# Specialist review brief: spectral counting of Robin candidates

Historical companion snapshot — September 15, 2026.
The mathematical text and review-status statements below describe that snapshot,
not the current consolidation status. Current scope and limitations:
https://arithmeticconnectome.com/robin
Internal source and audit filenames below identify private provenance records;
they are not public downloads. The reviewed circulation PDF is available at:
https://arithmeticconnectome.com/papers/robin-circulation.pdf
Contact: research@arithmeticconnectome.com

September 15, 2026. Circulation for mathematical and attribution review.

## Result submitted for scrutiny

The accompanying manuscript, *Spectral bounds for the number of Robin candidates*,
claims an eventual bound R(X)<=exp(L g(log X)), where

    R(X) = #{3<=n<=X : sigma(n)>=exp(gamma_E)n log log n},
    W(y) = inf_(t>=T*) [eta(t)log y + 2log t],
    P_0(y) = (log y/W(y))^(BC) exp(A_d W(y)^(B+1)/(log y)^B),
    g(y) = y sqrt(P_0(y))exp(-W(y)/2)/log y.

The generic claim assumes the manuscript's uniform zero-density and zero-free
inputs, W=o(log y), and W/log log y tending to infinity. Classical and VK
specializations use specified published inputs. The exact W stays inside the
exponential. The draft supplies no certified numerical cutoff or lower bound.

The potentially distinctive part is the weight-two integrated-tail estimate and
its composition with centered inclusion and classical benefit counting. The
counting method and moving benefit allowances are expressly credited to earlier
work. Originality of the analytic estimate and composition remains unresolved.

## Suggested reading order

Start with manuscript Section 3 (about the integrated tail), then Section 4
(the centered inclusion). Sections 2 and 5 complete the counting and aggregation.
Section 1 states all inputs and Section 7 records bounded literature comparisons.
The standalone tail note (internal record: spectral-tail-standalone-2026-09-15.md; not publicly available) provides a shorter
entry point; the full manuscript (internal record: spectral-robin-counting-manuscript-2026-09-14.md; not publicly available)
contains the complete argument relative to its named classical imports.

## Questions that would most improve this work

1. **Analytic correctness and uniformity.** Does Section 3 establish
   |I_theta(y)|<=C_I P_0(y)exp(-W(y))/log y with one fixed C_I? Please inspect
   the r-independent density integral, the substitution r=2A_d(W/log y)^B,
   and the handling of finite low zeros. An identified missing hypothesis or
   counterexample would be more useful than a general favorable verdict.
2. **Integrated explicit formula.** Is the passage from the once-integrated
   primitive to J_psi legitimate with the stated uniform remainder and
   dominated-convergence argument? Does the subsequent decreasing-envelope
   argument control the absolute value of the signed integral without silently
   asserting the same bound for the integral of the absolute error? Check the
   distinct weighted and unweighted prime-power corrections.
3. **Bridge and composition.** Does the exact centered Mertens identity, including
   the prime endpoint convention and nonpositive remainder, justify one positive
   H on the whole shell? Does the same H satisfy the counting conditions, and
   does the absolute-continuity argument supply the regularity required for
   global aggregation without importing a theta-window estimate?
4. **Existing results and significance.** Is this exact integrated-tail bound or
   the final counting rate already stated, or obtainable by a short standard
   adaptation? Please give a precise source and parameter map where possible.
   The draft does not treat unsuccessful searches as evidence of priority.
   Newer density inputs may improve the secondary factor; the chosen witnesses
   are sufficient examples, not claims of optimality.
5. **Circulation versus publication.** Assuming correctness, is the contribution
   substantial enough for a standalone paper, or better presented as a short
   note or an application of an existing theorem? Which exposition or source
   gaps should be resolved before a public preprint?

## Evidence supplied, and what it does not establish

Repeated user-run Claude audits found no defect in their checked written steps,
with report corrections preserved separately. They are model reviews of particular
snapshots, not independent human refereeing. Codex assisted with development,
formalization and manuscript maintenance.

Lean verifies the elementary reduction and a finite partition interface conditional
on named BT/ENS Lemma 1 inputs. The scoped build contains 28 local modules and 105
standard-axiom theorem reports. The spectral analytic chain, partition asymptotic,
centered inclusion and exact global specialization are not thereby verified.
At this September snapshot, the conditional implementation still awaited its
separate adversarial audit; this is not a current release-status statement.

Original Jutila/Ivic proofs and the remaining relevant Nicolas/EN sections have
not received a complete new audit in this project. The source ledger separates
located statements from proof-level checks. None of those literature limitations
is being resolved by claiming novelty here.

Please identify the manuscript version reviewed and distinguish mathematical
errors, missing source support, prior implication, and presentation suggestions.
A partial response focused on the first two questions would already be useful.