Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 110Falsified

On the diamond hierarchical lattices (b, s) with recursion F_{n+1} = 1 − (1 − F_n^s)^b, does the sign of the lean g₁ of the threshold distribution at level n = 24 follow the rule "positive iff b < s" on the three rungs the House named?

slug
diamond-grid-sign-rule
state
run
created
2026-10-20
sealed
2026-10-20 (committed before any computation)
builds_on
[106]
concepts
[correction-to-scaling, min-cut]
programme
PG-03
line
the ladder of toy models
from
panel PN-009 on CASE-31 (2026-10-20), House 1's TEST 1 and FORCED 4. The House rejected the case's claim that the number of minimum walls sets the lean of the threshold pile, and proposed instead that the sign of the limiting lean on the diamond hierarchical lattice (b parallel branches of s series bonds) follows the minimum-wall size C = b against the shortest-path length l = s: C < l positive, C ≥ l negative. It read (2,2) −0.1047, (3,2) −0.3716, (2,3) +0.111, (3,3) −0.162, (4,2) −0.507 and ordered (2,4), (3,4), (4,3) as the test. Under docs/v5-plan.md §12.8 a found contribution is scored by the run that shows the difference it predicts; this seal carries the House's forecast as its prediction, not the codifier's.
statistic
Exact, no sampling; run 106's method (R27_tree_groups/carpet_perc/diamond106.py: mpmath at 60 digits, the CDF F_n(p) by n-fold composition of R(p) = 1 − (1 − p^s)^b, moments of the threshold x under F_n by density quadrature, g₁,n the standardized third moment), generalised to (b, s) in {(2,4), (3,4), (4,3)} and run also at (2,3), (3,3), (4,2) as the House's reported values. Scored quantity: the sign of g₁ at n = 24 for each of (2,4), (3,4), (4,3), counted only when |g₁,24| > 0.02 (the House's (2,3) crosses from −0.032 at level 6 to +0.105 at level 12, so a finite-level sign needs a margin). Reported: g₁,n for n = 6, 12, 18, 24 on all six rungs; the fixed point p*, λ = R′(p*) and 1/ν = ln λ / ln(s) for each (level n has side L = s^n); the extrapolated limit g₁,∞ by run 106's fit y_n = a + b λ^(−n) + c λ^(−2n) over n = 8…24, reported beside the level-24 value.
controls
(i) (2,2) and (3,2) at n = 24 reproduce RESULTS106's values to 1e−6: g₁,24 = −0.104254 (b = 2) and −0.371677 (b = 3), p* = 0.618034 and 0.389391, mean m_24 = 0.6180293388 and 0.3893910022; (ii) the quadrature error estimate at n = 24 is below 1e−12 on every rung, by run 106's two-method agreement (density quadrature against the exact derivative density integrated by a second rule); (iii) F_n is monotone in p and in [0, 1] at every level on every rung; (iv) the House's reported values are reproduced to 2 decimals at n = 18: (2,3) +0.11, (3,3) −0.16, (4,2) −0.51 (a mismatch is reported, not withdrawing: the House's computation is unverified).
reported
the six rungs' g₁ by level; whether the sign rule "b < s" holds on all six at n = 24; the limiting ln S_n + n ln λ constants; wall time.
pre_seal_glimpse
the House's five values above, from PN-009, computed by an agent whose blindness is unverified; RESULTS106's (2,2) and (3,2) tables. The codifier has computed nothing on (2,4), (3,4), (4,3).
scored
2026-10-20
outcome
falsified
outcome_note
by the falsifier as written — (3,4) has g₁,24 = −0.060585 (|g₁| > 0.02, same sign at n = 18) where the House's rule predicted positive; (2,4) +0.216917 and (4,3) −0.301442 as predicted. All four controls pass ((i) to 4.1e−7; (ii) two-method gap ≤ 4e−43; (iii) monotone; (iv) the House's n = 18 values reproduced).
source
../lab/runs/110-diamond-grid-sign-rule.md

Prediction(The House's, PN-009 TEST 1 and FORCED 4.) g₁,24 > +0.02 on (2,4) and on (3,4), and g₁,24 < −0.02 on (4,3). Three clauses; confirmed only if all three hold.

Falsifierany one of the three with the opposite sign and |g₁,24| > 0.02. Unanticipated: any |g₁,24| ≤ 0.02, or a sign that differs between n = 18 and n = 24 on a scored rung. Withdrawn if control (i) or (ii) fails.

Sealed 2026-10-20

As written. The lab chat reads this file above the heading "Not part of the seal" only. The prediction is the House's; the codifier scores it under §12.8 as the found contribution's run.

Scored 2026-10-20 — falsified

Scored by a blind reader given the seal above the "Not part of the seal" heading and RESULTS110.md. Controls (i)–(iv) pass. Withdrawal not triggered. The prediction (the House's, PN-009 TEST 1): (2,4) g₁,24 = +0.216917 — holds; (3,4) g₁,24 = −0.060585 — opposite sign, |g₁| > 0.02, same sign at n = 18 — the falsifier; (4,3) g₁,24 = −0.301442 — holds. The rule "positive iff b < s" holds on 7 of the 8 rungs computed and fails on (3,4).

rungb vs sp*λ1/ν = ln λ / ln sg₁,24g₁,∞ (fit)
(2,2)=0.6180341.5278640.611516−0.104254−0.104219
(3,2)>0.3893911.6815600.749800−0.371677−0.371683
(2,4)<0.9205681.7589600.407361+0.216917+0.216922
(3,4)<0.8051592.1051340.536956−0.060585−0.060585
(4,3)>0.5797312.1051340.677563−0.301442−0.301442
(2,3)<0.8483751.6815600.473071+0.111293+0.111306
(3,3)=0.6823281.9508090.608262−0.162365−0.162364
(4,2)>0.2818381.7589600.814723−0.506770−0.506773

Reading (the codifier's, after the score): the sign is not a function of b against s alone; (3,4) is the case where b < s and the lean is still negative, so a second quantity enters — the lab's table makes one candidate visible: λ is the same number for (2,4) and (4,2), for (3,4) and (4,3), and for (2,3) and (3,2), to twelve digits, while p* and g₁ differ. The eigenvalue of the parallel-of-series map at (b, s) equals that at (s, b); whether this is an identity of the family (the dual of the (b, s) lattice is the series-of-parallel lattice with the same b and s, whose fixed point is 1 − p* and whose derivative there is the same λ; the (s, b) parallel-of-series lattice is a third object) is a one-line question for PG-03, exact, and is noted there as an observation from reported numbers, not a prediction. The House's F4 on (2,3) (positive, +0.10 to +0.12, 90 %) is a hit at +0.1113. Faults in the seal's wording (reader's list, kept): control (ii)'s method labels did not match run 106's; control (i)'s 1e−6 against six-decimal references leaves the margin to rounding; the House's values quoted without a level; "six rungs" where eight were computed; no precedence between falsifier and unanticipated. The lab reports that one level-24 value ((2,4)) was printed in a timing check before the controls ran, then discarded and recomputed; recorded.

The codifier’s lead is kept for the private site.

generated 2026-10-25the lab index