# The decoupling step at two excluded classes: what it actually is, what it costs, and the part of it that needs no proof > **RIDER 2026-08-30 (orchestrator, from `verify-0830-record-defects.md` §1, CONFIRMED to every digit on > independent code at x = 7..19).** PART 3 of `varE-theta2-step.js` builds the > shift group X2 by doubling the positive-h half of one pattern that is a > mirror image, not even. The corrected reading of §5 (lines 222-225): the > group sum δ·X2 reads −0.009235, 0.004509, 0.001365, 0.000736, 0.000394 at > x = 7..19 (2X2 = −0.5603 .. 0.1846, O(1) and falling; the column first > carried here, 0.094199 .. 0.007968, is withdrawn); δ·Xmix runs −0.034971 > .. −0.016835 over the same levels, Xmix = −0.49 ln y on the top three, the > slower of the two. **Applied 2026-09-05: the producer now sums both mirror > patterns, is re-embedded, and reproduces these values with x = 23 added > (δ·X2 = 0.000236, δ·Xmix = −0.013876); §5 above carries the corrected > numbers.** X, X1, X_dec and all ratios > are untouched; the §8 row's rung (MEASURED) is unchanged. *Staging note, 2026-08-28. Forward work on `varE-spectral.md` §2, one of the two open steps in that note; the other is its §5 limit theorem for the model, which this pass does not touch. Producer: `research/history/staging/varE-theta2-step.js` (embedded, `--check` clean, 1.7 s). Reads `varE-spectral.md` and its `.js` for the nine diagonal points, the exact `X_dec` at x = 7 and 11, and the level table (L = x#, y the largest prime below sqrt(L)). Recomputes nothing from `paper/variance-note.md`.* ## 0. Verdict, the open half first **The step is not proven and not bounded here.** No inequality is derived. What replaces it is a smaller open statement plus four new exact data points, and the smaller statement is still open. **The frequency-side hope in the task brief fails.** The local coefficients do not recombine into one shifted Fejer term. They recombine into `|Ehat_n(t)|^2` with `E_n = {r : n | r(r+2)}`, which expands into `3^omega(n)` distinct shifts carrying `4^omega(n)` total weight. Exactly two of those shifts are the clean `d = +-2` terms the brief predicted, and those two are indeed harmless. The remaining `3^omega(n) - 3` are CRT-mixed, and they are where the difficulty lives. **The measured coefficient is a near-cancellation.** The decoupling error, after the `delta` normalisation, is the sum of a `+0.1334/ln y` term and a `-0.1288/ln y` term (reading 4). The net `0.0046/ln y` is a difference of two numbers thirty times larger, so its value is not established even though its order is. **Nothing here touches the conjecture.** `variance-note.md` §4 already records that Var/E over a uniformly random window says nothing about the one anchored window the twin problem needs. This note is about whether a stated limit is the limit of the stated object, not about primes. With that said, what the pass produced: - **PROVEN, elementary, and checked at six levels.** The spectral sum is a real-space shift sum: `X = sum_{|h|= 7` the exact `W` splits into `3^omega` groups indexed by which of `p | h`, `p | h-2`, `p | h+2` each prime takes. The group where every prime takes `p | h` is the only one whose CRT class is 0, so Fact A evaluates it verbatim, with no equidistribution and no replacement. Its weight is the model's weight with `pi_p = 2/(p-2)` in place of `2/(p-1)`, because `(p-1)(p-4) + 2 = (p-2)(p-3)` exactly (§3, PART 1). - **MEASURED, exact, six levels, no Monte Carlo.** `X/X_dec` = 0.853427, 0.960382, 1.009802, 1.003895, 1.001448, 1.001312 at x = 7, 11, 13, 17, 19, 23. The corpus had the first two. `delta*(X - X_dec)` = -0.026118, -0.010572, +0.002907, +0.001268, +0.000502, +0.000477 (§4). - **MEASURED.** `delta*(X - X_dec)*ln y` is bounded on every level computed, and falling rather than settling: 0.01498, 0.00832, 0.00404, 0.00459 at x = 13..23. Bounded implies the error vanishes, so on this evidence the limit 0.45546 is not moved by the replacement (§5), and the evidence is for `O(1/ln y)` or better. Four levels exclude growth and nothing finer. - **Correction to `varE-spectral` §6a.** The model column there is Monte Carlo and is off by up to 0.0017 at x = 23, which is larger than any residual the note reports at x >= 13 (§6). ## 1. The replacement, restated in real space `K_L(t) = sum_{|h| < L} (1 - |h|/L) e(ht)` is the Fejer kernel, so X = sum_{nu != 0 mod M} What(nu) K_L(nu/M) = sum_{|h| < L} (1 - |h|/L) W(h) - L = sum_{|h| < L} (1 - |h|/L) (W(h) - 1), using `sum_{|h| 0` there, and that gap does not shrink with y. On `p >= 7` the ratio `W/V` is a convergent product times `prod_{p | h^2-4} (p-3)/(p-4)`, which is order one but not one. Whatever makes the totals agree is cancellation in the h-sum, not closeness of the summands. ## 2. What the frequency side actually gives Working the brief's step 1 through: with `nu_p = t (n/p)^{-1} mod p` the local coefficients multiply to prod_{p | n} hat f_p(nu_p) = |Ehat_n(t)|^2 / prod_{p|n} (p-2)^2, Ehat_n(t) = sum_{r in E_n} e(tr/n), E_n = {r mod n : n | r(r+2)}, so the exact conductor-n sum is `sum_{r, r' in E_n} T_n(r - r')` with `T_n(d) = sum_{|j| 2L` and `d = +-2`, `T_n(+-2) = T_n(0) - 2n/L + O(L)`, so those two lags each contribute a `(1 - 2/L)`-type copy of the unshifted term. They are the groups called `X2` below, and they are measured harmless. The clean recombination the brief hoped for does not extend past them: the remaining lags are CRT-mixed residues whose least representative is generically of size n, so their `T_n` values are not a fixed multiple of `T_n(0)` and their sum is exactly the equidistribution question the replacement was standing in for. Decoupling is not repaired by evaluating a shifted Fejer kernel. ## 3. The group that needs nothing Expanding the `p >= 7` product over the three cases gives X = X1 + 2*X2 + Xmix, with `X1` the group where every prime takes `p | h`, `X2` each of the two groups where every prime takes `p | h-2` or `p | h+2`, and `Xmix` the rest. Only `X1` has CRT class 0 at every prime, so only `X1` is a sum of `A_q(0)` terms and Fact A applies to it with nothing assumed. Its weight relative to the model's is fixed by one identity: (D_y / C_y) * prod_{7 <= p <= y} (1 + 2/((p-1)(p-4))) = 1, D_y = prod_{7 <= p <= y} p(p-4)/(p-2)^2, which holds prime by prime because `(p-1)(p-4) + 2 = (p-2)(p-3)`. PART 1 prints 1.00000000000000 at all six levels. Size-biased, `X1` carries `pi_p = 2/(p-2)` where the model carries `2/(p-1)`; the two differ in `E[ln n]` by `sum 2 ln p/((p-1)(p-2))`, which is O(1) against `ln L = 2 ln y`. So `X1` and the model share a `GD(2)` limit law at the same rung the model has it, HEURISTIC, inheriting `varE-spectral` §5's own unproven limit theorem, which is that note's second open step, rather than adding a new assumption. **What that buys.** The open statement is no longer "the local coefficients may be replaced by their means". It is "the shift groups `2*X2 + Xmix` contribute `o(1/delta)`". That is a statement about two explicit sums over shifts `h = +-2 mod q`, not about frequencies. ## 4. The error, exact, at six levels `delta*X` is Var/E. Both columns below are exact arithmetic, Kahan-summed, no Monte Carlo anywhere. | x | ln y | delta*X | delta*X_dec exact | corpus model (MC) | delta*(X - X_dec) | X/X_dec | |---|---|---|---|---|---|---| | 7 | 2.565 | 0.152075 | 0.178193 | 0.1785 | -0.026118 | 0.853427 | | 11 | 3.850 | 0.256267 | 0.266839 | 0.26744 | -0.010572 | 0.960382 | | 13 | 5.153 | 0.299499 | 0.296592 | 0.2975 | +0.002907 | 1.009802 | | 17 | 6.564 | 0.326772 | 0.325504 | 0.32579 | +0.001268 | 1.003895 | | 19 | 8.042 | 0.347302 | 0.346800 | 0.34637 | +0.000502 | 1.001448 | | 23 | 9.611 | 0.364320 | 0.363842 | 0.36214 | +0.000477 | 1.001312 | The sign change `varE-spectral` §2 inferred indirectly, from the model sitting above the data at x = 7 and 11 and below it from x = 13, is now direct: it sits between x = 11 and x = 13. The last two rows are 0.000502 and 0.000477, so the sequence is not monotone at the top; these are exact numbers, so that is arithmetic fluctuation between levels and not noise. x = 29 is out of reach for this method: `L/6 = 1.1e9` array entries, roughly 35 GB in the present layout. A segmented sieve would fit the memory and cost minutes, not seconds. ## 5. The 1/ln y split, and why the net coefficient is not established | x | delta*(X1 - X_dec)*ln y | delta*(X - X1)*ln y | sum | |---|---|---|---| | 7 | 0.07008 | -0.13708 | -0.06699 | | 11 | 0.10310 | -0.14381 | -0.04070 | | 13 | 0.13201 | -0.11703 | +0.01498 | | 17 | 0.13350 | -0.12518 | +0.00832 | | 19 | 0.13308 | -0.12905 | +0.00404 | | 23 | 0.13342 | -0.12883 | +0.00459 | Both halves settle. The group-1 excess settles near +0.1334 and the shift-group deficit near -0.1288, each stable in the third decimal over the top three levels, and each therefore consistent with an exact `c/ln y`. Their sum is the decoupling cost, and it is the difference of two quantities thirty times its own size. A one percent systematic error in either half would change it by half its value. So the defensible reading is the order, not the constant: the replacement error is `O(1/ln y) = O(2/ln L)` on every level computed, the same order as the model's own drift, hence it does not move the limit. Whether the net coefficient is 0.0046 or 0.002 or 0.008 is not settled by six levels, and nothing here needs it to be. The order is also the weaker of the two readings available: over x = 13..23 the net runs 0.01498, 0.00832, 0.00404, 0.00459, which falls rather than settles, so the evidence is for `O(1/ln y)` or better and the only failure mode these four levels exclude is growth (`redteam-0828-varE.md` §6). The two shift groups separately: `delta*X2` (each group, both mirror halves summed) reads -0.009235, 0.004509, 0.001365, 0.000736, 0.000394, 0.000236 at x = 7..23, so `2*X2` is `O(1)` and falling, and `delta*X2 = O(1/ln^2 y)`, one order better than needed. `delta*Xmix` runs -0.034971 .. -0.013876 and is the slower of the two, about `-0.49 ln y` in `Xmix` on the top levels. So the open statement narrows once more: it is about `Xmix`, the CRT-mixed lags, alone. ## 6. Two corrections this forces on `varE-spectral.md` - **§6a's model row is Monte Carlo.** Exact against quoted: 0.178193/0.1785, 0.266839/0.26744, 0.296592/0.2975, 0.325504/0.32579, 0.346800/0.34637, 0.363842/0.36214. The x = 23 gap is 0.0017, larger than any residual the note reports at x >= 13, so the residual column there is partly MC noise in the model, not structure in the data. With the exact model the residuals at x = 13..23 read +0.00291, +0.00127, +0.00050, +0.00048, all positive, where the note's read +0.0020, +0.0010, +0.0009, +0.0022. An independent rebuild reads +0.00291, +0.00130, +0.00050, +0.00063 at the same four levels (`redteam-0828-varE.md` §6). The last-digit differences are the measured column's own quantisation: `variance-note.md` §7 prints Var/E to four decimals, so that column carries +-5e-5, comparable to the residuals at x >= 29 (+0.00036, +0.00025, +0.00023). Residual structure at the top three levels is at the quantisation floor and is not signal. The same MC caution applies to the x = 41 forecast, which the exact method cannot reach and which a two-seed rerun at N = 2e7 puts at 0.402364 +- 0.000075 rather than 0.40184 +- 0.00075 (`redteam-0828-varE.md` §4). - **§2's framing.** "Replace `hat f_p(nu_p)` by `gamma_p`" overstates what has to be true. The `c = 0` group survives the replacement exactly, by PART 1's identity. What needs an argument is only the CRT-mixed lags. `variance-note.md` §7's X column agrees with exact recomputation to about five significant figures (4.6137 against 4.612929, 51.4047 against 51.400255, 121.4787 against 121.485245). Those trailing digits are rounding, not exactness, and the column should not be quoted past five figures. ## 7. Where the exact object sits in the literature `X = sum_{|h|