run 138 — Lab

Math Mapgenerated 2026-10-25 · a354234

PUBLIC BUILD

slug: the-comb-its-moving-step-and-a-fixed-one-at-6561
state: open
created: 2026-10-25
sealed: 2026-10-25 (committed before any H8 or H7 draw with these seeds)
builds_on: [133, 134, 137]
concepts: [fractal-dimension, symmetry, log-periodicity, invasion-percolation, fourier-duality]
programme: PG-03
line: the comb as a broken continuous symmetry (owner thread items 8, 11b, 16)
from: the reseal of run 137, which was withdrawn on 2026-10-25 because control (iii) gave integer targets with a 1e-6 tolerance in ln m_d (R28_reading/137/scorer137.md; OPS.md instance 25; TODO.md, "Reseal the H8 comb"). Run 137's readings are glimpses now and are declared below. Its scorer's lead, that two of the five groups of draws put the L = 6561 step back near 8^5, becomes clause D3. The drafter's premise is R27_tree_groups/carpet_perc/h8b_premise.py and its .log, which import h8_premise.py (run 137's premise) unchanged. The owner's rulings, 2026-10-25 10:32 (review page): seal it; D2 flipped, so "nearer P" (the owner's W2) is the prediction and "moves by 8" the falsifier. The codifier's rulings: D3 kept at 32,000 H8 draws; the strict combining rule; T's falsifier at 12.69; the lab may import h8b_premise.py; no control (iv) on 137's spent draws.
statistic: R27_tree_groups/carpet_perc/. The lab's script comb138.py is a copy of comb137.py (draws, read_counts, load, blocks, the controls' machinery and the 2187 statistic unchanged). Its L = 6561 step statistic and its (T) members are replaced by the two-window statistic and the two-column members, imported unchanged from h8b_premise.py (step_at2, scan, two_step, design_x, stat2); everything else comes from h8_premise.py as in run 137. perc_inv is the binary that run 137 used (md5 d886d608199da8ae1e350fea0e8ea600, from perc_inv.c unchanged); control (ii) checks it. Draws, full side only: H8 (graphs/H8.bin), 32,000 draws as eight jobs of 4,000 on 4 threads with perc seeds 138511, 138512, 138513, 138514, 138515, 138516, 138517, 138518, pooled in that order; the anchor, H7 (graphs/H7.bin), 32,000 draws as two jobs of 16,000 on 2 threads with perc seeds 138519, 138520, pooled in that order. Every clause is scored on these fresh draws only. Samples go to samples138/ with their .json sidecars. Blocks: 20 blocks of consecutive pooled draws. Notation: x = ln m; tooth k is the pair cos, sin(k pi x / ln P), whose frequency is k/(2 ln P) cycles per unit ln m (k pi / ln P radians); the fundamental is k = 2, the 3/2 tooth k = 3. (D) The step at L = 2187 is run 137's statistic unchanged (step_stat; ratio-1.02 bins from 32; search [ln 8192, ln 75000]; P = the anchor's L4 P-hat on run 134's 65 bins without a step column), giving m_d(2187). At L = 6561 the step statistic is the same local fit (bins with centres in [x_d - 4.0, x_d + 1.0], weight = count times a sin^2 taper over the window's first and last 0.2 in ln m, y = c0 + c1 (x - x_d) + c2 (x - x_d)^2 + teeth 1, 2, 3, 4, 6 at P, plus s q_j with q_j the bin-average of Phi((x - x_d)/0.04); dRSS = RSS without the q column minus RSS with it; the argmax over candidates with s < 0 on a grid of step 0.002 in ln m, refined by a parabola through the top three points), with P = the L4 P-hat on the (T) bins without a step column, run in two windows. N6, the fixed window: candidates in [ln 8^5 - ln 1.5, ln 8^5 + ln 1.5] (21,845 to 49,152), giving m_lo, dRSS_lo and s_lo. U6, the moving window: candidates in [ln (4 x 8^5), ln 500,000] (131,072 to 500,000), each local fit carrying one extra free column, the bin-average of Phi((x - ln m_lo)/0.04), giving m_up, dRSS_up and s_up. A step is detected in a window when its dRSS is at least 25 and its s at most -0.05. F-hat = m_up / m_d(2187). Errors on ln m_lo, ln m_up and ln m_d(2187): the 20-block jackknife (leave-one-out fits rescan) and the five-group split error, both reported, neither in a line. (T) Bins edges(384000) (ratio about 1.12, integer edges), cut below the first bin with fewer than 100 bursts; y = ln(count/width) at the bin's geometric centre, weight = count. Background columns 1, x, x^2, m/384000 and Legendre P_3 .. P_d of x mapped from [ln 32, ln 384000] to [-1, 1], plus two step columns, the bin-averages of Phi((x - ln m_lo)/0.04) and Phi((x - ln m_up)/0.04), each with its own free amplitude, each omitted only if it is below 0.05 in every bin of the cut window. Declared set: L4 (degree 4, teeth 1-4), L6 (degree 6, teeth 1-4), L6+6 (degree 6, teeth 1-4 and 6). Reported row, not a member: A4 (L4 with the two step columns and tooth 2 given u cos, u sin, u = 2(x - x_c)/(ln 384000 - ln 32)). Each fit scans P on 6.50 .. 10.00 step 0.02 (first grid point on ties) and is fitted at its own P-hat; v3 = (a3, b3) at that P-hat; C3 its 2 x 2 jackknife covariance from the 20 leave-one-block-out fits at the same P-hat, (19/20) sum (v_i - vbar)(v_i - vbar)'; chi2_3 = v3' C3^-1 v3, whose null law is Hotelling's (38/18) F(2, 18). In D2, P-hat_i is the declared member i's P-hat from this (T) fit. Drawing takes about 3.8 h (sizing).
pre_seal_glimpse: (1) All of run 137, read by the drafter: its seal and its lead, R28_reading/137/scorer137.md, RESULTS137.md and comb137.json. Its readings, which are now glimpses and enter no line as data. T read present (chi2_3 332.6, 24.9, 37.4 for L4, L6, L6+6 with one step column). The one-window step at L = 6561 was at 264,963 (dRSS 59.3, s -0.168), F-hat 7.4477, ln F-hat 2.007905, member P-hat 7.34, 7.34, 7.36. D2 was gated out at a median 7.34 against 7.30, with margins -0.028 to -0.030, nearer P. Two of the five groups of draws put the argmax at ln m 10.5427 (37,899) and 10.0713 (23,654), so the split error was 0.540 against a jackknife error of 0.0135. At 32,768 at L = 6561, dRSS was 14.19 and s -0.039. The 6561 comb-cancelled residual dipped -0.039 to -0.067 from 34,019 to 47,662 and recovered by 53,332. The fresh anchor sat at 35,576, and phi in the H8 window ran 8.72 to 0.99. (2) What the glimpse chose, and nothing else: the worlds simulated (combs at 7.40 to 7.55, whose two-column member P-hats reach 7.32 to 7.40; a fixed second step of -0.03 to -0.08; a quarter-size tooth), the stress factor 4.5 on phi - 1, the two windows (N6 is centred on 8^5 and holds both low fifths' positions), and the 32,000 H8 draws (set by D3's power, from worlds). Every line was set from simulated worlds, never from 137's draws. (3) Run 137's premise (h8_premise.py and .log) and its draft notes (R28_reading/drafts/138-review/notes.md); the stored per-draw counts of runs 133 and 134 through h8_premise's cache, exactly as run 137's premise read them. (4) No sample of run 137 was opened, no L = 729 or L = 6561 sample was read, graphs/H8.bin was not opened, and nothing was drawn. (5) The forecast, which carries no information about the outcome: T present (137's teeth were large in every member). D1 moves. D2 nearer P (137's F-hat 7.448 against member P-hats 7.34; in the worlds whose members read 7.34, the comb is near 7.55, where D2's prediction power is lowest, 0.70). D3 falsified or unanticipated, not present: 137's feature at 8^5 (s -0.039 at 32,768) is about the size of the dip the moved bulk leaves in N6 when no fixed step exists (median s_lo -0.042), and the noise-free argmax in N6 of every world without a fixed step sits at 37,978 to 39,303, beside 137's low fifth at 37,899.
seed_scan: {date: 2026-10-25, hash: 42d38e321beaa463, files: 5054}
source: ../lab/runs/138-the-comb-its-moving-step-and-a-fixed-one-at-6561.md

PredictionThe hypotheses under test, not the forecast (that is in pre_seal_glimpse). (T) Present: chi2_3 is at least 17.23 in every declared member. (D1) The step moves: a step is detected in U6 (dRSS_up at least 25 and s_up at most -0.05). (D2) It moves by the comb's own period: with the U6 step detected, ln F-hat is below the midpoint of ln P-hat_i and ln 8 for every declared member i, and below ln 8 - 0.025. (D3) A second step stays at a fixed burst mass near 8^5: in N6, dRSS_lo is at least 25 and s_lo at most -0.055.

Falsifier(T) chi2_3 is below 12.69 in at least one declared member: some declared background removes the tooth. (D1) No step is detected in U6 while N6 holds one (dRSS_lo at least 25 and s_lo at most -0.05): the step stayed where it was. (D2) With the U6 step detected, ln F-hat is at least ln 8 - 0.025: it moves by 8. (D3) dRSS_lo is below 10, or s_lo is above -0.05: no fixed step beyond the dip that the moved bulk alone leaves in N6. A clause that meets neither its prediction nor its falsifier is unanticipated. D2 is not scored if no step is detected in U6, and it then counts as unanticipated in the run outcome. Withdrawn if a control fails. Fit quality against the noise floor is reported and never withdraws. Run outcome (the combining rule): withdrawn if a control fails; otherwise falsified if any clause's falsifier fired; otherwise confirmed if all four clauses are confirmed; otherwise unanticipated. Reading table. T present: a 3/2 tooth that survives every declared background on the H8 window with both steps modelled. T falsified: a background the set already holds absorbs it. D1 moves: the top-end step grows with the system. D1 falsified: it is a burst-mass feature that stays. D2 nearer P: the moving step rides the bursts' own cutoff, which repeats by P. D2 falsified: it rides the block hierarchy (8 per level). D3 present: a second, fixed feature near the block mass 8^5 exists beside the moving one. D3 falsified: what sits near 8^5 at L = 6561 is no larger than the moved bulk's dip.

Statement

Run 137 asked three things of the comb at L = 6561 and was withdrawn on a defect of its own seal. Control (iii) gave its targets as integers but its tolerance as 1e−6 in ln m. At m ≈ 3.5 × 10⁴, one unit of m is 2.9 × 10⁻⁵ in ln, so a correct implementation could not pass. Its draws are spent, and what it read is now a glimpse. This draft asks the questions again on 32,000 fresh H8 draws and 32,000 fresh H7 anchor draws, with four changes.

Control (iii) gives every target as ln m at full precision, from a premise rerun. That rerun reproduces, bit for bit, the two values run 137's lab computed. A check in the premise asserts that each target's last printed place is finer than its tolerance. Run 137's integers fail that same check.
Two steps, two windows. In two of run 137's five groups of draws, the step at L = 6561 fell back near the L = 2187 position. So the step is now sought twice:
in a window fixed near 8⁵ (N6, 21,845 to 49,152);
in a window for the step that moves (U6, 131,072 to 500,000), whose fit carries a column for the N6 step.

Every tooth fit models both steps. In the worlds with a fixed step, run 137's one-window statistic locks onto the low feature in about 70% of replicas. 3. The clauses are restated for what 137 read. T is now predicted present. D1 asks whether a step appears in U6. D2 needs no gate: its prediction is "nearer P" and its falsifier is "moves by 8", and at the error H8 gives that falsifier has power 0.95 or more. D3 is new: is there a fixed step near 8⁵ at L = 6561 beyond what the moved bulk leaves there? 4. 32,000 H8 draws, not 16,000, because D3's falsifier needs it (below).

Resolution and sampling (stated before any power)

	L = 2187 (anchor)	L = 6561
tooth window in ln m	[ln 32, ln 48,000], 7.313	[ln 32, ln 384,000], 9.393
cycles of the fundamental in the window (P 7.30 / 7.45 / 7.55)	3.68 / 3.64 / 3.62	4.72 / 4.68 / 4.65
resolution, about one cycle per window	0.137 cycles (0.859 radians) per ln m	0.106 cycles (0.669 radians) per ln m
adjacent teeth, 1/(2 ln P) at P 7.45	0.249 cycles (1.564 radians) per ln m, 1.82 resolution units	the same, 2.34 resolution units
sampling step of the tooth bins in ln m (largest)	0.1210	0.1195
Nyquist	4.13 cycles (25.97 radians) per ln m	4.18 cycles (26.28 radians) per ln m
highest harmonic of the fundamental below Nyquist	8th (tooth index 16)	8th (tooth index 16)
highest tooth fitted	k = 6	k = 6

The step statistic uses bins of ratio 1.02: step 0.0198 nominal and 0.0377 largest, so Nyquist is 13.25 cycles (83.24 radians) per ln m, the 26th harmonic. Each candidate's local fit is 5.0 wide in ln m (2.49 cycles of the fundamental). N6 is 0.811 wide (0.40 cycles) and U6 is 1.339 wide (0.67 cycles).

The two steps sit about ln P apart, 1.99 to 2.08 in ln m, which is one period of the fundamental. So a step that moves by P lands at the same comb phase. D2's readings, P and 8, are 0.058 to 0.105 apart in ln m. Both of these are positions, set by the counts and by the step's sharpness (a width of 0.04, two bins), not frequencies. §9's limit does not bind them.

Seal: units under zoom

The ruler is zoom by b on burst size, m → b m, so x = ln m → x + ln b.

term	unit	under m → b m	closed how
y = ln(count/width)	log of a density	a power law shifts by a constant	the data, not a column
1, x, x², P₃ … P₆	(ln m)ⁿ	each mixes into lower degrees	as the family, one free coefficient per degree
m/384,000	a ratio of like quantities, pure	scales by b	alone: its coefficient absorbs b
the two step columns in each member, Φ(ln(m/m_lo)/0.04), Φ(ln(m/m_up)/0.04)	pure	unchanged: m_lo and m_up are measured on the same draws and ride the zoom	one free coefficient each
tooth k pair	tooth k, k/(2 ln P) cycles per ln m	its phase rotates by k π ln b / ln P radians	as a pair with two free coefficients
A4's u cos, u sin on tooth 2 (reported row)	tooth 2 times a pure u	u shifts, mixing into tooth 2	with tooth 2's pair
1, (x − x_d), (x − x_d)² in the step model	(ln m)ⁿ	unchanged when x_d rides the zoom	as the family
the other step's column in a U6 fit	pure	unchanged when m_lo rides the zoom	one free coefficient
the step Φ(ln(m/m_d)/0.04)	pure; its argument is the log of a ratio of like quantities	unchanged when m_d rides the zoom	one free coefficient s
F̂ = m_up / m_d(2187)	pure	invariant

The check passes for every member and for the two-window step model. Its largest residual is 2.0e−14 (h8b_premise.log, section Z). A passing check says which numbers are meaningful. It never says what they are worth.

Power at the declared size (h8b_premise.log section S; 500 replicas per row; 32,000 H8 draws against 32,000 anchor draws)

clause	outcome	world (estimate)	rate
T	falsified (some member < 12.69)	no tooth: 15 rows (combs 7.30 to 7.55, fixed step or not, φ × 4.5)	0.86 to 0.99
T	present	tooth at ¼ of M8's size (combs 7.30, 7.45, 7.55)	0.77 / 0.98 / 1.00
T	falsified	the same ¼ tooth	at most 0.09
D1	falsified (no step in U6, one in N6)	the step stays (combs 7.30, 7.45)	0.97
D1	moves	every moving world (16 rows)	1.00
D2	falsified (ln F̂ ≥ ln 8 − 0.025)	moves by 8 (combs 7.30, 7.55; fixed step or not; φ × 4.5)	0.95 to 0.96
D2	nearer P	moves by P: 7.30 / 7.45 / 7.55 / comb 7.55 with shift 7.45	0.98 / 0.87–0.92 / 0.70–0.80 / 0.96
D2	falsified	moves by P	at most 0.04
D3	falsified (s_lo above −0.05 or dRSS_lo < 10)	no fixed step (6 rows)	0.86 to 0.93
D3	present (dRSS_lo ≥ 25, s_lo ≤ −0.055)	fixed step −0.05 / −0.03	0.99–1.00 / 0.74–0.87
D3	present	no fixed step	0.02 to 0.03
D3	falsified	fixed step −0.03	at most 0.12

At 16,000 H8 draws, the size run 137 used, T, D1 and D2 keep their power: their falsifiers reach at least 0.95, 1.00 and 0.87. D3's falsifier, though, falls to 0.68 to 0.80, below 0.7 in the φ × 4.5 row.

With two step columns, the members' P̂ follow the comb: 7.12 to 7.20 at 7.30, 7.24 to 7.30 at 7.45, and 7.32 to 7.40 at 7.55. D2 needs no gate. Its falsifier, "moves by 8", keeps a power of 0.95 or more whatever the comb, and a comb near 7.55 only weakens the prediction.

Combining rule

The run's outcome is: - withdrawn if a control fails; - otherwise falsified if any clause's falsifier fired; - otherwise confirmed if all four clauses are confirmed; - otherwise unanticipated.

A clause that is not scored counts as unanticipated. D2 is not scored when no step is detected in U6.

Reported beside

None of these is scored: - For each member: P̂ with its jackknife error, the P̂ spread over the set, a3, b3, |A3/2|/|A1|, both step columns' amplitudes, φ, the floor and RSS/floor; and the reported row A4. - Alternative step columns: the set with run 137's single step column at the one-window m_d, and with no step column. - The step at each L: m_d(2187), m_lo and m_up, each with its step size, dRSS, jackknife and five-group split errors and the five groups' positions. - Run 137's one-window statistic on these draws. - dRSS and s at named masses: at 32,768 and at m_d(2187); and at 8 × m_d(2187) and P̂ × m_d(2187) in U6. - The fresh anchor: its m_d beside the stored 34,634 and run 137's 35,576. - Profiles: the comb-cancelled residual profile at both L, and the 6561 window's cutoff region.

Sealed 2026-10-25

As written. The lab chat reads this file above "Not part of the seal" only, plus comb137.py, h8_premise.py, h8b_premise.py, perc_inv.c and MANUAL.md's Lab conventions. It pins comb138.py's md5 before any draw, commits without --no-verify, and discloses memory in its first line (MANUAL, Lab conventions).

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

generated 2026-10-25the lab index