Finding 4: Atiyah 2018 §3.3 has a paper-direct step gap
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A failed-proof case study with the actual algebra. The gap was identified by the wider community at the time of Atiyah’s 2018 preprint; this post derives it explicitly from the paper’s own §2 axioms.
Setup (Atiyah 2018, paper-direct)
The paper introduces a function $T: \mathbb{C} \to \mathbb{C}$ called the Todd function with two key properties:
Property §2.6 (paper-direct, “weakly analytic”): \(T\{(1+f)(1+g)\} = T\{1+f+g\}\) The paper presents this as a linear approximation identity (this caveat will matter).
Property §2.7 (paper-direct): \(T(1+s) = T\!\left(1 + \tfrac{s}{2}\right)^2\)
The paper then claims a proof of RH by contradiction in §3.3.
§3.3 — the paper’s argument
Suppose $\zeta(b) = 0$ for some $b$ in the critical strip off the critical line. Define: \(F(s) := T\{1 + \zeta(s+b)\} - 1\)
The paper claims:
“Then $F(s) = 2F(s)$ in characteristic 0, hence $F \equiv 0$.”
From $F \equiv 0$ the paper derives $\zeta \equiv 0$, contradiction, so no such $b$ exists, hence RH.
The gap — explicit derivation
We work only from §2.6 and §2.7 as the paper presents them, and check whether $F = 2F$ follows.
Step 1. Apply §2.6 with $f = g = F$: \(T\{(1+F)(1+F)\} = T\{1 + 2F\}\) \(\therefore \quad T\{(1+F)^2\} = T\{1+2F\} \qquad (*)\)
Step 2. Apply §2.7 with $s = 2F$: \(T(1+2F) = T(1+F)^2 \qquad (**)\)
Step 3. Compose $(*)$ and $(**)$. Set $X := T(1+F)$:
- LHS of $(*)$: $T{(1+F)^2}$
- $(**)$: $T(1+2F) = X^2$
- RHS of $(*)$: $T(1+2F) = X^2$
So we obtain \(T\{(1+F)^2\} = X^2 \quad \text{and} \quad T(1+2F) = X^2.\)
Both equal $X^2$. The combined statement is \(X^2 = X^2,\)
a tautology. The derivation $F = 2F$ does not follow from §2.6 + §2.7 alone.
Why §2.6 cannot bear the weight
The paper itself frames §2.6 as a linear approximation (the “weakly analytic” qualifier in §2). Substituting $f = F$ where $F$ is not infinitesimally small is moving §2.6 outside its stated domain of validity.
This is Category C of the failed-proof catalog (Identity transplant): an equation valid only in a limited domain (here: linear approximation) used as an exact equality in proof.
A second gap in §3 — multi-valued inversion
Beyond the F=2F step, §3 also makes the inference: \(T(1+\zeta(s+b)) = 1 \quad \implies \quad \zeta(s+b) \equiv 0\)
But $T^{-1}(1)$ is generically multi-valued — the paper does not produce a uniqueness argument for the inverse $T^{-1}$. Without uniqueness, $T(1+w) = 1$ does not imply $w = 0$. This is Category D (Multi-valued inversion).
A third issue — §5 vs §3
§5 of the same paper concludes:
“The most general version of the Riemann Hypothesis will be an undecidable problem in the Gödel sense.”
— which sits awkwardly alongside the §3 “proof by contradiction”. The two cannot both be the author’s settled position. This is Category E (Self-acknowledged speculation alongside proof claim).
The 5 categories of failed RH proof manifestations
Atiyah 2018 manifests all five of the project’s failed-proof categories:
| Category | Description | Atiyah 2018 manifestation |
|---|---|---|
| A — Trivial circular | Spectrum/hypothesis trivially equivalent to conclusion | §3 derivation encodes the zero condition $\zeta(s+b) = 0$ as $F = 0$, not derived from independent structure |
| B — Reference circular | Core object’s well-definedness depends on unpublished work | $T(s)$ explicit form depends on Royal Society submission [2] (unpublished at preprint time) |
| C — Identity transplant | Equation valid only in limited domain used as exact in proof | §2.6 (linear approximation) used as exact in §3 (Step 1 above) |
| D — Multi-valued inversion | $f(g(s)) = 0 \implies g(s) = 0$ ignoring multi-valued inverse | $T(1+w) = 1 \implies w = 0$ without uniqueness argument |
| E — Self-acknowledged speculation | Paper contains “undecidable” alongside proof claim | §5 “undecidable in Gödel sense” + §3 “proof by contradiction” |
The framework as a prescriptive checklist
The 5-category framework is prescriptive (what to look for) rather than just descriptive (post-hoc analysis). For a reviewer or proof-writer working through any new RH proof attempt:
- A-check: Is any axiom, hypothesis, or definition 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 (e.g., approximations used as exact)?
- 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 triggers all five, which is the failure mode the framework was extracted from.
What this finding is not
- The §3.3 gap identification is not original to the project. The broader mathematics community (Tao, several others) noted the issue independently at preprint release time.
- The 5-category framework is the project’s contribution — as a structured checklist transferable to other proof attempts.
- This is not a personal critique of Atiyah. At the time of the 2018 preprint Atiyah was 89; the failure mode is structural, not personal. The Todd function itself (separate from the proof attempt) remains a mathematically interesting object.
- This is not RH progress — it documents a way RH proof attempts fail, not how they succeed.
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)
The project has not run this systematically. The framework is publication-ready as a standalone American Mathematical Monthly / Math Magazine–style expository contribution.
Refuting / strengthening this finding
If you can argue Atiyah 2018 §3.3 does derive $F = 2F$ from §2.6 + §2.7 alone, please email x2ever.han@gmail.com. Specifically: produce the missing intermediate step that this derivation does not see.
Or if you have a candidate failed proof that does not manifest any of the 5 categories — that would be a useful framework stress-test.