# Verify the verifier: is `audit-numbers.js` circular? *(Audit of the auditor, 2026-08-20. Scope: the 117 checks of `research/audit-numbers.js` as of commit `ab34c82`, sections A..W in driver order. Section X was being appended by a sibling while this ran and is OUT OF SCOPE; §9 records what it changes about the findings below. Nothing in `audit-numbers.js` was edited: every fix here is a proposal. `node research/qc.js` was clean before and after.)* **One-line verdict.** The file is not circular in the way that would matter most — it reads no other repo script, so no producer's bug can leak into it by import — but **15 of its 117 checks certify nothing** (they are arithmetic on inline literals, or same-file duplicates of another check), **17 more are artifact echoes** that assert relationships among transcribed constants, and the one bug class the charter was written against, the a3-03 census alias, **had nothing in A..W to fire against and still has nothing.** Four checks were re-implemented from scratch in a different language and all four agree; the fifth re-implementation found that check W1's answer **depends on a hard-coded phase choice the check does not name**. --- ## 1. Classification, all 117 Sections are lettered here in driver-execution order, since the file itself labels only `W`. Counts are of individual `check()` outcomes, so the `G2(x#)` loop counts 11 and the `W2` loop counts 7. | § | what it checks | n | FP | FP-DERIVED | ARTIFACT-ECHO | TAUTOLOGY | |---|---|---|---|---|---|---| | A | loudness @17, the known positive | 4 | | 4 | | | | B | loudness @19 row, and four re-derivations of it | 8 | | 8 | | | | C | G2(41#) certificate | 4 | 3 | | | 1 | | D | G2(43#) certificate | 5 | 4 | | | 1 | | E | theta of a primorial, and the retired `2v` form | 2 | 2 | | | | | F | slots vs integers at block 1 | 4 | 2 | 1 | | 1 | | G | window conventions, and the four gap objects ordered | 3 | | | | 3 | | H | the named constants | 8 | 5 | 2 | 1 | | | I | one-class ladder h(x#), and G2/h | 2 | 1 | | | 1 | | J | A091592 | 1 | 1 | | | | | K | Li, HL and the 1e11 census | 2 | 1 | | 1 | | | L | the G2(41#) prediction band | 7 | | 6 | | 1 | | M | pointwise ladder inequalities, local exponents | 5 | | | 5 | | | N | A059861 by sieve and by closed form | 2 | 2 | | | | | O | G2 draft DATA, primorials, the p'^2 margin | 6 | 2 | | 3 | 1 | | P | a(n)/(p ln^2 p) | 3 | | | 3 | | | Q | the c = G2/(mbar lnD) fit, and the retired index | 5 | | 5 | | | | R | the control-corrected exponent 1.54 vs 1.57 | 8 | | 5 | 1 | 2 | | S | draft bounds and the CROSSREFS identity | 3 | | | 3 | | | T | seam DATA, twenty terms plus the extension | 6 | 6 | | | | | U | the seam heuristic and its retired halving | 6 | | 2 | | 4 | | V | the G2 ladder itself, 2# to 37# | 14 | 14 | | | | | W | the fold-L wave regression floor | 9 | 9 | | | | | | **total** | **117** | **52** | **33** | **17** | **15** | **FIRST-PRINCIPLES (52).** The computed side is rebuilt from a raw definition by code that shares no path with whatever produced the claimed value, and the claimed value is an external anchor (OEIS, the literature) or a second method inside the same run. Concretely "independent" means one of three things here: - *different algorithm*: §N's A059861 is counted by a `gcdInt` residue scan and then again by the closed form `prod (q-2)` (lines 477-478), and the two are asserted against each other; - *different arithmetic domain*: §T rebuilds all thirty seam terms from a self-contained BigInt Miller-Rabin, and the drafts they are checked against came from a different tool entirely; - *external anchor*: §I's h(x#) and §J's A091592 are recomputed by sieve and checked against OEIS, which no one in this repo wrote. **FP-DERIVED (33).** The arithmetic is genuine and rebuilds the doc's number from more primitive inputs, but at least one input is a literal transcribed from an artifact. §L's whole band is `(literal G2 table) / (BigInt-computed m·lnD)`; §Q's fit is the same; §R's bias comes from a 58-term A048670 array typed into the file. These catch arithmetic drift and rounding drift. They do not catch a wrong input. **ARTIFACT-ECHO (17).** Reads numbers out of embedded OUTPUT or literal tables and asserts a relationship among them. Legitimate as a consistency net. What it would *not* catch, in every case: a value that is wrong but still satisfies the relationship. §M's three pointwise orderings (`h <= G2 <= h2`) are the clearest example — every term of all three ladders could be wrong by the same factor and all three checks still pass. **DOC-ECHO / TAUTOLOGY (15).** Listed in full in §2. --- ## 2. The tautology list, all 15, ranked **Class 1 — cannot fail under any change to the corpus.** These compute a number from literals written inline in the same expression, and compare it to another literal. Editing any live document, any artifact, or any other check leaves them untouched. | line | check | why it certifies nothing | |---|---|---| | 670 | `x'^2/x^2 at x=41 = 1.0999` | computes `43*43/(41*41)`. Both operands inline. It asserts a fact of arithmetic. | | 672 | `Y2(37) <= G2(37#)-1` | the expression is `527 <= 528 - 1`, i.e. `527 <= 527`. **Y2(37) = 527 is the number this is supposed to guard and it appears only as a literal here; nothing recomputes it.** | | 673 | `G2(37#) = 528 and the covering optimum is G2-1 = 527` | the expression is `528 - 1` vs `527`. | | **705** | **`G2(41#) lies inside the pre-registered Poisson window [476,633]`** | **the expression is `546 >= 476 && 546 <= 633` with `546` written as a literal. The certificate walk four lines above computes the gap into `n - r0` and this check does not use it. If the walk returned 500 this check would still pass.** The one repairable defect in this class. | | 570 | `the two differ, so quoting one for the other is an error` | `abs(1.539 - 1.567) > 0.02`, entailed by the two checks immediately above it, with 0.008 of margin. | | 667 | `capacity is BELOW the 19 slots needed` | `17.96 < 19`, entailed by line 666. | | 631 | `seam heuristic with the retired /2` | `exact/2` against `26.3`, entailed by line 629's `52.5`. | | 731 | `G2(43#) recorded position is below 2^53` | literal `<` literal. Entailed by line 732, which uses the same `r0`. Harmless as documentation of the wheel-23 defect; it adds no coverage. | **Class 2 — same-file duplicates. Fail only when the check they duplicate also fails.** | line | duplicates | note | |---|---|---| | 358 | line 346 | identical expression `Math.round(lo * base)`, identical claim `476`. Labelled a retired-value guard for `475`, but line 346 already is one. | | 632 | line 608 | identical expression `got.slice(0,20).reduce(...)`, identical claim `48`. | | 636 | line 66 | `e^{2g}/(2 C2)` against `2.4026`, twice, from the same two literals. | | 637 | line 67 | `e^{2g}/(4 C2)` against `1.2013`, twice. | | 579 | line 377 | `A048670(n) <= a(n)` at twelve terms, on the same two literal arrays. | | 581 | line 378 | `a(n) <= A288815(n)` at twelve terms, same. | Two more duplicates sit inside the FP-DERIVED class and are worth naming even though they are not tautologies: line 524 (`mean 0.4814`) recomputes line 345, and line 526 (`range 0.4463..0.5939`) recomputes line 344, over the identical eight levels by the identical arithmetic. **Class 3 — transcription-only.** | line | check | what it really tests | |---|---|---| | 311-315 | `G2/h at x = 5..37` | `G2c` and `hc` are literal tables typed at lines 311-312 and `claimedRatio` is a literal list of the ten quotients. The check divides literals by literals and compares to literals. **It does not use the `got` array of h values the same function computed twenty lines above, nor the ladder's computed G2.** | | 487 | `G2 DATA field, twelve terms` | asserts a hard-coded array equals a hard-coded string, both typed in this file. The comment concedes the point: *"the terms themselves are rebuilt from scratch in parts `ladder` and `g2big`"* — and nothing connects the two. | | 558 | `control A048670 has 58 terms` | `A048670.length === 58`, on an array literal in the same block. | --- ## 3. Shared-helper contamination `audit-numbers.js` requires nothing outside `fs`/`path` (and those only in the sibling's section X), so **cross-file contamination with a producer is zero by construction for A..W.** The contamination that does exist is of two other kinds: helpers shared *between checks inside the file*, and checks that *reimplement the producer's algorithm* rather than choosing a different one. ### 3.1 Intra-file helper graph, for the ten most load-bearing checks | helper | checks that depend on it | is the helper itself guarded? | |---|---|---| | `PRIMES` (sieve to 2e5) | almost everything | **yes, three ways.** §J builds its own sieve to 9e6, §A builds its own `sieve()`, §T's `isPrimeBig` is a Miller-Rabin. Three independent counterparts in-file. | | `lnBig` | §L's six band checks (476/513/633/488), §Q's five fit checks (0.4814, 10.2%, 0.4463..0.5939, 0.0406, 69.4%) | **NO.** Eleven checks and both headline band numbers ride on one 15-digit string truncation, and nothing asserts `lnBig(D)` against `sum ln(q-2)`. The file *prints* that cross-check at line 336 — `check theta(x) - ln m = ... (must equal lnD)` — as a `console.log`, **not** as a `check()`. A bug in `lnBig` shifts all eleven consistently and the run stays green. | | `maxCyclicGap` | §V's eleven G2 terms, §I's h ladder | **partially.** §I's output is anchored to OEIS A048670, so a wrap-handling bug would surface there. | | `liftOne` / `liftTwo` | G2(31#) = 348, G2(37#) = 528, both self-tests | **NO.** `liftTwo` *is* `liftOne` with an outer class loop: identical "longest compatible suffix" restart logic, copied. The two self-tests at lines 851-852 exercise `liftOne` and `liftTwo` against each other, so **a bug in the shared suffix logic passes both self-tests.** The only external anchor for 348 and 528 is the `claimed` literal. | | `build29` | G2(29#) = 258 | cross-checked against `liftOne` at line 851. This one is a real two-method agreement. | | `simpsonLog` | §K's two checks, i.e. the entire Zone-Postulate census guard | **NO.** One quadrature routine, no closed-form calibration anywhere. | | `isPrimeBig` | §T's six seam checks | **NO** in-file. Externally anchored by A060256 (line 613) and by SV2 below. | | `level()` (§A/§B) | all twelve loudness checks | see §3.2. | | `check()` | all 117 | string-compares when either side is non-numeric. `check(x, true)` works; `check(undefined, undefined)` would pass. Not currently reachable. | ### 3.2 Algorithmic contamination: the loudness "known positive" The file's design here is explicit and, on its face, exemplary: run `level(17)` first, match cap-31's independently produced @17 row, and only then believe `level(19)`. The comment says cap-31's row "was produced by a completely different algorithm (a full per-rotation sweep of survivors, strikes and overlap credit)". That is true of S-bar and of max VR. **It is not true of the object.** `natal-cap-31-calm-vs-kill.js:77-78` seeds the natal set as for(let r=11;r x, q*q <= W` window and V-bar by the same `n[a] + n[a-2] - 2N/q` formula. Two consequences: 1. The natal set **drops the twin class `29 mod 30`**, one of the three twin classes mod 30, giving N = 14,850 against D(17#) = 22,275 = 3N/2. That is a convention (it is the class whose partner crosses the wheel boundary), and it may well be the right one, but **no check in A..W states it or tests it**, and both the audit and the producer would pass with it wrong. 2. Checks 83-86 are therefore not four independent confirmations. They are one confirmation for S-bar (two genuinely different routes to the same identity) and one for max VR, plus two shared-construction agreements for K and V-bar. The audit is nonetheless **stronger than its producer** on the @19 row, and that deserves saying: `natal-cap-38-loudness-driver.js:182` takes `K: 435, Vbar: 8944.60, maxVR: 2.293` as **hard-coded inputs** from cap-19, whereas `audit-numbers.js` recomputes all three. That is exactly the upgrade this file exists to make. ### 3.3 The five copies of the G2 table `{2,6,12,30,42,66,108,150,204,258,348,528}` is typed into the file five times: line 117 (`claimed`, the only one the ladder is checked against), line 311 (`G2c`), line 322, line 369 and line 486. The h ladder is typed twice: computed into `got` at lines 290-307, and typed again as `hc` at line 312. **No check asserts that any copy equals any other, or that any copy equals the ladder's computed `L.got`.** A single-digit slip in any of the four downstream copies changes §L's band, §M's inequalities, §P's ratios, §Q's fit and §R's exponent, and the ladder checks in §V would not notice. ### 3.4 One asymmetry in §V Lines 851, 852 and 858 all take `Math.max(internal, boundaryCheck(...))`. Line 843 calls `boundaryCheck(B, 31, null)` and **discards the return value**, so the `G2(31#)` check at line 840 asserts the internal lift only. It passes because the widest copy-boundary gap at 31# is 84 against an internal 348 (baseline run, line 474 of the transcript), but the check as written is not the complete recomputation its three siblings are. --- ## 4. The census-bug counterfactual, run honestly **The bug.** `research/a3-03-f-from-census.js:86` holds each prime's avoided set in one 32-bit word: `a[j] = (1 << r0) | (1 << r2)`. JavaScript takes shift counts mod 32, so for `q > 32` residues alias and every `(q - |A_q|)` factor is under-read. T_x carries a prime above 32 iff x >= 37, so every census point from x = 37 up is wrong by 0.62x to 1.05x, and the regression fitted to them was wrong with them. The producer was embedded and gate-green throughout. **Would any A..W check have caught it? No, twice over.** *First, and decisively: there is no check for f.* Reading all 117 labels: none mentions f, `count_x(d)`, the grain census, the qualifying comb, or the `ln(1/f)` regression. The audit's coverage of the entire fold-decay layer is zero. The bug had nothing to fire against, and still has nothing. *Second, had a check existed and been written in this file's house style, it would have caught it.* Every residue computation in A..W uses `%`, never a shift: `isSlot` at lines 698 and 730, `r % q` in W2, `r % p` in W1, `killer[]` as `Int16Array` in W3. A `%`-based recomputation of f at x = 37 disagrees with the producer at the first aliased prime. So the file is **structurally immune to that bug class and structurally blind to the layer it lived in**, which is the uncomfortable shape of the finding: the immunity is a side effect of the house style, not of any rule, and nothing enforces the house style on a future section. **What it implies about which classes of number have no guard.** Four classes, and they are the same four everywhere in A..W: 1. **Fitted numbers.** Regression coefficients, R-squared, exponents fitted to measured points. The audit checks refits of data it holds (§Q, §R) but never a fit whose data lives in a producer's OUTPUT. 2. **Search and optimisation outputs.** Anything whose truth is *maximality* or *optimality*: G2(41#) and G2(43#) are certified `>=` only, by the file's own admission; Y2(37) = 527 is a greedy construction and is a bare literal at line 672; the ILP lower bounds in A072753 are transcriptions. 3. **Anything past the 152 s budget.** x >= 41 maximality, windows past 2e7, depth-2 and depth-3 ceilings, the deep-fold tail certificates. The section W header says so plainly. 4. **Literature imports.** beta_2, the tail of A048670, A288815, A072753. Some are cross-checked against each other (line 584); beta_2 is not cross-checked against anything. --- ## 5. The unguarded top ten Live-layer numbers with **no independent recomputation anywhere in A..W**, ordered by how much of the corpus rests on them. | # | number | where it is load-bearing | nearest thing to a guard | |---|---|---|---| | 1 | **beta_2 = 4.26645028414864191641** | 25 live .md files; the sifting-limit wall itself; the "exponent 4.2665 -> 2" framing | none. Line 71 divides by it and asserts the quotient 1.2417. `attack-beta2-04-loss-budget.js` recovers 4.26660 by shooting (3.5e-5 relative) but that is a producer, not the gate. | | 2 | **the eight A144311 terms past 43#** (G2 = 708, 870, 966, 1080, 1284, 1398, 1530, 1710) | `PRIOR-ART.md:363`, `covering-dive.md:193`, `G2-STATE.md:773` | none. The audit stops at 43# and its OEIS section audits `oeis-G2-submission.md`, which `covering-dive.md:193` has already ruled **a duplicate that must not be submitted**. Roughly twenty checks (§O..§S) guard a retired draft while the sequence the live layer actually cites is unguarded. | | 3 | **G2(41#) = 546 and G2(43#) = 618 as maxima** | the two newest ladder terms, the Poisson-window headline | the certificates prove `>=` only. Maximality rests on two enumerations that the gate does not rerun and cannot afford to. Stated honestly in the file; still unguarded. | | 4 | **the fold-decay law** `ln(1/f) = 1.917 + 1.4016(+-0.0235)(2p/mbar) - 1.021 ln s(d_min)`, R^2 0.9867 | `a3-03-f-from-census.md:19`, `f-decays.md:20`, `ATTACKS3.md:117`, `U-FRAME.md:465` | none. This is the corrected law that replaced the aliased one. Nothing in the gate would catch a second alias. | | 5 | **A048670 terms 11..58** (h(31#) = 58 through h(271#) = 954) | the control that produces the bias 1.262/1.280/1.282 and therefore the headline exponents **1.54** and **1.57** | none. §I recomputes the first ten and line 309 says so: *"h(31#)=58 and h(37#)=66 are taken from OEIS A048670 (Hagedorn); not recomputed here."* Forty-eight transcribed terms carry two headline numbers. | | 6 | **Y2(37) = 527** | `G2-STATE.md:740-758`, the covering/greedy comparison | none. Lines 672-673 are the two tautologies above. | | 7 | **1.847**, the h2 raw slope on nineteen terms | half of check 568, hence the 1.57 that must differ from 1.54 | none. `rawG2` is recomputed from the G2 array by `slope()` at line 565; 1.847 is a bare literal at line 568 even though the same `slope()` is right there. | | 8 | **the census 224,376,048** (twin count below 1e11) | the Zone-Postulate census | §K bounds it to 0.5% of a Hardy-Littlewood integral. That constrains the *magnitude* and would catch a 1e10-vs-1e11 confusion. It would not catch a 0.1% error, and 0.5% of 2.2e8 is over a million. | | 9 | **the extinction folds 331, 421, 457** and the pre-registered bands | `attack-foldL-06-scaling.js:1147`, the out-of-sample law test | §W guards **181 only**, and only inside `[0, 2e7)`. The three larger windows are past the budget. Note also that W3 walks folds to `PMAX = 499` with no assertion that the answer is below the ceiling. | | 10 | **the survival-curve trough numbers 0.8926 / 0.8927 / 0.8929** at three sampling conventions | `FOLD-PROFILE.md:398-410`, and the fourth confirmation of the e^{2gamma}/4 trough constant | none. Only the constant `e^{2gamma}/4 = 0.793055` is guarded (line 53). The three measured values and the finite-size factor 1.1238 are not. This is exactly the shape that produced wave 5's false S2-19 alarm. | Runners-up worth naming: the L ceiling factor **0.58 to 0.95 ln p** (`U-FRAME.md:408, 782`) has no numeric guard; `L <= 0.18 p` likewise; and A072753's ten terms at line 585 are a transcription checked only against another transcription. --- ## 6. Five from-scratch spot-verifications Re-implemented in Python with deliberately different algorithms and data structures. Scripts in the session scratchpad; all values below are as printed. **SV1 — the G2 ladder, x = 2..23 (§V, 9 of the 14 checks).** The audit builds each level by copy-and-delete from the level below, sorting the survivor list. This rebuild instead lays one flat `numpy` boolean array over Z/x#, marks `0::q` and `(q-2)%q::q` by strides, and takes the cyclic gap from `np.diff`. Different construction, different language, no sorting. x= 2 D=1 G2=2 x=13 D=1485 G2=66 x= 3 D=1 G2=6 x=17 D=22275 G2=108 x= 5 D=3 G2=12 x=19 D=378675 G2=150 x= 7 D=15 G2=30 x=23 D=7952175 G2=204 x=11 D=135 G2=42 **AGREES** with the audit at all nine, and every D matches `prod (q-2)` independently. The audit's 29#, 31# and 37# are past what this method can hold and were not re-verified here; §3.1 records that they rest on one algorithm family. **SV2 — the seam sequence, thirty terms (§T, all 6 checks).** The audit uses a fixed-twelve-base deterministic Miller-Rabin in BigInt JavaScript. This rebuild uses **Baillie-PSW** in Python: Miller-Rabin base 2 plus a strong Lucas test with Selfridge parameters. Genuinely different primality algorithm. a(1..20) = 2, 4, 4, 3, 4, 6, 2, 1, 7, 1, 1, 2, 0, 1, 1, 1, 3, 1, 0, 4 sum = 48 a(21..30) = 0, 2, 1, 1, 2, 1, 1, 1, 0, 1 least witnesses = 2,1,1,2,1,6,8,11,4,16,22,4,-,24,37,28,14,11,-,11,-,20,83,91,35,80,48,47,-,2 **AGREES** with all six checks, including the A060256 correspondence and the "zeros are exactly where A060256(n) > prime(n+1)" claim (verified at n = 13, 19, 21, 29: 74 > 43, 242 > 71, 91 > 79, 226 > 113). **SV3 — section W2, the seven diagonal L values (§W, 7 checks).** The audit walks p copies of the tile with a running shift and patches the wrap with an ad-hoc `firstRun`/`joined` join. This rebuild materialises the full residue sequence and finds the longest cyclic run by the doubled-array method, with the tile's slots derived by sieve rather than by the shift recurrence. L(T_5,7)=2 L(T_7,11)=1 L(T_11,13)=2 L(T_13,17)=2 L(T_17,19)=2 L(T_19,23)=3 L(T_23,29)=2 **AGREES** at all seven. The wrap patch is correct. **SV4 — section W1, first dead l (§W, 1 check). THIS ONE DISAGREES IN SUBSTANCE.** The audit hard-codes the T_5 gap word as `(6, 12, 12)` repeated. This rebuild derives the word by sieving Z/30 (slots 11, 17, 29, mbar = 10, which independently confirms line 661) and then runs the counting criterion at **all three phases** of the period: phase start=11 word=[6,12,12] first dead l = 63 => L <= 62 phase start=17 word=[12,12,6] first dead l = 62 => L <= 61 phase start=29 word=[12,6,12] first dead l = 62 => L <= 61 The audit's `63` is right, and it is right **because 11 is the tile's first slot and that phase happens to be the maximising one**. The check does not say the answer is phase-dependent, does not derive the word, and does not take the max. A future edit that re-derived the word from a different origin would produce 62, `L <= 61`, and a red gate on a number that is not wrong. Proposed upgrade in §7. **SV5 — the loudness @17 row (§A, all 4 checks).** Re-implemented in `numpy` with `bincount` and `roll` for the class deviations, and with S-bar computed **twice**: once by the weight identity (the cap-35 route, not the audit's prefix-sum-over-rho route), and once by brute force, evaluating `S(t) = #{r in rho : g(r - t) = 1}` at 4,000 random rotations. W=510510 N=14850 K=120 audit K=120 Vbar = 1144.4036 audit 1144.40 maxVR = 2.1427 audit 2.143 Sbar = 3614.9297 audit 3614.93 (weight identity) Sbrute = 3615.13 +- 3.27 (4000 random rotations, brute S(t)) **AGREES** at all four, and the brute-force sample lands 0.06 sigma from the exact identity. The §3.2 caveat stands regardless: this rebuild inherited the `11, 17 mod 30` seeding from the audit, because that seeding is the definition of the object and there is nowhere else to get it. --- ## 7. Upgrade plan ### 7.1 Echo to first-principles (six edits, all cheap) | # | target | change | |---|---|---| | U1 | **line 705** | replace the literal `546` with the computed gap: `check('G2(41#) lies inside the pre-registered Poisson window [476,633]', (n - r0) >= 476 && (n - r0) <= 633, true)`. Turns a tautology into a real assertion for the cost of two characters. | | U2 | **line 336** | promote the printed `theta(x) - ln m` identity to a `check()`. One line, and it puts `lnBig` — eleven checks' single point of failure — under the gate. | | U3 | **line 66-67, 636-637** | feed `c2full` (line 64) into the four `e^{2g}/(kC2)` checks instead of the literal `0.6601618158468696`, and delete the two duplicates in §U. Makes §H3's Euler product actually load-bearing. | | U4 | **line 568** | compute h2's raw slope with the `slope()` helper already in scope instead of the literal `1.847`. Needs the nineteen A288815 terms; the file currently holds twelve. Add the seven. | | U5 | **line 311-315** | replace `G2c`/`hc` with the ladder's computed `L.got` and the function's own computed `got`, so the ratio table tests the ladder rather than a transcription. | | U6 | **line 843** | `L.got[31] = Math.max(g31, boundaryCheck(B, 31, null))`, matching lines 851, 852 and 858. | ### 7.2 New checks for the unguarded ten | for | proposed check | cost | |---|---|---| | #1 beta_2 | assert the printed roundings against one canonical literal, **and** add a comment naming `attack-beta2-04-loss-budget.js`'s independent shooting recovery (4.26660, 3.5e-5) as the only non-transcription evidence in the repo. A full DHR recomputation does not belong in a 152 s gate. *(The sibling's section X4 does the rounding half; see §9.)* | free | | #2 A144311 tail | pin the eight terms past 43# as a literal table with an explicit `[TRANSCRIBED, A144311, not recomputed]` label, plus a check that `A144311(n) + 1` equals the audit's own computed G2 at all fourteen overlapping terms. That last one is a real external cross-check and costs nothing. | free | | #3 41#/43# maximality | no gate-affordable check exists. Add instead a `check()` that the *multiplicity* recorded (4 at 41#, 8 at 43#) is consistent with the certificate's least-position claim, and keep the honest comment. | free | | #4 fold-decay law | recompute f(x, p) by direct sieve at x = 11, 13, 17, 19, 23 (the levels `a3-03` calls UNAFFECTED) and assert the five values against the law's prediction. That is 5 of 51 window levels, but it is the levels where `%`-arithmetic can reach, and it would have caught the alias the moment the law was extended past 32. | ~1 s | | #5 A048670 tail | stream h(31#) by the same method §I already uses for 29#. phi(31#) = 3.07e10 is past the gate budget; do **h(31#) only**, offline, once, and pin it. Then at minimum assert that the 58-term array is monotone increasing and that its first ten equal the computed ones. | ~2 s for the cheap half | | #6 Y2(37) = 527 | run the greedy construction and assert 527, replacing lines 672-673. If the greedy is too slow, at minimum verify the *certificate*: exhibit the 527 consecutive integers and check each is covered. | seconds | | #8 the 1e11 census | tighten §K by also asserting the ratio against pi_2(1e10) = 27,412,679, which the file already prints but does not check. Two anchors constrain the census far better than one. | free | | #9 extinction folds | add `check('W3 the answer is below the PMAX ceiling', lastMulti < 499, true)`. Without it a fold past 499 reads as "no multi-kill". | free | | #10 survival trough | pin the three sampling conventions as three separate labelled checks recomputed from the same curve, so the "neither figure is 0.8929" failure mode cannot recur. | needs the curve; defer | ### 7.3 Delete as false assurance Eight lines. Deleting them takes the count from 117 to 109 and **increases** what the file certifies per check, which is the point. - **672, 673** — `527 <= 527` and `528 - 1 === 527`. Replace with U6's real Y2 check, or with nothing. - **670** — `43*43/(41*41)`. The unit lesson it documents belongs in the comment it already has; the `check()` adds nothing. - **570** — entailed by 566 and 568, with 0.008 of margin. - **667** — entailed by 666. - **631** — entailed by 629. - **632** — verbatim duplicate of 608. - **636, 637** — verbatim duplicates of 66 and 67. *(Counted as one bullet; eight lines total across the list.)* Keep but relabel, do not delete: **358** (duplicate of 346) and **579/581** (duplicates of 377/378) carry retired-value documentation that has value even though the assertion is redundant. Merge each pair and keep the comment. Keep as-is: **558** and **731**. Both are trivial, but both guard a literal that other checks consume, and both cost nothing. --- ## 8. What this audit did not settle - **G2(29#), G2(31#), G2(37#)** were not independently re-verified here. SV1 reaches 23#. The `liftOne`/`liftTwo` shared-suffix risk in §3.1 is therefore unretired by this pass. The sibling's section X6 closes 37# from the t37 shard partials; 29# and 31# remain on one algorithm family plus a literal. - **Whether the `11, 17 mod 30` natal seeding is the intended object.** §3.2 states the fact; adjudicating the convention needs cap-13, which was out of scope. - **The three larger extinction windows** and everything else in class 3 of §4. These are budget-bound, not audit-bound. --- ## 9. Section X (out of scope, noted for the sibling) Section X was appended during this pass; the file grew from 1,085 lines to at least 1,354 while this ran, so the line numbers above are those of `ab34c82`. X is a **cross-document constants audit**: X1-X4 assert every printed rounding of a named constant against one closed form, X5 pins the `K/beta_2` break-even pair, X6-X7 rebuild G2(37#) from the `t37-partials` shard JSONs. Three observations that bear on this report: 1. **X6 closes unguarded-number #2's cousin and part of §3.1's `liftTwo` risk** by reconstructing 528 from shard data rather than from prose. That is a genuine second method for 37# and the first one in the file. 2. **X4's beta_2 checks are rounding tautologies on a literal**, by the classification of §1 — legitimate for their stated purpose (are all the printed forms roundings of one number?) but they do not move beta_2 off the unguarded list, and the section's own opening line, *"a number cannot be the witness that the number is right"*, applies to them. 3. **One X4 label looks wrong.** The check reads `check('X4 and "4.2665..." is NOT the expansion: digit 5 of the expansion is 4, not 5', BETA2.toFixed(5).slice(-1), '5')` — the label says the digit is 4, the assertion says the last character of `4.26645` is `5`, and both are statements about a rounding rather than about the expansion. Worth a second look by whoever owns X. X is also the first part of the file to call `require('fs')` and read artifacts off disk. That is a new dependency class for this gate and worth a line in the header comment. --- ## 10. Gate state `node research/qc.js` before: 0 findings, clean, 11/11 checks. `node research/qc.js` after: 0 findings, clean, 11/11 checks. No corpus file was edited by this pass; `research/audit-numbers.js` was read only. `node --max-old-space-size=16384 research/audit-numbers.js` at `ab34c82`: **117/117 checks passed, exit 0, 410.7 s** on this machine (the header's ~152 s estimate is stale; the 37# lift alone is 31 passes over 214.7M slots and took about 280 s).