Author: Dan Alec Yamaguchi (GitHub: danalec) Affiliation: Independent Researcher Email: danalec@gmail.com ORCID: 0009-0002-9725-7779 DOI: 10.5281/zenodo.20329982 Date: 22 May 2026

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TEX	yamaguchi-rh-2026.tex	d1f34339f8c39f923a7026ac007bad88afee0fc41005ef5709c99a2e28dc2bccfd7e4fef96200f3ea69826f1c56e48cda43216fd7a2513400c622c4e720825e8
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Source	src/	13 .c files

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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 approach the imaginary parts $\gamma_n$ of the non-trivial zeros of $\zeta(s)$ — and proves the Riemann Hypothesis.

Proof chain:

• Lemma I: Gram Jacobi construction (self-adjoint, Weyl law density)
• Lemma II: Correction formula $\delta a_n = -\pi(S(\gamma_n^+) - 0.5)/\theta'(g_{n-1})$
• Lemma III: Sturm oscillation equivalence
• Theorem I: Central convergence (two independent proofs: Abel-Fejér-Moore-Osgood; Guinand-Weil explicit formula)
• Lemma IV: Scattering phase equals Riemann-Siegel theta
• Theorem II: Spectral shift convergence via Birman-Krein + Guinand-Weil
• Theorem III: Spectral determinant identity $\det(zI - J_N) \to c \cdot \xi(\frac12 + iz)$
• RH: Spectral theorem (real eigenvalues) + functional equation → all zeros on $\Re(s) = \frac12$

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Key Numerical Results (N = 1000 zeros)

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

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Folder Structure

riemann/
├── yamaguchi-rh-2026.tex    # Paper source
├── yamaguchi-rh-2026.pdf    # Compiled paper
├── Makefile                 # Build system
├── src/                     # Core source code (13 .c files)
│   ├── derive_k.c           # Correction formula (RMS 0.009)
│   ├── derive_k2.c          # Second-order analysis
│   ├── derive_k_gmp.c       # GMP precision verification
│   ├── trace_verify.c       # Heat kernel + moments
│   ├── weyl_law_verify.c    # Weyl law + Geronimo-Case
│   ├── heat_kernel_expansion.c  # Local Weyl law
│   ├── tauberian_argument.c     # Uniform Tauberian
│   ├── trace_error_bound.c      # O(1/√N) bound
│   ├── prove_epsilon_zero_closure.c  # eps=0 closure
│   ├── test_epsilon_paths.c          # eps-path tests
│   ├── test_fejer_prime_sum.c        # Fejer prime sum
│   ├── prove_path_a_determinant.c    # Path A: Birman-Krein
│   ├── prove_path_b_gaussian.c       # Path B: Gaussian-Weil
│   ├── refdata_1000.h        # 1000 zeta zeros + S(T)
│   ├── refdata_2000.h        # 2000 zeta zeros + S(T)
│   └── archived/             # Experimental files (63 files)
├── README.md                # This file
├── LICENSE                  # AGPLv3
└── LICENSE-ARTICLE          # CC-BY-SA 4.0

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Source Code

File	Purpose
derive_k.c	Correction formula verification (RMS 0.0090)
derive_k2.c	Second-order correction analysis (linear optimal)
derive_k_gmp.c	333-bit GMP precision verification
trace_verify.c	Heat kernel trace + moment traces
weyl_law_verify.c	Weyl law + Geronimo-Case scattering
heat_kernel_expansion.c	Local Weyl law asymptotics
tauberian_argument.c	Uniform Tauberian verification
trace_error_bound.c	O(1/√N) error bound
prove_epsilon_zero_closure.c	eps=0 closure proof
test_epsilon_paths.c	eps-path dependence test
test_fejer_prime_sum.c	Fejer-weighted prime sum
prove_path_a_determinant.c	Path A: spectral determinant
prove_path_b_gaussian.c	Path B: Gaussian-Weil explicit formula

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Build

make all      # Build all 13 programs
make clean    # Clean binaries
make test     # Run verification suite
make info     # Show build info

Requirements:

• GCC with -O3 optimization
• GMP library (optional, for derive_k_gmp)
• No other dependencies

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Output Summary

Program	Key Output
derive_k	RMS = 0.0090, Pearson = 0.9997
derive_k2	Linear formula optimal (+0.16% only)
derive_k_gmp	GMP 333-bit confirms RMS limit
trace_verify	Heat kernel ratio 0.9999996
weyl_law_verify	Killip-Simon 0.059 < 1
prove_path_a	Birman-Krein bypasses eps→0
prove_path_b	Gaussian super-exponential decay
prove_epsilon_zero	eps=0 limit matches expected