bibkey: cavalcante2026collectivetail authors: Claudemir de Souza Cavalcante year: 2026 title: A Collective Tail Criterion for the Prime Exponents of a Hypothetical Least Counterexample to Robin’s Inequality doi: 10.5281/zenodo.20733714 url: https://doi.org/10.5281/zenodo.20733714 claim: The working paper combines least-counterexample structure with a collective exponent-defect budget and obtains explicit necessary valuation ranges; it does not prove Robin’s inequality or supply a FIB-to-multiplicative transport. strata_touched: [] license: CC-BY-4.0 triage: anchor

Collective exponent tails for a hypothetical Robin counterexample

The source is Claudemir de Souza Cavalcante, A Collective Tail Criterion for the Prime Exponents of a Hypothetical Least Counterexample to Robin’s Inequality, Zenodo record 20733714, 17 June 2026. The deposited PDF is 9 pages and has SHA-256 dd586ea45b57aecfbb05f74c11387f42d11bab791ea8777d6946094bbc78172b2. The source is reference input; it was not independently formalized here.

The collective budget

Assume that $c>5040$ is the least integer violating Robin’s strict inequality in that range. For

$$D(n)=\sum _{p\mid n}-\log (1-p^{-{\nu }_p(n)-1}),$$

the paper combines the initial-prime support and non-increasing exponents of a superabundant least counterexample with its explicit upper estimate for $n/\phi (n)$. It obtains

$$D(c)<\log \left(1+\frac{0.0094243}{31.03^3}\right)<\frac{94243}{298775737270}.$$

The displayed decimal is replaced by the rational bound for the finite certificates. The source uses Vega’s lower bound on the largest prime factor and the verified primorial range to ensure that the relevant prime intervals are in the support of $c$.

Tail criterion and supplied cutoffs

For a prime $q$, $r\ge 1$, and $q\le X\le P^{+}(c)$, define

$$T_r(q,X)=\sum _{q\le p\le X}-\log (1-p^{-(r+1)}).$$

If $T_r(q,X)$ exceeds the budget above, monotonicity of the exponent vector forces ${\nu }_p(c)\ge r+1$ for every prime $p\le q$. Exact integer certificates in the source give

$${\nu }_p(c)\ge 4(p\le 61),{\nu }_p(c)\ge 3(p\le 479),{\nu }_p(c)\ge 2(p\le 233911).$$

These are necessary conditions under the least-counterexample hypothesis. The paper explicitly leaves the remaining exponent patterns compatible with the collective budget; no contradiction and no all-integer Robin bound is claimed.

Boundary for the FIB route

The result is multiplicative: it uses ${\nu }_p(c)$, initial prime support and the same integer’s divisor-sum factorization. A five-window address $[null,2,3,25,5]$ records additive Fibonacci inclusion and does not imply any of these $p$-adic valuations. Existing modular observations can test the necessary divisibility conditions on the same generated integer; see the combined-cutoff application. Such a filter does not recover the complete exponent vector, establish superabundance, or pay the signed Robin tail. The source supplies stronger conditional valuation inputs, rather than a new FIB estimate.