Lemma 4 — Failed Proof Categories
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Process lemma — a 5-category framework for systematic critical reading of failed RH proofs. The Atiyah 2018 case study (see Finding 4) manifests all five.
Background
Failed RH proofs are 반면교사 — by analyzing past failure modes, future spectral / analytic / arithmetic candidates can avoid the same traps.
The 5 categories
Category A — Trivial Circular
Spectrum identification is trivially equivalent to the $\zeta$-zero condition.
- Detailed templete: see Lemma 1 — Spectral Candidate Circularity Check.
- Examples: BBM 2017, Sierra 2007/2016 ($\psi_z(0) = -\zeta(z)$ boundary condition).
Category B — Reference Circular
The core object’s well-definedness depends on another paper that is unpublished or unverified.
- Example: Atiyah 2018 → paper [2] (Royal Society submission, unpublished at preprint time). Atiyah’s $T$ function is referenced via [2] without explicit construction in the 2018 preprint itself.
- The proof’s foundation is in a citation, not in the paper at hand.
Category C — Identity Transplant
An equation is defined to hold only in a limited domain (e.g., linear approximation), but the proof uses it as an exact equality.
- Example (Atiyah 2018):
- §2.6 (paper-direct, “weakly analytic” — i.e., linear approximation): $T{(1+f)(1+g)} = T{1+f+g}$
- §3 uses this as exact equality with $f = g = F$ (where $F$ is not small) to derive $F = 2F$.
- The substitution moves §2.6 outside its stated domain of validity.
Category D — Generic Multi-valued Inversion
In a step of the form $f(g(s)) = 0 \implies g(s) = 0$, the inverse $f^{-1}$ is treated as single-valued without a uniqueness argument, despite being generically multi-valued.
- Example (Atiyah 2018): §3 derives $\zeta \equiv 0$ by inferring $T(1+w) = 1 \implies w = 0$, but $T^{-1}(1)$ is generically multi-valued (Atiyah’s Todd function is described as a polynomial of bounded degree, and bounded-degree polynomials have multiple preimages).
Category E — Self-acknowledged Speculation
The paper itself contains a disclaimer like “the most general case is undecidable” alongside an actual “proof” claim. The two cannot both be the author’s settled position.
- Example (Atiyah 2018):
- §3 claims a proof by contradiction.
- §5 says: “The most general version of the Riemann Hypothesis will be an undecidable problem in the Gödel sense.”
- Either the §3 proof is correct (contradicting §5) or §5 is correct (contradicting §3). Both cannot be settled positions.
Evaluation protocol — applying the framework
For any new RH proof / spectral candidate, run five checks:
- A-check: Is any axiom or hypothesis trivially equivalent to the conclusion the paper wants to draw?
- B-check: Does any core object require unpublished or unverified external work?
- C-check: Are any equations used outside their stated domain of validity?
- D-check: Are any function inversions performed without a uniqueness argument?
- E-check: Does the paper contain self-acknowledged disclaimers that contradict the proof claim?
When 2 or more categories trigger on the same proof attempt, that is strong evidence of structural failure.
Atiyah 2018 paper-direct deep verification (5/5 manifestations)
Following an explicit deep read of Atiyah 2018 §1–§5, all five categories manifest:
| Category | Atiyah 2018 manifestation |
|---|---|
| B — Construction undefined | $T(s)$ explicit form is paper-direct absent (depends on Royal Society submission [2]) |
| C — Property inconsistency | §2.6 (logarithm-like multiplicative→additive) + §2.7 ($T(1+s)=T(1+s/2)^2$ exponential constraint) + polynomial degree $k(K)$ — the three are inconsistent unless $T \equiv 1$ |
| C/D — Proof step ambiguity | §3.3’s “$F(s) = 2F(s)$” statement is paper-direct not produced by §2.6 + §2.7 alone (see derivation in 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 is artificial (Hirzebruch Todd polynomial + von Neumann fusion speculative) — not a naturally arising analytic object |
→ Paper-direct 5/5 categories all manifested in Atiyah 2018, self-contained.
§3.3 corrected derivation — the actual content
Working only from §2.6 and §2.7 as the paper presents them:
- Apply §2.6 with $f = g = F$: \(T\{(1+F)^2\} = T\{1+2F\}\)
- Apply §2.7 with $s = 2F$: \(T(1+2F) = T(1+F)^2\)
- Compose. Set $X := T(1+F)$: \(T\{(1+F)^2\} = T\{1+2F\} = X^2\)
Both expressions equal $X^2$. The combined statement is $X^2 = X^2$ — a tautology. The paper-direct “$F = 2F$” derivation is not produced by §2.6 + §2.7 alone.
The full step-by-step derivation with discussion is in Finding 4.
Why this framework might be useful
The framework is prescriptive (what to look for) rather than just descriptive (post-hoc analysis). For any reviewer or proof-writer working through a new RH proof attempt, the 5-question checklist runs in minutes:
- A: trivially circular?
- B: reference circular?
- C: identity transplant?
- D: multi-valued inversion?
- E: self-acknowledged contradiction?
If 2+ categories trigger, the proof attempt is structurally suspect. Atiyah 2018 triggers all five — the framework was extracted from that case study.
Where the framework could go
Systematic application to:
- de Branges’ RH proof series — multi-decade attempts, with partial retractions
- Amateur preprint catalog (vixra archive of failed RH attempts)
- Historical Hilbert program failures — more nuance: those weren’t “failed proofs” in the same structural sense, but Category A/B are present in some early attempts
The project has not run the systematic application. The framework is publication-ready as a standalone American Mathematical Monthly / Math Magazine expository contribution.
What this is not
- Not a guarantee that any proof passing all 5 checks is correct (necessary but not sufficient).
- Not a personal critique of Atiyah or any particular author. The failure modes are structural, not personal.
- Not a contribution to mathematics proper — it is a contribution to the practice of evaluating mathematical proof attempts.
Reading order
- Application case study: Finding 4 — Atiyah 2018 §3.3 step gap.
- Detailed Category A template (Spectral Candidate Circularity): Lemma 1.
- Reporter’s manager-mode review of the methodology paper: Reporter Flag — Cycle Protocol over-claim.