bibkey: wolford2025biroot
authors: Isaac Wolford
year: 2025
title: "Combinatorial and Gaussian Foundations of Rational Nth Root Approximations: Theorems and Conjectures"
doi: 10.48550/arXiv.2508.14095
url: https://arxiv.org/abs/2508.14095v3
claim: "Conjecture 3.4: the generalized binomial biroot converges for all positive real x,c and every positive integer n."
strata_touched:
  - D5/S3/ArithSums/WolfordBinomialBiroot
license: citation-only
triage: anchor

Generalized binomial biroot

Wolford, arXiv:2508.14095v3, Conjecture 3.4 states:

For any $x>0$, positive integer $n$, and $c>0$, $$\underset{m\to \infty }{\lim }{\beta }_m^n(x,c)=\sqrt[n]{x}$$ where ${\beta }_m^n(x,c)$ is defined as above.

The definition in Section 3.1.3 is

$${\beta }_m^n(x,c)=\frac{{\sum }_{k=0}^{\lceil m/n\rceil }{x}^k{c}^{m-nk}\left(\genfrac{}{}{0ex}{}{m}{nk}\right)}{{\sum }_{k=0}^{\lceil m/n\rceil -1}{x}^k{c}^{m-nk-1}\left(\genfrac{}{}{0ex}{}{m}{nk+1}\right)}.$$

Both bounds are inclusive. Powers of positive c have integer exponents. The source explicitly sets binomial coefficients to zero when the lower index exceeds the upper index. The positive-index denominator contains its strictly positive k=0 term. Index m=0 is outside the positive-index limit. The positive real nth root is x raised to the reciprocal real n.

Locator

DOI: 10.48550/arXiv.2508.14095; URL: https://arxiv.org/abs/2508.14095v3.

Primary source

The v3 source archive is https://arxiv.org/src/2508.14095v3. Its SHA-256 is 4ad7f2346693d8cdb2c686f90416dd1a1d56229dace1c7ae84203209692d8a8a. The member main.tex has SHA-256 3402df742475b710e44e388927ebd5f8fd81b8537a67c4dc55ef9a1f70a49156. The definition is at line 339, the conjecture at lines 357–361, the statement that no general proof is attempted at line 363, and the explicit zero-extension convention at line 448. The arXiv metadata identifies v3 as the version of 2025-11-16; the first submission is from 2025-08-15. These are source version identifiers.

Related source and scope

Shunia and Prunescu, Short Recurrences and Straight-Line Programs for Primes, arXiv:2610.13365v1, Appendix A.3, proves its integer-root term under integer radicand and root-index restrictions. Its coefficient multisection and root-of-unity expression are related mathematics; its stated result is not the arbitrary-positive-real Wolford assertion. The archive SHA-256 is 6a26773124ae8a9277b857c7fbb738857270bc37d7174a1b154502ad0852d496; its main.tex SHA-256 is f886d4a8c2469506f460576095973089e7d2df80908b1978310e49f3a0eb152b.

The checked primary and nearby source contain no proof of the exact full assertion. This bounded source comparison establishes neither worldwide openness nor priority. Loogle and LeanSearch access is denied; Reservoir catalog availability does not establish a filtered theorem search. The proof uses the retained finite stochastic mixing supplier; no third-party theorem or paper proof is copied into this module.

Attribution

The quoted conjecture and definition are attributed to Isaac Wolford. This note is bibliographic and citation-only; it grants no license to redistribute the original paper or its figures. The Lean proof is a separate repository derivation.