C_3c: improved lower bound 1.6747338950414058 on a 147-point support

[CoolRmal]

Improves the lower bound for $C_{3c} = SD({0,1,2,\infty};-1)$ from the currently certified $1.674733895020824993\ldots$ (#93, 95-point support) to

C_3c >= 1.674733895041405870063135756722213999136383713818148696811828

a gain of $2.06 \times 10^{-11}$. Markdown changes only.

The certificate

An explicit distribution on 147 points of $\mathbb{Z}^2$ with exact rational weights at common denominator $10^{320}$, summing to exactly 1, with $X-Y$ injective on the support. Certificate and verifiers:

• gist.github.com/CoolRmal/5368357cd781d7e5c676c9d68ad24d22, pinned revision 62621d2

H(X-Y)                          = 1.90118266043426947934028635251294...
H(X) = H(Y) = H(X+Y) = H(X+2Y)  = 1.13521477415805507651...

All four constrained entropies agree to 33 digits, so no constraint is slack. Because $X-Y$ is injective on the support, $H(X-Y)$ is the entropy of the distribution itself. The four heaviest points, carrying 85% of the mass, are $(-1,1)$, $(-1,-1)$, $(-3,1)$, $(1,-1)$; the support spans $x \in [-45,43]$, $y \in [-31,39]$.

Since the weights are exact rationals, every pushforward is exact and only the logarithms are inexact; those are enclosed in interval arithmetic rather than rounded, and the reported bound is the exact floor truncation of the lower endpoint.

Verifying

Needs only mpmath, no other dependency:

python3 check_cert.py certificate_3c_147pt.json
python3 reproduce_3c.py certificate_3c_147pt.json

The two scripts were written independently of each other; the second is 25 lines using only the standard library and mpmath, and recomputes the value from the integer weights alone.

Method, and a remark that may be of wider interest

At an interior optimum with all four constraints active, stationarity gives a product law $p_g = \prod_j m_j(L_j(g))^{\mu_j}$ with no normalising constant, where $m_j$ is the pushforward of $p$ under $L_j$. Newton on $(\log p, \mu)$ with the analytic Jacobian then converges quadratically. Candidate points to add are scored exactly: adding $g$ gains $(1-\sigma)\varepsilon^*$ where $\sigma$ is the total multiplier mass of the directions in which $g$ opens a fresh value, so a gain exists iff $\sigma < 1$. Column generation against that criterion terminates when no lattice point in the window has positive gain, which closes this support at 147 points.

The remark: the existing certificates on this page are not converged. Most of the improvement here comes from solving an existing basin exactly rather than from a new configuration. The same effect applies to 3b, where re-solving the published 13-point support of #92 without changing a single point raises it by $4.07 \times 10^{-35}$, to $1.778988841420693549805193887477289475110008577468\ldots$ I have not included that here, to keep this PR to one constant, but I am happy to open it separately if useful.

Floating-point search is adequate only to locate basins, not to rank them: re-solving 187 distinct supports exactly promoted a basin that scores worse in double precision. An enumeration of all 1333 five-point cores in a $5\times5$ window, screened and then solved exactly, found nothing better.

Disclosure

Prepared with AI assistance (Claude). The certificate is machine-checkable by the two linked scripts, and all references and numbers here were verified before submission. The bound remains well under the proven upper bound of $7/4$.

🤖 Generated with Claude Code