Improve lower bound for the Fourier Entropy-Influence constant to 6.514326913930565372 (n=17 certified construction)

[463464q435q43]

Target: teorth/optimizationproblems, problem 71a (constants/71a.md) —
add lower-bound table row with reference tag [MI2026b], plus the reference
entry given at the end. Upgrades our previous submission #94.

---

Verify — 60 seconds

One file, no other downloads (the certificate is inlined in the script):

pip install mpmath          # only dependency (tested 1.3.0)
curl -sLO https://zenodo.org/records/21113935/files/check_c71_n17.py
python3 check_c71_n17.py

Expected output (exact):

selftest Maj3:  I = 3/2 (want 3/2), H brackets 2: PASS
selftest beta(1/2) in [6.454783716562, 6.454783716563] (want 6.45478371656..., < 6.4547837166): PASS
SELFTEST PASS
n=17 terminal (headline record, logic-monotone): n=17, balanced (fhat(0)=0): True, I = 261/128, monotone = True, sha256 binding OK, 2578 nonzero coefficients
  C = H/(I-1) certified in [6.514326913930565372650625175956..., 6.514326913930565372650625175957...]
  claimed (floor-truncated)  6.51432691393056537265062517595609726535914349523745739524537
  headline (18-digit floor of lower endpoint)  6.514326913930565372
  claimed <= lower endpoint: True;  lower endpoint > 6.4547837166 (> beta(1/2)): True
  CERTIFIED
ALL CERTIFIED

Runtime: under five seconds. The script self-tests on known anchors (Maj₃ and
an independent recomputation of Hod's $\beta(1/2)$), then certifies the bound;
every verdict is an exact rational comparison of interval endpoints. Exit 0
iff certified. --dps 240 reruns at doubled precision (same digits).

Deep check (optional, ~minutes): the public audit bundle
(DOI 10.5281/zenodo.21113935,
file fei_c71_n17_bundle.zip) replays the full from-scratch adversarial
audit:

python3 audit_ridge_fourthline_n17.py certs/ridge_n17_g4_best2_iter14.json
# expect exit 0, "ALL 18 CHECKS PASSED", stdout identical to the bundled run log

What changed

	value	source
our merged submission #94	$> 6.4901128435233943$	explicit monotone function, $n=14$
published record	$> 6.4547837$	Hod 2017, Thm 4.4
this PR	$\ge 6.514326913930565372$	explicit monotone function, $n=17$, same certificate format as #94

constants/71a.md only: one lower-bound table row, the "best established
range" line, and one reference entry. Margin over the #94 certificate's
upper endpoint: $+0.02421407040717101297$ (exact-rational comparison,
floor-truncated). Margin over Hod's $\beta(1/2)$:
$> 0.05954319736780681883$ (exact rational against an independently
recomputed 41-digit enclosure). Like #94, the bound holds even restricted to
monotone functions.

Summary

An explicit balanced, logic-monotone Boolean function on $n = 17$ variables
(certificate pins the exact integer Fourier spectrum, 2,578 nonzero
coefficients, and the SHA-256 of the full 131,072-entry truth table), with

• exact total influence $I = 261/128$ (so $I-1 = 133/128$ exactly), by two
  independent routes (spectral and Fourier-free edge-boundary count);
• certified spectral entropy $H \ge 6.7687928090059780825197902218918823147872$
  (40-digit floor; two independent routes, both enclosure width $<10^{-100}$,
  combined outward);
• exact balance ($65{,}536$ TRUE points, $\hat f(\emptyset) = 0$ as an integer
  identity) and exhaustive logic-monotonicity (0 violations over all
  1,114,112 covering pairs, two independent routes).

The only theorem used is O'Donnell–Tan (arXiv:1304.1347, Lemma 5.1; Hod's
Prop 1.2): for balanced $g$ with $H[g]>0$,
$C_{71} \ge H[g]/(I[g]-1)$. Exact rational division gives

$$C_{71} ;\ge; 6.51432691393056537265062517595609726535914349523745739524537$$

(59-digit floor of the exact lower endpoint; the headline is its 18-digit
floor truncation).

This is one batched upgrade ($n=14 \to 17$; internal intermediates subsumed)
and is submitted as the terminal value of this construction family: a
dedicated saturation campaign over multiple independent ascent mechanisms
found zero candidates above the certified bar, so we do not expect to follow
this PR with small increments from the same family.

Verification chain (summary)

Four independently written verification lines, all passing: the producing
certificate pipeline; an independent checker (different Fourier and influence
routes); an independent exact rescore; and a from-scratch adversarial audit
(18 checks: dual WHT/influence/entropy routes, independent $\beta(1/2)$
recomputation with certified tail, exhaustive monotonicity by two routes,
digit-law verification of every quoted string, and 9 negative tamper controls
all correctly rejected). The audit re-ran reproducibly from the public bundle
alone. One formatting anomaly (stored 59-digit strings nearest-rounded at the
final digit by the v1 formatting path) was caught by the audit and is
disclosed rather than silently fixed; every digit string in this PR comes
from the audit's re-derivation, and the anomaly is ~55 orders of magnitude
below every margin quoted here.

Proposed page edit

constants/71a.md, "Known lower bounds" table — append:

| $>6.514326913930565372$ | [[MI2026b](#MI2026b)] | Explicit balanced **logic-monotone** function on 17 variables via the [[OT2013](#OT2013)] amplification rule $C \ge H/(I-1)$; exact influence $261/128$; certified by exact-rational spectrum + interval arithmetic; replayable certificate, see PR. The function is monotone, so the same bound holds even restricted to monotone functions. <a href="#MI2026b-bound">[MI2026b-bound]</a> |

References — append (the existing [MI2026] entry for the n=14 bound is kept
intact; this PR adds a distinct [MI2026b] tag):

- <a id="MI2026b"></a>**[MI2026b]** Mosaic Intelligence ([@111111](https://x.com/111111)). *A certified n=17 lower bound for the Fourier Entropy-Influence constant.* [Certificate archive](https://doi.org/10.5281/zenodo.21113935), submitted to this repository (2026).

The "best established range" paragraph is updated to headline the new value
(the monotone-restriction remark still holds and is kept).

Attribution

Amplification rule: O'Donnell–Tan ([OT2013]). Record framework and the
$\beta$ benchmark: Hod ([Hod2017]). New-content tag: [MI2026b], Mosaic
Intelligence (@111111) — same attribution as our
submissions #92/#93/#94/#95.

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.