[Karnaugh33]

This addresses the fixed-call-budget question in
PR6049 and
Discussion6052
for that construction's minimal power-of-two rank alphabet. It adds a scoped
proof, two exact checkers and explicit counterexamples to broader claims.
The selected result and all five framework assumptions are unchanged.

Let h,N>=1 be integers. Let A_j,B_j be nonnegative integer call counts on
ranks 2^j, 0<=j<=2h+2, with at most N calls per profile, including shared
paid calls. Put F(p)=sum_j(B_j-A_j)2^(jp). Assume zeros of multiplicity at
least h at p=0 and p=1, and two distinct interior zeros alpha<beta with F>0
between them. The endpoint conditions include equal count and rank mass.
Then the losing-window width w=beta-alpha satisfies

w > 2*6^h/(N*ln(2)).

The constant is asymptotically sharp, including irrational quadratic interior
roots. For h=1 this is w>12/(N ln2) on {1,2,4,8,16}. N counts each profile
separately; using a combined two-profile budget doubles the numerator.

A stronger bound uses S=6^h/2 and Z=floor((floor(N/S)-1)/2):

w >= log_2((Z+1)/(Z-1)),
w > 2*6^h/((N-S)*ln(2)).

Every eligible input has Z>=2 and N>S. The discrete bound is attained at
every N=S*(4*n^2-1), n>=3, by the residual
(n*x-n-1)*((n-1)*x-n), proving exact finite-budget optima on that infinite
sequence. We do not claim attainment at every budget.

For h=1,N=1000, an exhaustive exact census finds 8,158 eligible quadratics,
including 7,259 with irrational roots. The minimum width is log_2(81/80),
attained by 60x^2-161x+108 (987 calls per side) and 72x^2-161x+90 (969 calls).
The independent shifted-coordinate enumeration agrees.

The scope counterexamples are part of the result. Seven ranks with simple
endpoint zeros (h=1 with larger support) admit widths Theta(N^(-3/2)), so no
positive universal constant/N bound survives there. This does not contradict
the h=2 seven-rank theorem, which requires double endpoint zeros. Further
exact examples show why the two-interior-crossing condition and the stated
rank alphabet cannot be dropped from the sampling consequence.

Artifacts are in research/fixed-budget-hidden-windows/:

• README.md: precise claim, limits, credit and open next questions.
• PROOF.md and SCOPE-COUNTEREXAMPLES.md: uniform arguments and boundary cases.
• verify.py, independent_check.py and test_verify.py: exact finite checks.
• RESULT.json, independent-check.json and VALIDATION.json: saved replay results.
• SOURCES.json and two inert exports: exact predecessor provenance.
• replay.py, LICENSE and notices: bounded reproduction and retained attribution.

From the repository root:

python3 -I -B research/fixed-budget-hidden-windows/replay.py

This runs both verifiers and all 26 test methods in normal and optimized modes,
serially, with a 30-second timeout per command. It compares both JSON outputs
with the saved receipts. Acceptance uses integer/rational arithmetic, certified
enclosures and exact Sturm root counts. The package writes no files and needs
no network, Git, third-party dependency or upstream-code execution.

Runs used CPython 3.14.7. Python 3.10 syntax/API compatibility was inspected,
not runtime-tested. The fixed-precision interval helper is conservatively
incomplete at extreme valid inputs; this limit is documented. Finite tests
corroborate the written uniform proof; they do not replace it. No complete
selected-supplier replay or repository-wide suite was run, and the standalone
package does not register a generic research-package CI hook.

Scientific base: 721c038.
PR6049 source: 2646b6d.
SOURCES.json records exact paths, public URLs, Git blob IDs and SHA256 hashes.
The compact checker authenticates the manifest and exported blob bytes; it
does not bundle the full offline commit-tree chain or claim publication ancestry.

zxv's PR6049 supplies the hidden-window construction, endpoint framework and
minimal-support result. Its credited predecessors include
Nick Petalas's PR5174
and Ali Romman's root-budget work. Those results and the classical tools are
credited separately from the budget bounds, attained cases and boundary
counterexamples here. No global novelty claim is made. Apache-2.0 and upstream
notices are retained.

Prepared with substantial OpenAI Codex assistance, including GPT-6 Astra and
separate AI proof/implementation review. AI review contributed the stronger
discrete argument and scope counterexamples. No independent human review,
proof-assistant acceptance or maintainer approval is claimed.

Please review the endpoint/degree hypotheses, per-profile normalization,
discrete inequality and equality case, census completeness, and seven-rank
counterexample family. This is a partial answer for a specific arithmetic
class, not a native supplier, new kappa, or general arbitrary-rank theorem.
The remaining questions concern attainability at other budgets and sharp
separation for the larger-support class. An archival merge alone should not
be treated as scientific acceptance.