Improve problem 3a lower bound to 1.1835129324... (exact-count certificate)#95
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Improve problem 3a lower bound to 1.1835129324... (exact-count certificate)#95463464q435q43 wants to merge 4 commits into
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Result
This PR improves the lower bound for problem 3a (the Gyarmati–Hennecart–Ruzsa
sum-difference constant) to:
(100-digit floor-truncated string available in the certificate artifact.
Every quoted string is a valid lower bound; no nearest rounding anywhere.)
Margin over the published record (1.1740744, [G2026]):
+0.0094385324218615106564747894020784326947 (exact, the record value being
exactly rational) — for scale, the published 2025→2026 progression advanced
+0.001.
Construction
Same capped digit-family construction as [G2026], at stronger parameters:
base B = 33, depth d = 420, cap T = 1392.
with digit sum at most T; the GHR lemma bound is
1 + log(|U−U|/|U+U|)/log(2 max(U)+1).
dynamic programming (no floats anywhere); the full integers are in the
attached artifact
gh3a_b33_d420_T1392_counts.json(gist).
sits ~0.0085 below its own family's asymptote; the base-33 alphabet above
has asymptote V∞ = 1.1855230 (cap ratio τ* = 3.29707), and larger d climbs
toward it (here τ = 1392/420 = 3.31429).
A second, fully independently audited rung of the same family is included as
a smaller exhibit for the reviewer: d = 120, T = 384, certifying
C_3a >= 1.1795410574186522379964980688629718533566(artifact
gh3a_b33_d120_T384_counts.jsonin the same gist; its serialrecount takes ~1h vs. days for d = 420). It is strictly superseded by
the headline. (The d = 240 and d = 320 rungs that headlined earlier revisions
of this PR,
C_3a >= 1.1823357212262927829173650427903744879117andC_3a >= 1.1830053419750735511771153534053862079772, remain in the gist withtheir own verification artifacts; both are strictly superseded.)
Verification
All counting is exact integer DP; the value is certified as an exact-rational
log enclosure (quoted digits are floor truncations). Each of the three
integers was recomputed by independently written implementations:
algorithm), and independently reproduced digit-for-digit by the audit-side
recount below.
pairs; a CRT/modular DP with exact reconstruction (different arithmetic);
and an independent dual disjoint 57-bit prime-set CRT lattice DP
(different author, different code, different arithmetic; 38 primes per set,
the two disjoint prime sets reconstructing the same integer).
a second, independently written parallel implementation. In addition, the
independent CRT layer certifies a proven upper bound S_relax >= |U+U| with
relative gap 2.8e-190; the rigorous lower bound computed from S_relax alone
(audit code only, no producer numbers) already reproduces all 40 headline
digits.
crt_verify_d420.json(same gist) —PASS on all checks, beats the record and the d = 120 rung in exact rational
arithmetic.
|U−U|^11 > |U+U|^11 * (2 max(U)+1)^2independently establishesC_3a > 1 + 2/11 = 1.1818... > 1.1740744 with a single big-integer
comparison (no logs needed).
no-carry boundary holds exactly and the DP counts are exact).
Replay (standalone checker
independent_check_3a.pyin the gist; stdlibonly, from-scratch implementations, two |U−U| algorithms cross-checked
internally):
python3 independent_check_3a.py selftestchecks every counter againstbrute-force enumeration on small instances (seconds).
Attribution
Construction family due to [G2026] (the current record holder's method,
optimized within and beyond its published parameters). New-content tag:
[MI2026], Mosaic Intelligence (@111111) — sameattribution as our submissions #92/#93.
AI assistance disclosure
This is a fully AI-derived result: the construction was found and certified by
Mosaic Intelligence's automated search-and-verification system, and the
submission text was AI-prepared. All numerical results and references were
independently re-run and verified before submission.