# The kill shadow's drift amplitude: a derived finite-y factor, and a floor the record printed four times too small **2026-08-19, TODO item 5's last open number.** Pre-registration: [shadow-amplitude-prereg.md](shadow-amplitude-prereg.md), **committed alone** (`6c49f5f`) before either producer existed — the first git-provable pre-registration in this thread. Producers, both formally embedded: [`../../shadow-amplitude-01-derivation.js`](../../shadow-amplitude-01-derivation.js) (0.1 s, pure computation, no measured datum in it) and [`../../shadow-amplitude-02-measure.js`](../../shadow-amplitude-02-measure.js) (17.4 s). The ten cluster depths and slot counts are **cited** from `../../shadow-buchstab-02-instrument.js` and `../../adversary-wave2-01-shadow.js` under the standing compute rule, never re-measured. **No live document was edited. Proposals only.** **VERDICT: PARTIALLY EXPLAINED**, by the pre-registration's own rule — C1, C3 and C4 pass, C2 fails at one cluster of ten, C5 is clear. The amplitude is not a defect of the law. What is left is stated in §5 with its size and its sign. --- ## 0. The scorecard | criterion | result | |---|---| | **C1** amplitude: 1 inside the 2σ interval of (measured drift)/(predicted drift) | **PASS** on all three binnings with the correction; **FAIL** on two of three without it | | **C2** every cluster within 2σ of the corrected prediction, mean z in [−1,+1] | **FAIL** — nine of ten inside, `y ~ 12000` at z = −2.12; mean z −0.401 | | **C3** the derived `1/K` equals the independently measured `λ_pre` | **PASS, 10 of 10** | | **C4** is the local pair density approaching HL, or oscillating about it? | **PASS** — oscillating; 4 of 10 clusters above 1, mean 0.31σ | | **C5** kill (surviving trend, or one-sided `λ_pre − 1/K`) | **not tripped** — trend 1.98σ against a 2σ bar, 0 clusters one-sided | Amplitude ratios, before and after, with the exact-`N_dist` 2σ intervals: | binning | uncorrected | 2σ | corrected | 2σ | |---|---|---|---|---| | ten clusters, `y ~ 1000 → 26000` | **2.294** | [1.034, 3.555] ✗ | **1.449** | [0.653, 2.245] ✓ | | ladder, `y = 1009 → 25409` | **2.648** | [1.018, 4.278] ✗ | **1.879** | [0.722, 3.036] ✓ | | `y ≥ 2000` | 1.186 | [0.221, 2.151] ✓ | 0.825 | [0.154, 1.496] ✓ | ## 1. The correction is an identity, and it has no free parameter `shadow-buchstab-02-instrument.js` normalises every count by the **exact** tile density `δ(y) = ½ ∏_{2y}(1 − 1/(p−1)²)` and `C₂` the twin-prime constant: δ(y) = 2 C₂ M(y)² / A(y) (identity, checked to 2e-14) δ_∞(y) = 2 C₂ e^{−2γ} / (ln y)² K(y) := δ(y)/δ_∞(y) = (e^γ M(y) ln y)² / A(y) and `δ_∞·ρ` is the exact Hardy–Littlewood local pair density — at `u = 2`, `δ_∞ · e^{2γ}/4 = 2C₂/(2 ln y)² = 2C₂/(ln x)²`. Therefore measured(y) = λ(y) · B_x(y) / K(y), `λ` being the true local pair density over its HL form. **The corrected prediction is `B_x(y)/K(y)`.** `K < 1` at every level reached, so the correction is positive, which is the sign of the observed residual. It is level-wide, so it cannot disturb D3 (which passed 10 of 10) nor the coefficient `2 − 1/ln 2` (a first moment *inside* the band). `K` is almost entirely the Mertens partial product: at `y = 1009` its two halves are `(e^γ M ln y)² = 0.992885390` and `1/A = 1.000126148`, so the twin-constant tail carries under 2% of the correction, and by `y = 22901` that tail is `1.000003971`. This is precisely route (a) of the task — the second-order term of the pair-Mertens product over the prime range the band sieves — and it is the whole of the derived term. Frozen corrected predictions (prereg §2, reproduced by producer 01): | y0 | B_x | 1/K | **B_corr** | measured | res | |---|---|---|---|---|---| | 1000 | 0.83365 | 1.00967 | **0.84171** | 0.85004 | +0.00833 | | 1400 | 0.83192 | 1.00728 | **0.83797** | 0.84124 | +0.00327 | | 2000 | 0.83025 | 1.00584 | **0.83510** | 0.83285 | −0.00225 | | 2900 | 0.82886 | 1.00520 | **0.83317** | 0.83212 | −0.00105 | | 4200 | 0.82756 | 1.00429 | **0.83111** | 0.82830 | −0.00281 | | 6000 | 0.82629 | 1.00218 | **0.82809** | 0.82690 | −0.00119 | | 8500 | 0.82512 | 1.00163 | **0.82647** | 0.82741 | +0.00094 | | 12000 | 0.82401 | 1.00223 | **0.82585** | 0.82325 | −0.00260 | | 18000 | 0.82279 | 1.00207 | **0.82449** | 0.82315 | −0.00134 | | 26000 | 0.82177 | 1.00138 | **0.82290** | 0.82279 | −0.00011 | ## 2. The measurement half: routes (a) and (b) agree, 10 of 10 Below `y²` a level-`y` slot is a twin prime and nothing else (a `y`-rough composite needs two factors above `y` and so exceeds `y²`), so the pre-band `[y²/2, y²)` is scored against the `u ≤ 2` branch `e^{2γ}/u²`, which is exact HL with **no independence conjecture in it**. Pooled over every level of every cluster — a measurement this corpus had never made — it gives `λ_pre := measured_pre/pred_pre`, which the derivation says must equal `1/K`. It does, within 2σ at all ten clusters: 1.00748 against 1.00967 at `y ~ 1000`, 0.99983 against 1.00163 at `y ~ 8500`, 1.00096 against 1.00138 at `y ~ 26000`. A derivation and a measurement that never touch the same integers return the same finite-`y` factor. The identity behind it is checkable from raw counts and is: at `y = 1009` the pre-band's measured depth 0.83771 over its prediction 0.82998, times `K`, is **1.00226**, and the window's 3684 twin primes over the Hardy–Littlewood sum 3675.7 is **1.00226**. The sieve count and the prime count agree to five decimals. (The pre-band's slot count equals its twin-prime count exactly: 81 and 81 at `y = 101`, 3684 and 3684 at `y = 1009`.) ## 3. The record's counting floor is four to six times too small `shadow-buchstab-02` prints `se(Poisson) = measured/√Ntot` on the **pooled** slot total. The pooling oversamples: at `y0 = 2900` the 53 bands `[y², 2y²]` cover one interval `[6538249, 17558738]`, and `Ntot = 2219418` counts **61931** distinct pairs — a 35.84× oversample. A pair `n` is sampled iff `n` is `y_lo`-rough for the *smallest* cluster level with `n < 2y_lo²` (roughness decreases in `y`, so the smallest covering level is the weakest test), which makes the exact count one segmented pass. | y0 | Ntot pooled | N_dist exact | oversample | se printed | se true | |---|---|---|---|---|---| | 1000 | 266735 | 12981 | 20.55× | 0.00165 | 0.00746 | | 2000 | 1417269 | 42351 | 33.46× | 0.00070 | 0.00405 | | 2900 | 2219418 | 61931 | 35.84× | 0.00056 | 0.00334 | | 8500 | 1945202 | 248328 | 7.83× | 0.00059 | 0.00166 | | 26000 | 1779677 | 1779677 | 1.00× | 0.00062 | 0.00062 | The understatement is worst — 4.5× to 6× — at exactly the clusters where the residual is largest. This is a defect in the record's error column, not in its measurement. Poisson is used here as a **floor**, not an estimate: prime pairs in long intervals are not Poisson and the residual correlation between overlapping bands can only widen the bar. ## 4. The record's own candidate explanation is the wrong half `shadow-buchstab.md` §4 reads the small-`y` excess as "the local pair density is still approaching its Hardy–Littlewood form at `x ~ 10⁶`". Measured directly, `λ_twin = λ_pre · K` — the local twin density over its HL form at each cluster's own scale — reads 0.99784 1.01926 1.00574 0.99912 1.00229 0.99934 0.99820 1.00078 0.99833 0.99959 **above 1 at 4 of 10 clusters, mean 0.31σ, and its largest excursion is at `y ~ 1400`, not at the smallest scale.** The local pair density oscillates about its HL form at these scales; it is not walking up to it. What has not reached its asymptote is the **fair share the instrument divides by**, and that is derivable rather than empirical. The §4 reading should be replaced, not softened. The same correction dissolves §4's other claim. Subtract `1/K` from the eleven points of `shadow-buchstab-02` part (D) and the column reads +0.00922 +0.01130 +0.00107 +0.00105 +0.00474 −0.00144 −0.00376 −0.00292 +0.00209 −0.00217 −0.00036: it changes sign three times and is not a decay at all. The apparent eleven-fold fall from `y ~ 1000` to `y ~ 2000` was a smooth 1.7-fold fall in `1/K` plus scatter of the size §3 gives. "Far too steep a decay for a `1/ln y` or `1/ln²y` term" was true and pointed the wrong way: the derived term is not a power of `1/ln y` at all, it is the prime-counting fluctuation inside `M(y)`. ## 5. What is left, and why the label is PARTIALLY EXPLAINED Two residues survive the correction, and they point opposite ways. - At `y ~ 1000` the correction leaves **+0.00833 ± 0.00746** (1.12σ). It is gone by `y ~ 2000` (−0.00225 ± 0.00405). So the steep small-`y` excess is explained to 1.1σ and there is no term left to name there. - Across the eight clusters at `y ≥ 2000` the inverse-variance mean of the corrected residual is **−0.00079 ± 0.00043**, a **1.85σ over-correction**, and the corrected residual's trend on `ln y` is **−0.523 ± 0.264 = 1.98σ**, which missed C5's 2σ bar by a hair. One cluster, `y ~ 12000`, sits at z = −2.12 and is what fails C2. Both are consistent with zero at 2σ and neither is worth naming yet. The honest arithmetic: **45.1% of the missing drift is the derived `1/K`**, and the remaining −0.00844 is 1.13σ of the endpoint clusters' own floor. Under the pre-registration's rule DERIVED needs C1 **and** C2 **and** C3 **and** C4, so the label is PARTIALLY EXPLAINED. Grading it higher would repeat exactly the error `adversary-wave2.md` §1 correction 1 caught in the parent record. What would settle the remainder is more **distinct** pairs. At `y ~ 1000` that means a wider pool, not a longer run: the 36 bands there already exhaust the `x`-range they cover, and 12981 pairs is all the arithmetic in `[776161, 2495378]` contains. A ±40% pool at each centre, or centres chosen so the unions do not overlap, would roughly halve every floor in §3. ## 6. Proposed disposition of TODO item 5 The item's remaining open number is answered as far as the data can answer it. Proposed: **item 5 leaves the file**, with these carried into the record it points at. 1. `shadow-buchstab.md` §2's drift law gains its amplitude clause: `0.793055·(1 + (2 − 1/ln2)·ln2/ln y + …)/K(y)`, with `K` defined as in §1 and the note that `1/K` is the pair-Mertens partial-product factor and supplies 45.1% of the drift the raw law under-predicts. 2. `shadow-buchstab.md` §4's reading is **replaced**: the local pair density oscillates about HL at these scales (4 of 10 clusters above, mean 0.31σ); what has not converged is the fair share. 3. `shadow-buchstab-02-instrument.js`'s `se(Poisson)` column is **wrong by 4× to 6×** at every pooled cluster and should be recomputed on distinct pairs, or labelled as what it is. This is the correction with the widest blast radius: any conclusion in the corpus that priced a pooled-band residual against `measured/√Ntot` priced it against the wrong floor. 4. The verdict label on the kill shadow stays **SHAPE-ONLY** as `adversary-wave2.md` set it. Nothing here promotes it; what changes is that the amplitude gap now has a derived first term and a measured error bar instead of an unexplained factor 2.6. Absence: no prior treatment of this correction to a band-averaged Buchstab depth is claimed. `../../SEARCH-CONVENTIONS.md` §1 carries no owning convention for the shadow constant, and the pieces used here — Mertens' third theorem's error term, the singular-series partial product — are textbook and are cited as such, not as findings.