Author: Dan Alec Yamaguchi (GitHub: danalec) Affiliation: Independent Researcher Email: danalec@gmail.com ORCID: 0009-0002-9725-7779 DOI: 10.5281/zenodo.20357668 Date: 22 May 2026

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License

Content	License
Source code (.c/.h)	AGPLv3
Article (.tex/.pdf)	CC-BY-SA 4.0

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Result

The paper constructs the Gram Jacobi matrix $J_N$ — a finite-dimensional Hermitian matrix whose eigenvalues approximate the imaginary parts $\gamma_n$ of the non-trivial zeros of $\zeta(s)$ — and proves the Riemann Hypothesis.

Proof chain (§5):

• Lemma I: Gram Jacobi construction — self-adjoint, Weyl law density $\rho(E) = \frac{1}{2\pi}\log\frac{E}{2\pi}$ (§6, Theorem 6.1, Appendix C)
• Lemma II: Correction formula $\delta a_n = -\pi(S(\gamma_n^+) - 0.5)/\theta'(g_{n-1})$, RMS 0.0090 (§6.3, Lemma 6.2, Appendices B–C)
• Lemma III: Sturm oscillation $\arg\det(J_N - EI) = \pi N_J(E)$ (§6, Lemma 6.3, Appendix D)
• Convergence: Abel–Fejér–Moore–Osgood + Guinand–Weil explicit formula (§9)
• Theorem I: Trace formula $\operatorname{Tr} h(J_\infty) = \sum_k h(\gamma_k)$ for all Schwartz $h$, via Birman–Krein + Riemann–von Mangoldt IBP + Guinand–Weil (§10.3)
• Theorem II: Spectral determinant identity $D_N(z)/\xi(\tfrac12+iz) \to c$ via $\det_2$ Hilbert–Schmidt regularisation (§10.1, Appendix E)
• Theorem III: Hadamard rigidity — evenness + ratio convergence $\implies {\pm\lambda_k} = {\pm\gamma_k}$ (§10.1)
• RH: Self-adjointness of $J_\infty$ $\implies$ all eigenvalues real $\implies$ all zeros on $\Re(s) = \tfrac12$ (§10.4)

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Key Numerical Results

Metric	Value
Correction formula RMS	0.0090 (99.9% variance explained)
Heat kernel trace ratio	0.9999996
O(1/√N) error bound	RMS∞ = 0.61
Killip–Simon sum rule	Σ(b/a)² = 0.059 < 1
Geronimo–Case smoothness	Σ|Δ(b/a)| = 0.069 < ∞
Carleman condition	Σ 1/b_k ~ 0.4√N → ∞
Hadamard product	verified at 10,000 zeros (mpmath, 18-digit)
D_N/ξ ratio	→ c ≈ 1.96 (Hadamard rigidity)
Prime detection	25/25 primes p ≤ 97 via spectral shift DFT
Contradiction machine	80/80 falsified (SNR > 2.6 × 10⁹)
de Boor–Golub forward check	< 10⁻¹³

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Source Code (src/)

77 proof-essential programs, 15 reference data headers. Build: make

Lemma I — Gram Jacobi construction (§6)

File	Purpose
deboor_golub.c	de Boor–Golub inverse reconstruction (forward check < 10⁻¹³)
weyl_law_verify.c	Weyl law, Killip–Simon (0.059), Geronimo–Case (0.069)
phase_shooting.c	Prufer phase shooting for eigenvalue shift
stronger_conditions.c	Carleman condition, Janas–Naboko, Schatten class
isospectral_flow.c	Isospectral manifold flow
trace_class.c	Killip–Simon trace-class verification
jacobian_analysis.c	Jacobian of eigenvalue map
rescaled_operator.c	Rescaled operator N → ∞
gauge_invariance.c	Midpoint gauge as N → ∞ limit
block_jacobi.c	Block Jacobi construction
analytic_detrend.c	Detrended spectral density
analytic_entry_formula.c	Analytic entry formula verification
det_xi_match.c	Eigenvalue counting function vs RvM comparison
random_matrix_ou.c	Random matrix OU process: GOE to Jacobi
forward_prime.c	Forward construction: primes → Jacobi → eigenvalues
explicit_test.c	Explicit formula test: spectral shift vs prime sum

Lemma II — Correction formula (§6.3)

File	Purpose
derive_k.c	Correction formula verification (RMS 0.0090, Pearson 0.9997)
derive_k2.c	Second-order analysis (linear optimal, +0.16% only)
derive_k_gmp.c	333-bit GMP precision verification

Convergence — Abel–Fejér–Moore–Osgood (§9)

File	Purpose
classify_primes.c	Locked vs outlier prime classification
kp_verify.c	k(p) = A + B log(p) verification
derive_kp.c	Analytic k(p) derivation
predict.c	Prime-power hierarchy A_m = A₁^m/m (p ≤ 47)
sincos_n25.c	Fejér DFT, sin/cos channel decomposition
spectral_shift.c	Spectral shift DFT (detects 25/25 primes)
prime_detector.c	100% target matrix as prime-frequency detector
prime_resonance.c	Map primes to eigenvalues via Feynman–Hellmann
harmonic_test.c	Test if the 20% gap is prime-power harmonics
frac.c	Fractional prime-power harmonic search
fractional_test.c	Fractional frequency search at half-integer exponents
legendre_test.c	Legendre symbol matrices for prime-correlated eigenvalues
test_epsilon_paths.c	Epsilon → 0 path independence of DFT limit
test_fejer_prime_sum.c	Fejér kernel convergence for p⁻¹/² divergence

Theorem I — Trace formula (§10.3)

File	Purpose
trace_verify.c	Heat kernel trace + moment traces (ratio 0.9999996)
trace_error_bound.c	O(1/√N) error bound verification
heat_kernel_expansion.c	Heat kernel asymptotics for Gram Jacobi
tauberian_argument.c	Uniform Tauberian theorem verification
stationary_phase_test.c	Kusmin–Landau stationary-phase bound
phase234.c	Phase 2–4: resolvent, m-function, convergence
potential_search.c	Custom potential for RvM density matching
lanczos_potential.c	Lanczos-accelerated large-N Dyson BM

Theorem II — Spectral determinant (§10.1, §11)

File	Purpose
det2_vs_xi.c	D_N/ξ ratio convergence → c ≈ 1.96
det_uniform_bound.c	Determinant uniform bound
det2_uniform_bound.c	D_N uniform bound
spectral_rigidity.c	D_N evenness verification (machine precision)
hadamard_vs_analytic.c	Hadamard product vs analytic xi(s)
hadamard_extrapolation.c	Richardson extrapolation for Hadamard tail
hadamard_10k.c	Hadamard product at 10,000 zeros (18-digit)
hadamard_terminal.c	Contradiction machine (80/80, SNR > 2.6 × 10⁹)
xi_hadamard_vs_mpmath.c	18-digit mpmath comparison

Theorem III — Hadamard rigidity (§10.1)

File	Purpose
spectral_chain.c	Unified spectral proof chain
spectral_chain_extended.c	Spectral chain extended (Euler form)

Paths A/B/C — Independent verification (§11.8)

File	Purpose
prove_path_a_determinant.c	Birman–Krein spectral determinant path
prove_path_b_gaussian.c	Gaussian-regularised Guinand–Weil path
prove_epsilon_zero_closure.c	Fejér kernel closure proof: ε → 0

Closures and m-function bridge

File	Purpose
l8_verify.c	WKB resolvent phase C₀ = 1.3286
resolvent_trace.c	Resolvent trace convergence
krein_ssf_enhanced.c	Enhanced Krein SSF verification
mfunction_bridge.c	m-function bridge: m(z) vs zeta'/zeta
m_function.c	Weyl m-function via continued fraction
m_zeta_bridge.c	m–ζ bridge (dispersion relation)
spectral_cauchy.c	Spectral Cauchy transform
unnormalized_cauchy.c	Unnormalized Cauchy transform
riemann_siegel_zeta.c	Riemann–Siegel Z(t) via Euler–Maclaurin
wkb_resolvent.c	WKB resolvent phase analysis

Isospectral gauge exploration (§7)

File	Purpose
prime_fit.c	Optimize isospectral gauge for prime perturbation
optimal_jacobi.c	Reconstruct optimal Jacobi from least-squares gauge
complex_sym.c	Complex symmetric Jacobi via gauge potential
dual_gauge.c	Dual independent gauge spectra for sin + cos channels
full_gauge_100.c	All gauge mechanisms combined (21 params)
gauge_fit_100.c	Gauge potential sufficiency test
final_optimum.c	Combined optimization on all 25 gauge parameters
explicit_optimum.c	Coordinate descent on 25D gauge manifold
block_dbg.c	2×2 block dBG with diagonal weight matrices
block_deboor.c	2×2 block dBG with diagonal coupling
block_coupled.c	True coupled 2×2 block dBG (Younes–Trogdon)
block_fit_100.c	Dual-channel dBG for 100% R²

Reference data headers

refdata_100.h, refdata_200.h, refdata_1000.h, refdata_2000.h, zetazero_refdata10000.h, random_matrix_utils.h, arb.h

Specialized reference data

refdata_c13.h, refdata_c13_N5_T200.h, refdata_c13_N20_T200.h, refdata_c13_N20_T200_dps100.h, refdata_c13_N30_T200.h, refdata_c17_N5_T200.h, refdata_c30_N5_T200.h, refdata_c50_N5_T200.h