Bond percolation on the b = 2 diamond hierarchical lattice (each bond replaced at every level by two parallel paths of two bonds; level n has 4^n bonds and "size" L = 2^n). The spanning probability at level n for bond probability p is exactly R^n(p) with R(p) = 1 − (1 − p²)² = 2p² − p⁴, so the distribution of the spanning threshold x over random bond orderings has CDF F_n(p) = R^n(p) exactly. Computing F_n to high precision for n = 1 … 24 and its moments — mean m_n, ln S_n (S the standard deviation), skewness g₁,n — do the corrections to scaling of the moments obey the tied form exactly?

slug: diamond-rung-threshold-moments
state: run
created: 2026-10-16
sealed: 2026-10-16
builds_on: [102, 105]
concepts: [fractal-dimension, correction-to-scaling]
programme: PG-03
line: the ladder of toy models, walk item of 2026-10-15
from: the owner's thread of 2026-10-15 (norms, discrete scale invariance, the comb of Mellin poles) and the literature note docs/walk/2026-10-15-fractal-ladder-literature.md, whose reader derived, unverified, that on the diamond hierarchical lattice the correction exponent is locked to 1/ν. The tied form (ω = 1/ν) has been falsified twice on the carpet (runs 100, 102). This seal asks whether the tie is exact on the first solvable rung of the ladder, so that the walk up the ladder has a known starting point.
statistic: Exact arithmetic (mpmath, 60 digits): p* = (√5 − 1)/2, λ = R′(p*) = 4p*² ≈ 1.5279, 1/ν = ln λ / ln 2. For each n, the moments of x from F_n by quadrature on [0, 1] with F_n = R∘…∘R. Fit, for n ≥ 8, the three sequences to a + b·λ^{−n} + c·λ^{−2n}, and read the ratio of successive corrections. The scored quantity: the empirical correction exponent ω_m of each moment, defined by (m_{n+1} − m_∞)/(m_n − m_∞) → 2^{−ω} with m_∞ the fitted limit (for ln S, the correction to ln S + n ln λ; for g₁, to g₁,∞).
controls: (i) m_n from quadrature agrees with m_n from the exact derivative density F_n′ integrated by a second method to 12 digits; (ii) p* and λ recovered from F_n's fixed point to 30 digits; (iii) the known ν ≈ 1.635 (arXiv:2202.09436 as cited by the reader) reproduced; (iv) F_n(p) monotone in p and in [0,1] for every n.
reported: g₁,∞ and S_∞ to 10 digits; the coefficients b, c per moment; the same for a second map, the b = 3 diamond (three parallel paths of two bonds), to see whether the tie survives a change of b.
outcome: falsified
outcome_note: by the falsifier's first clause as written — ln S's ratio at n = 16 is 0.656167 against 1/λ = 0.654508, 1.66 × 10⁻³ over the 10⁻³ line; the mean's is 5.1 × 10⁻⁵ off and g₁'s 6.7 × 10⁻⁴; the deviation is the next term of a power series in λ^−n, which the seal's precision could not resolve (the seal's fault, scored as written)
source: ../lab/runs/106-diamond-rung-threshold-moments.md

PredictionOn this rung the tied form is exact: ω = 1/ν for all three moments, with the ratio of successive corrections equal to 1/λ to within 10⁻⁶ by n = 16. One clause. Reported, not scored, and written down now: the limiting skewness g₁,∞ is negative (R is concave at p*, R″(p*) = 4 − 12p*² ≈ −0.58, so the map compresses the upper side of the threshold distribution more than the lower), with |g₁,∞| between 0.1 and 0.6; the limiting ln S_n + n ln λ converges to a constant with corrections in powers of λ^{−n} only; and no term in the moments' expansion is log-periodic in n (the moments are analytic functions of λ^{−n} through the Koenigs linearisation of R at p*), so the comb of 2026-10-15 has zero harmonics on this rung for moments-versus-size, and the place to look for it is the free-energy-like sums over levels, not the threshold moments.

Falsifierthe ratio of successive corrections for any moment differs from 1/λ by more than 10⁻³ at n = 16 with the fit converged (the tie is not exact even here), or the corrections are not a power series in λ^{−n} (a second exponent appears that is not a multiple of 1/ν). Unanticipated: the ratio approaches 1/λ but more slowly than λ^{−n} (a marginal operator). Withdrawn if the quadrature does not converge to 12 digits by n = 24 (control (i)).

Sealed 2026-10-16, 23:55 Phoenix, before any computation (codifier). Written in chat while the Mac was unlinked and sent to the owner as a file at that minute (the send is the timestamp); the computation was launched in a lab agent given the statistic and controls only, after the send; this commit is the first the repository sees of it, and the agent had not reported when it was made. The number 106 is taken at this commit.

Scored 2026-10-17 — `falsified`

Scored on 2026-10-17, 00:10 Phoenix, from R27_tree_groups/carpet_perc/RESULTS106.md (bb9ae73), computed by a lab agent given the statistic and controls only. A scoring reader, given the seal's clause and falsifier and the lab's numbers without the lead, read the outcome as falsified by the first clause and raised six wording faults, listed below; it lacked the n = 16 rows and inferred two of its moment verdicts from trends — the rows, read here, correct one of them (g₁ is inside the line at n = 16, not over it). Scored as written.

Controls. (i) the two quadratures agree to at least 44 digits at every n; (ii) p* = 0.618033988749894848… and λ = 1.527864045000420607… to 30 digits; (iii) ν = 1.63528, the literature's 1.635; (iv) F_n monotone in [0, 1], no violation. Not withdrawn.

The scored quantity, at n = 16 (ratio of successive corrections against 1/λ = 0.654508497):

moment	r₁₆	r₁₆ − 1/λ	ω₁₆ (1/ν = 0.611516)
mean	0.654457683	−5.1 × 10⁻⁵	0.611628
ln S	0.656167473	+1.66 × 10⁻³	0.607864
g₁	0.655180413	+6.7 × 10⁻⁴	0.610036

The clause asked for 10⁻⁶ at n = 16: not met by any moment. The falsifier's first clause fires on any moment over 10⁻³: ln S is over it. Falsified, as written.

What the falsifier measured. The corrections are a power series in λ^−n — the free-exponent fit finds the next term at λ^{−3n} (k = 3.03, 2.82, 2.89 for the three moments), the three-term residuals are 10⁻⁷, and r_n − 1/λ at n = 16 is the size of the (c/b) λ^−n term the fit truncates (about 2 × 10⁻³ for ln S). So the ratio was over the line because the seal's tolerance (10⁻⁶) and its test point (n = 16) were set without checking the decay rate of the sequence the statistic was built on: at n = 16 the next correction is 10⁻³, and a free fitted limit adds an error (3 × 10⁻⁸ in the mean's a against the exact p*) that moves the ratio by 10⁻³ at n = 23 and makes the mean's ratio overshoot. The falsifier fired on the instrument. The seal is scored falsified all the same: the clause was the codifier's to write, and it was written wrong. OPS instance under "Do not read a trend off an underpowered run" and a new candidate rule: a seal's tolerance is checked against the statistic's own decay before sealing (one instance; the bar is two).

Reported, not scored. g₁,∞ = −0.1042 at b = 2: negative as written, and inside the 0.1–0.6 band by 0.004; at b = 3, g₁,∞ = −0.3717 with R″(p*) = +1.23, so the seal's reason for the sign (concavity of R at p*) is wrong while the sign held on both rungs. lim S_n λ^n = 0.39155 (b = 2), 0.38075 (b = 3). The cosine term's frequency sits at the edge of every scan (aliasing at integer n, as the seal said it would); with f restricted its amplitude is 5.7 × 10⁻⁴ to 1.9 × 10⁻³ of the leading term — consistent with no log-periodic term in the moments and unable to show it. On the b = 3 rung the ratios approach 1/λ = 0.594686 the same way (mean 0.595071, ln S 0.577969, g₁ 0.600123 at n = 23). The lead's content — one analytic map, one eigenvalue, the tie exact in the sense that every correction is a power of λ^−n — is what the numbers show, and is not what the clause scored.

The reader's six wording faults, kept as a record: no precedence between a falsifier clause and the unanticipated clause when both are literally met; "fit converged" undefined; "the fitted limit" names no fit; the scored quantity is fragile to the fitted limit's error; the n = 16 rows were not in the reader's excerpt; "a multiple of 1/ν" has no tolerance. All six are the seal's.

What is next, if resealed. The same computation with the exact limits where they are known (p* for the mean), the ratio taken at the n where the truncated term is below the tolerance, and the tolerance set from the λ^−n decay; or the scored quantity changed to the coefficients of the λ^−n series themselves. Resealing is the programme's call (PG-03, the ladder).

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

generated 2026-10-25the lab index

Scored 2026-10-17 — falsified

Scored 2026-10-17 — `falsified`