# The spectral form pushed to a limit: two unproven steps, a closed constant, and why 0.611 was an artefact > **RIDER 2026-08-30 (orchestrator, from `verify-0830-record-defects.md` §1.3 item 5 and > `attack-0830-varE-identification.md`).** "TWO open steps" in this note's > ledger and §0 reads ONE: `varE-limit-theorem.md`'s ledger states "Step 2 of > the two is CLOSED: the decoupled model's limit is exactly Pr[GD(2) > u] with > NO correction term", and the identification step IS the θ = 2 replacement > error (`varE-theta2-step.md` §1). Whether that theorem survives review is > its own HELD status. *Staging note, 2026-08-28. TODO item 9, second pass. Producer: `research/history/staging/varE-spectral.js` (embedded, `--check` clean, 3.2 s). Reads `research/history/staging/varE-asymptotic.md` (the sibling pass of the same day) and `paper/variance-note.md` §§6-7. The nine exact diagonal points are read from `variance-note.md` §7 and are recomputed nowhere here.* ## 0. Verdict, disconfirming half first **Two steps are unproven, and together they are the gap.** The evaluation below replaces the local Fourier coefficient $\hat f_p(\nu)$ by its mean over the $p-1$ nonzero frequencies, which is an identity at one excluded class and false pointwise at two, which is the comb's case; where the error can be computed exactly it is 15% at $x = 7$ and 4% at $x = 11$ (reading 1), and nothing here proves it shrinks. Separately, the passage from the decoupled sum to $\Pr[GD(2) > u]$ is a limit theorem about the model that is asserted and not derived here: it needs the convergence of $\ln n/\ln y$ to $GD(2)$, which is routine and unwritten, and it needs the $n \le L$ band to vanish, which §4 states as $O(1/\ln W)$ and does not evaluate. Everything downstream is HEURISTIC and stands or falls with both. **And no computed level reaches the constant.** The model itself sits at 0.402 at $x = 41$ against its own limit 0.4555, so the constant rests on the second step above and not on data. **Nothing here touches the conjecture.** Var/E is a statement about a uniformly random window. `variance-note.md` §4 already records that this says nothing about the one anchored window the twin problem needs. With that said, what the pass produced: - **PROVEN, and newly checked.** The spectral form $X = \sum_{\nu \ne 0 \bmod M} \hat W(\nu) K_L(\nu/M)$ reproduces the sibling's brute-forced $X(210) = 4.612929$ to $2.9\times10^{-7}$, but only with the CRT twist $\nu_p = \nu\,(M/p)^{-1} \bmod p$. Without it the sum is 4.441347 while the total-mass check $\sum_{\nu\ne0}\hat W = 1/\delta - 1$ still passes, so that check does not catch the omission (§1, reading 2). - **PROVEN given the decoupling step.** The decoupled spectral sum collapses to a single expectation over a random divisor, $\delta X = \mathbb{E}\big[\{L/n\}(1-\{L/n\})\,n/L\big]$ with each prime $p \le y$ present independently with probability $(p-\alpha_p)/(p-1)$ (§3). Fact B falls out of it, not into it. - **HEURISTIC.** $\lim \operatorname{Var}/\mathbb{E} = \lambda_2(u)$, the tail of the generalized Dickman law $GD(2)$, and at the diagonal $u = 2$ $$\lambda_2(2) \;=\; 1 - e^{-2\gamma}\Big(\tfrac92 - 4\ln 2\Big) \;=\; 0.45545648,$$ equivalently $\lim X/\ln^2 W = \lambda_2(2)/\kappa = 0.410$ (§5). - **MEASURED, 22 points, no fitted parameter.** The model reproduces the nine diagonal points to $\le 0.0029$ for $x \ge 13$, §6's six-point $u$-sweep at $y = 401$ to $\le 0.018$, and §6's seven-point $u = 2$ level sweep to $\le 0.005$ (§6). The §6a model column is Monte Carlo; against an exact model column the residuals at $x = 13..23$ are $+0.00291$, $+0.00130$, $+0.00050$, $+0.00063$ (`redteam-0828-varE.md` §6), all positive, and at $x \ge 29$ they sit at the quantisation floor of the measured column. The model is close and is not exact. - **REFUTED, as an inference, on both halves of the protocol.** §7's argument that the limit is near 0.611 because $a + b/\ln\ln W$ beats $a + b/\ln W$ does not survive a control. Run on the model, a sequence that tracks the measured data to 0.003 and whose limit is 0.455456 by construction, the same in-sample protocol returns intercept 0.6164 from $1/\ln\ln W$ and 0.4468 from $1/\ln W$ on an exact model column (`redteam-0828-varE.md` §4): same winner, same margin, same two numbers the data give (0.6108 and 0.4454). The frozen out-of-sample half, which this note's own control does not run, fails on the control too: fitted on $x = 13..31$ and frozen, $1/\ln\ln W$ forecasts the model at $x = 37$ with residual $-0.00039$ against $1/\ln W$'s $+0.00276$. Neither the in-sample winner nor the forecast separation is evidence about the limit on this range (§7). ## 1. The spectral form, restated exactly, and checked Notation as in `variance-note.md` §7 (Corollary 3). Let $\alpha_p$ be the number of comb classes surviving at $p$: $\alpha_2 = \alpha_3 = 1$, $\alpha_5 = 2$ (the teeth 11, 17 mod 30 are 1, 2 mod 5), $\alpha_p = p-2$ for $7 \le p \le y$; so $\delta = \prod_{p\le y}\alpha_p/p$. Write $W(d) = \prod_{p \le y} f_p(d)$ with $$f_p(d) \;=\; \frac{p\,\rho_p(d)}{\alpha_p^{2}},\qquad \hat f_p(0) = 1, \qquad \hat f_p(\nu \ne 0) \;=\; \frac{\big|\sum_{a \in E_p} e(\nu a/p)\big|^{2}}{\alpha_p^{2}} \;\ge\; 0,$$ where $E_p$ is the excluded set: $E_p = \{0,-2\}$ for $p \ge 7$, giving $\hat f_p(\nu) = (2 + 2\cos(4\pi\nu/p))/(p-2)^2$, which is the sibling's §4 local factor. Then, with $K_L$ the Fejér kernel, $$\boxed{\ X(L) \;=\; \sum_{\nu \ne 0 \bmod M} \hat W(\nu)\,K_L(\nu/M), \qquad \hat W(\nu) = \prod_{p \le y} \hat f_p\big(\nu\,(M/p)^{-1} \bmod p\big). \ }$$ **The twist is not decoration.** Coding $\hat W(\nu)$ as $\prod_p \hat f_p(\nu \bmod p)$ gives 4.441347 at $x = 7$ against the true 4.612929, a 3.7% error, while leaving $\sum_{\nu\ne0}\hat W(\nu) = 29.333333 = 1/\delta - 1$ intact, because the twist permutes frequencies within each conductor. With the twist the sum agrees with the sibling's brute force to $2.9\times10^{-7}$ (PART 0). Any future spectral computation in this corpus should carry this check. The two exact facts the sibling proved are used unchanged: $\sum_{\nu \ne 0 \bmod q} K_L(\nu/q) = r_q(q-r_q)/L$ with $r_q = L \bmod q$ (Fact A), hence $q \mid L$ contributes exactly zero (Fact B). ## 2. The decoupling step, named and priced Define the mean local coefficient $$\gamma_p \;:=\; \frac{1}{p-1}\sum_{\nu \ne 0} \hat f_p(\nu) \;=\; \frac{p/\alpha_p - 1}{p-1} \;=\; \frac{p-\alpha_p}{\alpha_p\,(p-1)},$$ so $\gamma_p = 2/((p-1)(p-2))$ for $p \ge 7$, $\gamma_5 = 3/8$, $\gamma_2 = \gamma_3 = 1$. **DECOUPLING (unproven).** Replace $\hat f_p(\nu_p)$ by $\gamma_p$ at every $\nu_p \ne 0$. Two things are worth saying about this step before anything is built on it. 1. **It is exact at one excluded class.** If $|E_p| = 1$ then $\hat f_p(\nu \ne 0) = 1/\alpha_p^2$, a constant, and the replacement changes nothing. That is the rough-integer case, and it is why the corresponding asymptotic is a theorem in print (§9). At $|E_p| = 2$ the coefficient $(2+2\cos(4\pi\nu/p))/(p-2)^2$ ranges over $[0, 4/(p-2)^2]$ with mean $2/((p-1)(p-2))$, a factor $(p-2)/(2(p-1)) \approx 1/2$ below its maximum at every prime. **This one factor per prime is the missing log** the sibling's §3 could not find: bounding by the maximum gives the weight $4^{\omega(q)}$ and $\sum_{q\le Q}4^{\omega} \asymp Q\ln^3 Q$, hence the refuted $X \asymp \ln^3 W$; using the mean gives $2^{\omega(q)}$ and $\sum_{q \le Q} 2^{\omega} \asymp Q \ln Q$, hence $\ln^2 W$. 2. **Its error is measured only twice.** Full divisor enumeration (PART 2) gives $X_{\rm dec} = 5.405185$ against $X = 4.612929$ at $x = 7$, and $15.694986$ against $15.0751$ at $x = 11$: true/model $= 0.8534$ and $0.9605$. Two points, both at the bottom of the range, both with the model *above* the truth. At $x \ge 13$ the model sits slightly *below* the measured data (§6), so the error changes sign somewhere in $11 < x < 13$ and no monotone statement about it is available. The heuristic reason to expect the step to be harmless asymptotically: the Fejér weight concentrates on $|\nu| \lesssim M/L$, and the conductors carrying the mass have $\ln q \approx \ln L$ (§4), while $M/L$ has 194188 digits at $x = 31$ (`natal-cap-16` reading 2). Equidistribution of $\nu \bmod q$ over a range astronomically longer than $q$ is exactly what the step needs. It is not proven, and no attempt is made here to prove it. ## 3. What the decoupled sum evaluates to Under the replacement, $\prod_p \hat f_p$ becomes $\prod_p\big[\gamma_p + (1-\gamma_p)\mathbf 1[\nu_p = 0]\big]$. Expanding over the subset $e$ of primes where the second term is taken and using Fact A with $n = M/e$, $$X_{\rm dec} \;=\; \frac{1}{L}\,\mathbb{E}_\Theta\big[\,r_n(n-r_n)\,\big], \qquad r_n = L \bmod n,$$ where $\Theta$ makes each prime $p \le y$ a factor of $n$ independently with probability $\gamma_p$. Two exact consequences: - $\mathbb{E}_\Theta[n] = \prod_{p\le y}\big(1+\gamma_p(p-1)\big) = \prod_{p\le y} p/\alpha_p = 1/\delta$, checked at $x = 7$ (PART 1). - Size-biasing by $n$ preserves independence and turns $\gamma_p$ into $\pi_p = \gamma_p \alpha_p = (p-\alpha_p)/(p-1)$, the number of excluded classes divided by $p-1$. Hence $$\boxed{\ \frac{\operatorname{Var}}{\mathbb{E}} \;\approx\; \mathbb{E}\Big[\Big\{\tfrac{L}{n}\Big\}\Big(1-\Big\{\tfrac{L}{n}\Big\}\Big)\frac{n}{L}\Big], \qquad \Pr[\,p \mid n\,] = \pi_p = \frac{p-\alpha_p}{p-1}\ \text{independently}. \ }$$ Fact B is now automatic: $n \mid L$ forces $\{L/n\} = 0$. The diagonal's arithmetic input is inside the formula rather than a correction to it. ## 4. The conductor groups, and where the growth lives Read off the three groups directly from $r_n(n-r_n)/L$: | group | value of the statistic | total after $\times\,\delta$ | |---|---|---| | $n \mid L$ (all prime factors $\le x$, since $L = x\#$) | exactly 0 | 0, exactly | | $n \le L$, $n \nmid L$ | $\le n^2/4L$ | $O(1/\ln W)$: moves the drift, not the limit | | $n > L$ | $r_n = L$, statistic $= n - L$ | the whole limit | The task's guess for the third row is right: for $n > L$ the statistic is exactly $n - L$, which after size-biasing is $(1 - L/n)$. The second row is the band near $L$; bounding it by $n^2/4L$ and summing $2^{\omega(n)}$ gives $O(\ln W)$ before the $\delta$, so it is a $1/\ln W$ correction and is **not** evaluated here. That is the reason the drift *coefficient* below is not derived even though the limit is. The growth rate follows without any further work. Because $\mathbb{E}_\Theta[n] = 1/\delta$ exactly and $\delta \ln^2 W \to \kappa = 1.109905$ (sibling §2, PROVEN), and because the fraction of that expectation sitting above $L$ is bounded away from 0 and 1, $X$ is $\asymp \ln^2 W$ with an order-one constant. So of the three shapes the task asks about, the answer is the first: $X = c\ln^2 W\,(1 + O(1/\ln W))$, never $\ln^2 W \ln\ln W$ and never $\ln^3 W$. The corrections enter through $\ln y$-sized offsets in $\ln n$ against $\ln L$, all of relative size $1/\ln y = 2/\ln W$, so **the drift family is a series in $1/\ln W$**, which is the principled reason the sibling's §8 row recorded as missing. ## 5. The limit, and the constant $\ln n = \sum_{p \le y}\mathbf 1_p \ln p$ with $\Pr = \pi_p$. Since $\pi_p = 2/(p-1)$ for $p \ge 7$, Mertens gives $\mathbb{E}[\ln n] = 2\ln y + O(1)$ and $\operatorname{Var}[\ln n] = \ln^2 y\,(1+o(1))$: the threshold $\ln L = u \ln y$ sits at the mean when $u = 2$, and the distribution does not concentrate. Its Laplace transform is $$\mathbb{E}\big[e^{-s\ln n/\ln y}\big] \;\longrightarrow\; \exp\Big(\theta\!\int_0^1 \frac{e^{-sw}-1}{w}\,dw\Big),\qquad \theta = |E_p| = 2,$$ the **generalized Dickman law** $GD(\theta)$, with density $f_\theta$ satisfying $f_\theta(t) = e^{-\theta\gamma}t^{\theta-1}/\Gamma(\theta)$ on $(0,1]$ and $t f_\theta'(t) = (\theta-1)f_\theta(t) - \theta f_\theta(t-1)$ beyond. And $(1 - L/n)^{+} \to \mathbf 1[\ln n > \ln L]$. Hence $$\lim \frac{\operatorname{Var}}{\mathbb{E}} \;=\; \lambda_\theta(u) \;:=\; \Pr\big[GD(\theta) > u\big], \qquad u = \frac{\ln L}{\ln y},\ \ \theta = \text{excluded classes per prime}.$$ For $\theta = 2$, $f_2(t) = e^{-2\gamma}t$ on $(0,1]$ and $f_2(t) = e^{-2\gamma}\big[t(3-2\ln t) - 2\big]$ on $(1,2]$, so $$\lambda_2(2) \;=\; 1 - e^{-2\gamma}\Big(\tfrac12 + 4 - 4\ln 2\Big) \;=\; 1 - e^{-2\gamma}\Big(\tfrac92 - 4\ln 2\Big) \;=\; 0.45545648,$$ closed form and numeric delay-equation solution agreeing to $10^{-8}$ (PART 3), with $\int f_2 = 1$ and mean $= 2$ to eight decimals as they must be. Equivalently $\lim X/\ln^2 W = 0.45545648/1.109905 = 0.410$. The tail at the other exponents, which §6 of the note can test: | $u$ | 0.6 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | |---|---|---|---|---|---|---| | $\lambda_2(u)$ | 0.9433 | 0.8424 | 0.6571 | 0.4555 | 0.2820 | 0.1577 | **The passage above is the second open step.** Both arrows are asserted rather than derived here: the convergence of $\ln n/\ln y$ to $GD(2)$ is routine and is nowhere written down in this note, and $(1 - L/n)^{+} \to \mathbf 1[\ln n > \ln L]$ needs §4's $n \le L$ band to vanish, which §4 states as $O(1/\ln W)$ and does not evaluate. The constant is no stronger than that step (`redteam-0828-varE.md` §0). **Convergence, checked.** The model's leading part at $u = 2$ climbs 0.32116, 0.35378, 0.37271, 0.38660, 0.39357, 0.39838 over $\ln y = 6.9 \ldots 16.7$, and its own $a + b/\ln y$ fit returns 0.45285 against the closed form 0.455456 (PART 4b). So the model does converge where it is claimed to, and it does so slowly: at $x = 41$ it is still 0.053 short. ## 6. Against the data: 22 points, no fitted parameter The model has no free parameter. Its inputs are $\alpha_p$, $y$ and $L$. **6a. The nine diagonal points** (`variance-note.md` §7), model by Monte Carlo, $N = 4\times10^5$, s.e. $\approx 7\times10^{-4}$: | $x$ | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |---|---|---|---|---|---|---|---|---|---| | measured | 0.1521 | 0.2563 | 0.2995 | 0.3268 | 0.3473 | 0.3643 | 0.3774 | 0.3876 | 0.3958 | | model | 0.17850 | 0.26744 | 0.29750 | 0.32579 | 0.34637 | 0.36214 | 0.37695 | 0.38740 | 0.39566 | | residual | $-0.0264$ | $-0.0111$ | $+0.0020$ | $+0.0010$ | $+0.0009$ | $+0.0022$ | $+0.0005$ | $+0.0002$ | $-0.0006$ | Lead with the bad part: at $x = 7$ and $x = 11$ the model is wrong by 17% and 4% relative, exactly the decoupling error of §2, and at $x \ge 13$ the seven residuals average $+0.0009$, roughly 3 sigma of the pooled MC noise. The model is not exact anywhere. It is within 0.0022 everywhere from $x = 13$ up. **6b. §6's $u$-sweep at $y = 401$**, a different object (set $A$, all three mod-30 houses, $\alpha_5 = 3$) and a different question. Model against the note: | $u$ | 0.6 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | |---|---|---|---|---|---|---| | note §6 | 0.845 | 0.685 | 0.477 | 0.290 | 0.157 | 0.076 | | model | 0.8543 | 0.7025 | 0.4842 | 0.2927 | 0.1576 | 0.0764 | The note's empirical regularity 1, $\ln(\operatorname{Var}/\mathbb{E}) \approx -(0.24u^2+0.13u)$, is a two-parameter fit to a curve the model returns with none, and the model's normalization $\lambda_2(0) = 1$ is the note's own observed $\to 1$ as $u \to 0$. **6c. §6's $u = 2$ level sweep**, model against the note: | $y$ | 97 | 199 | 401 | 797 | 1009 | 1499 | 2003 | |---|---|---|---|---|---|---|---| | note §6 | 0.251 | 0.281 | 0.290 | 0.303 | 0.307 | 0.317 | 0.321 | | model | 0.2541 | 0.2810 | 0.2944 | 0.3061 | 0.3095 | 0.3196 | 0.3232 | ## 7. What this does to the 0.611 reading `variance-note.md` §7 concludes that the limit is near 0.611 because, on the frozen-coefficient forecast at $x = 37$, $a + b/\ln\ln W$ beat $a + b/\ln W$ by a factor ten, and because it refits with a stable intercept. The sibling's §5 already showed the fit sets are misdescribed and that two more forms fit as well with intercepts 0.5366 and 0.7198, and recorded that no principled reason to prefer one intercept was known. Here is one, and it is a control rather than an argument. The nine model values of §6a form a sequence whose limit is $0.455456$ **by construction**, and which tracks the measured data to 0.003 over the range where the data live. Running the note's own protocol on it (fits on $x = 13..37$), with the model column rebuilt exactly at $x = 13, 17, 19$ and by Monte Carlo at $N = 2\times10^{7}$ elsewhere (`redteam-0828-varE.md` §4; this note's own PART 6b column is Monte Carlo at $N = 4\times10^{5}$ and reads 0.6151 and 0.4463, inside its own noise): | form | intercept on the model | bias vs 0.455456 | intercept on the measured | measured minus bias | $\|c\|_1$ | gap vs its bound | |---|---|---|---|---|---|---| | $a + b/\ln W$ | 0.4468 | $-0.0086$ | 0.4454 | 0.4540 | 2.60 | $0.0014 \le 0.0076$ | | $a + b/\ln\ln W$ | 0.6164 | $+0.1609$ | 0.6108 | 0.4499 | 6.91 | $0.0056 \le 0.0201$ | | $a + b/\sqrt{\ln W}$ | 0.5402 | $+0.0848$ | 0.5366 | 0.4518 | 4.96 | $0.0036 \le 0.0144$ | | $a + b/\ln W^{1/3}$ | 0.6337 | $+0.1783$ | 0.6279 | 0.4496 | 7.43 | $0.0059 \le 0.0216$ | | $a + b\cdot$(inert Mertens share) | 0.7281 | $+0.2727$ | 0.7198 | 0.4471 | 9.83 | $0.0083 \le 0.0286$ | | $a + b/(\ln W\ln\ln W)$ | 0.4219 | $-0.0336$ | 0.4211 | 0.4546 | 2.03 | $0.0008 \le 0.0059$ | | $a + b/\ln\ln^2 W$ | 0.4833 | $+0.0278$ | 0.4809 | 0.4531 | 3.48 | $0.0023 \le 0.0101$ | **The frozen half, run on the same control.** `variance-note.md` §7 leans hardest on the other protocol: fit on the early points, freeze the coefficients, forecast $x = 37$. The control above does not run it; `redteam-0828-varE.md` §4 does, fitting on $x = 13..31$: | form | model forecast at $x = 37$ | model residual | data forecast | data residual | |---|---|---|---|---| | $a + b/\ln W$ | 0.39281 | $+0.00276$ | 0.39259 | $+0.00321$ | | $a + b/\ln\ln W$ | 0.39596 | $-0.00039$ | 0.39567 | $+0.00013$ | The model's own value at $x = 37$ is $0.395567 \pm 0.000106$, so the $-0.00039$ is resolvable. On a sequence whose limit is 0.455456 by construction the frozen $1/\ln\ln W$ forecast beats $1/\ln W$ by a factor 7, in the same direction and at the same ordering as the data's factor 25. So §7's out-of-sample argument carries no information about the limit either, and the refutation covers both halves of the protocol rather than one. Three readings, in order of how much they are worth. 1. **The winner is the same and the intercept is wrong.** On the control, $1/\ln\ln W$ wins in-sample (rms $8.99\mathrm{e}{-4}$ against $2.26\mathrm{e}{-3}$) and its intercept misses the known limit by $+0.161$. So on data of this shape and this range, "the $1/\ln\ln W$ form fits best" carries no information about the limit. This is MEASURED, and it refutes the §7 inference as a method, not the number 0.611 as a fact. 2. **Every form's bias transfers.** Each form's intercept on the model sits within 0.009 of its intercept on the measured data, across a range of intercepts from 0.42 to 0.73. That is the sense in which the control is the right control. 3. **Bias-corrected the seven forms agree, and the agreement is mostly forced.** Removing each form's control bias from its measured intercept gives 0.4540, 0.4499, 0.4518, 0.4496, 0.4471, 0.4546, 0.4531: a spread of 0.0075 around $\lambda_2(2) = 0.455456$, against an uncorrected spread of 0.3062. That spread is not evidence for the constant. The intercept is a linear functional $a = \sum_i c_i y_i$ with $\sum_i c_i = 1$, so the corrected column is $0.455456 + (a_{\rm data} - a_{\rm model})$ and its width is bounded a priori by $\|c\|_1 \cdot \max_i |{\rm data} - {\rm model}|$. With $\max|{\rm data} - {\rm model}| = 0.00291$ and $\|c\|_1$ between 2.03 and 9.83 that bound runs 0.0059 to 0.0286, and every observed gap sits inside it (`redteam-0828-varE.md` §4). So the last column restates that the model tracks the data pointwise, and it is neither an independent measurement of the constant nor a check that could have failed. ## 8. Item 9, and the registered $x = 41$ forecast **What item 9 gets.** A candidate limit with a mechanism and a closed form, $\lambda_2(2) = 1 - e^{-2\gamma}(9/2 - 4\ln 2) = 0.45546$, at HEURISTIC rung; a derived reason the drift family is $1/\ln W$ and not $1/\ln\ln W$; an explanation of the sibling's refuted $\ln^3 W$ as the mean-versus-maximum of one local coefficient; and a control that explains where 0.611 came from. What it does not get is a proof: §2's replacement is open and so is §5's limit theorem for the model, and item 9 needs both closed. **$x = 41$.** The model predicts $0.402364 \pm 0.000075$ (MC, $N = 2\times10^{7}$, two seeds, `redteam-0828-varE.md` §4). `var41-prereg.md` registers $[0.4013, 0.4040]$; the model is inside it and near its middle. So the tenth point does not test the model either, and the sibling's finding stands unchanged: $x = 41$ retires the one-term $1/\ln W$ form (0.39941) and separates nothing else. The surviving forecasts 0.40296, 0.40309, 0.40408, 0.40500 are now joined by a fifth, 0.40236, which is the only one carrying a derivation rather than a fit. ## 9. Prior art: this is Gorodetsky's convention at two classes Run the same construction with **one** excluded class per prime and it gives $\lambda_1(u) = e^{-\gamma}\int_u^\infty \rho$, numerically 0.663124, 0.438541, 0.218559, 0.093970, 0.035636, 0.012095 at $u = 0.6 \ldots 3$ (PART 3). That is Gorodetsky's $\lambda(u)$ for the variance of $y$-rough integers in short intervals (*Math. Z.* **308** (2024) no. 4, Paper 59, arXiv:2111.00853, Thms 1.1/1.3), and $\mathbb{E}[\{L/n\}(1-\{L/n\})\,n/L]$ is his $\sum_n g_y(n)\{H/2n\}(1-\{H/2n\})$ in the corpus's variables. Two consequences, one of them a caution. - The $\theta = 1$ case of §5 is **published and proven, and the identification is an exact identity rather than a numerical match.** Every $n$ is even, since $\pi_2 = 1$; writing $n = 2n'$ turns the decoupled sum into his (1.5) and (1.6) term by term, checked at $(y,H) = (13,210), (13,2310), (29,30030)$ to relative $1.7\times10^{-16}$. His Lemma 1.5 gives $V_{q_y}(H) = M(H,y)$ exactly for a primorial window, which is this corpus's window, with no hypotheses, and his Theorem 1.3(1) then gives $M \sim H P_y \lambda(u)$ for $y \ge \exp((\log\log H)^{5/3+\varepsilon})$, which the diagonal satisfies (`lit-dickman-variance.md` §1, `redteam-0828-varE.md` §3). So at $\theta = 1$ the whole chain from the exact finite-level variance to $\lambda_1(u)$ is a published theorem about exactly this object. Its value to $\theta = 2$ is limited and should be stated as such: the reason §2's step is invisible at one class is that the local coefficient is constant there, $\hat f_p(\nu \ne 0) = 1/(p-1)^2 = \gamma_p$ identically, and that says nothing about $\theta = 2$, where the step is the gap. - **This route is not novel.** The owning convention is "the variance of integers without small prime factors in short intervals" with coordinate $u = \log H/\log y$, exactly as `research/SEARCH-CONVENTIONS.md` records it via `import-rough-anatomy.md` §8. What is not in print, as far as this pass knows, is the two-class value $\lambda_2$, and this pass did not search for it. The Gorodetsky paper has since been read at source (`lit-dickman-variance.md` §1, `redteam-0828-varE.md` §7): it is $\kappa = 1$ only, with zero occurrences of `tuple`, `twin`, `admissible`, `generalized`, `dimension` and `Poisson` in its full text. Whether a successor states the $k$-tuple version is still **NOT CHECKED**, with four channels owed, and it should be checked before anything is claimed as new. Aryan (arXiv:1302.2296) is the nearest neighbour found so far: his Lemma 1.2 is this object at general tuple size, with an upper bound and no asymptotic (`lit-dickman-variance.md` §2). ## 10. Defects noticed in passing - `paper/variance-note.md` §7's closing sentence ("the live hypothesis is $\lim = 0.611$ ... the 0.44 reading is demoted") rests on a model comparison that returns 0.6164 on a control whose limit is 0.455456, and on a frozen forecast separation that the same control reproduces while missing its own limit. The §7 sentence should be reversed, not softened. Its replacement sentence is drafted in `applied-0828-varE.md` §5; the live document is unedited. - `paper/variance-note.md` abstract ("reduces the open question to the value of a single constant, near 0.611") carries the same claim into the abstract. - The sibling's `varE-asymptotic.md` §8 row "the intercept is not localized to 0.611 ... a principled reason to exclude ... NO. No such reason is known" now has a reason on both halves of the protocol, and that note carries it. - Any spectral computation in this corpus that indexes $\hat W$ by $\nu \bmod p$ rather than $\nu (M/p)^{-1} \bmod p$ is wrong by a permutation and will still pass a total-mass check. `natal-cap-29-sigma-plateau.js` STEP 2 was **not** audited for this here. ## 11. What would falsify this, and whether the check has run | claim | rung | falsifier | has the check run | |---|---|---|---| | $X = \sum_{\nu\ne0}\hat W(\nu)K_L(\nu/M)$ with the CRT twist | PROVEN | disagreement with brute force | YES, $x = 7$, $2.9\mathrm{e}{-7}$ | | $\delta X_{\rm dec} = \mathbb{E}[\{L/n\}(1-\{L/n\})n/L]$, $\pi_p = (p-\alpha_p)/(p-1)$ | PROVEN given decoupling | a level where the divisor sum disagrees with the MC | YES, $x = 7$ and 11, exact enumeration vs MC inside 1 s.e. | | the decoupling step is asymptotically harmless | CONJECTURED | a level where true/model stays away from 1 | PARTLY. Exact only at $x = 7$ (0.8534) and $x = 11$ (0.9605); at $x \ge 13$ only indirectly, against the measured data, at $\pm 0.002$ | | $\lim \operatorname{Var}/\mathbb{E} = \lambda_2(u)$, $\lambda_2(2) = 0.45546$ | HEURISTIC, two open steps (§2's replacement, §5's limit theorem) | the model failing at a new level, or a proof that either step shifts the limit | NO. No level beyond $x = 37$ exists, and the model is 0.053 short of its own limit at $x = 41$ | | $\lambda_1 = $ Gorodetsky's $\lambda$ | PROVEN: an exact identity with his (1.5)/(1.6), and Lemma 1.5 makes the primorial window his own case | a $(y,H)$ where the two sums differ | YES, three pairs to relative $1.7\mathrm{e}{-16}$, and the paper is read at source (`redteam-0828-varE.md` §3) | | the drift family is a series in $1/\ln W$ | HEURISTIC | a derived $\ln\ln$ term in the model's expansion | NO. The $n \le L$ band that carries the $1/\ln W$ coefficient is not evaluated | | §7's fit-comparison does not estimate the limit, on either half of its protocol | MEASURED (control) | a control matching the data as well and reproducing its own limit in-sample AND on the frozen forecast | YES; in-sample PART 6b, the control matches to 0.003 and misses by $+0.161$; frozen forecast `redteam-0828-varE.md` §4, residual $-0.00039$ on a sequence whose limit is known | | the model's $x = 41$ value 0.402364 | MEASURED (forward, MC) | a measured $r(41)$ outside $0.4024 \pm$ a few $10^{-3}$ | NO. Var(41) priced and declined, TODO item 2 | | this is novel | NOT CLAIMED | Gorodetsky or a successor stating the $k$-class $\lambda_k$ | NO. Not searched. Do not call it new before that search | The one line worth carrying out of this pass: the corpus should stop quoting 0.611 and start quoting $\lambda_2(2) = 0.45546$ as a **heuristic** candidate with two open steps, together with the control that shows why 0.611 appeared, on both halves of the protocol that produced it, and why no amount of further levels of the present kind will settle it.