# Morning report, 2026-08-16: what the overnight run settled

All three tracks finished. One headline result, one reversal of a claim our
own script made, and four artifacts that finally have real output.

## 1. β(41) = 0.8455, and a fifth consecutive residual collapse

`natal-cap-37-at41-march.js`, 6.16 hours over eight shards.
Log: `~/Files/primeoire-runs/at41/at41-run2.log`.

Custody first, because it is what makes the number worth anything. GATE 0
reproduced @7 through @29 digit-for-digit. GATE 1 returned
S(31) = 283,449,187 exactly and GATE 2 returned S(37) = 7,998,394,865
exactly, both through the same sharded code path that then produced @41, and
both on the newly wheel-optimised engine. 133 sampled @41 survivors were
checked individually and are all genuine twin pairs (13-base Miller-Rabin,
deterministic to 3.317e24, plus a strong-Lucas second opinion).

At W = 41# = 304,250,263,527,210:

> N = 5,666,163,252,750 = 2·∏(p−2) exact
> S(41) = 256,725,962,834
> E(41) = 303,627,067,641.7
> **β(41) = 0.8455**

Against the four forecasts frozen before the run: classical plus persisted
residual 0.8459 misses by −0.0004, classical-raw 0.8449 by +0.0006,
free-linear 0.8443 by +0.0012, pinned-linear 0.8488 by −0.0033 and stays
dead. The classical residual sequence is now

  +0.0046, +0.0026, +0.0016, +0.0010, **+0.0006**  at @23, @29, @31, @37, @41,

five consecutive collapses, each one forecast in advance of its run. The
zero-knob series (e^{2γ}/4)(1 + 2/lnW + 6/ln²W) has no rival description at
any computed level.

Two honest limits. The remaining gap to the conjectured limit e^{2γ}/4 is
0.0525, which is 87 times the deepest residual, so the limit is still
extrapolation-supported rather than observed. And @41 is the end of the road
for this engine: @43 has W = 1.308e16, above 2^53, which is precisely where
the CRT anchoring product stops being exact. The @43 forecast is on record
anyway at 0.8393 raw, 0.8399 with the persisted residual.

First survivors at @41 are (17442827, 17442829) onward, all above
√W ≈ 17,442,771, as Lemma 3 of the anchored note requires.

## 2. Lemma V: our own script's headline was wrong, and the verdict reverses

`sift-limit-lemmaV.js`, 234 seconds.
Log: `~/Files/primeoire-runs/lemmaV/lemmaV-full.log`.

**What stands.** The mean square is exactly computable and machine-verified
three independent ways: the identity T = H·M + R to 2.4e-13 over full
periods, the closed form against brute force at seven configurations, and a
spectral form agreeing to 5e-14. The plateau is real: 2⟨ρ²⟩ is H-free, so the
mean square saturates rather than growing with the window. The measured
cancellation exponent is γ ≈ 0.15 to 0.21 against a provable 0.61.

**What broke.** The S6 section compares the sieve mean square against an
elementary second moment, and its `elementaryVariance` is wrong by about
three orders. The tell was in its own output: Var/E² printed as 0.500, 0.500,
0.500, 0.5002, 0.5006 across seven (z, H) pairs spanning a decade in both.
Brute force over every window position of the period gives the truth, and it
also agrees with `paper/variance-note.md`, which measures this count as
strictly sub-Poisson:

| z, H | claimed Var/E² | true Var/E² | true Var |
|---|---|---|---|
| 13, 1315 | 5.006e−1 | 5.884e−4 | 3.4751 |
| 19, 3806 | 5.002e−1 | 2.365e−4 | 6.5212 |
| 19, 12360 | 5.000e−1 | 3.252e−5 | 9.4586 |

**The conclusion reverses.** With the true elementary bound, the sieve mean
square does not beat the elementary second moment. It loses, by about 1.4×,
at every point brute force can reach: 8.249e−4 against 5.884e−4 at z=13,
3.722e−4 against 2.365e−4 at z=19 u=2.8, and 4.658e−5 against 3.252e−5 at
z=19 u=3.2. The printed "sieve/elem = 0.002 … 0.000" and every "elementary
needs H > …" line in S6 are artifacts. The function is flagged in place.

The sieve does scale better in H, about H^−2.0 against H^−1.7, so it
overtakes eventually, near H ≈ 4e4 at z=19 on the measured trend. So the two
methods are genuinely comparable rather than one dominating, which is a much
less exciting statement than the script printed and is the true one.

**The number that actually matters for the route, and it is not encouraging.**
S6's last block does not touch the broken function and stands. Conditional on
an unproven Gaussian maximal law for the sawtooth, all-positions positivity
needs a window of length H > z^θ with

  θ = 2.159, 2.183, 2.159, 2.252, 2.308, 2.345  at z = 19, 23, 29, 31, 41, 47.

The Gap Reformulation needs an exponent below 2. This sits above 2 and is
drifting upward with z, not downward. Read plainly: even granting the maximal
law, this route as currently set up does not reach the finish line, though
2.16 to 2.35 would still beat both our published 4.2665 and the full
decoupling target 2.649 if it could be made unconditional.

## 3. The artifact chain: four scripts now have real output

Logs in `~/Files/primeoire-runs/chain/`.

- **natal-cap-36 default** (36.5 s). Six levels of G30_agg. The decay-law
  shortcut is now refuted harder than yesterday: fits give R² of 0.44, 0.26,
  0.33, and the sequence is "flat within its own spread, and non-monotone
  twice". Mean 0.1101 over @13..@29, spread 0.0313.
- **natal-cap-36 --at29** (781 s). Confirms both deep levels certified:
  G30_agg(23) = 0.0945 and G30_agg(29) = 0.1176, each with ΣCov < 0 and
  exactly one prime out of 1,739 and 7,863 showing Cov > 0.
- **natal-cap-35 default** (101 s). Reproduces the four-level X-channel table
  including the monotone deficit (X̄−X(0))/S̄ = −0.2287, +0.0134, +0.1568,
  +0.2225. It takes no level argument, so @23 was not reached.
- **natal-cap-34 verify** (56 s). 88,520 checks passed; T₄ at @13 agrees with
  cap-32 on every layer, and T₂ = 46,186.769663306746 reproduces cap-27's
  printed value by an independent Theorem-5 path.

## 4. What this changes

- `paper/anchored-note.md` now carries the tenth level and the fifth collapse.
- TODO item 0 needs rewriting around the θ > 2 finding. The mean-square work
  is done and verified; what it bought is smaller than we thought.
- TODO item 4 stays refuted, now with six levels behind the refutation.
- The provenance warning in `WAVE7-RESULTS-2026-08-15.md` can be retired for
  cap-34, cap-35 and cap-36, whose numbers are now reproduced from clean runs.

## 5. Still not done

The wrap engine at @17 (item 6) remains unmeasured and unlaunched, for the
reason given last night. Var(41) was not computed, so the z column of the
anchored note's §3 table has no entry at @41. Neither is blocking anything.