# Pre-registration: scanstat-t37 — the seventh exact level of the moving-sum exponent **Sealed before any producer for this pass was written, and committed alone**, with message `prereg: scanstat-t37` and nothing else in the commit. `git log --stat` is the proof. Every number in §1, §2 and §3 was computed from **already embedded** corpus figures by a throwaway script kept outside the repository, so no producer file existed when the predictions were fixed. The producer recomputes each of them from the same published inputs and aborts on any mismatch, so the freeze is machine-checkable after the fact as well as commit-ordered. The target is `T₃₇`: `D = ∏_{5≤p≤37}(p−2) = 217929355875` slots over `W = 37# = 7420738134810` positions, which is 1150 times the work of the T₂₉ run that `import-scanstat-04-score.js` did in 38.9 s. ## 0. What this registers, and the T₃₁ disclosure `import-scanstat.md` §6 named T₃₁ as the unreached sixth level. A sibling pass is computing T₃₁ right now under `research/scanstat2-*` and its own sealed pre-registration, `scanstat2-prereg.md` (commit `cde163a`). **Its result is not available to this file and is not used anywhere in it.** The fit here therefore runs on the **five committed levels T₁₃, T₁₇, T₁₉, T₂₃, T₂₉ only.** That creates one asymmetry worth fixing in advance rather than arguing about later. By the time T₃₇ lands, T₃₁ will exist, and a six-level refit will be available and will look more informative than the five-level one. Refitting after seeing the target is exactly the move pre-registration exists to forbid. So this file pre-commits to reporting **both**, in this order and with these labels: 1. **`H*₅(T₃₇) = 0.381254`, the pre-registered prediction, band below.** This is the only number the verdict in §2 is read against. 2. **`H*₆(T₃₇)`, POST-HOC, computed after T₃₁ was known.** Reported for interest, never as the criterion, and always carrying the label. Its sensitivity is small and is registered here so a later reader can check nothing was tuned: `dH*₆/dH(T₃₁) = 0.723`, so a T₃₁ landing one standard error (0.008) off the line moves `H*₆(T₃₇)` by 0.006, a little over half the width of the band below. If T₃₁ lands exactly on the five-level line at 0.359402, the six-level law is numerically identical to the five-level one and `H*₆ = H*₅`. ## 1. Prediction: H(T₃₇) ### Inputs, all embedded, none recomputed `H` at a level is the OLS slope of `ln sd_m` on `ln m` over the frozen grid `M = [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64]`, where `sd_m` is the exact standard deviation of `slot[i+m] − slot[i]` over `i ∈ Z/D`. | level | D = ∏_{5≤p≤x}(p−2) | ln D | H (published) | ln c (published or refit) | sd₁ (published) | |---|---|---|---|---|---| | T₁₃ | 1485 | 7.303170 | 0.2661 | 2.64446 | 12.0326 | | T₁₇ | 22275 | 10.011220 | 0.2804 | 2.81034 | 14.5863 | | T₁₉ | 378675 | 12.844434 | 0.3001 | 2.93725 | 17.1618 | | T₂₃ | 7952175 | 15.888956 | 0.3216 | 3.05202 | 19.4656 | | T₂₉ | 214708725 | 19.184793 | 0.3367 | 3.13670 | 21.4409 | `H`, `ln c` and `sd₁` at T₁₃, T₁₇, T₁₉ are the embedded output of `research/import-scanstat-03-prereg.js` (1). `H` and `sd₁` at T₂₃ and T₂₉ are the embedded output of `research/import-scanstat-04-score.js` (1) and (2); `ln c` at those two levels is not printed there, so it is refit here from the twelve published `sd_m` values in the same output blocks, which are given to three decimals on quantities of size 20 to 90, a rounding of order 1e-5 relative. The four decimals printed for `H` are the inputs used, so this refit is reproducible by anyone holding only the embedded record. Nothing here reads a tile. Target constants, exact: `D = 217929355875`, `W = 7420738134810`, `ln D = 26.107437`, `mbar = W/D = 34.05112`, `√(2 ln D) = 7.2260`. ### The rule OLS of `H` on `ln D` over **all five** committed levels. The frozen three-point law of the first pass was `H = 0.220511 + 0.006140 lnD`; the five-point refit is ``` H = 0.220795 + 0.006146 * lnD s = 2.4468e-3, se(slope) = 2.6083e-4 residuals (T13..T29): +4.191e-4 -1.925e-3 +3.620e-4 +3.150e-3 -2.006e-3 ``` which is bit-for-bit the same law the sibling pass froze for T₃₁, from the same five published rows. Evaluated at `ln D = 26.107437`: > **H\*₅(T₃₇) = 0.381254** The superseded three-point law lands at 0.380811 at the same point, so the two fits differ by 4.4e-4 here and this test cannot separate them. That is registered as a limitation, not as a result. ### The band Same construction as the sibling pass and as the adversarial review of the first pass: the 95% **confidence band on the fitted line**, `ŷ ± t_{n−2,0.975} · s · √(1/n + (x₀−x̄)²/Sxx)`, with `n = 5` and `t₃ = 3.182446`. > **95% confidence band on the line at T₃₇: [0.369866, 0.392641]** (half-width > 0.011387, `se_mean = 0.003578`) > > 95% prediction band, for reference only, not the criterion: > **[0.367459, 0.395049]** The extrapolation is 6.92 units of `ln D` past T₂₉, twice the reach of the T₃₁ step, so the band is wider than T₃₁'s (half-width 0.008625) even with the same five points behind it. ## 2. The criterion, and what each outcome supports Let `Ĥ` be the measured OLS slope at T₃₇ on the same grid, with its own regression standard error `se(Ĥ)` (0.0096 at T₂₃ and 0.0080 at T₂₉, so 0.008 or a little less is expected). - **INSIDE [0.369866, 0.392641]**: the linear-in-`ln D` rule is a *description* that holds over seven exact levels spanning `ln D` from 7.30 to 26.11, four of them out of sample and three blind, across a 7.4e12-position tile. It stays a description and not a law: §1(d) and §1(e) of `import-scanstat.md` still forbid extrapolating it to `H = 0.5`, since the true `m`-dependence is a crossover with no fitted scale and the duality forces the exponent to turn over at `m = D/2`. - **OUTSIDE the band**: the linear form was a coincidence of a short ladder. Five points on a smooth monotone series admit many curves, and this is the longest extrapolation attempted. A miss falsifies the specific functional form, not the finding that `H < 0.5`. - **Either way**, the `√m` kill stands or falls only on whether `Ĥ` is far from 0.5. **`√m` is refuted at T₃₇ if `Ĥ + 3·se(Ĥ) < 0.5`.** This is registered separately because it is a different claim from the shape of `H(ln D)`, and the first pass conflated their strengths once already. - **Direction, registered separately.** `H` has risen at every level so far. **`Ĥ` is predicted to exceed 0.3367**, the T₂₉ value; a non-increase would be a qualitative break and is a bigger event than a band miss. ## 3. Secondary, weak, and registered as weak The same five-level refits on `ln c` and on `ln sd₁`: ``` ln c = 2.379851 + 0.041107 * lnD s = 3.5883e-2 ln c*(T37) = 3.453049 -> c* = 31.5966 95% confidence band on the line: [3.286050, 3.620048] -> [26.7371, 37.3394] ln sd_1 = 2.177678 + 0.048380 * lnD s = 4.5636e-2 ln sd_1*(T37) = 3.440754 -> sd_1* = 31.2105 95% confidence band on the line: [3.228366, 3.653142] -> [25.2384, 38.5957] ``` Both bands are ±18% to ±24%, so these predictions can barely be failed and are registered only so that a wild miss is on the record. `sd₁` is a property of the gap multiset, not of the word's order, and there is no reason on file to expect it linear in `ln D`. The two frozen model tables at T₃₇, on `maxsum_m = m·mbar + sd_m·√(2 ln D)` with Model A `sd_m = c* m^{H*}` and Model B `sd_m = sd_1* √m`: | m | A: sd_m | A: maxsum_m | B: sd_m | B: maxsum_m | |---|---|---|---|---| | 1 | 31.597 | 262.4 | 31.210 | 259.6 | | 2 | 41.154 | 365.5 | 44.138 | 387.0 | | 3 | 48.034 | 449.2 | 54.058 | 492.8 | | 4 | 53.602 | 523.5 | 62.421 | 587.3 | | 6 | 62.562 | 656.4 | 76.450 | 756.7 | | 8 | 69.815 | 776.9 | 88.277 | 910.3 | | 12 | 81.486 | 997.4 | 108.116 | 1189.9 | | 16 | 90.932 | 1201.9 | 124.842 | 1446.9 | | 24 | 106.133 | 1584.1 | 152.900 | 1922.1 | | 32 | 118.436 | 1945.5 | 176.553 | 2365.4 | | 48 | 138.235 | 2633.3 | 216.233 | 3196.9 | | 64 | 154.260 | 3294.0 | 249.684 | 3983.5 | Model A's row at `m = 1` predicts 262.4 against the known exact answer 528, which prices the tail factor's failure rather than the exponent's: `excess/sd_m` is 9.039 at `m = 1` at T₂₃ and 10.628 at T₂₉, not `√(2 ln D)`. **No claim is made that these maxsum tables are close.** What is registered is the comparison, unchanged from the first pass: **A′ = `sd₁·m^{H*}` must beat B′ = `sd₁·√m` on the ln-RMS of `excess_m = maxsum_m − m·mbar` over the frozen grid**, both anchored at the measured `sd₁(T₃₇)`, which is unknown until the run. ## 4. Custody gates the producer must pass before it reports anything 1. **Slot count = 217929355875 exactly.** This is the Copying Theorem product `∏_{5≤p≤37}(p−2) = 3·5·9·11·15·17·21·27·29·35`, and the run counts slots independently by streaming, so the two must agree to the digit. Written out: `3·5 = 15`, `·9 = 135`, `·11 = 1485` (T₁₃), `·15 = 22275` (T₁₇), `·17 = 378675` (T₁₉), `·21 = 7952175` (T₂₃), `·27 = 214708725` (T₂₉), `·29 = 6226553025` (T₃₁), `·35 = 217929355875` (T₃₇). 2. **`maxsum₁(T₃₇) = 528 = G₂(37#)`**, the twelfth exact ladder entry, as carried by `research/exact-g2-ladder.js` (x = 37, pos = 544899485411) and OEIS A144311. Note what this gate is worth on its own: the ladder's maximality at 37# rests on a filter argument, `maxsum_THRESH(T_v) < G₂`, while this run scans every one of the 217929355875 gaps. A pass is an **independent exhaustive maximality certificate** for the ladder entry; a value above 528 would refute the ladder and the OEIS term, and a value below 528 would mean the engine is broken, since 528 is exhibited by trial division. 3. **`Σ_i gap_i = W = 7420738134810`**, i.e. `mbar·D = W` exactly, in integers. 4. **`Σ_i (slot[i+m] − slot[i]) = m·W` exactly for all twelve m**, in exact integer arithmetic, since each gap is counted `m` times on the cycle. This is a per-`m` gate, and it is the one that catches a sharding error: a dropped or double-counted window fails it at every `m` at once. 5. **The same engine, run first at T₂₃ and T₂₉, must reproduce the embedded columns of `research/import-scanstat-04-score.js` exactly**, both `maxsum_m` and `sd_m` to the three decimals printed there, at all twelve m, and must return `maxsum₁` = 204 and 258 and `D` = 7952175 and 214708725. This is a deliberate recomputation of embedded artifacts, permitted under the standing compute rule as validation of a success before it is extended, and it is the only way to check a new engine against a known answer. 6. **Sharded and unsharded must agree at T₂₃ and T₂₉**, on every printed digit of every column, before the sharded run at T₃₇ is trusted. Any gate failing aborts the run. A partial green is a defect. ## 5. The compute plan, registered in advance Registered here so that no part of it is a post-hoc choice. **Grid.** `M = [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64]`, unchanged from the first pass. `m_max = 64`. **Engine.** The T₂₉ lever streams the slot sequence through a ring buffer of depth 64 and never materialises the gap word. T₃₇ is 1150 times that work, so the engine is rewritten with the same interface and three changes: the small primes up to 13 are carried as a 30030-position candidate wheel of 1485 offsets instead of being sieved, so only the primes 17 to 37 are marked and only 1485 of every 30030 positions are ever examined; the segment is sized to stay inside cache; and the moment accumulators are exact, the sums in integers below 2^53 and the sums of squares in BigInt, since `Σ v²` at `m = 64` reaches 1.0e18 and would lose 2.4% to Float64 rounding at this depth. Statistics are recovered as `Var = (D·Σv² − (Σv)²)/D²` in exact integers, so the reported `sd_m` carries no accumulated error. **Sharding, and the overlap argument.** The position range `[0, W)` is cut into `N` contiguous ranges on multiples of 30030. Shard `s` **owns** the windows whose starting slot lies in `[start_s, end_s)`, so every one of the `D` windows is owned by exactly one shard and none is owned twice. A window of `m ≤ 64` slots reaches at most 64 slots past its start, so shard `s` continues streaming past `end_s` until it has emitted exactly **64 further slots**, and scores only the windows it owns. The overlap is stated in slots rather than positions on purpose: it is exact and needs no bound on the gap, whereas a position-length overlap would need `maxsum₆₄ < overlap`, which is the very quantity being measured. The last shard's continuation wraps to position 0 with `W` added, which is the cyclic completion the unsharded engine does with its `head` array. `maxsum` and `minsum` combine by max and min over shards, `Σv` and `Σv²` by addition, and gate 4 checks the combination. **Target.** N chosen from the measured throughput of the T₂₃ and T₂₉ validation runs so the wall time is at most 12 h at `nice -n 15`, on a 10-core machine already running other jobs. ## 6. Not predicted here - The T₃₇ `maxsum_m` values themselves, beyond `maxsum₁ = 528` which is a gate and not a prediction. The tail factor is unexplained (`import-scanstat.md` §6) and any maxsum prediction inherits that. - `se(Ĥ)` at T₃₇. It has fallen from 0.0096 to 0.0080 across the last two levels and is expected to fall again, but no number is registered. - Anything about T₃₁. Its verdict belongs to `scanstat2-prereg.md`, and this file's criterion does not move whatever T₃₁ returns. ## 7. Producers Added in a second commit, after the seal, and changing no number, no rule and no criterion above: the file names of the code that carries this out. `git log --stat` separates the two commits, and the first one contains this file alone. - [`../../scanstat-t37-01-engine.js`](../../scanstat-t37-01-engine.js) — the engine (module, no output of its own): wheel in candidate-index space, exact moments, sharding by slot ownership. - [`../../scanstat-t37-02-validate.js`](../../scanstat-t37-02-validate.js) — gates 5 and 6: the engine against the embedded T₂₃ and T₂₉ columns, parsed from `import-scanstat-04-score.js` rather than retyped, plus the forced BigInt-flush check and the T₃₇ throughput projection. - [`../../scanstat-t37-03-shard.js`](../../scanstat-t37-03-shard.js) — one shard. - [`../../scanstat-t37-04-run.js`](../../scanstat-t37-04-run.js) — the driver: forks the shards, combines the partial moments, runs gates 1 to 4, and scores this file's frozen predictions after re-deriving every one of them from the embedded inputs and aborting on any mismatch.