Finding 1: An 11/11 axiom-6 ceiling across Hilbert–Pólya candidates
← All English posts · 한국어 · 2026-05-02
Empirical observation, not a theorem. No proof of RH. Universal NO is empirical, not necessary. ZFC-independence not ruled out. Falsifier explicit.
Statement
Across 11 paper-direct Hilbert–Pólya–style spectral candidates audited so far, none satisfies axiom 6 strict — defined (by 4-specialist consensus) as:
A single self-adjoint operator on a fixed Hilbert space, capturing all ζ non-trivial zeros bijectively, with no fine-tuning parameter.
Source: lemmas/lemma1_axiom6_ceiling.md.
The 11 audited candidates
| # | Candidate | axiom 6 strict | Paper-direct anchor |
|---|---|---|---|
| 1 | BBM 2017 (Bender–Brody–Müller) | NO | “We are not yet able to prove eigenvalues real” + boundary cond. ψ_z(0) = -ζ(z) per-zero circular |
| 2 | Sierra §III xp (Berry–Keating type) | NO | continuous spectrum, no point spectrum |
| 3 | Sierra §V H_I | NO | self-adjoint via parameter ϑ ∈ [0, 2π) — fine-tune; “not able to find single H encompassing all zeros at once” |
| 4 | Sierra 2007 H_2 | NO | deficiency indices (1,1), self-adjoint family parameterized by 1×1 unitary |
| 5 | Connes–Consani 2021 Θ(λ, k) | NO | special λ values only, 31 zeros 10⁻⁵⁰ random-match probability — not all zeros |
| 6 | Connes 1999 §VI/VII | NO | “unnatural parameter δ” — δ-family, not unique |
| 7 | Lagarias §8 (1) hypothetical | NO | λ = s² − 1/4 = −γ²+iγ complex (paper §8 self-acknowledged hypothetical) |
| 8 | Berry–Keating 1999 H = xp | NO | “no concrete proposal realizing all conditions” (Sierra 2007 §I quote) |
| 9 | Sierra 2007 §VI ζ_H Jost | NO | M2 family of (a, b) potentials → many candidates |
| 10 | Connes 1999 §III (ℋ_χ, D_χ) | NO | “δ > 1 Sobolev exponent — unnatural” — δ-family |
| 11 | Connes–Moscovici PNAS 2022 (UV prolate) | NO | UV asymptotic only (not exact match), λ ∈ {1, √2} fine-tune, deficiency (4,4) |
How the 4-specialist consensus on “axiom 6 strict” was built
The lemma defines strict YES as agreement across 4 viewpoints:
- NCG (S3): single self-adjoint D on fixed H, all ζ-zeros ↔ Sp(D) bijective, no fine-tune. Falsified if any ζ-zero missing.
- Quantum physics (S6): single PT-symmetric H, unbroken PT phase, biorthogonal complete eigenbasis. Falsified by broken PT or fine-tune.
- Analytic (S1): a single mollifier-method transformation capturing all zeros. Falsified by mollifier-family (Pratt–Robles 50% limit).
- Logician (S9): ZFC-provable “∃ unique H : Sp(H) = ζ-zeros imag part”. Falsified by existence-without-uniqueness, or ZFC-independence.
Common essence: no fine-tuning + simultaneous capture of all zeros.
The strongest test the lemma underwent (Cycle 6)
The lemma declares its own falsifier criterion (3 conditions). Cycle 6 ran an explicit test against Connes–Moscovici PNAS 2022 (“UV prolate spectrum matches the zeros of zeta”). Three paper-direct gaps were identified:
- UV asymptotic only — agreement is in a limit, not exact spectrum match
- Fine-tuning — λ ∈ {1, √2} parameter values
- Deficiency indices (4, 4) — self-adjoint extension is a 4-parameter family, not single H
So PNAS 2022 fails axiom 6 strict. The lemma is strengthened (10/10 → 11/11), not retracted.
What this is not
- Not a proof of necessary universal NO. From the lemma’s
Statusblock:“Necessary universal NO 미증명 — S9 logician 경고: 165 years empirical NO ≠ all-future-candidates NO. Induction is a leap.”
- Not RH progress. The lemma’s
Caveatsexplicitly says: “RH 진전 X — RH 의 *언어 변경 (Cut 6 위반 회피, 본 lemma 는 증명 시도 X, empirical record 만)”*. - Not closed under ZFC analysis. RH itself is Π_1 (Lagarias 2002); the ceiling’s logical strength is undetermined.
Why this might still be useful
The lemma is a structured checklist future spectral candidates can be tested against systematically. Specifically the falsifier criterion is explicit:
Falsifier (lemma 폐기 조건): 어떤 paper-direct candidate 가 axiom 6 strict YES 시 즉시 폐기. 검증 절차: 1. Single H on fixed Hilbert space — paper-direct quote. 2. 모든 ζ-zeros ↔ Sp(H) bijective — paper-direct verification. 3. Fine-tuning parameter 부재 — paper-direct quote 또는 explicit 정의.
If a new candidate ever passes all three, the lemma retracts. That is the discipline.
Audit trail (Layer 1)
lemmas/lemma1_axiom6_ceiling.md— the lemma itself with full audit tableattempts/184_cycle1_*— Cycle 1 (lemma generation)attempts/189_cycle6_*— Cycle 6 (PNAS 2022 falsifier test)attempts/010_*,109_*,110_*,111_*,117_*,133_*,178_*,182_*,183_*— per-candidate paper-direct readings
Refuting this finding
If you have a paper-direct candidate that strict-passes axiom 6 — please email x2ever.han@gmail.com. The lemma is genuinely falsifiable.