bibkey: cavalcante2026blockexchange authors: Claudemir de Souza Cavalcante year: 2026 title: A Block-Exchange Valuation Bound for Superabundant Numbers and Robin’s Inequality doi: 10.5281/zenodo.21725356 url: https://doi.org/10.5281/zenodo.21725356 claim: The paper gives an exact 92-versus-47 prime-block exchange that raises the uniform exponent-two cutoff for a hypothetical least Robin counterexample to 7,608,793; it remains a necessary condition and does not prove Robin’s inequality or identify a FIB multiplicative source. strata_touched: [] license: citation-only triage: anchor

Block exchange and the current valuation cutoff

The source is Claudemir de Souza Cavalcante, A Block-Exchange Valuation Bound for Superabundant Numbers and Robin’s Inequality, Zenodo record 21725356, 31 July 2026. The deposited PDF is 9 pages and has SHA-256 e792a805a09dc2f64b5a5fa2e198dc3a230b6fbe412a03c674c430c589c40c4b. This is a source-scope record, not an independent proof audit or Lean verification.

The exchange lemma

For a superabundant number

$$n=\prod _{p\le P}{p}^{a_p},a_2\ge a_3\ge \cdots \ge a_P\ge 1,$$

suppose a supported prime $q$ has exponent one. Every supported prime at least $q$ then has exponent one. The source exchanges $k$ consecutive primes $u_i$ beginning at $q$ for $\ell$ larger supported primes $v_j$. If

$$\prod _i{u}_i<\prod _j{v}_j,$$

and

$$\prod _i\left(1+\frac{1}{u_i(u_i+1)}\right)\prod _j\frac{v_j}{v_j+1}>1,$$

the exchanged integer is smaller and has larger abundancy, contradicting superabundance.

Exact supplied bound

Using $k=92$ consecutive primes beginning at $M=7,608,793$ and the $\ell =47$ largest primes below

$$B=29,582,000,000,000,$$

the source checks both exchange inequalities after clearing denominators by exact integer arithmetic. It follows that every superabundant $n$ with $P^{+}(n)>B$ satisfies

$${\nu }_p(n)\ge 2(p\le 7,608,793).$$

Vega’s lower bound on the largest prime factor of a hypothetical least Robin counterexample among integers greater than 5040 places that source beyond $B$, so the same cutoff is a necessary condition for such a counterexample. The paper compares it with the earlier direct cutoff $5,438,903$ and with the collective-tail cutoff $233,911$.

The fixed 92-versus-47 certificate is sharp only for that chosen block: its abundancy comparison reverses when the starting prime is advanced to the next prime. This is not a global optimality statement about all possible block exchanges.

Boundary for the FIB route

The argument is a local exchange in the multiplicative exponent vector. The FIB five-state window is an additive Zeckendorf inclusion state; its symbols do not provide $P^{+}(n)$, ${\nu }_p(n)$, or the superabundance hypothesis. Necessary divisibility filters can nevertheless use the existing modular observations of the same finite source. A complete Robin comparison still requires the actual multiplicative data and its signed remainder.

Application through the existing FIB CRT interface

The FIB volume, §205.4, already combines forced divisors of $N_{r,g}=1+F_{r}g$ into one CRT class and counts that class in a multiplier interval. §207.3 already records the complete-core inverse candidate and the conditions for actual realization. Reuse those interfaces with the valuation cutoffs above and the collective-tail cutoffs, setting

$$M=\prod _{p\le 61}{p}^4\prod _{61<p\le 479}{p}^3\prod _{479<p\le 7608793}{p}^2.$$

For a prime index $r\ge 7$, put $V=F_r$ and $I_r=[\lceil V/10\rceil ,\lfloor V/5\rfloor ]\cap \mathbb{Z}$. Conditional on accepting both external valuation results, if, for some $g\in I_r$, the integer $N_{r,g}>5040$ is the globally least Robin counterexample in that range, then $M\mid N_{r,g}$. The existing CRT interface therefore requires

$$\gcd (V,M)=1,g\equiv -V^{-1}(\mathrm{mod}M).$$

If the gcd condition fails there are no such candidates. Otherwise the number satisfying this necessary congruence in $I_r$ is at most $\lfloor (|I_r|-1)/M\rfloor +1$. In particular $|I_r|\le M$ leaves at most one candidate. The same source has $N_{r,g}\le 1+V^2/5$, so $M\mid N_{r,g}$ also requires $V^2\ge 5(M-1)$. These exact integer conditions avoid treating rounded rank scales as certified boundaries.

This is a parameter application of existing CRT and size comparisons, not a new estimate. Its low-rank scales are already inside the finite Robin ranges recorded in Axler’s Lemma 2.3 and, conditional on its external certificates, the Polak verification, also applied to this family in FIB §258.2. It adds no Robin-safe range. Excluding an integer from being the globally least counterexample does not establish Robin for that integer; neither does retaining a congruence candidate establish a violation. No independent complete proof audit of either Cavalcante source or Lean verification of this application is claimed.