[JeetuSK0808]

Adds C_89b, the Ramsey multiplicity constant of $K_5$ (the minimum density of monochromatic $K_5$ over 2-colourings of $K_n$), as a second member of the family of C_89 ($K_4$). It also records a new certified upper bound, $C_{89b} \le 0.0016870052$*. Markdown only: constants/89b.md, README rows (the existing 89 row is relabelled 89a), and a "Recent progress" line.

side	bound	source
upper	$1/512$	Erdős 1962 (random colouring; conjectured sharp)
upper	$0.001769$, $0.001720$	Thomason 1989, 1997 (as summarised by PPSS)
upper	$2320651 \cdot 2^{-24} \cdot 3^{-4} < 0.001708$	Parczyk–Pokutta–Spiegel–Szabó, FoCM 2024, Thm 1.2 (192-vertex Cayley graph)
upper	$0.0016870052$*	this PR, details below
lower	$0.001524$	PPSS 2024, Thm 1.2 (flag algebras)

The floating-point flag algebra value $0.001557$ of Kiem–Pokutta–Spiegel (arXiv:2312.08049), which they did not certify, is mentioned under "Additional comments" only.

The new bound. I start from the PPSS 192-vertex graph (Zenodo 10.5281/zenodo.6602512, CC-BY-4.0) read as a step graphon: red between blocks joined by an edge, blue inside blocks. That reproduces their value. Projected gradient descent then moves all 18,528 block-pair densities, inside-block ones included, until the line search stalls; the result is rounded to multiples of $2^{-20}$. The density is computed exactly over every pair of blocks: the integer numerator $N(R) + N(2^{20} - R)$, with $N(R) = \sum_{i,j} R_{ij} \operatorname{tr}((\mathrm{diag}(R_i \circ R_j) R)^3)$, modulo 11 primes, combined by CRT, plus a check prime. The evaluator shares no code with the search, reproduces PPSS's $2320651/1358954496$ exactly, and agrees with brute force on small cases. The result is

$107218716425177032911811117405568916263805081691166740697/63555655070790142733837367909977622463036348875198349443072 = 0.0016870051344094\ldots$,

that is $2.07 \times 10^{-5}$ below PPSS. Certificate, checker and search code are at JeetuSK0808/tao-constants-certificates@82a6e4d (python3 exact5.py c89b_stepgraphon_m192_D20.json.gz, a few minutes). It is asterisked as self-hosted.

The lint script from earlier PRs reports no new warnings against main. I numbered the page 89b because it belongs to the 89 family; renumbering is fine.

AI use: prepared with Claude Code (Anthropic), which located and checked the sources, drafted the page, and wrote the search and the exact evaluator. Disclosed in the page's Contribution notes as well.

🤖 Generated with Claude Code

[carlostoledo1891]

Independent replay of the K5 certificate, offered as the second party the new rule asks for.

I re-derived the monochromatic K5 density of the 192-block step graphon in c89b_stepgraphon_m192_D20.json.gz (sha256 19c43354…, at 82a6e4de, byte-identical at 98b6ee34) with two implementations that share no code with each other or with exact5.py/k5.py, which I kept as text and did not run: one in Python (Fraction; the count as a trace on each block pair's common support), one in JavaScript (the count modulo ten 25-bit primes, lifted by the Chinese remainder theorem). Both give

107218716425177032911811117405568916263805081691166740697 / 63555655070790142733837367909977622463036348875198349443072
= 0.00168700513440941956548558…

the fraction the page states, to the last digit; the printed 0.0016870052 is that value rounded up at ten places, so the bound holds. Controls: raising one block-pair value by 2⁻²⁰ moves the value in both implementations; a printed bound one unit lower fails; W ≡ ½ gives 2⁻⁹ exactly; and the PPSS 192-vertex graph (Zenodo 6602512) read as a 0/1 graphon gives its published 2320651/1358954496. The step from a step graphon's density to an upper bound on C_89b is the standard random colouring, which I read and did not re-derive.

Record, pins and both programs: https://github.com/carlostoledo1891/cert-machine (certs/graphon-replay.json; corpus/optconst-queue/c89-2026-10-27; instruments/optq/graphon_k5.py and graphon-second.js), and the page https://carlostoledo.co/reports/optconst-queue.html#graphons. I have no connection to the author. (One note for the README: the descent moved 1,248 block pairs to two values, not all 18,528 entries; the bound is unaffected.)