# Do the actual Mobius signs cancel in the small-common-divisor kernel? **Twin-prime infinitude remains OPEN and nothing here moves it.** This note records one finite measurement. Its calibration is MEASURED. A ratio measured on seven dyadic scales can refute a hope (if the actual coefficient shows no advantage at all) or lend heuristic weight to one; it cannot show a power saving, cannot establish (21) of [grouped-divisor-moment.md](grouped-divisor-moment.md), and cannot move the sufficient margin of [RESEARCH-HANDOFF.md §3](RESEARCH-HANDOFF.md). ## 0. Pre-registered falsifier and controls, written before the run Fixed before executing [kernel-sign-control.js](kernel-sign-control.js). **Falsifier.** Let the ratio be rho_k(x) = |X_small(actual)| / |X_small(random draw k)|, k = 1..8. For each draw k separately, fit the least-squares slope of log2 rho_k against log2 x over the dyadic points that share one value of J0 = floor(x^(1/20)). Report the mean slope over the eight draws and its sample standard deviation. *If mean slope + sd >= 0, the measurement gives no heuristic support for Mobius-sign cancellation in the small-j kernel along this route, and this note says so first.* If mean slope + sd < 0, the note may say the measurement is **consistent with** extra cancellation over the measured range; it may never say the measurement implies, shows or establishes it, and it must state the finite range and the fact that a decreasing finite ratio is compatible with no asymptotic saving. The same statistic is reported for the whole moment Mfrak and for the absolute-value control (control 1), which has no sign structure at all and is therefore the upper reference, not a null. **Negative controls, all required to fire.** 1. *Complete periods are detected.* In a configuration where the class of nonzero pairs with c dividing theta\*R is nonempty, dropping those pairs must change the kernel value. In the production boxes the class is proved empty by an exact inequality, and that is reported as a measured share of zero, not as an absence of the class from the derivation. 2. *Conjugation is load-bearing.* Replacing the conjugate of the second factor by the factor itself must destroy the reality of X_small: the unconjugated ordered-pair sum must have a nonzero imaginary part. 3. *Both endpoints are shifted under reciprocity.* The left-orientation kernel built with the reciprocal shift must agree with the native kernel, and the variant built without the shift must disagree. 4. *All-ones coefficients reproduce the zero-frequency size.* With b_u = 1 the equal-frequency class of (5) must dominate the moment, its ordered-pair count must respect the counting bound 8\*N\*A\*(1+log(2\*min(N,A))), and its mass must respect the majorant B²\*max|c_h|²\*max|F|\*|I|\*(pair count). **Identity checks.** The class decomposition is computed by an exact divisor sieve, and it is checked against literal ordered-pair enumeration at small sizes before any production number is read. ## 1. What is measured, exactly The box is (delta,nu) = (8/25,9/20) of [grouped-divisor-moment.md §6](grouped-divisor-moment.md) and [RESEARCH-HANDOFF.md §5](RESEARCH-HANDOFF.md). For x = 2^j: * left expanded interval I = (M,2M] with M = floor(x^(14/25)); every integer m in I is summed, so |I| = M; * right original interval J = (E,2E] with E = floor(x^(9/20)) and cutoff W = Z = max(2,floor(x^(1/20))); * **top** expanded box ell in (E\*Z, 2E\*Z] (so ell is of order x^(1/2)) and **low** expanded box ell in (E,2E], with u = ell/g in (N,2N]; * g in {1,2}, theta = 2/g, sigma = -1, native endpoints z0 = x/2 and z = x; * harmonic band H = [A,2A) with A = round(x^(3/50)) (the top band) and A = 1; * c_h = -W(h/(T+1))/(2 pi i h) with Vaaler's W of [endpoint-fourier.md (7)](endpoint-fourier.md) and T = 4\*round(x^(3/50)). The coefficient is the actual one, b_u = A_right(g u), built from (13) at s = 0 with that J and W. Two structural facts about it at these finite sizes shape everything below. * On the **top** box no ell lies in J, so A_0 vanishes identically there and A_1(ell) reduces to the prime-power sector alone. Since floor(x^(1/20)) = 2 for every x in the feasible range, that sector has the single prime power r = 2, and A_1(ell) = -mu(ell/2)\*log 2 for even ell, zero for odd ell. The top-box coefficient is therefore exactly a Mobius sign times a constant, supported on the even u. This is the sharpest possible form of the sign question and simultaneously a finite-size degeneracy: the r-sector of (13) is not exercised beyond r = 2. * On the **low** box A_0(ell) = mu(ell) and A_1(ell) = -mu(ell)\*log ell on the whole box, with no prime-power term. This is the pure Mobius sector. The three coefficient choices share one support: actual b_u; |b_u| (control 1); and |b_u| times an independent seeded sign, eight draws (control 2). A fourth, b_u = 1 on the whole box, is the all-ones control (iv). **Classes.** For an ordered pair put j = (u1,u2), u_i = j\*ell_i, c = lcm(u1,u2) = j\*ell_1\*ell_2 and R = h1\*ell_2 - h2\*ell_1. The four classes are R = 0 (equal rational frequencies h1/u1 = h2/u2), nonzero R with c | theta\*R (complete periods), nonzero R with j > J0, and X_small of (21), the nonzero R with j <= J0, where J0 = max(1,floor(x^(1/20))). X_small is computed by an exact divisor sieve, not by pair enumeration: with f = 1_{n<=J0} \* mu in the Dirichlet sense, sum_{d|n} f(d) = 1_{n<=J0}, so the whole j <= J0 class equals sum_m sum_d f(d)\*|G_d(m)|², where G_d(m) restricts the u-sum to multiples of d. The script verifies that identity numerically for every n <= 2N in every configuration, and checks the resulting class values against literal ordered-pair enumeration at small sizes. Two classes are proved empty on every production configuration rather than assumed away: * a nonzero R with c | theta\*R needs theta\*|h1\*u2 - h2\*u1| >= u1\*u2, and theta\*max(h)\*2N < (N+1)² holds throughout, so the complete-period class is empty and its measured share is exactly 0; * an equal-frequency pair with h/u = p/q in lowest terms has j = q\*gcd(t1,t2) >= q, and min q over the atoms exceeds J0 throughout, so no equal-frequency pair has small j. Hence the sieve's j <= J0 class is exactly X_small of (21). ## 2. Result **Falsifier verdict, negative in all four families.** The pre-registered slope of log2(|X_small(actual)|/|X_small(draw k)|) against log2 x, fitted over the six scales j = 20,22,24,26,28,30 that share J0 = 2, has mean and sample standard deviation over the eight draws * -0.0570 +- 0.1741 (top box, g=2, A~x^(3/50)), * 0.2088 +- 0.2072 (top box, g=2, A=1), * -0.1443 +- 0.2237 (top box, g=1, A~x^(3/50)), * -0.0813 +- 0.3024 (low box, g=1, A~x^(3/50)). Mean slope + sd >= 0 in every case. By the rule written in section 0, **the measurement gives no heuristic support for Mobius-sign cancellation in the small-j kernel along this route.** It does not refute such cancellation; it fails to detect any, in a regime described below where it would be hard to see. The whole moment shows no advantage either. Mfrak(actual) divided by the mean of the eight random-sign moments is 1.027, 1.038, 1.020, 1.021, 1.019, 1.014 across j = 20..30 on the top box at g=2, with slope -0.0024 +- 0.0035 — the actual coefficient's moment is if anything a per-cent or two **larger** than a random-sign one. That is expected: the equal-frequency class carries between 97.66% and 102.58% of Mfrak in every row, and on the diagonal the sign of b_u cannot matter. The descriptive aggregate in TABLE 4 was computed after the pre-registered fit and is not the pre-registered test. Over the 27 configurations with a nonempty small-j class, |X_small(actual)| has mean rank 5.15 among itself and its eight draws, against the null expectation 5.00 with null standard deviation 0.50 for that mean; 12 of 27 configurations put the actual value below its draw median, against the null expectation 13.5. Both distribution-free statistics sit at the null: in this kernel and at these sizes the actual Mobius signs behave like one more independent sign draw. Two ratios in TABLE 4 need reading carefully rather than at face value. The geometric mean of |X_small(actual)|/mean|X_small(random)| is 0.604, which is **not** evidence of an advantage: the numerator is one value and the denominator an average of eight, so by Jensen's inequality that ratio is below one under the null as well. The geometric mean of |X_small(actual)|/|X_small(|b|)| is 0.316, comparing two single values, and that one does say something — against the all-positive control, which has no sign structure at all, the signed coefficient's kernel is smaller by roughly a factor of three. Signs help against no signs. Mobius signs do not help against random signs, which is the question asked here. Growth rates. d log2 Mfrak / d log2 x is 0.982, 1.001, 0.979 and 0.990 in the four families, against budget slopes of 1.388, 1.325, 1.426 and 1.495, and the generic N³ exponent 1.500. The measured moment therefore tracks the diagonal term and sits far below its Weil budget: at j=30 the top box at g=2 measures 2.6265e+6 against a budget of 2.5676e+12. d log2 |X_small| / d log2 x is 0.684, 0.929, 0.455 and 0.585, all well under the 36/25 = 1.440 that (21) asks for — but that is the growth of the actual finite value in a regime where the moment grows like x^1 rather than x^1.5, and it is not an estimate of the asymptotic exponent. All four negative controls fired, and the sieve identity agrees with literal ordered-pair enumeration on all five class quantities in four configurations. The endpoint matters: at j=22, g=2, A=2 the native z = x gives Mfrak = 1.1513e+4 and X_small = -8.7460e+1, while z = 3x/4 gives Mfrak = 3.5937e+3 and X_small = -5.1390e+1. Numerical headroom is adequate: the divisor sieve cancels by between 137.1x and 26201.6x, and the worst-case double-rounding floor relative to |X_small| is 2.48e-7. The output block below is copied mechanically out of the binding written by `node research/qc/embed.js research/kernel-sign-control.js`; the deterministic artifact is [kernel-sign-control.json](kernel-sign-control.json). ``` KERNEL SIGN CONTROL — actual A_right(gu) against |A_right| and seeded random signs box delta=8/25 nu=9/20: M=floor(x^(14/25)), I=(M,2M]; E=floor(x^(9/20)), Z=max(2,floor(x^(1/20))) top expanded box ell~E*Z (~x^(1/2)), low expanded box ell~E; u=ell/g; native z0=x/2, z=x; sigma=-1 c_h = -W(h/(T+1))/(2 pi i h) with Vaaler W and T=4*Atop; J0=max(1,floor(x^(1/20))); draws=8 dyadic scales j: 18,20,22,24,26,28,30 control (iii) reciprocity: 6552/6552 shifted-endpoint kernels equal the native kernel; 6547 unshifted ones differ identity check — divisor sieve against literal ordered-pair expansion: x=4096 g=1 A=2 J0=2 N=48 M=24: 900 ordered pairs (R=0: 32, j<=J0: 656, periods: 0); 5 quantities agree x=4096 g=2 A=2 J0=2 N=24 M=24: 900 ordered pairs (R=0: 32, j<=J0: 768, periods: 0); 5 quantities agree x=8192 g=1 A=1 J0=1 N=64 M=24: 361 ordered pairs (R=0: 19, j<=J0: 0, periods: 0); 5 quantities agree x=8192 g=2 A=3 J0=2 N=32 M=24: 3249 ordered pairs (R=0: 59, j<=J0: 2808, periods: 0); 5 quantities agree control (i) complete periods: detector N=6 A=8 M=20 has 62 period pairs of 2304; dropping them moves the nonzero j<=J0 kernel from -1.4484e-2 by -8.7191e-3 control (ii) conjugation: at j=18 g=2 A=2 the conjugated X_small = 1.2476e+1 is real; the unconjugated ordered-pair sum is 1.0960e+0 + 9.8785e+0i control (iv) all-ones coefficients, equal-frequency domination (top box, g=2, A=Atop): j=18 N=274 M=1082: Mfrak(ones)=2.0698e+3, R=0 share 108.30% (actual A1 on the same box: 97.66%); 638 equal-frequency ordered pairs vs the (5) count bound 1.046e+4; class majorant 1.3215e+4 j=20 N=512 M=2352: Mfrak(ones)=8.4549e+3, R=0 share 105.53% (actual A1 on the same box: 100.72%); 1194 equal-frequency ordered pairs vs the (5) count bound 1.955e+4; class majorant 5.3760e+4 endpoint sensitivity at j=22 g=2 A=2: z=x gives Mfrak=1.1513e+4 X_small=-8.7460e+1; z=3x/4 gives Mfrak=3.5937e+3 X_small=-5.1390e+1 TABLE 1 — the moment Mfrak. n per row: |I|=M values of m, nz_u divisors u with A_right(gu)!=0, |H|=A harmonics j box g A J0 N M nz_u atoms Mfrak(A1) Mfrak(|A1|) Mfrak(random) mean +- sd Mfrak(A0) budget 18 top 2 2 1 274 1082 166 332 7.3248e+2 6.8433e+2 6.9160e+2 +- 3.0227e+1 0 2.7434e+7 18 top 2 1 1 274 1082 166 166 6.4526e+2 6.3136e+2 6.3327e+2 +- 2.4117e+1 0 1.4672e+7 18 top 1 2 1 548 1082 166 332 4.5876e+2 4.0989e+2 4.1148e+2 +- 2.6359e+1 0 1.1659e+8 18 low 1 2 1 274 1082 166 332 9.3168e+4 9.5638e+4 9.0466e+4 +- 2.1651e+3 2.5817e+3 2.2695e+9 20 top 2 2 2 512 2352 310 620 2.8491e+3 2.8632e+3 2.7749e+3 +- 8.8428e+1 0 1.7708e+8 20 top 2 1 2 512 2352 310 310 2.4951e+3 2.5581e+3 2.5526e+3 +- 3.6176e+1 0 9.1907e+7 20 top 1 2 2 1024 2352 310 620 1.7516e+3 1.6635e+3 1.6226e+3 +- 4.1720e+1 0 7.3217e+8 20 low 1 2 2 512 2352 310 620 4.7756e+5 4.5954e+5 4.5061e+5 +- 7.9240e+3 1.0883e+4 1.7703e+10 22 top 2 2 2 955 5113 580 1160 1.1513e+4 1.1286e+4 1.1094e+4 +- 1.5851e+2 0 1.1386e+9 22 top 2 1 2 955 5113 580 580 1.0201e+4 9.7556e+3 1.0217e+4 +- 1.7055e+2 0 5.7571e+8 22 top 1 2 2 1910 5113 580 1160 6.9926e+3 6.6882e+3 6.5697e+3 +- 1.8351e+2 0 4.5936e+9 22 low 1 2 2 955 5113 580 1160 2.2462e+6 2.3044e+6 2.2094e+6 +- 2.7588e+4 4.2780e+4 1.3526e+11 24 top 2 3 2 1782 11113 1086 3258 4.0971e+4 4.1335e+4 4.0167e+4 +- 2.8406e+2 0 9.6319e+9 24 top 2 1 2 1782 11113 1086 1086 4.2035e+4 4.1403e+4 4.1928e+4 +- 2.6285e+2 0 3.6114e+9 24 top 1 3 2 3564 11113 1086 3258 2.5025e+4 2.3886e+4 2.3995e+4 +- 3.9785e+2 0 4.9396e+10 24 low 1 3 2 1782 11113 1086 3258 4.7417e+6 4.9622e+6 4.7228e+6 +- 3.6470e+4 7.5531e+4 1.3409e+12 26 top 2 3 2 3326 24154 2019 6057 1.6409e+5 1.6684e+5 1.6067e+5 +- 1.2699e+3 0 6.1991e+10 26 top 2 1 2 3326 24154 2019 2019 1.6576e+5 1.6211e+5 1.6486e+5 +- 6.1213e+2 0 2.2667e+10 26 top 1 3 2 6652 24154 2019 6057 9.8117e+4 9.7491e+4 9.7055e+4 +- 5.2774e+2 0 3.1865e+11 26 low 1 3 2 3326 24154 2019 6057 2.2656e+7 2.3198e+7 2.2418e+7 +- 9.9463e+4 3.1060e+5 9.9968e+12 28 top 2 3 2 6208 52498 3772 11316 6.5649e+5 6.6411e+5 6.4439e+5 +- 2.5795e+3 0 3.9904e+11 28 top 2 1 2 6208 52498 3772 3772 6.4971e+5 6.5000e+5 6.5260e+5 +- 4.8264e+3 0 1.4232e+11 28 top 1 3 2 12416 52498 3772 11316 3.9324e+5 3.9245e+5 3.8809e+5 +- 2.2935e+3 0 2.0558e+12 28 low 1 3 2 6208 52498 3772 11316 1.0817e+8 1.0971e+8 1.0632e+8 +- 5.5324e+5 1.2878e+6 7.3804e+13 30 top 2 3 2 11585 114104 7045 21135 2.6265e+6 2.6425e+6 2.5895e+6 +- 9.2785e+3 0 2.5676e+12 30 top 2 1 2 11585 114104 7045 7045 2.5865e+6 2.5943e+6 2.5860e+6 +- 4.9483e+3 0 8.9366e+11 30 top 1 3 2 23170 114104 7045 21135 1.5830e+6 1.5714e+6 1.5512e+6 +- 3.9659e+3 0 1.3258e+13 30 low 1 3 2 11585 114104 7045 21135 5.0683e+8 5.1466e+8 4.9943e+8 +- 9.6734e+5 5.2910e+6 5.3984e+14 TABLE 2 — X_small of (21) and the four ordered-pair classes as shares of Mfrak (actual coefficient A1) j box g A J0 X_small(A1) X_small(|A1|) mean|X_small(rnd)| rms(rnd) rank |Xs|/Mfrak R=0 share periods j>J0 share sieve cancel 18 top 2 2 1 1.2476e+1 -2.7383e+1 1.7663e+1 2.2758e+1 5/9 1.70% 97.66% 0 (empty) 0.64% 223.3x 18 top 2 1 1 5.9706e+0 -2.0867e+1 1.7869e+1 2.2632e+1 2/9 0.93% 99.62% 0 (empty) -0.54% 419.4x 18 top 1 2 1 CLASS EMPTY (0 ordered pairs with j<=J0) 0.00% 96.08% 0 (empty) 3.92% - 18 low 1 2 1 -2.6841e+3 -2.1502e+3 1.7930e+3 2.0180e+3 7/9 2.88% 101.35% 0 (empty) 1.53% 137.1x 20 top 2 2 2 -5.6170e+1 -3.1269e+0 4.9625e+1 6.8690e+1 7/9 1.97% 100.72% 0 (empty) 1.25% 176.3x 20 top 2 1 2 -6.2734e+1 1.2603e+1 3.2602e+1 3.8338e+1 8/9 2.51% 102.58% 0 (empty) -0.07% 140.1x 20 top 1 2 2 -3.5223e+1 -9.7056e+1 3.8619e+1 4.5630e+1 5/9 2.01% 100.77% 0 (empty) 1.24% 222.3x 20 low 1 2 2 8.9228e+3 -3.2127e+3 7.2101e+3 9.2315e+3 6/9 1.87% 98.10% 0 (empty) 0.04% 183.3x 22 top 2 2 2 -8.7460e+1 -2.5726e+2 1.0290e+2 1.1652e+2 5/9 0.76% 99.99% 0 (empty) 0.77% 482.6x 22 top 2 1 2 4.1716e+0 -3.3853e+2 1.2118e+2 1.3856e+2 1/9 0.04% 100.20% 0 (empty) -0.24% 9005.0x 22 top 1 2 2 -1.4487e+2 -3.2459e+2 1.6337e+2 2.0516e+2 6/9 2.07% 101.12% 0 (empty) 0.96% 229.1x 22 low 1 2 2 -2.9883e+4 2.2984e+4 1.7080e+4 2.1191e+4 8/9 1.33% 101.42% 0 (empty) -0.09% 278.8x 24 top 2 3 2 2.1394e+2 6.1840e+2 2.2236e+2 2.4632e+2 6/9 0.52% 99.49% 0 (empty) -0.02% 736.4x 24 top 2 1 2 -7.0618e+0 -3.7162e+2 2.3043e+2 2.5233e+2 1/9 0.02% 99.57% 0 (empty) 0.44% 22937.6x 24 top 1 3 2 4.9533e+2 -7.3278e+2 2.3957e+2 2.8023e+2 8/9 1.98% 97.82% 0 (empty) 0.20% 252.7x 24 low 1 3 2 -1.7659e+4 1.9170e+5 1.9369e+4 2.3995e+4 6/9 0.37% 100.44% 0 (empty) -0.06% 1048.6x 26 top 2 3 2 6.9576e+2 3.8908e+3 8.5581e+2 1.0362e+3 5/9 0.42% 99.63% 0 (empty) -0.05% 952.5x 26 top 2 1 2 1.1776e+3 -3.1025e+3 6.0521e+2 7.4798e+2 8/9 0.71% 99.75% 0 (empty) -0.46% 569.6x 26 top 1 3 2 1.9849e+1 -6.1772e+2 6.2540e+2 7.1588e+2 1/9 0.02% 99.90% 0 (empty) 0.08% 26201.6x 26 low 1 3 2 1.2276e+4 6.0113e+5 1.0617e+5 1.4022e+5 2/9 0.05% 100.34% 0 (empty) -0.40% 7549.5x 28 top 2 3 2 5.1793e+2 4.7708e+3 1.8142e+3 2.1933e+3 3/9 0.08% 99.96% 0 (empty) -0.04% 5383.8x 28 top 2 1 2 -4.0222e+3 -2.8246e+3 3.3869e+3 4.3404e+3 7/9 0.62% 100.52% 0 (empty) 0.10% 690.7x 28 top 1 3 2 -6.3190e+2 -3.6992e+3 1.4288e+3 1.7076e+3 4/9 0.16% 100.13% 0 (empty) 0.03% 3476.8x 28 low 1 3 2 6.1727e+5 1.7666e+6 3.9718e+5 4.5860e+5 7/9 0.57% 99.52% 0 (empty) -0.09% 747.0x 30 top 2 3 2 -1.1644e+4 5.1029e+3 7.5491e+3 8.5113e+3 8/9 0.44% 100.35% 0 (empty) 0.09% 1005.5x 30 top 2 1 2 3.0114e+3 1.3199e+4 4.6759e+3 5.2863e+3 3/9 0.12% 99.86% 0 (empty) 0.03% 3819.9x 30 top 1 3 2 2.2917e+3 -1.1430e+4 3.3113e+3 4.3940e+3 5/9 0.14% 99.88% 0 (empty) -0.02% 4028.0x 30 low 1 3 2 4.5448e+5 9.2797e+6 7.5063e+5 1.1253e+6 5/9 0.09% 99.71% 0 (empty) 0.20% 4986.1x rank = position of |X_small(actual)| among itself and the 8 seeded draws, smallest first; 1/9 would mean the actual coefficient is the most cancelled of the nine worst-case double-rounding floor over these rows, relative to |X_small|: 2.48e-7 TABLE 3 — exponent fits and the pre-registered falsifier (primary fit: the points sharing one J0) top box, g=2, band A~x^(3/50); fit over j=20,22,24,26,28,30 (n=6 scales, J0=2) d log2 Mfrak / d log2 x: actual 0.982 |b| 0.984 random 0.984 +- 0.004 budget 1.388 generic N^3 exponent 1.500 d log2 |X_small| / d log2 x: actual 0.684 random 0.741 +- 0.174 ((21) needs an exponent below 36/25 = 1.440) FALSIFIER, slope of log2(actual/random) per draw: Mfrak -0.0024 +- 0.0035 X_small -0.0570 +- 0.1741 ratio actual/random Mfrak: 1.027 1.038 1.020 1.021 1.019 1.014 ratio actual/random |X_small|: 1.132 0.850 0.962 0.813 0.285 1.542 ratio actual/|b| Mfrak: 0.995 1.020 0.991 0.984 0.989 0.994 |X_small|: 17.963 0.340 0.346 0.179 0.109 2.282 VERDICT: mean slope + sd >= 0 — NO heuristic support for Mobius-sign cancellation in the small-j kernel here top box, g=2, band A=1; fit over j=20,22,24,26,28,30 (n=6 scales, J0=2) d log2 Mfrak / d log2 x: actual 1.001 |b| 1.001 random 0.998 +- 0.002 budget 1.325 generic N^3 exponent 1.500 d log2 |X_small| / d log2 x: actual 0.929 random 0.720 +- 0.207 ((21) needs an exponent below 36/25 = 1.440) FALSIFIER, slope of log2(actual/random) per draw: Mfrak 0.0022 +- 0.0021 X_small 0.2088 +- 0.2072 ratio actual/random Mfrak: 0.977 0.998 1.003 1.005 0.996 1.000 ratio actual/random |X_small|: 1.924 0.034 0.031 1.946 1.188 0.644 ratio actual/|b| Mfrak: 0.975 1.046 1.015 1.023 1.000 0.997 |X_small|: 4.978 0.012 0.019 0.380 1.424 0.228 VERDICT: mean slope + sd >= 0 — NO heuristic support for Mobius-sign cancellation in the small-j kernel here top box, g=1, band A~x^(3/50); fit over j=20,22,24,26,28,30 (n=6 scales, J0=2) d log2 Mfrak / d log2 x: actual 0.979 |b| 0.987 random 0.988 +- 0.004 budget 1.426 generic N^3 exponent 1.500 d log2 |X_small| / d log2 x: actual 0.455 random 0.599 +- 0.224 ((21) needs an exponent below 36/25 = 1.440) FALSIFIER, slope of log2(actual/random) per draw: Mfrak -0.0095 +- 0.0038 X_small -0.1443 +- 0.2237 ratio actual/random Mfrak: 1.080 1.064 1.043 1.011 1.013 1.021 ratio actual/random |X_small|: 0.912 0.887 2.068 0.032 0.442 0.692 ratio actual/|b| Mfrak: 1.053 1.046 1.048 1.006 1.002 1.007 |X_small|: 0.363 0.446 0.676 0.032 0.171 0.200 VERDICT: mean slope + sd >= 0 — NO heuristic support for Mobius-sign cancellation in the small-j kernel here low box, g=1, band A~x^(3/50); fit over j=20,22,24,26,28,30 (n=6 scales, J0=2) d log2 Mfrak / d log2 x: actual 0.990 |b| 0.994 random 0.994 +- 0.002 budget 1.495 generic N^3 exponent 1.500 d log2 |X_small| / d log2 x: actual 0.585 random 0.666 +- 0.302 ((21) needs an exponent below 36/25 = 1.440) FALSIFIER, slope of log2(actual/random) per draw: Mfrak -0.0043 +- 0.0019 X_small -0.0813 +- 0.3024 ratio actual/random Mfrak: 1.060 1.017 1.004 1.011 1.017 1.015 ratio actual/random |X_small|: 1.238 1.750 0.912 0.116 1.554 0.605 ratio actual/|b| Mfrak: 1.039 0.975 0.956 0.977 0.986 0.985 |X_small|: 2.777 1.300 0.092 0.020 0.349 0.049 VERDICT: mean slope + sd >= 0 — NO heuristic support for Mobius-sign cancellation in the small-j kernel here TABLE 4 — descriptive aggregate over every configuration with a nonempty small-j class (post-hoc summary added after the pre-registered fit in TABLE 3; it is not the pre-registered test) configurations n = 27 (7 dyadic scales x 4 boxes, minus the empty-class row at j=18) mean rank of |X_small(actual)| among the nine values: 5.15 null expectation 5.00, null sd of this mean 0.50 configurations with |X_small(actual)| below the median of its 8 draws: 12 of 27 (null expectation 13.5) geometric mean of |X_small(actual)| / mean|X_small(random)|: 0.604 geometric mean of |X_small(actual)| / |X_small(|b|)|: 0.316 geometric mean of Mfrak(actual) / mean Mfrak(random): 1.0213; 3 of 27 configurations below artifact: kernel-sign-control.json (7 dyadic scales, 28 configurations, 8 seeded draws each) MEASURED ONLY: no power saving, no asymptotic rate and no twin margin follows; (21) and the global margin remain OPEN ``` ## 3. Limitations Lead with these; they are larger than the result. 1. **The reachable regime is not the regime (21) is about.** The moment grows like x^0.98 to x^1.00 here and is 97.66% to 102.58% equal-frequency class. X_small carries 0.02% to 2.88% of it. The Weil term N³ that sets the 3/2 budget is nowhere near dominant at these sizes, so a sign effect inside it would be measured against a diagonal a hundred times larger. 2. **J0 = floor(x^(1/20)) is 2 across the whole feasible range** (and 1 at j=18, where the g=1 top box's small-j class is empty because that box's support is entirely even). "Small common divisor" is literally "gcd 1 or 2" here, not a band. 3. **Z = floor(x^(1/20)) is also 2**, so the prime-power sector of (13) has one prime power. The top-box coefficient collapses to -mu(ell/2)\*log 2 and the low-box coefficient to -mu(ell)\*log ell. Repeated primes and multiple r values, which (13) admits and the derivation must handle, are not exercised. Reaching floor(x^(1/20)) >= 3 needs x >= 2^60, far outside this engine. 4. **One box, one orientation, one twist.** (delta,nu) = (8/25,9/20) only; native orientation and sigma = -1 only; s = 0 only, so no divisor twist; two harmonic bands rather than the full logarithmic stack; T = 4\*round (x^(3/50)) rather than the derivation's ceil(x^(2tau)max(1,MN/x)). The one endpoint sensitivity run moved Mfrak from 1.1513e+4 to 3.5937e+3. 5. **Eight draws.** In almost every row the spread of |X_small| across draws is comparable to |X_small| itself, so a single row carries little power; the aggregate rank is the strongest available reading and it sits at the null. 6. **Finite scaling never proves an asymptotic power saving**, and it never disproves one. Six or seven dyadic points spanning a factor of 64 in x cannot separate x^1.44 from x^1.50. ## 4. What would change the reading * A derivation. The only thing that settles (21) is a matched estimate with its coefficient interface, norms and error budget, as [RESEARCH-HANDOFF.md §6](RESEARCH-HANDOFF.md) specifies. This measurement cannot supply one and does not narrow the space of possible ones. * A configuration where the nonzero off-diagonal actually dominates the finite moment. That needs the harmonic band A comparable to N, which the actual box does not give at reachable x; a synthetic box with A ~ N would measure a different object and would have to say so. * More draws (say 64) rather than more x-scales: the spread across draws, not the number of scales, is what limits the current statistic. * A finding that the sieve identity or an emptiness certificate is wrong. Both are checked here — the identity against literal enumeration, the certificates by exact inequality — but a defect in either would invalidate every class share above. ## 5. Validation [kernel-sign-control.js](kernel-sign-control.js) is deterministic and seeded (mulberry32, base 0x5EED0000, seed a fixed function of draw index, scale, g, band and box), writes [kernel-sign-control.json](kernel-sign-control.json), and carries an embedded output block bound to its own code hash. It checks the Dirichlet identity sum_{d|n} f(d) = 1_{n<=J0} for every n <= 2N in every configuration, agrees with literal ordered-pair enumeration on the moment, the pair expansion, its reality, the R=0 class, the j<=J0 class and the j>J0 class in four small configurations, and asserts the two emptiness certificates and the (5) counting and majorant bounds in every production configuration. The negative controls are asserted, not merely printed: an inert control aborts the run. These checks certify their finite scope only. The embedded tail carries a `forced` stamp dated 2026-09-06. The reason is recorded here rather than left to guesswork: the first binding of this script, made minutes earlier in the same session and never committed, predated TABLE 4. Adding that descriptive aggregate changed the output, the embed guard refused to replace a bound tail whose output had changed, and the re-bind used the tool's documented override. The tool's own comparison reports 0 of 546 figures in the replaced block not reproduced: no measured figure changed, only lines were added.