run 137Settled
../lab/runs/137-the-comb-and-its-step-at-6561.mdPredictionThe hypotheses under test, not the forecast (that is in pre_seal_glimpse). (T) The full side has no 3/2 tooth of its own. T is one-sided: only its falsifier is scored, and its "absent" cell, chi2_3 below 7.50 in every declared member, is reported and is not a confirmation. (D1) The step moves with the system: ln F-hat is above half of ln P-hat_i for every declared member i. (D2) It is a block two levels below the host (8^(n-2)), so it moves by 8: ln F-hat is above the midpoint of ln P-hat_i and ln 8 for every declared member i; scored only if the median of the three members' P-hat is at most 7.30.
Falsifier(T) chi2_3 is at least 17.23 in every declared member: a tooth that no declared background removes. Anything else: T is reported, not scored. (D1) ln F-hat is below half of ln P-hat_i for every declared member: the step is a fixed burst mass. (D2) ln F-hat is below the midpoint of ln P-hat_i and ln 8 for every declared member: the step moves by the comb's own period. Members that straddle a D line: unread for that clause. D1 and D2 are unscorable (reported, not scored) if the step is not detected at either L; D2 is reported, not scored, if its gate fails. Withdrawn if a control fails. Fit quality against the noise floor is reported and never withdraws. Reading table. T present: a 3/2 tooth that survives every declared background on the H8 window with the step modelled. T absent: reported only; on the estimated data-like worlds it is reached in at most about a third of replicas even with no tooth. T split: nothing is read. D1 fixed: the drop is a burst-mass feature that does not grow with the system. D2 nearer P: the drop rides the bursts' own cutoff. D2 nearer 8: it rides the block hierarchy.
The comb is a ripple in how often invasion bursts of each size occur. On the Sierpinski carpet H it repeats each time burst size grows by a factor P of about 7.3. Two things about it are open.
First, the carpet allows ripples only at whole multiples of the basic rate, yet runs 133 and 134 saw one at one and a half times it: the 3/2 tooth. At L = 2187 the window of burst sizes holds only about 3.6 cycles of the fundamental, and on so short a record a slow swing in the background and a tooth at 3/2 look alike. L = 6561 adds about one cycle.
Second, the burst-size density has a sharp step down at L = 2187, at 1.057 × 8⁵ (5.5 jackknife errors above 8⁵). Above the step there are about 17% fewer bursts per unit size than the power law below predicts, then a plateau to about 100,000, then a recovery before the real cutoff. A block of the carpet two levels below the host holds 8⁵ cells at this size. If the step is that block, at L = 6561 it moves by 8 to about 277,000. If it rides the bursts' own cutoff, it moves by P to about 253,000. If it is a fixed burst mass, it stays.
This draft draws 16,000 fresh H8 draws and 32,000 fresh H7 draws. It reads the step's position at both sizes with one declared statistic, and it models the step inside every tooth fit.
| L = 2187 (run 134) | L = 6561 (this draft) | |
|---|---|---|
| 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 7.2 / 7.46 / 7.55) | 3.70 / 3.64 / 3.62 | 4.76 / 4.67 / 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 7.46 | 0.249 cycles (1.563 radians), 1.82 resolution units | the same, 2.34 resolution units |
| sampling step in ln m (largest bin, integer edges) | 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, 0.0377 largest; Nyquist 13.25 cycles per ln m, the 26th harmonic). Its local window is 5.0 in ln m, 2.49 cycles of the fundamental. D2's two readings sit 0.058 to 0.105 apart in ln m. That is a position, and its resolution is set by the counts and by how sharp the step is, not by the window's length.
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, x^2, P_3 .. P_6 | (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 |
| step column in each member, Phi(ln(m/m_d)/0.04) | pure | unchanged, because m_d is measured on the same draws and rides the zoom | one free coefficient |
| tooth k pair | tooth k; frequency k/(2 ln P) cycles per ln m | the pair rotates its phase by k pi 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 |
| step Phi(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; 1.00 of it leaves its span under b = 8 if m_d is held | one free coefficient s |
| F-hat = m_d(6561)/m_d(2187) | pure | invariant |
The check passes for every member and for the step model, with the largest residual 2.0e-14 (h8_premise.log, section Z). A passing check says which numbers are meaningful. It never says what they are worth, so it supports none of the values predicted above.
| clause | outcome | world (estimate) | rate |
|---|---|---|---|
| T | present (falsifier) | tooth of M8's size, or half of it | 1.00 |
| T | present | no tooth (seven rows) | 0.00 to 0.01 |
| T | absent (reported) | no tooth, data-like worlds | 0.06 to 0.36 |
| T | absent (reported) | no tooth, model exact | 0.92 |
| D1 | fixed | step stays | 1.00 |
| D1 | moves | step moves by 7.30, 7.55 or 8 | 0.99 to 1.00 |
| D2 | nearer P | P 7.30 / 7.32 (scored: gate passes 1.00) | 0.96 / 0.92 |
| D2 | gate | comb and shift both 7.55 | passes 0.00: reported |
| D2 | nearer 8 | moves by 8 | 0.99 |
| D2 | nearer P | moves by 8 | 0.01 |
Not scored: T's "absent" and "split" cells; the reported row A4 (with the step column); the set without the step column on the H8 window and on [32, the last H8 edge at most m_d/1.25]; per member P-hat with its jackknife error, the P-hat spread over the set, a3, b3, |A3/2|/|A1|, the step column's amplitude, phi, the floor and RSS/floor; m_d at each L with its step size, dRSS, jackknife and five-group split errors; dRSS and the step size at m = 32,768 and at 8 x m_d(2187) and P-hat x m_d(2187) at L = 6561; the fresh H7 anchor's m_d beside the stored 34,634; the comb-cancelled residual profile at both L (as the premise prints it for 2187); and the 6561 window's cutoff region.
As written. The lab chat reads this file above "Not part of the seal" only, plus comb134.py, h8_premise.py, perc_inv.c and MANUAL.md's Lab conventions. It commits through the hook (never --no-verify) and runs with memory off (OPS, instances 22 and 23).
withdrawn (control (iii) fails as written; the fault is the seal's)A blind scorer read the seal above the line and RESULTS137.md (a825940) and nothing else (R28_reading/137/scorer137.md). Scores are as written.
Controls. (i) and (ii) pass. (iii) fails. The seal gives the anchor target as the integer 34,585 and requires agreement to 1e−6 in ln m_d. The lab's value, 34,585.470, differs by +1.36e−5. The premise log's own value, the one the integer rounds, is the 34,585 printed at h8_premise.log:89, and the seal never gave it at full precision. An integer target cannot carry a 1e−6 tolerance in ln at m ≈ 3.5e4, where one unit is 2.9e−5. The lab read the check "to rounding" under an RFI nobody answered; the scorer applied the seal as written. The run is withdrawn on a defect of the codifier's seal, not of the lab's work.
Readings, not scores (the draws are now seen, so nothing here can be tested on them): - T would read falsified (tooth present): χ²₃ is 332.6 (L4), 24.9 (L6) and 37.4 (L6+6), all above 17.23, with the step column in every member. - D1 would read confirmed (the step moves): ln F̂ = 2.0079 (F̂ = 7.448), above half of ln P̂_i by about 1.01 for every member. - D2 is gated out: the median P̂ is 7.34, above the 7.30 line. Its margins (−0.028 to −0.030) fall on the "nearer P" side, the owner's W2. This is a reading only. - Two of the five groups of draws put the L = 6561 step back near the L = 2187 position (m ≈ 23,700 and 37,900), against 265,000 for the pooled draws. The split error is 0.54, against a jackknife error of 0.0135. That suggests two features: one fixed near 8⁵ and one that moves. This is the codifier's lead for the reseal.
Owed: a reseal with control (iii)'s targets at full precision and fresh seeds, after a seal-lint check that a target's digits can carry its tolerance (OPS instance 25).
Theme instance (added 2026-10-25, records). None: the run is withdrawn and its numbers are readings, not scores. The reason is filed at themes/symmetry.md §2.1k; the instance waits for the reseal.
generated 2026-10-25the lab index