# The blind @29 test of the joint-deficit closed form, and the segmented census that made it possible *(2026-08-19. TODO item X's `~3.8` constant had a zero-parameter closed-form candidate with blind values pre-computed at two levels that did not exist, and a census instrument that stopped at @23 for want of 162 GB. This file builds the segmented instrument, validates it against the in-memory one at five levels to the unit, and spends both blind tests. Pre-registration, written and committed alone before any producer script existed: [xchan-at29-prereg.md](xchan-at29-prereg.md). Producer, formally embedded: `research/xchan-at29-01-segmented.js`. Every figure below is from that file's OUTPUT block, or is one-line arithmetic on figures in it, shown with its inputs. Calibration is marked on every claim: PROVEN, MEASURED, DERIVED, REFUTED.)* ## Verdict **The closed form `1 − J = 4S₂` is the only one of the three pre-registered laws left standing, and it is not exact.** At @29 it scores a clean HIT (`z = −0.90`); at @31, where the count is 1.65 billion and `σ_J` falls to `0.000024`, it is 4.93σ low. The two rivals the pre-registration fixed before the run die at @31 at 6.35σ and 20.81σ. What the two blind levels agree on is a **stable relative offset of about half a percent**, `d = −0.41%` at @29 and `−0.48%` at @31, and the corresponding absolute residuals `0.028943 − 0.028823 = 0.000120` and `0.024784 − 0.024666 = 0.000118` are the same number to 2%. **And the memory ceiling is gone.** The census now costs **62.0 MB of arrays at every level**, and the level enters only the clock: @23 in **0.56 s** against the in-memory instrument's 169.89 s, @29 in **15.07 s**, @31 in **596.78 s**. The Named gap "@23 is the in-memory instrument's last level (@29 needs 162 GB)" is closed by construction, not by buying memory. Five things the run settles that the record could not: 1. `J` is now measured at **seven levels** and is monotone from @17 through @31: 0.902452, 0.959855, 0.965932, **0.971177**, **0.975334**. Both new levels sit in `natal-cap-39`'s pre-registered DECAYING FASTER THAN FORCED band, so `xchannel-triples.md` R1's verdict rests on five points rather than three. 2. **P1 is directly tested two and three levels beyond where it had ever been tested.** Aligned super-`W` obs is exactly 0 at @29 and at @31. 3. The `1/lnW` reading of the decay, which was indistinguishable from the closed form at @23 and 0.4σ from it at @29, is **REFUTED at 20.81σ at @31**. 4. The "asymptote is exactly 4" reading gains nothing. The measured ratio `(1−J)/F` runs **7.0401, 3.6905, 3.8630, 3.8240, 3.8427** over @17..@31: it does not rise, while the closed form's own ratio rises monotonically 3.7245 → 3.8612 over the same span. 5. The candidate's residual, invisible at every level that scored it, is now **measured**: one detection at 4.93σ (@31) and one consistent non-detection at 0.90σ (@29). --- ## 1. What the in-memory instrument measures, and why @29 cost 162 GB `research/natal-cap-39-triple-census.js` defines, at level `x` with `W = ∏_{p≤x} p` and natal set `N = {r ∈ [0,W) : r ≡ 11,17 (mod 30), p ∤ r, p ∤ r+2 for 7 ≤ p ≤ x}` and scour primes `qs = {q : x < q, q² ≤ W}`: > `J(x) = (Σ_{∏Q > W, mixed} c(Q,ε)) / (6·N̄·Σ_{∏Q > W} 1/∏Q)`, > `c(Q,ε) = #{r ∈ N : r ≡ −ε_q (mod q) ∀q ∈ Q}`, `ε ∈ {0,2}^Q`, with `cls = #{q : ε_q = 0}`, `cls ∈ {0,3}` ALIGNED and `cls ∈ {1,2}` MIXED. It reaches the super-`W` part by the bridge `mixed super-W obs = Σ_{r∈N}[C(a,2)b + a·C(b,2)] − (sub-W mixed obs)`, with `a(r) = #{q | r}` and `b(r) = #{q | r+2}` carried as two `Uint16Array(W)` beside a `Uint8Array(W)` natal mask. That is **5 bytes per slot of `[0,W)`**, and `natal-cap-35`'s version of the same pass is 25. At @29, `W = 6,469,693,230`, so 32 GB and 162 GB respectively — the number `xchannel-triples.md` §7 records, and the reason the series stopped. The bridge's second term is the other half of the price: the sub-`W` census enumerates 4,517,592 triples at @23 and scans `A` at stride `∏Q` across the whole of `[0,W)`, which is random access over the resident array by construction. ## 2. The segmented instrument, and its two levers `research/xchan-at29-01-segmented.js`. Two changes, and neither approximates anything. **(L1) Exchange the order of summation.** Summing over `r` on the outside instead of over triples: > `mixed super-W obs = Σ_{r∈N} #{(q₁ W}` > ` + Σ_{r∈N} #{(q₁ ∈ D_a(r), q₂ W}` with `D_a(r) = {q ∈ qs : q | r}` and `D_b(r) = {q ∈ qs : q | r+2}`. Every `r` contributes through its own two divisor lists and nothing else, so the census is a sum of a bounded function of the local factorisation over a sieved interval — exactly the shape the corpus's segmented-sieve lever handles, and the same shape `research/import-scanstat-04-score.js` streams over these same 6.47e9 positions for T₂₉ with a depth-64 ring buffer and no materialised gap word. **[MEASURED] The lists are short**: `max |D_a| = 5, max |D_b| = 6` at @29 and `6, 6` at @31, against the code's capacity of 7, which it checks. **(L2) Sieve the compressed slot line.** Every `r ∈ N` is `≡ 11` or `17` (mod 30), so only `W/15` of the line can matter. Indexing slots by `s = 2k + t` with `r = 30k + (t ? 17 : 11)`, a scour prime `q` marks one class on one side at `r ≡ CRT(class, side) (mod 30q)` — every `2q`-th slot. The marking cost falls by the full factor 15. At @29 that is **431,312,882 slots and 994,345,351 scour marks** where the uncompressed line would need 1.49e10. **Exactness at the one inequality.** The kernel compares `q₁q₂q₃` to `W`. At @31 that product reaches 9.0e16 and is not exact in a double, so it is never formed: `U[m] = ⌊W/q_m⌋` is precomputed with an integer correction (the division is not trusted), and the test is `q₁q₂q₃ > W ⟺ q₁q₂ > U[m]`, valid because every scour prime exceeds `x` so `q_m ∤ W` and `W/q_m` is never an integer. `q₁q₂ ≤ 2.0e11` is exact. **The denominator is computed as a tail, never as a difference.** `Σ_{∏Q>W} 1/∏Q` is accumulated directly over the super-`W` triples by a monotone pointer on the third prime, so no cancellation enters it. Its consistency with the other route is printed anyway: `miss + sub = e₃` agrees with the elementary symmetric function to eight decimals at every level (0.25113952 at @29, 0.33095269 at @31), and the triple counts sum to `C(K,3)` exactly (80,993,070,011 at @29; 8,812,286,043,484 at @31). **Memory.** `A` (1 byte) + `a` (1) + `b` (1) + two `Uint16` divisor lists of capacity 7 (28) = **31 bytes per slot on 2²¹ slots = 62.0 MB, at every level**. The level enters only the segment count: 206 at @29, 6,376 at @31. ## 3. The validation gate — the instrument reproduces the in-memory one exactly Prereg §5 fixed this before the producer was written: no @29 number is reported unless the new instrument reproduces `natal-cap-39`'s embedded OUTPUT to the unit at @11, @13, @17, @19 and @23. Fourteen integers and three rounded reals per level, 85 comparisons in all. | level | `N̄` | `B₃` split (3,0)/(2,1)/(1,2)/(0,3) | sub-`W` triples | sub-`W` mixed | **mixed super-`W`** | (2,1)/(1,2) | `Σ_{∏Q>W}1/∏Q` | `J` | |---|---|---|---|---|---|---|---|---| | @11 | 90 | 0/2/0/0 | 0 | 0 | **2** | 2/0 | 0.006909 | 0.536070 | | @13 | 990 | 12/89/70/11 | 71 | 16 | **143** | 77/66 | 0.023518 | 1.023632 | | @17 | 14850 | 848/2729/2713/847 | 5246 | 2243 | **3199** | 1622/1577 | 0.039784 | 0.902452 | | @19 | 252450 | 27377/85713/85449/27391 | 162982 | 97097 | **74065** | 37139/36926 | 0.050943 | 0.959855 | | @23 | 5301450 | 879225/2779604/2779107/879385 | 4517592 | 3751046 | **1807665** | 904409/903256 | 0.058834 | 0.965932 | **[MEASURED] GATE: 5 reference levels checked, 0 mismatches.** Every integer, and `Σ_{∏Q>W}1/∏Q`, the CRT denominator and `J` at the record's own printed precision. The two instruments share no data structure: one enumerates triples and scans a resident `W`-wide mask, the other sieves a compressed slot line in 62 MB segments and never enumerates a triple in the numerator at all. Two further identities hold at every level including the two new ones: `sub + sup = B₃` class by class (INTERNAL IDENTITY PASS), and the counted `N̄` equals `2∏_{7≤p≤x}(p−2)` (143,139,150 at @29; 4,151,035,350 at @31). ## 4. The price, and what it now costs to go up a level | level | slots | scour marks | segments | census | level total | |---|---|---|---|---|---| | @23 | 14,872,858 | 30,527,907 | 8 | 0.55 s | **0.56 s** (natal-cap-39: 169.89 s) | | @29 | 431,312,882 | 994,345,351 | 206 | 13.02 s | **15.07 s** | | @31 | 13,370,699,342 | 33,742,377,278 | 6,376 | 544.68 s | **596.78 s** | Whole file, seven levels: **612.6 s**. The difference between the census column and the level column is the denominator's `K²` pass, 2.05 s at @29 and 52.10 s at @31. **[MEASURED] The factor at @23 is 303.** The in-memory census needed 169.89 s and 1.203 GB there (`xchannel-triples.md` §3); this one needs 0.56 s and 62.0 MB, and gets the same integers. The @31 leg cost 596.78 s, which is why the prereg's conditional @31 clause was executed rather than deferred. **Timings are the only thing here that moves.** The file was run twice end to end on a machine shared with four concurrent siblings, and the two runs have the **same normalised output hash** (`out-sha256 0c5f57087dd86f49…`): every integer, every ratio and every verdict is identical, and only the wall clock differs. ## 5. @29 and @31, scored against the pre-registration The predictions are `import-stein.md` §3.2's, adopted verbatim in prereg §2 and hard-coded in the producer. **[MEASURED] Custody first:** `4S₂` recomputed here from the level's own prime list is **0.028943** at @29 and **0.024784** at @31, agreeing with the record's blind values to the printed digit. The prime lists are `K = 7863` and `K = 37534`, which are the record's own counts, running 31..80429 and 37..447829 inside the intervals (29, 80429] and (31, 447829] it names, and `F` likewise reproduces at 0.7537% and 0.6419%. | level | mixed super-`W` obs | CRT | `J` | `1 − J` | `σ_J` | predicted `1−J` | **z** | **d** | |---|---|---|---|---|---|---|---|---| | @19 | 74,065 | 77,162.70 | 0.959855 | 0.040145 | 0.003527 | 0.041122 | −0.28 | −2.38% | | @23 | 1,807,665 | 1,871,421.20 | 0.965932 | 0.034068 | 0.000718 | 0.033678 | +0.54 | +1.16% | | **@29** | **53,660,192** | **55,252,747.16** | **0.971177** | **0.028823** | **0.000133** | **0.028943** | **−0.90** | **−0.41%** | | **@31** | **1,653,241,687** | **1,695,051,393.52** | **0.975334** | **0.024666** | **0.000024** | **0.024784** | **−4.93** | **−0.48%** | (The @19 and @23 rows are this instrument's own reproduction of the record; the record scores them −0.29 and +0.60 from its rounded `1−J`.) **PRE-REGISTERED VERDICTS, printed by the producer's own classifier.** - **@29: TEST 1 `z = −0.90` → HIT. TEST 2 `d = −0.41%` → TIGHT.** Combined verdict, prereg §3: **HIT**. The J-band clause does not fire. - **@31: TEST 1 `z = −4.93` → MARGINAL. TEST 2 `d = −0.48%` → TIGHT.** Combined verdict, prereg §3: **SURVIVES WITH A MEASURED OFFSET — right to `d`, not right.** The J-band clause does not fire. **[MEASURED] R1 — the offset is the finding, and it is one detection, not two.** `d` is −0.41% at @29 and −0.48% at @31, and the absolute residuals `0.028943 − 0.028823 = 0.000120` and `0.024784 − 0.024666 = 0.000118` agree to 2%. But `σ_J` is 0.000133 at @29, so **@29 does not detect the offset** — it is consistent with it at 0.90σ and equally consistent with zero. The detection is @31's alone, at 4.93σ. The honest statement is: *one level measures a residual of 1.18e−4, a second level is consistent with the same residual and cannot resolve it, and the two levels that fitted the closed form had error bars 0.003527 and 0.000718, respectively 30 and 6 times too wide to have seen it.* No closed form is proposed for the residual here; two points, one of them a non-detection, do not license one. **[REFUTED] R2 — the two rival laws, fixed in prereg §4 before the run.** | law | @29 prediction | z at @29 | @31 prediction | z at @31 | |---|---|---|---|---| | N1, the candidate `4S₂` | 0.028943 | **−0.90** | 0.024784 | **−4.93** | | N2, constant multiple of `F` fixed at @23 | 0.029145 | −2.43 | 0.024818 | −6.35 | | N3, `1 − J ∝ 1/lnW` anchored at @23 | 0.028990 | −1.26 | 0.025165 | −20.81 | Prereg §4 registered in advance that @29 could not separate these — N3 sits 0.4σ from the candidate there — and that @31 could. It did. **N3 is refuted at 20.81σ and N2 at 6.35σ, and the candidate is nearer than either at both blind levels.** The closed form is the last law standing, and it is also refuted as exact. **[MEASURED] R3 — the ratio does not rise, and "asymptote exactly 4" gets no support.** The measured `(1−J)/F`, on the producer's own `F`: | x | 17 | 19 | 23 | **29** | **31** | |---|---|---|---|---|---| | measured `(1−J)/F` | 7.0401 | 3.6905 | 3.8630 | **3.8240** | **3.8427** | | `4S₂/F` | 3.7245 | 3.7804 | 3.8189 | **3.8399** | **3.8612** | The closed form's ratio rises monotonically toward 4 by construction; the measured one falls from @23 to @29 and rises again to @31, inside a spread of 0.04. **None of that shape is resolved**: the @23 ratio's own bar is ±0.081 (`xchannel-triples.md` R3), which swallows the whole spread. What IS resolved is the vertical offset at the two sharp levels, where `σ_J/F` is 0.000133/0.007537 = 0.018 and 0.000024/0.006419 = 0.0037: the measured ratio sits 0.0159 below the formula's at @29 and 0.0185 below it at @31, which is the primary test's 0.90σ and 4.93σ read in another column. So the "asymptote exactly 4" clause in `import-stein.md` §3.2 is a property of the formula and not a measured trend, and the one thing the measurement does say about it is that the formula runs high. **[MEASURED] R4 — `J` itself, seven levels, monotone from @17.** 0.536070, 1.023632, 0.902452, 0.959855, 0.965932, **0.971177**, **0.975334**, with increments from @17 of +0.057403, +0.006077, **+0.005245**, **+0.004157**. Both new levels classify DECAYING FASTER THAN FORCED under `natal-cap-39`'s pre-registered bands (0.945 < J ≤ 0.995), and the monotonicity clause holds. The deficit is still large in its own units: at @31 the shortfall is 1,695,051,393.52 − 1,653,241,687 = **41,809,706.52 coincidences, more than 1,000 σ_J from zero**. Decaying is not absent, and two more levels do not make it absent. **[PROVEN, now MEASURED twice more] R5 — P1 holds at @29 and @31.** Aligned super-`W` obs is **exactly 0** at both, by direct enumeration of the divisor lists rather than by the bridge subtraction that produced the @19 and @23 zeros. **[MEASURED] R6 — the two orientations stay symmetric.** (2,1) and (1,2) split 26,827,582 / 26,832,610 at @29 (0.019% apart) and 826,645,240 / 826,596,447 at @31 (0.006% apart), against 27,626,373.58 and 847,525,696.76 each. The deficit is not carried by one orientation at either new level. ## 6. What this does not show - **No derivation.** `import-stein.md` §3.3 killed the Chen–Stein route (`b₃/(b₁+b₂) = 73.4` at @23) and §3.4 refuted the super-`W` decomposition route. Nothing here revives either. The closed form is still a formula that fits, now known to be accurate to about half a percent and no better. - **The residual is not identified.** One 4.93σ detection at one level. Any closed form fitted to it now would be fitted to a single number. - **N2 is not fully separated from N1 by structure, only by value.** A constant multiple with the constant re-fitted at @29 or @31 rather than at @23 would track the measurements as well as `4S₂` does; what @31 refutes is the specific constant the prereg fixed at @23. - **The `σ_J` error model is inherited, not derived.** These counts are exact arithmetic; `√obs/CRT` is the record's own Poisson proxy for their fluctuation, and every z here depends on it. A different fluctuation model moves the 4.93σ. - **`j = 4` is still untouched**, and the steelman is stated at `i + j ≥ 3`. Its missing mass at @23 is 0.040035, and the same instrument would reach it. - **No σ_X, no rotation ensemble, no `m`-decomposition.** This file computes the joint census and nothing else; `natal-cap-39`'s other columns are not reproduced here and are not claimed. - **No live document carries any of this.** `TODO.md` item X, `paper/anchored-note.md`, `research/moire-theorems.md`, `import-stein.md` and `xchannel-triples.md` were not touched. §7 proposes the item X edit; it is not applied. ## 7. Proposed update to TODO item X (report only, not applied) Three edits, and one Named gap to close. **(a) The closed-form clause.** Replace *"the `~3.8` constant now has a ZERO-PARAMETER closed-form candidate ... landing at −0.29σ (@19) and +0.60σ (@23) with asymptote exactly 4 ..., and blind values pre-computed at @29 ... and @31 .... so the derivation stays the item's open analytic target, now with a target formula and two blind tests waiting"* with: > **BOTH BLIND TESTS SPENT, 2026-08-19** (`research/history/staging/xchan-at29.md`, > pre-registered in `xchan-at29-prereg.md` and committed alone before the > producer existed). `1 − J = 4S₂` measures **0.028823 at @29 against the > predicted 0.028943 (z = −0.90, HIT)** and **0.024666 at @31 against 0.024784 > (z = −4.93, d = −0.48%)**. It is the **only survivor of the three laws the > prereg fixed**: a constant multiple of the forced scale fixed at @23 dies at > 6.35σ and `1 − J ∝ 1/lnW` at 20.81σ, both at @31. It is also **refuted as > exact** — the two blind levels share a relative offset of −0.41% and −0.48%, > absolute residuals 1.20e−4 and 1.18e−4, detected at 4.93σ at @31 and only > consistent at 0.90σ at @29. **"Asymptote exactly 4" is untested and gains > nothing**: the measured ratio runs 7.0401, 3.6905, 3.8630, 3.8240, 3.8427 over > @17..@31 and does not rise, while the formula's rises 3.7245 → 3.8612. The > derivation remains the item's open analytic target and now has to produce the > half-percent offset as well as the leading term. **(b) The `J` series.** *"the mixed super-W joint deficit runs J = 0.9025, 0.9599, 0.9659 at @17/@19/@23"* becomes *"...0.9025, 0.9599, 0.9659, 0.9712, 0.9753 at @17/@19/@23/@29/@31, monotone at all five, all three post-@17 levels in the DECAYING FASTER THAN FORCED band, and P1 (aligned super-W ≡ 0) now directly enumerated at @29 and @31 as well."* **(c) Named gaps.** Strike *"@23 is the in-memory instrument's last level (@29 needs 162 GB)"* and replace with: > The census is segmented (`research/xchan-at29-01-segmented.js`): **62.0 MB of > arrays at every level**, @23 in 0.56 s against the in-memory instrument's > 169.89 s and to the same integers, @29 in 15.07 s, @31 in 596.78 s. The memory > ceiling is gone and the clock is what is left: @29 → @31 cost a factor of 40 > for a factor of 31 in `W`, so @37 is an overnight run rather than an > impossibility (extrapolated, not priced). **(d) The first move.** *"decide whether δ's sign flip is finite-size or structure before any segmented @29 rewrite"* is spent: the rewrite exists and was cheaper than the decision it was gated behind. The cheap next instrument named there, *"j = 4's own census, missing mass 0.040 at @23"*, is unchanged and is now reachable at @29 and @31 by the same code path. ## 8. Files touched - `research/history/staging/xchan-at29-prereg.md`: new, committed **alone** before the producer existed (`27dc709`, "prereg: xchan-at29"). - `research/xchan-at29-01-segmented.js`: new. Tail embedded with `node research/qc/embed.js --timeout 1800 research/xchan-at29-01-segmented.js`. **The file carries a PRESENTATION NOTE in its header and the reason is on the record inside it.** Each blind level was computed once before the canonical embed run, with these predictions and these bands already hard-coded and the pre-registration already committed, so neither figure was ever produced by an unclassified run. @29 was first computed on a revision differing only in two OUTPUT lines (the custody line, and a corrected scour-mark work count, which is printed and not used); @31 was first computed on the present revision, before the default level list gained 31. No band, rule, threshold, prediction or computed quantity changed at any point, and the embedded block reproduces both first runs digit for digit. The file was embedded twice: the first embed's header quoted the @31 probe's own census time, which the canonical run does not reproduce because the machine is shared with four concurrent siblings, so the figure was replaced by a pointer to the block's own price and the file re-ran. - `research/history/staging/xchan-at29.md`: this file. No live document was edited. ## 9. Gate **Before:** `node research/qc.js` TOTAL = **0**. **After:** the count **oscillates between 0 and 6** depending on the minute it is read, and every finding in it is `uncited-script`. `node research/qc.js --full` reports **117/117 checks passed** and fails the gate only on whatever that TOTAL happens to be. **Reporting "0 before, 0 after, unchanged" would be true of a number and false of the fact.** The tree is shared with four concurrent siblings, as `xchannel-triples.md` §9 describes, and the finding set moved four times while this ran: six (five siblings' scripts and this session's `research/xchan-at29-01-segmented.js`, waiting for the report that cites it), then four (one sibling's `fdecay-deep` family), then zero, then four again as that sibling's tree changed under the reading. **Since this file was written, none of the findings has been this session's**: it cites its own producer, and the `refs`, `quotes`, `crosslinks`, `scripts`, `transfers`, `calibration`, `absence`, `sourcing`, `embeds`, `provenance` and `search-convention` checks are all clean on both files this session wrote. `embeds` in particular is clean after the numbered READINGS were appended below the OUTPUT banner, which is what confirms the `code-sha256` binding was not disturbed by them. `node research/qc/embed.js --check --timeout 1800 research/xchan-at29-01-segmented.js`: **code-sha256 matches, out-sha256 matches**, on a third full execution of the file.