# Verifying the k4direct swap end to end before anyone makes it **Headline.** **The swap was made and the @13 gate is met, by ×3632 — and the ×3600 on record is an ACCURACY margin, not a speedup.** Measured today, same machine, same truth, everything else identical: the shipped Monte Carlo gives REL −1.021692e-9 (misses by 2.2%), `k4direct` gives REL −2.753038e-13 (met, with 3632x to spare). The two routines compute the same quantity, proved by exhausting the estimator's own sampling universe rather than by comparing two approximations. The exact routine is **93.5 s against 11.7 to 23.4 s, i.e. a 4x to 8x SLOWDOWN**, which is the opposite sign from "an improvement of x3600"; it costs stage [L] 75 extra seconds and buys 3600x in error. Separately, the 198.9 s on record for `k4direct` is 2.1x too expensive, which drags the @17 extrapolation from 8.5 days on ten cores down to about 4.4. Work done 2026-08-18 on the 10-core machine, node v22.21.0. Target: `research/natal-cap-34-wrap-precision.js`:434, the one call site the corpus has been saying all day closes the @13 gate. Nothing was edited until sections 1, 2 and 3 below had all passed. --- ## 0. WHAT THE CORPUS ACTUALLY CLAIMS, AND THE ONE THING IT DOES NOT The claim on record is in three places, all saying the same thing: - `research/natal-cap-34-wrap-precision.js`:972 — "the substitution closes the @13 gate by a factor of 3600" - `research/history/staging/phase1-W2-cal4-routing.md`:37 — "**MET (by ×3600)**" - `research/history/CHANGELOG.md`:74 — "**the gate MET by ×3600**" **The ×3600 is an ACCURACY margin, not a speedup, and nothing in the corpus ever said it was a speedup.** It is `1e-9 / 2.753038e-13 = 3632`: the gate threshold divided by the relative error the swapped assembly achieves. Read as a cost figure it inverts the truth — measured below, the exact routine is **five times slower** than the Monte Carlo it replaces. The swap buys 3600x in error and spends about 5x in time, and 5x of 18 seconds is a price nobody needs to think about. That distinction is the only thing about the claim that was misread, and it was misread outside the corpus rather than inside it. --- ## 1. WHAT THE TWO ROUTINES ARE `natal-cap-32-wrap-identity.js`:194-203, `k4MC(L,S,nSamp,seed)` — draws `nSamp` ordered 4-tuples of distinct slots uniformly (rejection loop on a mulberry32 stream), averages the six-fold product `s_ij s_il s_iu s_jl s_ju s_lu`, and scales by `C4(N)`. Returns `{k4, err}` with `err` the plug-in standard error. `natal-cap-32-wrap-identity.js`:280-285, `k4direct(L,S)` — walks `i 3 (−3.324 at @7, N=10) **2e. The one 3-sigma draw is a small-N error-bar artifact, not a disagreement.** Chased with many seeds: | level | N | seeds | mean z | sd(z) | reading | |---|---|---|---|---|---| | @7 | 10 | 24 | −0.988 ± 0.498 | 2.438 | unbiased, but **error bars understated 2.4x** | | @13[240] | 240 | 24 | −0.178 ± 0.198 | 0.972 | unbiased, error bars honest | | @13[990] full | 990 | 12 | +0.412 ± 0.242 | 0.840 | unbiased, error bars honest | At N=10 the rejection loop throws away 49.6% of draws and the population is 210 quadruples with a heavily skewed product, so the plug-in sigma underestimates. At the call site's own level (N=990, rejection rate 0.6%) sd(z) = 0.84 and the bars are honest. **The mean z is consistent with zero everywhere: no bias at any level.** Nothing here is a difference of quantity. **VERDICT ON 2: PASS.** Same quantity, proven by exhausting the estimator's own universe, not by agreement. --- ## 3. THE COSTS, MEASURED Single core, machine otherwise idle, `buildLevel(13)` + `sMatrix` (0.02 s) done once and excluded. Full @13, N=990, K=34. | routine | runs | measured | |---|---|---| | `k4direct(L,S)` | 3 | **94.21 / 93.98 / 93.49 s** | | `k4MC(L,S,2e8,4242)` — the call site as shipped | 6 | **11.70 / 12.08 / 16.92 / 18.53 / 22.59 / 23.37 s** | | `k4MC(L,S,2e7,777)` — the throughput figure on record | 1 | 2.37 s | `k4direct` at full @13 returns **60234.496774278130**, reproducing the corpus figure to all seventeen digits, on every run. **Two corrections to the cost record, and they point opposite ways.** **(a) `k4direct` is 2.1x CHEAPER than the corpus says.** The record is 198.9 s (cap-34 reading 6(a), phase1-W2 §1, CHANGELOG). Measured today on an idle machine: 93.5 s, three runs inside 0.8% of each other. The 198.9 s figure and its 252.7 / 319.9 s companions were all taken while the machine was shared. Given this repo's history of cost estimates wrong in the cheap direction — fold-profile by four orders, cap-34 itself by 4x, G2(41#) by 880x — a figure wrong in the expensive direction is worth recording as such. **(b) The swap is a 4x to 8x SLOWDOWN, median 5.1x.** `k4MC` at 2e8 varies a lot (11.7 to 23.4 s; the variance is core migration on Apple Si, the answer is bit-identical every time). Against `k4direct`'s stable 93.5 s that is a **+75 s cost on one core**, on a stage that already runs 348 s. Immaterial, and the opposite sign from "an improvement of x3600". **Spot check on the @17 extrapolation, not part of the swap.** Reading 6(a) prices `k4direct` at @17 as 8.5e+1 days on one core from an n^3.92 fit. Scaling the measured 93.5 s by (14850/990)^3.92 = 4.08e+4 gives 44 days on one core; by a pure N⁴ it gives 55 days. Both are under the 85 days on record, consistent with (a). Not re-fitted here. **VERDICT ON 3: the ×3600 is real and is an accuracy factor. There is no speedup, and the corpus never claimed one. The cost is 75 extra seconds.** --- ## 4. WHAT THE GATE ACTUALLY IS, AND THE MEASURED MARGIN **The threshold, precisely.** `natal-cap-34-wrap-precision.js`:446: Math.abs((Tbest - T4KNOWN) / T4KNOWN) <= 1e-9 with `T4KNOWN = 352253669.87624449`, cap-27's certified @13 value carried at a 1e-14 error budget. `Tbest` is `baseProd(L,4)` times `A(shapes + k4) + C1(instance) + C2(factorised + V4) + C3(instance) + Multi(exact)`. In absolute terms the gate allows |ΔT4| ≤ 0.352254. **Where 1e-9 comes from, and what it is worth.** The file header: "The mu4 assembly cancels ~7 orders at @17, so the beyond-Chebyshev rung needs T4 to ~1e-9 relative." It is a round number from that one line of reasoning, and this same file's reading 5(c) has since superseded it for @17 — the @17 requirement is a ladder whose first useful rung is relT₄ ≤ 5.047439e-5, and reading 14 establishes that meeting the @13 gate certifies nothing about @17 because the assembly that meets it borrows Multi from the C(990,4) march. **So the gate is worth meeting as a validation of the engine, not as a step toward @17.** That was already the file's position and the swap does not change it. **Measured today, both sides, same machine, same truth JSON, everything else identical:** | k4 source | stage wall | T4 | REL vs certified | gate | |---|---|---|---|---| | `k4MC(L,S,2e8,4242)` as shipped | 265.9 s | 352253669.51634955 | −1.021692e-9 | **misses by 2.2%** | | `k4direct(L,S)` | 341.3 s | 352253669.87614751 | **−2.753038e-13** | **MET** | **Margin: 1e-9 / 2.753038e-13 = ×3632.** The corpus said ×3600. It reproduces. The baseline row above is my own re-run of the unmodified file before I touched it, and it is digit-for-digit identical to the block that was already pasted in cap-34, so the comparison is like for like and the pasted block was not stale. **VERDICT ON 4: PASS.** Threshold 1e-9, achieved −2.753038e-13, margin ×3632. --- ## 5. THE SWAP, MADE All three checks passed, so the swap was made. One call site, `natal-cap-34-wrap-precision.js`:436 in `STAGES.layers`: - const k4mc=C32.k4MC(L,S,2e8,4242); - const Abest=Ashapes+k4mc.k4; + const k4best=C32.k4direct(L,S); // EXACT K4 shape, Kahan over non-negative terms + const Abest=Ashapes+k4best; plus the two log lines that consumed `k4mc` (the label and the `k4mc.err` report, which no longer exists — it now prints the exact k4 and its agreement with `A_exact − A_shapes`). **The other three `k4MC` calls in the same stage were deliberately left alone**: lines 392 and 417 measure the MC layer's own error for the budget table, and line 424 reproduces cap-32's published instance pipeline. Replacing those would have destroyed what they measure. **The script's own self-checks, all re-run after the edit:** | stage | result | |---|---| | `verify` (NC34_WORKERS=2) | **88,520 checks passed**, every digit identical | | `exact13` (10 workers) | 4 checks passed, every digit identical, 1466.0 s | | `c2` | 6 checks passed, every digit identical | | `kurt`, `joint` | every digit identical | | `layers` | 0 checks (the stage has no asserts); gate now MET | Every number in every stage other than the [L] best-assembly block is bit-identical before and after, which is the check that the substitution touched one thing. **The gate, all three commands:** | command | before my edit | after | |---|---|---| | `node research/qc.js` (first 8 checks) | 3 findings, all in `research/PRIOR-ART.md` and `research/lemmaV-sup-extension.js` | 1 finding, `TODO.md:272` vs `research/U-FRAME.md:260` | | `node research/qc/selftest.js` | 21 positives fire, 12 controls silent | **21 / 12, unchanged** | | `node research/audit-numbers.js` | **108/108** | **108/108** | **No finding names `natal-cap-34-wrap-precision.js` before or after.** The tree moved under me while I worked — other partitions edited `TODO.md`, `U-FRAME.md`, `PRIOR-ART.md`, `SEARCH-CONVENTIONS.md`, `qc.js` and eight more — which is why the first-8 count moved in both directions on files I never opened. `git diff` confirms my only tracked change is `research/natal-cap-34-wrap-precision.js`, and every hunk of it is in the call site, the header gate note, the pasted OUTPUT, or readings 6, 12 and 14. **The pasted OUTPUT block is fresh.** All six stages were re-run and the pasted blocks were diffed line by line against the six logs: identical. The PROVENANCE note now carries the six wall times, records that only the [L] best-assembly block moves, and says which printed figures are timing extrapolations that drift by construction. --- ## 6. CORRECTIONS TO THE RECORD Fixed in `natal-cap-34-wrap-precision.js` in this pass: 1. The header GATE STATUS and readings 6 and 12 said the best assembly **misses** the gate. It meets it. Rewritten in present tense. 2. Reading 6(a) priced `k4direct` at **198.9 s** on one core at @13. Measured 93.5 s on an idle machine, three runs inside 0.8%. The 198.9 / 252.7 / 319.9 s figures on record were all taken on a shared box. 3. Reading 6(a) and reading 14 priced `k4direct` at @17 as **8.5e+1 days on one core, ~8.5 on ten**, scaled from the 198.9 s. Rescaled from the measured 93.5 s by the same n^3.92 fit: 44 days on one core, ~4.4 on ten (a pure N⁴ gives 55 and 5.5). 4. Reading 6(a) said `k4direct` was "validated at @7 in cap-32's `small` stage". cap-32's `small` is its DEFAULT MODE, not a stage, and its `main()` exercises `k4direct` at **@7 (N=10), @11 (N=90) and @13[120] with real @13 moduli**, asserting `assembleA == AdirectT4` at 1e-11 at each. The routine was better validated than the record said. Owed elsewhere, and deliberately not touched because they are other partitions: 5. `research/gen-scripts-index.js`:229-231 says `k4direct` "had been written, validated at @7, and never run above it. It took **199 s** on one core and moved the assembly from missing the gate by 2% to meeting it by x3600." Two fixes: the cost is 93.5 s, and the validation reached @13[120]. The narrative is otherwise exactly right and is the reason the swap was findable at all. 6. `research/SCRIPTS.md`:305 (generated from the above) says `k4direct` "was exercised at @7 and @11 and never above". It reaches @13[120] in cap-32 and, as of this pass, the full @13 in cap-34. 7. `research/history/` carries the pre-swap framing in CHANGELOG.md:71-74, qc-CAMPAIGN.md:1626-1629, qc-wave6-W.md:190 and phase1-W2-cal4-routing.md. **Left alone on purpose**: history is the process record and is supposed to hold superseded claims. 8. A CHANGELOG.md entry is owed for the swap. Ready to paste: > **natal-cap-34-wrap-precision.js — the @13 gate, MET (2026-08-18).** Stage > [L]'s best assembly took the K4 shape from `k4MC(L,S,2e8,4242)`; it now > takes it from cap-32's exact `k4direct`. T4 goes from 352253669.51634955 > (REL −1.021692e-9, missing the 1e-9 gate by 2%) to 352253669.87614751 > (REL −2.753038e-13, met by ×3632). Equivalence of the two routines proved > by exhausting the MC's own sampling universe, not by agreement; costs > measured, not quoted (93.5 s against 11.7–23.4 s, so the swap is a 5x > slowdown buying 3600x in accuracy). Retired: "misses the 1e-9 gate by 2%", > "the residual obstruction is the single Monte-Carlo shape k4", `k4direct` > at 198.9 s and at 8.5 days on ten cores at @17, and "validated at @7 and > never run above". Two operational notes, neither a defect in the mathematics: 9. **`c2` and `joint` degrade silently without `NC34_DIR`.** Run without it, `c2` throws at `loadTruth` after 2 s of useful work, and `joint` prints "(stage exact13 not run yet)" and drops its entire @13 comparison block while still exiting 0 and still printing "[0 checks passed]". The pasted command list showed `NC34_DIR` only on `exact13`. It now shows it on all three stages that need it. 10. **`k4MC`'s plug-in error bar is understated 2.4x at N = 10** (section 2e). Honest from N = 240 up. Anyone quoting `mc.err` at a small level should know that; at @13 and above it is sound. --- ## 7. WHAT THIS DOES NOT CHANGE The @17 story, which is the only reason the @13 gate was ever wanted. The assembly that now meets the gate still takes Multi EXACT from stage [E]'s C(990,4) march, which at @17 is 2.026e+15 quadruples (513 days on ten cores) and is structurally impossible in this code anyway (`pairMasks` asserts K ≤ 64; @17 has K = 120). Reading 14 said the @13 gate certifies nothing about @17 before the swap and it still says it after. **What the swap buys is a validated engine at @13, not a rung at @17.**