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Riemann Hypothesis Campaign 2026 — Complete Index

CIPHER Research · Jean-Paul Niko · April 7–8, 2026

Status: CLOSED. Seven approaches attempted. Seven structural obstructions identified. Two theorems proved. Zero proofs of RH.

The One-Sentence Problem

Construct a canonical, finite-information operator from primes whose spectrum is rigid enough to determine individual eigenvalues (zeros) without importing them.

Proved Results

Result Statement Page
Theorem A \(\partial_\sigma \ln\lvert\xi\rvert = 0\) at \(\sigma = 1/2\) Entropy Valley
Theorem B \(\partial^2_\sigma \ln\lvert\xi\rvert > 0\) at \(\sigma = 1/2\) Entropy Valley
Theorem C Zero-free region: \(\beta - 1/2 > \delta/\sqrt{2}\) Entropy Valley
Equivalence RH \(\iff\) monotonicity of \(\lvert\xi\rvert\) Entropy Valley
Reduction RH \(\iff\) \(\delta_{\max} < 1\) Jensen
\(\dot\Sigma > 0\) Entropy monotonicity (Lindblad/Spohn) Master Ref v4
\([Q, \Sigma] = 0\) Entropy is BRST-closed Claim B

The Seven Walls

Round Approach Wall Page
1 L² norms Rigged Hilbert space RH-002
2 Entropy-weighted norms ω/Tate/Goldilocks Claim B
3 Equivariant Lefschetz Non-compact, wrong fixed points Lefschetz
4 Average → pointwise = Lindelöf = RH Jensen
5 GL universality = Montgomery = RH GL Universality
6 \(\mathbb{F}_1\) Hodge Infinite genus → Castelnuovo vacuous F₁ Condensate
7 Kato-Rellich Singular measure jump, not operator perturbation Post-Mortem

Structural No-Go Results

Result Statement Page
Norm no-go No weight makes ω unitary + recovers ζ + regularizes selectively Norm-Free Attack
Topology no-go Lefschetz inapplicable (non-compact, wrong fixed points) Lefschetz
Algebra no-go Castelnuovo vacuous for infinite genus F₁ Condensate
Perturbation no-go Smooth → exact is not an operator perturbation Post-Mortem

The Meta-Obstruction

Any framework producing large continuous spectra, infinite-dimensional state spaces, or relying on global averaging cannot isolate zeros sharply enough. The successful strategy must preserve discrete arithmetic structure and achieve spectral rigidity from finite local data.

Key Insights

Insight Description Page
Entropy Valley Critical line is transverse minimum of \(\lvert\xi\rvert\) Entropy Valley
Compensation Mechanism Zeros cluster where primes conspire Jensen
\(C = \text{Fr}_1\) Instantiation = Frobenius at \(q = 1\) F₁ Condensate
AEI = Arakelov Action-Entropy Identity = metric at infinity F₁ Condensate
Jacobi from primes Operator exists, Carleman holds, but eigenvalues don't match Post-Mortem

All Pages (Chronological)

  1. CIPHER-2026-RH-002: The Riemann Hypothesis from Weil Unitarity — Original proof (Round 1, killed)
  2. RH Metaplectic Attack — Architecture page (updated to CLOSED)
  3. Adelic-BRST Bridge — Path 3 framework (Round 2)
  4. Claim B: Entropy-Weighted Inner Product — Weighted norm (Round 2, killed)
  5. Equivariant Lefschetz Attack — Topology (Round 3, killed)
  6. Norm-Free Attack Post-Mortem — No-go results from Rounds 1-3
  7. Entropy Valley Theorems — Theorems A, B, C (proved)
  8. Jensen + Entropy Dominance — Analytic framework (Round 4)
  9. GL Universality and Gap Bound — Moments framework (Round 5)
  10. F₁ Condensate — Algebraic geometry pivot (Round 6, killed)
  11. Campaign Post-Mortem — Final document with all 7 rounds
  12. BuildNet Dispatch — Adversarial review dispatch

COG Ledger

Agent COG
@D_GPT 30,000
@D_Gemini 22,500
Total 52,500

The Next Attack

Entry point: Local spectral reconstruction from oscillatory perturbations of measures.

Required: A canonical, finite-information operator from primes with spectral rigidity. Not blocked by any known theorem. Sits at the intersection of orthogonal polynomial theory, inverse spectral problems, and analytic number theory.

Bounty: 100,000 COG for a proof of RH. Open indefinitely.