# Custody wave three: the two stopwatches are gone, and the last judgement cases are bound **Counter: 12 hand-pasted tails to 3.** Nine of the twelve are now bound by `research/qc/embed.js`. The three that remain are the three priced over budget, and each has a recorded reason below. Every file touched here was verified with `node research/qc/embed.js --check` after its embed, and `node research/qc.js` stayed at TOTAL 0 for everything in this wave's scope throughout. The classes this wave had to clear were the ones wave two left because they need a reader: two tails whose content depended on a WALL CLOCK, one tail that asserted a table its code had never printed, and seven tails carrying hand-written prose or a second invocation's output inside the OUTPUT region, where an embed would silently destroy it. --- ## 1. The two stopwatches Both are fixed the same way: the gate that decided what the tail contained is now a property of the arithmetic, not of the machine. **`research/a3-09-histogram-operator.js`.** The fold diagonal ran to `p <= 160` and broke out when a level cost more than 33 wall-clock seconds, so the number of points in its two regressions was set by machine load. Wave two measured the consequence: a loaded machine stops at `p = 137` and reads `0.767` and `0.729` where the pasted tail reads `0.770` and `0.738`. The gate is now `const DIAGONAL_MAX_P = 139`, stated in the code with the reason. **139 rather than 137 is the load-bearing choice.** It is the level the 2026-08-16 run actually reached, it is the last affordable one (the inclusion-exclusion roughly doubles per prime step, so `p = 139` costs about 75 s of the script's 198 s and `p = 149` would add about 150 s more), and it makes the diagonal 31 points on every machine. **Both fitted constants reproduce exactly**: `all 31 points (p=7..139): a=-0.235 b=1.2992 R2=0.99292 -> tail ~ exp(-2p'/(0.770*mbar))` and `p>=37 only (23 points): a=-0.452 b=1.3550 R2=0.99355 -> tail ~ exp(-2p'/(0.738*mbar))`. So no document in the corpus carries a stale constant from this file, and the forecast in TODO item 2 and in the thirty-first pass entry — that the constants would move — is superseded rather than confirmed. The alternative bound, 137, would have moved them for no gain. One further change was needed to make the block re-checkable. The diagonal's last column is the level's cost in seconds and was printed as a bare number, so `qc/tailfmt.js` could not see it as volatile and every re-run would have failed `--check`. It now prints with an `s`, the header column is renamed `s` to `cost`, and `tailfmt` normalises it. The 14 figures the guard flagged as missing on the first attempt were exactly those 14 cost values; nothing else in the block moved, and a line-by-line diff of the old and new blocks shows only timing brackets and the replaced stopping line. **`research/a3-01-misalignment-ledger.js`.** Its fold progress lines printed whenever 30 wall-clock seconds had passed, which is why wave two saw `copy 9/31`, `18/31` and `27/31` vanish on a loaded machine. The gate is now `const PRINT_EVERY_KTH_COPY = 9`, tested on the copy index. At fold 31 that is copies 9, 18 and 27 of 31, so the 2026-08-16 tail reproduces digit for digit (`1807708938`, `3615417901`, `5423126831` at the three copies, and `G2=348 at 4 site(s)`). Folds 11 through 29 now print one to three progress lines of the same kind, which is new output and destroys nothing. The embed took no `--force`: every figure of the legacy block is in the new run. The banner's own prose about the original run's cost moved into the file header, verbatim and quoted, because an embed header can only state the run it made. ## 2. `genealogy.js`: the table was right, the limit was not The tail asserted a six-entry convergence table with no code behind it and closed it at `0.41625`. The code now computes the table — `delta(p) = D(p)/p#` accumulated as `prod (1 - 2/q)/2`, sieved to 9973 — and prints it. **All six entries reproduce exactly**: `0.3253 (13), 0.3661 (31), 0.4007 (97), 0.4093 (499), 0.4140 (1999), 0.4150 (9973)`. Nine of eleven figures were sourceless and all nine are now sourced, which is the good half of the finding. The limit is not. `2*C2*e^{-2gamma} = 1.3203236316937248 * e^{-1.1544313298} = 0.4162145328309681`, so the fifth decimal is 1 and not 5. The code has always printed `0.41621`; only the paste and the header comment said `0.41625`. This is the single figure the embed guard refused on, and the embed was forced after the old block was quoted here: ``` // OUTPUT (2026-08-14): // p=5: slots=3 orphans=0 edge-children=3 (Seam Lemma: 3) // p=7: slots=15 orphans=0 edge-children=5 (Seam Lemma: 5) // p=11: slots=135 orphans=0 edge-children=9 (Seam Lemma: 9) // p=13: slots=1485 orphans=0 edge-children=11 (Seam Lemma: 11) // delta*ln^2(p): 0.3253 (13), 0.3661 (31), 0.4007 (97), 0.4093 (499), // 0.4140 (1999), 0.4150 (9973) -> 0.41625 (2*C2*e^{-2gamma}) ``` The file also had no READINGS banner. It has one now, with the custody note in it. **`0.41621` is the bound truth from this file onward.** The other sites that carry `0.41625` are listed in §5; they are not edited here. ## 3. Prose and second invocations, moved out of OUTPUT The rule applied in every case: the OUTPUT region is a recording of one run, so anything a reader wrote and anything a *different* invocation produced moves above it or into READINGS, verbatim, with a provenance line naming the invocation that does produce it. Nothing was deleted, and nothing was reworded except to indent it and put it in quotation marks, which is necessary because `qc/tailfmt.js` finds the OUTPUT banner by scanning backwards for the last line matching `// OUTPUT`, and a moved banner line below the real one would capture the tail. | file | what was inside OUTPUT | where it went | what the tail is now | |---|---|---|---| | `attack-beta2-04-loss-budget.js` | the entire eight-section summary, hand-written from a 531-line run; nothing printed it | READINGS 0, with the roundings it performs named | the 531-line run, 9.8 s | | `a3-08-adjacent-pairs.js` | section `[6b]`, the T31 leg from a separate detached `--t31` run of 2,620 s with 28 of 31 progress lines elided | READINGS 0, invocation named | the default run, 89.6 s | | `a3-10-lower-tightness.js` | the `deep31` and `deep37` legs, two further modes that `process.exit(0)` before the main leg | READINGS 0b, both invocations named | the main leg, 27.6 s | | `attack2-rankin2d.js` | sections (A) to (F), a hand summary of the default run and of `--deep` | READINGS 0, both invocations named | the default run, 168.3 s | | `natal-cap-32-wrap-identity.js` | a hand summary of `small`, `at13` and `at17`, the last fanning out to 6 + 6 workers | READINGS 0, all three invocations named | `small`, 50.4 s | | `a3-02-diagonal-f.js` | two invocations pasted together, `fast ` and `deep` | see below: unified, not moved | one run of mode `all`, 268.1 s | **`a3-02-diagonal-f.js` is the one composite that dissolved.** Its mode `all` runs the fast leg and the deep leg in one process, and PHASE C' takes the deep leg's five results from the command line, so a single invocation reproduces both halves and PHASE D then recomputes from scratch the five numbers PHASE C' was handed. That makes the tail self-checking. **A precision finding came out of it.** Passing the deep results as the block prints them (`1.844e-2`, `32.2105`) reproduces every table cell but moves PHASE E's two `ln(1/f)` intercepts by 0.001, to `1.357` and `1.601` against the pasted `1.358` and `1.602`. Passing them at full precision — `f = 114848070/6226553025 = 0.018444887490538957`, `mbar = 200560490130/6226553025 = 32.21051668952904` — reproduces `1.358` and `1.602` exactly. So the 2026-08-17 run was fed the exact values, the pasted block is right, and the invocation is now recorded in the file header where the next reader will find it. No value moved. ## 4. The three left, and why | file | cost | reason | |---|---|---| | `natal-cap-33-overnight.js` | ~38 min | four separate run modes (`run1|run2|run3|smoke`), priced over budget by TODO item 1 and untouched here. A concurrent wave is working the over-budget tier (`custody-overnight.md`); this file belongs to that wave, not this one. | | `natal-cap-34-wrap-precision.js` | ~35 min over six stages | a six-command chain sharing a scratch directory through `NC34_DIR`, with `exact13` alone at 1466 s. `embed.js` binds one invocation and this needs six in order. Its OUTPUT header already carries a full provenance declaration naming all six commands and their costs, which is the honest form for a tail of this shape. | | `natal-cap-37-at41-march.js` | 6.16 h | the `@41` march itself, pasted byte for byte out of three log files that live outside the repository at `~/Files/primeoire-runs/at41/`. Not reproducible inside a migration budget. Its header already declares every block's source log, date and elision. | ## 5. What the corpus still carries **`0.41625` for `2*C2*e^{-2gamma}`, five live sites.** The value is wrong in the fifth decimal at every one of them, and `audit-cross-document-constants.md` already found this and called it "one wrong digit, uniformly, which is precisely the class a cross-document check can never catch". Nothing below is edited by this wave. | line | what it says | |---|---| | `paper/moire-primes.md`:266 | `2C₂e^{−2γ} = 0.41625…` | | `paper/beta2-note.md`:183 | `with 2C₂e^{−2γ} = 0.41625…` | | `paper/beta2-note.md`:185 | the numerical verification sentence, `0.41625` | | `research/sift-limit-attack.md`:201 | `G₂ ≲ (2F/0.41625)·pₙ·ln²pₙ` | | `research/history/staging/attack-block-08-secondmoment.md`:352 | `G2 <~ (2F/0.41625) p_n ln^2 p_n` | `research/attack-beta2-04-loss-budget.js`:183 prints `(the 0.41625 that maxgap-law.md carries)`. `maxgap-law.md` no longer carries it, so that string is stale in two ways at once and is a code change, not a document one. **`0.770` and `0.738` need no change anywhere.** Outside `a3-09` itself the only occurrences are `TODO.md`:73 and `CHANGELOG.md`:856, and both are forecasts that these constants WOULD move once the stopwatch was replaced. They did not, because the bound was set at 139. Those two sentences are now wrong in their prediction and should be retired when TODO is next pruned. ## 5b. The nine, verified Every one re-ran under `node research/qc/embed.js --check` after its embed and returned `code-sha256 matches` / `out-sha256 matches`. For the two stopwatch files that is the whole point of the wave: a3-09's 31-point diagonal and a3-01's `copy 9/31 | 18/31 | 27/31` lines came back identical on a second, differently loaded run. | file | invocation the tail records | wall | |---|---|---| | `a3-01-misalignment-ledger.js` | `FOLDS=7,11,13,17,19,23,29,31 node --max-old-space-size=12000 …` | 115.1 s | | `a3-02-diagonal-f.js` | `node --max-old-space-size=8000 … all 0.018444887490538957 32.21051668952904 6226553025 348 4` | 268.1 s | | `a3-08-adjacent-pairs.js` | `node research/a3-08-adjacent-pairs.js` | 89.6 s | | `a3-09-histogram-operator.js` | `node --max-old-space-size=8000 …` | 197.7 s | | `a3-10-lower-tightness.js` | `node --max-old-space-size=4096 …` | 27.6 s | | `attack-beta2-04-loss-budget.js` | `node research/attack-beta2-04-loss-budget.js` | 9.8 s | | `attack2-rankin2d.js` | `node research/attack2-rankin2d.js` | 168.3 s | | `genealogy.js` | `node research/genealogy.js` | 0.1 s | | `natal-cap-32-wrap-identity.js` | `node research/natal-cap-32-wrap-identity.js` | 50.4 s | ## 6. Riders for the next wave - **`tailfmt.locate` scans BACKWARDS for the last `// OUTPUT`.** Any prose moved into READINGS that begins a line with that word silently steals the tail. Five files in this wave needed the moved banner indented and quoted for that reason. - **The `readings-not-traceable` advisory rises when this work is done right.** It went 137 to 139 across the wave, and the increase is entirely material that moved from OUTPUT into READINGS with a provenance declaration. The advisory counts figures a reading quotes that the block does not print; a declared second invocation is exactly that, by construction. It is not a defect and should not be driven down by deleting the declarations. - **A bare number that is really a duration breaks `--check` forever.** Print it with an `s`. `tailfmt`'s volatile list matches `\d+(\.\d+)?\s*(ms|s|sec|secs| seconds)` and nothing else. - **The gate showed 5 `uncited-script` findings during this wave**, all from untracked scripts written by a concurrent import wave (`fdecay-deep-00-core`, `import-interp-01`, `import-interp-02`, `scanstat2-02-crossover` and one more). None are this wave's and none were touched.