Improve C_3a exact-count lower bound to 1.1873823054 (budgeted semigroup-mask certificate)

[barnettcaleb1]

Improve C_3a exact-count lower bound to 1.1873823054 (budgeted semigroup-mask certificate)

This PR improves the finite/exact-count lower bound for constant 3a (the Gyarmati–Hennecart–Ruzsa sum–difference constant) from 1.1835129324 [MI2026] to

C_3a ≥ θ(U) > 1.1873823054 (full certified value 1.187382305438637309089264177964),

via the GHR finite-set lemma applied to an explicit finite set U. This is a certified value at a finite depth, not a limit: it updates the unstarred README cell value. The asterisked limit-type entries (1.187326127925948* [Num2026] and 1.19102809* [K2026]) are a different tier and remain in place; note however that the finite value certified here exceeds the [Num2026] limit.

Construction

U = { Σ_{i=0}^{255} a_i · 379^i : a_i ∈ M, Σ_i a_i ≤ 11923 }, where M = ⟨24,26,36,39⟩ ∩ [0,189]

is the numerical-semigroup mask from [K2026]'s progression (|M| = 99, B = max M = 189, base Q = 2B+1 = 379, carry-free). This is the budgeted (Gerbicz–Zheng-style) truncation of that mask family at m = 256 digits; its limit value under [K2026]'s bound is 1.1893936243, which deeper truncations approach.

Certified quantities

All computed by digit-string dynamic programming (U itself has max U ≈ 10^659; nothing is enumerated), every quantity bounded in the safe direction:

quantity	direction	value	method
N ≥ |U+U|	upper	log₁₀ N ≈ 493.80620	exact big-integer DP: #{(s_i) ∈ (M+M)^256 : Σs_i ≤ 23846} (necessity: elements of U have digit-budget ≤ L)
T ≤ |U−U|	lower	log₁₀ T ≈ 617.50319	counts difference strings with Σκ(d_i) + |Σd_i| ≤ 23846, κ(d) = min{a+b : a,b ∈ M, a−b = d}; each such string is realized by an explicit pair in U×U (minimal witnesses: Σα = (Σκ+Σd)/2 ≤ L, Σβ = (Σκ−Σd)/2 ≤ L); C DP with FE_DOWNWARD directed rounding and exact power-of-two rescaling
D ≥ |U−U|	upper	log₁₀ D ≈ 619.00469	exact big-integer DP (necessity: Σκ(d_i) ≤ 2L); verifies the GHR admissibility condition |U−U| ≤ D < q = 2·max U + 1 in exact integer arithmetic
max U	exact	log₁₀ q ≈ 660.13164	greedy most-significant-digit assignment (optimality proof in the certificate; cross-checked by DP and enumeration on small cases)

θ = 1 + log(T/N)/log(q) evaluated at high precision with outward rounding (ln T floored, ln N and ln q ceiled).

Verification

• Pure-integer record proof (no floating point anywhere): the certificate integers satisfy T¹⁰⁰⁰⁰ > N¹⁰⁰⁰⁰ · q¹⁸⁷³, hence θ(U) > 1 + 1873/10000 = 1.1873 > 1.1835129324, checked in exact big-integer arithmetic (int_check.py). The 30-digit value above additionally uses directed-rounding Decimal logarithms with explicit ulp margins.
• verify.py recomputes everything from the four semigroup generators — mask, alphabets, κ table, all DPs, final θ with outward rounding (~10 min single-machine run; log included).
• The two exact counts (N, D) were reproduced bit-for-bit by an independent CRT computation over 71 31-bit primes using a different algorithm and code base (adversarial review pass), in addition to the package's own two-prime cross-check.
• The C difference-count DP (tdp.c) was validated against exact references in all three of its arithmetic regimes: exact (m ≤ 6, equality at every threshold including binding-corridor edge cases), rounding-active (m = 8: certified-lower at all 1601 thresholds), and rescale-active (m = 16: certified-lower at all 2001 thresholds); output is thread-count-invariant and a fresh compile reproduces the archived DP output byte-for-byte.
• Brute-force validation on small analogs where U can be enumerated exactly (a ⟨3,5⟩ mask in base 15 and the real mask at m = 2, 3): every bound direction, witness realizability, and greedy max-U optimality confirmed (validate.py).
• Independent replication in pure exact integer arithmetic (independent-verification/): a second implementation of the difference-count lower bound (numpy int64 with certified floor-halving rescaling — no floating point, no rounding-mode dependence), written without reference to tdp.c and validated exactly against enumeration on 13 cases. It certifies weaker-but-record-beating values at the same parameters (e.g. θ ≥ 1.18455 at m = 128 with no floats), and brackets the primary pipeline from below at every depth tested, consistent with its coarser tail handling.
• Scan ladder (best certified θ per depth): m=32: 1.17866, m=64: 1.18315, m=128: 1.18582, m=192: 1.18684, m=256: 1.1873823054.

Verification package: https://github.com/barnettcaleb1/c3a-exact-count-certificate

Changes in this PR

• constants/3a.md: one new row in the lower-bound table (chronological append) + one reference entry.
• README.md: 3a cell 1.1835129324 (1.19102809*) → 1.1873823054 (1.19102809*); one line added to Recent progress.

AI use disclosure

Per the contribution guidelines: this submission was prepared with substantial AI assistance (Anthropic Claude) for code, computation, and drafting, directed and reviewed by the human contributor, who takes responsibility for the claims. The certificate is fully machine-checkable independent of its provenance via the included verify.py.