Lemma 4 — Failed Proof Categories

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Process lemma — 실패 RH proofs의 systematic critical reading 5-category framework. Atiyah 2018 case study가 5개 모두 manifest (Finding 4 참조).

Background

실패 RH proofs는 반면교사 — 과거 failure mode 분석으로 향후 spectral / analytic / arithmetic 후보가 같은 trap 회피.

5 categories

Category A — Trivial Circular

Spectrum identification이 trivially $\zeta$-zero condition과 동치.

• 상세 templete: Lemma 1 — Spectral Candidate Circularity Check.
• 예시: BBM 2017, Sierra 2007/2016 ($\psi_z(0) = -\zeta(z)$ boundary condition).

Category B — Reference Circular

핵심 객체의 well-definedness가 다른 paper에 의존, 그 paper가 unpublished or unverified.

• 예시: Atiyah 2018 → paper [2] (Royal Society 제출, preprint 시점 미발표). Atiyah의 $T$ function이 [2]에 의존, 2018 preprint 자체에 explicit construction X.
• Proof의 foundation이 citation에, paper 자체에 없음.

Category C — Identity Transplant

Equation이 limited domain (예: linear approximation)에서만 성립한다고 정의됐는데, proof에서 exact equality로 사용.

• 예시 (Atiyah 2018):
  • §2.6 (paper-direct, “weakly analytic” — 즉 linear approximation): $T{(1+f)(1+g)} = T{1+f+g}$
  • §3가 이를 $f = g = F$ ($F$는 작지 않음)로 exact equality 사용 → $F = 2F$ 도출.
  • Substitution이 §2.6의 stated domain of validity 외부.

Category D — Generic Multi-valued Inversion

$f(g(s)) = 0 \implies g(s) = 0$ 형태 step에서, inverse $f^{-1}$가 generically multi-valued임을 무시하고 single-valued처럼 취급.

• 예시 (Atiyah 2018): §3가 $T(1+w) = 1 \implies w = 0$를 inferring해서 $\zeta \equiv 0$ 도출. 그러나 $T^{-1}(1)$은 generically multi-valued (Atiyah의 Todd function = polynomial of bounded degree, multiple preimages).

Category E — Self-acknowledged Speculation

Paper 자체가 “the most general case is undecidable” 같은 disclaimer + “proof” claim 양립 시도. 둘 다 author의 settled position일 수 없음.

• 예시 (Atiyah 2018):
  • §3가 proof by contradiction claim.
  • §5: “The most general version of the Riemann Hypothesis will be an undecidable problem in the Gödel sense.”
  • §3 proof 맞으면 §5와 모순, §5 맞으면 §3와 모순. 둘 다 settled position 불가능.

평가 protocol — framework 적용

새 RH proof / spectral candidate에 5 check 적용:

1. A-check: Paper의 axiom/hypothesis가 결론과 trivially 동치?
2. B-check: 핵심 객체가 unpublished/unverified external work 의존?
3. C-check: Equation이 stated validity domain 외부에서 사용?
4. D-check: Function inversion이 uniqueness argument 없이 수행?
5. E-check: Paper의 self-acknowledged disclaimer가 proof claim과 모순?

2개 이상 trigger → 구조적 failure 강한 evidence.

Atiyah 2018 paper-direct deep verification (5/5 manifestations)

Atiyah 2018 §1–§5 deep read 시 5 categories 모두 manifest:

Category	Atiyah 2018 manifestation
B — Construction undefined	$T(s)$ explicit form paper-direct 부재 (Royal Society [2] 의존)
C — Property inconsistency	§2.6 (logarithm-like multiplicative→additive) + §2.7 ($T(1+s)=T(1+s/2)^2$ exponential) + polynomial degree $k(K)$ — 셋이 $T \equiv 1$ 외엔 inconsistent
C/D — Proof step ambiguity	§3.3의 “$F(s) = 2F(s)$” statement는 §2.6 + §2.7 만으로 paper-direct 도출 X (derivation은 Finding 4)
E — Self-contradiction	§3 proof by contradiction + §5 “RH undecidable in Gödel sense” — paper-direct self-acknowledged inconsistency
A/B — Not naturally arising	$T(s)$ construction artificial (Hirzebruch Todd polynomial + von Neumann fusion speculative) — 자연스럽게 등장하는 analytic object 아님

→ Paper-direct 5/5 categories all manifested in Atiyah 2018, self-contained.

§3.3 corrected derivation — 실제 내용

Paper의 §2.6, §2.7만 사용:

• §2.6 with $f = g = F$: \(T\{(1+F)^2\} = T\{1+2F\}\)
• §2.7 with $s = 2F$: \(T(1+2F) = T(1+F)^2\)
• 합성. $X := T(1+F)$: \(T\{(1+F)^2\} = T\{1+2F\} = X^2\)

양변 $X^2$ 일치. 합 statement는 $X^2 = X^2$ — tautology. Paper-direct “$F = 2F$” derivation은 §2.6 + §2.7만으로 도출 X.

전체 step-by-step derivation은 Finding 4.

본 framework 가치

Framework는 prescriptive (무엇을 찾을지) — descriptive post-hoc 분석 아님. Reviewer/proof-writer가 새 RH proof attempt 검토 시 5-question checklist 분 단위:

1. A: trivially circular?
2. B: reference circular?
3. C: identity transplant?
4. D: multi-valued inversion?
5. E: self-acknowledged contradiction?

2+ categories trigger → 구조적 의심. Atiyah 2018은 5개 모두 trigger — framework가 그 case study에서 추출됨.

Framework 향후 적용

Systematic application 후보:

• de Branges’ RH proof series — 다년 attempts, partial retractions
• Amateur preprint catalog (vixra archive of failed RH attempts)
• Historical Hilbert program failures — nuance: 같은 structural sense의 “failed proofs”는 아니지만 Category A/B 일부 early attempts에 존재

프로젝트는 systematic application 미수행. Framework는 American Mathematical Monthly / Math Magazine 류 standalone expository contribution publication-ready.

아닌 것

• 5 check 모두 pass하는 proof가 옳다는 보장 X (necessary, sufficient X).
• Atiyah 또는 어떤 author 개인 비판 X. Failure modes는 structural, personal X.
• 수학 contribution 아님 — 수학 proof attempt 평가 *practice 에 대한 contribution.

Reading order

• 적용 case study: Finding 4 — Atiyah 2018 §3.3 step gap.
• Detailed Category A template: Lemma 1.
• Reporter’s manager-mode review: Reporter Flag — Cycle Protocol over-claim.

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