slug: bar-lev-2026-universal-hyperbolic-bound-refutation bibkey: barlev2026blockretrieval doi: 10.48550/arXiv.2603.17154 url: https://arxiv.org/html/2603.17154v2 triage: theorem motivation_gids:

• D5/S3/Resource/VandermondeHyperbolicRefutation.result

The universal hyperbolic retrieval bound

Problem

Bar-Lev, arXiv:2603.17154v2, Section V-B, Conjecture 1, asserts s1/E1+s2/E2<=1 for every finite field, every positive two-file partition s1+s2=k with max(s1,s2)>=2, every length n>=k, and every rank-k generator. Each physical column index is drawn independently, uniformly and with replacement. Ei is the expectation of the minimum time at which the sampled-column span contains the entire coordinate file. Duplicate indices remain separate draws. Preregistration #11729 fixes this complete statement and a nonzero-column refutation criterion.

Motivation

The assertion proposes a universal tradeoff between complementary files under arbitrary linear encoding. A single valid counterexample decides this all-generators assertion. A systematic positive subcase or a projected-rank bound does not decide it.

Gap

The mathematical certificate uses the actual iid expectation bridge; there is no assumed recurrence or chosen expectation definition. The escape audit is unfinished. Scoped compilation reports IE-C050, unclassified_form, E5.unsaturated_definition at the mirrored Reg source’s predicate; current accepted binding evidence for this source is not obtained. The compiled bridge, law variation and sensitivity do not establish declared_validated. The concrete registration obstacle and missing binding evidence are tracked in https://github.com/the-omega-institute/trureturing/issues/11765. Normal admission, freezing, rendering and merged delivery are separate obligations. No merged settlement KPI is claimed.

Route

Over ZMod 11, take k=3, (s1,s2)=(1,2), n=20. Ten physical columns are (1,0,0). The other ten are (1,x,x^2) for x=1,...,10. Every column is nonzero. Parameters 0,1,2 give three independent Vandermonde columns, so the matrix has rank three.

For a finite set S of observed parameters, three distinct parameters span the ambient space. When S has at most two elements, choose a,b with S contained in {a,b}. The functional y2-(a+b)y1+ab*y0 annihilates every sampled column. Its value at the first basis vector is ab; its value at the last basis vector is one. Consequently the first whole file is recovered exactly when zero is observed or at least three distinct parameters are observed. The second whole file is recovered exactly when at least three distinct parameters are observed.

For iid prefixes with at most two allowed types, count two-element supersets and singleton supersets of the observed set. If the allowed type set has size m, its failure indicator is the sum of pair-container indicators minus m-2 times the sum of singleton-container indicators, plus choose(m-1,2) times the empty-prefix indicator. Actual finite cylinders have probability equal to the total physical-index mass of the container raised to the prefix length.

The resulting exact failure tails, including the empty prefix, are

45(1/10)^t-80(1/20)^t+36*0^t

and

10(11/20)^t+45(1/10)^t-9(1/2)^t-90(1/20)^t+45*0^t.

Summing these tails through the frozen retrieval_time_probability_bridge gives the actual expectations E1=34/19 and E2=767/171. Thus 1/E1+2/E2=26201/26078=1+123/26078>1.

Falsifier

The certificate would fail if the generator were rank deficient, if either whole-file recovery characterization failed, if any physical index multiplicity were lost in the iid law, or if the exact failure-tail sum did not give these actual stopping-time expectations. Changing the sampling law, information coordinates or conjecture scope cannot repair such a failure.

Evidence

The formal target is D5/S3/Resource/VandermondeHyperbolicRefutation.result : not claim. The sole public mathematical theorem is the full refutation. Its field-specific negative derivation, probability and geometry proofs are local to result, and it directly uses the frozen actual retrieval bridge and coordinate-file definitions. The intended admission basis is open-problem-resolution under #11729, with a conservative bind-only proof-shape classification under the strict normalization rule. No escape-witness admission or bind-only companion theorem is claimed.

The mirrored Reg source uses the enrolled counterexample template, extracting a finite-field witness and its negative predicate from the full source theorem using classical not_forall. Its bridge retains the full negated assertion, with law variation and slot sensitivity. These proofs and the mathematical certificate have axiom closure propext, Classical.choice, and Quot.sound. Scoped compilation reports E5.unsaturated_definition at predicate; no current accepted or validated binding certificate is claimed. The mathematical theorem is not a registration-completion claim.

Triage

The counterexample mechanism is the high sampling mass of the repeated singleton column together with many independent quadratic-curve types. The complementary plane waits for rank three, but its reciprocal-time penalty does not offset the singleton advantage. The witness is nonsystematic in the fixed file coordinates; a row-basis change would also change those files and would not establish the original assertion.

The source’s file-dedicated bound and restricted systematic results are not contradicted. Claims about the limiting-region conjecture require their own source-faithful field and length quantifiers; this result does not assert an independently formalized settlement of Conjecture 3.

ASSUMED-UNVERIFIED

The original and restricted follow-up retain the arbitrary-generator conjecture. The bounded literature and supplier searches found no dominating full settlement. This is not an exhaustive priority certificate. Canonical producer output, current registration binding evidence, normal admission, freezing and merged delivery are not established by the mathematical source alone.