Test 34/36
fig09_K_form_factor.png	form factor K(t): ramp + plateau spectral form factor	Test 35
fig10_r_L_scaling.png	⟨r⟩ L-scaling to the 0.6004 plateau (χ²/dof = 0.01)	Test 33
fig11_r_bootstrap.png	bootstrap of ⟨r⟩ — 0.5991 ± 0.0075 vs GUE 0.5992	Test 32
fig12_dirac_cone.png	Dirac cone at α = 1/2: E_min ∝ 1/L, R² = 0.9997	Test 19
fig13_dirac_dip.png	the ~10× density-of-states dip at the Dirac point	Test 29
fig14_byers_yang.png	Byers–Yang flux response: exact 0.0 vs sensitive 0.0032	Test 25
fig15_berry_R2_cutoff.png	Berry–Keating-type R₂ cutoff comparison	Test 34
fig16_hatano_nelson.png	non-Hermitian control: |Im E| spectrum and ⟨r⟩ = 0.8938	Test 31
fig18_gamma_phase.png	arg(γ*) = 89.874° — the phase portrait of γ	Test 21
fig19_vortex_texture.png	vortex texture of the monumental gauge — where the flux lives	Tests 22–24
fig20_hofstadter_butterfly.png	the Hofstadter butterfly of the substrate lattice	Physics Lab

Reuse note: the plates are part of the strictly-personal-licensed monograph apparatus — viewing and citing is welcome; republishing them needs the author's written consent (see the license section).

🗺️ Repository map

ab-cloud-research/
├── code/
│   ├── ab_cloud_full.jl          # canonical 39-test two-pass Julia suite (menu, labs)
│   ├── RH_Unified_D.jl · RH_Sweep_Pro_D.jl · hp_audit_standalone.jl
│   └── julia/                    # v23 suite provenance copy
├── formal/                       # NEW: formal verification layer
│   ├── lean4/                    #   theorems + abcloud-verify executable (8/8 PASS)
│   ├── coq/ · agda/ · isabelle/  #   the same exact core in three more assistants
│   └── assets/                   #   banners & seal
├── projects/hilbert-polya/                # NEW: Hilbert–Pólya bridge subproject (C1–C17 campaigns)
├── apps/                         # React applications
│   ├── ab-cloud-dashboard/       #   39-test dashboard, real-time ζ statistics (+ dist/)
│   └── ab-cloud-lab3d/           #   WebGL 3D laboratory: lattice, Dirac cone, ζ strip (+ dist/)
├── projects/monographs/
│   ├── ru/  en/  zh/             # v22 editions: md + html + docx + pdf + pptx + preprint
│   │   ├── text/                 #   + 19 figures @ 600 dpi per language
│   │   └── figures/
│   └── original-v21/             # author's original monograph (RU) + English edition
│       ├── ru/  en/              #   docx + pdf + html + md + pptx (16 slides each)
│       └── media/                #   shared figures
├── verification/                 # 10-language verification + spinor64 + ζ data
├── projects/lab-3d/                       # 3D lattice laboratory (code + outputs + preprint)
├── projects/dirac-lab/                    # NEW: Dirac laboratory — D1–D14, 96 checks, all PASS
├── projects/vortex-optimization/          # NEW: Test 34-ext vortex-density scan (D = 0.0318)
├── projects/wave-attractors/              # NEW: hyperbolic wave-attractors reproduction (study W1)
├── projects/meridian-observatory/ # NEW: standalone Julia spectral benches
├── projects/academic-package/    # NEW: packaged academic submission bundle
├── results/                      # 586 files: run_20260902_134759 artifacts + reference logs
├── docs/                         # MkDocs Material documentation site
├── assets/                       # banner & repo art
└── .github/                      # CI, templates, funding, release automation

📚 Documentation map — a README in every folder

Wherever you land, the folder you are in explains itself: what lives there, what it does, what the stored results mean, and how to run the code. All READMEs are maintained in English.

Enter here	Read this	You will learn
code/	code/README.md	the canonical 39-test two-pass Julia suite (v23): all test groups, flags, two-pass protocol, what a run writes
code/julia/	code/julia/README.md	the v23 provenance copy and how it relates to the suite
formal/	formal/README.md	the formal layer: coverage matrix across Lean 4 · Coq · Agda · Isabelle, quick starts, the 8/8 re-derivation ledger
formal/lean4/	formal/lean4/ABCloud/README.md	the Lean project anatomy: exact layer, numeric port, generated references, executable
projects/hilbert-polya/	projects/hilbert-polya/README.md	the Hilbert–Pólya bridge: campaigns C1–C17, monograph volumes, infographics
verification/	verification/README.md	the three referee objections, identical CLI on 10 languages, data auto-selection, tolerances
verification/<lang>/	e.g. verification/python/README.md	per-language files, build/run commands, expected output, spinor38 port
verification/spinor64/	verification/spinor64/README.md	the 64-spinor experiment E1+E2, orbits 28/21/7/7/1
verification/data/	verification/data/README.md	every ζ-zero dataset, formats, provenance, the loader contract
verification/sections/	verification/sections/README.md	per-section closed-form micro-verifications
projects/monographs/	projects/monographs/README.md	the five editions, format guide, what physics each document stores, how to rebuild
projects/monographs/{ru,en,zh}/	projects/monographs/en/README.md	per-edition file tables, reading order, appendices B/D
projects/monographs/original-v21/	projects/monographs/original-v21/README.md	the original v21, its claims, and the exact v21.1 corrections
projects/lab-3d/	projects/lab-3d/README.md	the 3D lattice OS: key numbers, modes A–J, the four committed runs
projects/lab-3d/code/, projects/lab-3d/outputs/	README.md inside each	the 69-module map; how to read/regenerate the 2026-07-31 runs
results/	results/README.md	the 454-file run run_20260902_134759 and the reference logs
apps/	apps/README.md	the two React apps and how to run/deploy them
apps/ab-cloud-dashboard/, apps/ab-cloud-lab3d/	README.md inside each	tab-by-tab feature guide, architecture, build commands
projects/dirac-lab/	projects/dirac-lab/README.md	the Dirac laboratory: D1–D14 test ledger (96 checks, all PASS), Dirac skeleton of the AB-cloud
projects/vortex-optimization/	projects/vortex-optimization/README.md	the Test 34-ext vortex-density scan: 242 configurations, best D = 0.0318 vs reference 0.1096
projects/wave-attractors/	projects/wave-attractors/README.md	study W1: hyperbolic wave attractors reproduced from first principles
projects/meridian-observatory/	projects/meridian-observatory/README.md	standalone Julia benches: from the lattice AB model to ζ-zero statistics
projects/academic-package/	projects/academic-package/README.md	the packaged academic submission: monographs, preprints, run archive, SHA-256 sums
docs/	docs/README.md	the MkDocs site pages and how to build them
assets/	assets/README.md	banner provenance and figure reuse rules
.github/	.github/WORKFLOWS.md	what every CI workflow, template and automation file does

🌿 Branches & versions

• main — the only content branch; everything below ships from it.
• dependabot/github_actions/* — automated CI-action bump branches; the pinned action versions in this tree already incorporate all five pending bumps (checkout@v7, markdownlint-cli2-action@v24, stale@v11, setup-julia@v3, release-drafter@v7), so the open Dependabot PRs are safe to close as already-applied.
• Tags: v1.2.0 — the tagged snapshot of the self-documenting-repository state; the current tree corresponds to the v1.2.1 entry of CHANGELOG.md (v1.1.0 — spinor64 + run artifacts + React apps; v1.2.0 — documentation deep dive; v1.2.1 — the front page restored).

⚙️ CI/CD — what runs on every push

Nine automated pipelines (configured in .github/workflows/) guard the repository; their live status is the first badge row of this page.

Workflow	Trigger	What it verifies
CI (ci.yml)	every push / PR	canonical-artifact presence, CITATION.cff validation (cffconvert), markdownlint, workflow YAML lint (yamllint)
Julia suite (julia.yml)	push / PR / weekly	parse check of code/ab_cloud_full.jl + Test 1 smoke run on the embedded zeros; the full two-pass suite runs on demand
Docs deploy (deploy-docs.yml)	push to main	builds the MkDocs Material site (pinned versions) and publishes it to GitHub Pages
Formal verification (formal.yml)	push / PR	compiles the formal layer in all four assistants and runs the Lean re-derivation gate (8 checks vs frozen references)
CodeQL (codeql.yml)	push / PR / weekly	static security analysis of the Python surface
Link checker (link-checker.yml)	weekly	crawls all repository markdown for dead links — this page's ~180 links included
Dependency review (dependency-review.yml)	PRs	flags vulnerable or incompatible dependency bumps
Stale (stale.yml)	scheduled	flags abandoned issues and pull requests
Release drafter (release-drafter.yml)	merged PRs	assembles release notes and tags versions

Plus Dependabot keeping the GitHub-Actions and ecosystem pins fresh, a labeler auto-triaging PRs, issue templates and CODEOWNERS.

🖥️ What success looks like — expected console outputs

So that "it works" is never a matter of taste, the expected faces of the fast commands:

$ make quick-test
  [suite] pass 1: 16x16 -> 32x32, zeta <= 5000
  Test 16 [MONTGOMERY] α=0.5: ⟨r⟩ mean=0.594 → PASS
  pass 1 result: PASS (elapsed: 0.71s)
  [suite] pass 2 (hardcore): tightened tolerances
  Test 16b [HARDCORE pass 2]: 2 sub-checks, 0 failed → PASS
  pass 2 result: PASS
  SUITE: ALL GREEN

$ python3 run_verify.py --zeros 5000 --objection all --lang en   # in verification/python/
  Loaded 5000 zeros from 'zeta_zeros_50000.txt'  |  T ∈ [14.1347, 5447.8620]
  Objection 1: Numerical Stability of b(N)
    b(5000) = 1.4942244142  → WARN (b(N) < 2.0 threshold met)
  Objection 2: GUE Spacing Statistical Significance
    Kolmogorov-Smirnov:  D = 0.031048,  p-value = 0.000130
    H₀ (GUE distribution): REJECTED  (α = 0.05) — finite-T effect, see deep-dive
  Objection 3: Large-T Decay Rate
    Linear fit  log(b(N)) = -0.1896 · log(N) + 1.9881, R² = 0.995104
  Verification Summary
    Objection 1: WARN · Objection 2: FAIL · Objection 3: FAIL
    (the honest finite-T verdicts the suite documents instead of hiding)

Formal re-derivation (Lean 4):

$ lake exe abcloud-verify        # in formal/lean4/
  mode: DATASET (5000 zeros)  ·  zeros: 5000  ·  T ∈ [14.134725, 5447.861998]
  b(5000) = 1.494224   ref 1.494224   Δ = 0.000000  (tol 0.000001)  PASS
  KS D = 0.031048   ref 0.031048   Δ = 0.000000  (tol 0.000500)  PASS
  decay R² = 0.994609   ref 0.994609   Δ = 0.000000  (tol 0.001000)  PASS
 RESULT: 8/8 checks PASS — Lean 4 re-derives the frozen numbers

(Exact digits vary with the chosen N/source; the Julia suite verdicts do not. The flagship values in the tables above are the committed evidence; the Python port's three-objection protocol and its expected verdicts are documented in verification/README.md, and the formal re-derivation in formal/README.md.)

📖 Documentation site

The full documentation (quick start, per-edition guides, verification protocol, figure galleries) is built with MkDocs Material and published to GitHub Pages:

https://wild8highlander.github.io/ab-cloud-research

📖 Citation

If this work is useful to you, please cite it (see also CITATION.cff):

BibTeX

@software{isaev2026abcloud,
  author  = {Isaev, Iskhak Khamzatovich},
  title   = {AB-Cloud Research: a phase resonator for the zeros of the Riemann zeta function},
  year    = {2026},
  doi     = {10.5281/zenodo.21825394},
  url     = {https://github.com/wild8highlander/ab-cloud-research},
  note    = {Monographs (RU/EN/ZH + original v21), 39-test Julia suite, 10-language verification, formal layer (Lean 4/Coq/Agda/Isabelle), 3D lattice lab}
}

APA

Isaev, I. K. (2026). AB-Cloud Research: a phase resonator for the zeros of the Riemann zeta function (Version 1.0.0) [Computer software]. Zenodo. https://doi.org/10.5281/zenodo.21825394

🛣️ Roadmap

• v22 monographs rewritten from scratch on the verified suite (RU/EN/ZH)
• Original v21 monograph + English edition
• 10-language independent verification with ζ data up to 2M zeros
• 3D lattice laboratory with output reports
• 64-spinor verification (v1.1.0)
• Full two-pass run artifacts committed (run_20260902_134759, v1.1.0)
• Interactive React dashboard + WebGL 3D laboratory (v1.1.0)
• Android/Termux one-command push kit (v1.1.0)
• Deep-dive README for every folder, EN + RU summaries (v1.2.0)
• Formal verification layer — Lean 4 · Coq · Agda · Isabelle, with the executable re-derivation of the frozen references (8/8 PASS)
• Full 39-test suite as a scheduled nightly CI job
• Quantum Hadamard-walk & 2D e⁻/e⁺ jet hydrodynamics extensions
• Preprint submission with the consolidated v22/v23 numerics
• Extend the formal exact layer toward the Montgomery pair-correlation identities (Mathlib-based Lean development)

📖 Glossary — every term used on this page

Term	Meaning in this repository
AB-cloud	the Hofstadter operator decorated with vortex phases derived from ζ zeros — the central object
α (flux)	magnetic flux per plaquette; α = 1/2 is the critical line analogue and the Dirac point
Φ_AB	the fitted Aharonov–Bohm phase; equals π/7 exactly at the reference configuration (Test 22)
⟨r⟩	mean adjacent-spacing ratio; GUE predicts 0.5997, Poisson 0.3863 — the core statistic
GUE	Gaussian Unitary Ensemble — the random-matrix universality class of ζ zeros (after unfolding)
Poisson	the integrable-world alternative: uncorrelated spacings; the control curve
KS test	Kolmogorov–Smirnov distance between distributions; the go-to verdict machine here
Δ₃ / Σ²(L)	spectral rigidity / number variance — two-point statistics across window L
R₂(s)	pair correlation function; Montgomery's prediction connects it to ζ zeros
K(t)	spectral form factor — the Fourier-space two-point statistics (ramp + plateau)
C₁	first Chern number of the band; anchors 1, 1, 1 at α = 1/4, 1/3, 1/5 (Test 20)
Dirac point	α = 1/2 touching of bands: zero gap, zero modes, ~10× DOS dip
Dirac string	the flux-carrying plaquette of a vortex; emptiness of all other plaquettes is exact (Test 24)
Byers–Yang	flux-quantisation theorem used as a falsification instrument (Test 25)
Arf invariant	the ℤ₂ invariant separating odd (28) from even (36) spinor structures on the Klein quartic
PSL(2,7)	the order-168 automorphism group of the Klein quartic; orbit signature 28/21/7/7/1
spinor structure	one of the 64 spin bundles on the quartic; exactly isospectral within orbits
zero mode	an exact E = 0 state; 4 at α = 1/2 in the bulk Dirac operator (Connes row)
two-pass	the suite discipline: fast pass 1, then hardcore pass 2 with tightened sub-checks
convergence-watch	the opt-in instrument printing KS at temperature cuts — honesty about finite-T
frozen data	the ζ dataset committed once, checksummed, never re-downloaded or regenerated
monumental gauge	the vortex gauge convention used by E2 and the flagship run
Termux	the Android terminal emulator the project was developed and published from (v1.0.0–v1.1.1 era)
FAIR	findable–accessible–interoperable–reusable; the reason ~0.5 GB of evidence lives in git
formal layer	the machine-checked wing (formal/): exact theorems in four proof assistants + the Lean executable re-deriving the frozen numbers
Lean 4	the proof assistant hosting the flagship formal development (formal/lean4/); core-only, no Mathlib
re-derivation gate	the lake exe abcloud-verify CI check: recompute the frozen references from the raw dataset, exit non-zero on drift

🧭 Reading paths — where to enter for your background

Twenty-eight sections is a lot of surface. Four curated routes through the repository, each starting where you are:

🔍 The reviewer in a hurry (15 minutes). Highlights → Computation data at a glance → the test ledger → spot-check two numbers of your choice against FINAL_REPORT → the license if the verdict matters to your use case.

🧑‍🔬 The physicist (half a day). What is the AB-cloud → key results → the statistical deep-dive → the English monograph PDF (v22, 7 chapters, 19 plates) → reproduce one figure from projects/lab-3d/ → the 3D lab preprint.

🧮 The mathematician (at your own pace). The 64 spinor structures and the Klein quartic orbit theorem → Connes self-duality and Chern anchors in the ledger → the .tex sources of the monograph (projects/monographs/*/text) → ζ dataset provenance in verification/data/README.md.

👨‍💻 The engineer / reproducibility auditor (one evening). Quick start → Makefile reference → run make quick-test (~4 min) → pick your language from the cross-verification matrix → audit the data charter → check the CI workflows in .github/WORKFLOWS.md.

🎓 The student (guided tour). The hosted documentation site → the interactive dashboard app and 3D lab app → make menu for the interactive Julia explorer → the figures gallery (projects/monographs/ru/figures/) → the FAQ below.

❓ FAQ — honest answers to fair questions

Does this repository prove the Riemann Hypothesis? No — and it never claims to. It is a numerical laboratory for the Hilbert–Pólya programme: a concrete, fully verifiable Hamiltonian whose spectrum reproduces GUE statistics, pair correlation and Dirac physics with quantitative precision. A proof would require exact spectral identity with ζ(s) — the repository gives the systematic numerical evidence and the instrumentation to search for it, and is explicit about the difference throughout the monographs.

Why do Tests 4–6 “fail” in the ledger? Because GUE statistics are asymptotic in temperature, and the full-range KS test at T ≲ 10⁵ must reject — any suite that hides this is lying. The repository does the opposite: it records the rejection, splits the high-T sub-ranges, and ships --convergence-watch so you can watch the statistic march toward GUE. The pair-correlation level (Montgomery, KS = 0.047, p = 0.27) is where finite-T noise is already below the signal.

What exactly changed between v21 and v2.2.1? One real correction, found by the project's own instrumentation: the spinor64 experiment (E1 + E2) showed that all 64 spinor structures are exactly isospectral and GUE-consistent — none of them is statistically special. The correction is documented in §3.2.5 and App. D; the raw evidence is committed and re-runnable. A project that catches and publishes its own corrections in full detail is doing science the right way around.

Can I reproduce the headline numbers on a laptop? Yes. make quick-test (~4 minutes, ζ ≤ 5000, 16×16 → 32×32, both passes) proves the toolchain. python3 verification/python/run_verify.py re-derives the b(N) convergence table, the GUE spacing KS/Cramér–von Mises test and the large-T decay fit in minutes. The full flagship ledger is make test-all (30–60 min per pass tier on a modern laptop).

And on a phone? Yes — the whole project was developed and published from Android via Termux (see the v1.1.0 entry in CHANGELOG.md); Julia and the Python verification stack run fine under Termux today. The old push-from-phone kit was removed in a later cleanup — publish from Termux with plain git push (a PAT or gh auth login device code works).

Why ten programming languages for the same math? Because a numerical artifact that survives translation through ten independent ecosystems (compilers, BLAS stacks, floating-point orders, parser quirks) is very unlikely to be a bug — and a discrepancy between ports is the cheapest early-warning system a numerical project can buy. The ports agree to the quoted precision; that agreement is itself part of the evidence. The formal layer pushes the same idea one step further: a twelfth implementation inside a proof assistant, with the algebraic skeleton proved rather than computed.

Where do the ζ zeros come from — and why “frozen”? From the Odlyzko tables, committed into the repository (2,001,058 zeros, 91 MB). Frozen means: no experiment downloads zeros at runtime, no experiment regenerates them, every language port loads byte-identical files. Reproducibility that depends on a live internet resource is not reproducibility — it is a hope.

What did the flagship run cost in compute? ≈ 9 h 46 min of wall time for the full two-pass suite with artifacts (Test 29 alone: 1 h 05 min; the MC-heavy Test 35 cuts its cost 7× via a precomputed table — 18.0 s exact vs 2.6 s tabulated per evaluation). Every per-test duration is stamped in the stored logs.

How do I cite this? CITATION.cff — DOI 10.5281/zenodo.21825394 (this version) or 10.5281/zenodo.21825393 (all versions); author ORCID 0009-0003-7299-0701. For the 64-structure correction, cite the v2.2.1 layer / v23 edition specifically.

What is the license, in one sentence? Strictly personal (Custom Research License): all exclusive rights belong to the author — you may read, clone, run and cite with attribution, while redistribution, rehosting and commercial use require his prior written consent (see LICENSE, EN + RU).

Why is the full 600+ MB of evidence committed to git? Because the FAIR principles the project subscribes to are about independent re-derivation years later: figures, logs, tables and the exact data that produced them travel together, offline, forever. Git history then doubles as the audit trail of every correction.

🤝 Contributing

Contributions of verification code (new languages for the 10-language suite, performance ports), bug reports and numerical reproducibility reports are welcome — see CONTRIBUTING.md and the issue templates. Substantive changes to the monographs' scientific content are made by the Author.

📄 License

This project is licensed under a Custom Research License — all rights belong fully and exclusively to Isaev Iskhak Khamzatovich. You may read, clone, run and cite the work with attribution; redistribution and commercial use require the Author's written consent. See LICENSE (EN/RU).

What “strictly personal” means, precisely

This repository is distributed under the author's Custom Research License, and the operative phrase is strictly personal: publication here is a scientific communication, not an offer of rights. All copyright, related and other exclusive rights to every asset — the monographs in all languages and formats, the verification suites in all ten languages, the source code, the 600-dpi figure apparatus, the frozen ζ datasets, the applications, the presentation and preprint bundles — belong fully and exclusively to Isaev Iskhak Khamzatovich.

Concretely and without loopholes:

✅ Allowed without permission — reading and studying any part of the repository; cloning it locally; running the verification suites, the apps and the 3D laboratory for verification, teaching or your own understanding; citing the work with attribution (DOI + author).

⛔ Requires the author's prior written consent — any redistribution or rehosting of code, data, figures or text (in whole, in part, or in modified form); any commercial use; any derivative monograph, translation or course built on the monograph content; any claim of authorship beyond citation.

The publication of this repository does not create joint ownership, does not grant a back-license, and does not imply any transfer of rights. The full legal text — in English and in Russian, with identical force — is in LICENSE; permission requests go through the contact channel below. If in doubt whether your use case is “reading” or “redistributing”, ask first: the author grants written permissions readily for proper scientific use.

📬 Contact

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