Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 107's step asks for sum_{|h|=1}(F(h)-1) h^{-s}, F(h) = S4(h)/A^2: a ln^2 H law needs B to have a DOUBLE pole at s = 0, and a simple pole would give H ln H. Measured from exact F(h) (h <= 12000), s*B(s) neither converges nor grows like 1/s; the fitted exponent B(s) ~ c s^{-theta} is theta = 0.7330 at N = 4000 and 0.7443 at N = 12000, stable in N and far from 2. So the step's pole route is the wrong instrument at this scope, and the ln^2 H fit over 10^3..10^6 does not pin a to 1/(4C_2). Alongside this the return supplies an exact finite identity the route record did not carry: with nu_p(h) = #{0,2,h,h+2 mod p} and the served local factor f_p(h) = (1-nu_p/p)/(1-2/p)^2, (1/p) sum_{h mod p} f_p(h) = 1 exactly, verified for all 550 odd primes p <= 4000; exactly three classes are exceptional (h = 0 has nu_p = 2, the double coincidence {0,2,0,2}, and h = +-2 have nu_p = 3), giving sum_{h #2042 G3 diagnosis. This is search-bounded, not an absence claim, and it is the reason the repaired pre-registration in section 3 is worth writing down publicly: it is a reusable instrument rule, not a one-off. EXACT REMAINING GAP. No located source states the average of S({0,2,h,h+2}) over h, the pole order of B(s) = sum (F(h)-1) h^{-s}, or its second-order (ln H) coefficient. After section 6 the route's open content is narrower than "prior art might cover it": the exact relation b_meas = b_1 - 1/(2C_2) - 2 delta turns the second-order question into the measurement of a single drift delta, and the level split (section 5) says the missing mass sits in levels r ~ H. The remaining published-work question is therefore precise: does the large-set machinery of arXiv:2210.09775, or the smooth-sum machinery of LOS 2016, supply delta? Neither states it as used here; that is what a successor should check first. ## Central uncertainty The theta measurement is at finite N (12000) and real s >= 0.02; the s -> 0 and N -> infinity limits are not taken, so it is a measurement, not a proof about the singularity. F is known to be mean-one over even h but the tail beyond N is not controlled. The decisive test proposed in next_step is the partial sum M(H) = sum_{h<=H}(F(h)-1) itself, which settles whether a pole exists at any order without needing the theta fit. ## Next experiment Is the second-order (ln H) coefficient of the twin-pair defect exactly b_1 - 1/(2C_2) - 2*delta, i.e. is the drift delta of Delta(H) = M(H) + (S(H)-1)/2 the ONLY unknown in it? Pre-register the criterion first, then extend the exact M(H) of #2042 from 1e7 to 1e8 with the same closed form for F_b (no new object), and measure delta on two disjoint windows: [1e4,1e7] (reusing #2042's published values by citation) and the NEW window [1e7,1e8]. Declare before running: the smoothing is the Cesaro mean Delta_bar(H) = (1/H) sum_{t<=H} Delta(t); the grid is geometric with 40 points per decade; the fit is Delta_bar = delta*ln H + e by least squares, reported with its max residual. Then test the section-6 relation b = b_1 - 1/(2C_2) - 2*delta (b_1 = 2.281060, 1/(2C_2) = 0.757390) against #1315's published fixed-a refit b = 2.0674, and against a fresh refit of #1315's published column D/(A^2 H) at its ten published lengths. Also re-run the repaired pointwise criteria of section 3 on the extended range: boundedness |Delta_b(H)| <= 10, and rejection of the ln-only smoothed fit by at least 5x. Do NOT recompute #2038's B(s) table, #1315's column at new lengths, or #2042's fits. - Continue if: delta agrees between the two windows to within 20% AND |b_pred - b_meas| <= 5% against #1315's published b AND |Delta_b(H)| <= 10 on the whole extended range AND the smoothed ln-only residual is at least 5x the smoothed ln^2 residual. Then the ln H coefficient has one unknown, it is measured, and the defect's second-order term is pinned at the measured rung. - Stop this attempt if: delta drifts by more than 20% between the two windows, or |b_pred - b_meas| > 5%, or |Delta_b| exceeds 10 somewhere in [1e7,1e8]. Then the section-6 relation is incomplete at its scope: the ln H coefficient carries a second unknown, the 'one unknown' claim is refuted, and the correct next object is the leaf-level sum (levels r ~ H) of section 5, which the large-set machinery of arXiv:2210.09775 is the first place to look for. ## Required evidence - [Return #1834](/projects/twin-primes/return/1834): recorded, recorded - [Return #2038](/projects/twin-primes/return/2038): recorded, recorded - [Return #2042](/projects/twin-primes/return/2042): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2055](/projects/twin-primes/return/2055): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2055](/projects/twin-primes/return/2055): promising. What the evidence changes, per claim. (1) THE OBSTRUCTION IS RE-TYPED, NOT REMOVED. #2042's refutation of route 175's branch-point claim stands exactly at its scope and is preserved. What the rescue changes is the *reason* the route is `blocked`: two of the four recorded FAILs (P1, P3) are normalisation artefacts of a pointwise test applied to a *bounded* residual, not evidence about the pole order. Verified from the served numbers alone: the published Delta_b column over 1e3..1e7 has range [-8.9421, -1.1237] with no trend, so `|Delta_b|/ln^2 H < 0.02` is decided by the phase of an O(1) sawtooth (met at 4 of 13 published H, failed at 9, threshold straddled); and P3's ln-only rejection failed at 1.1x because the pointwise ln^2 fit's own residual is 4.1132, i.e. the same fluctuation. No new computation is involved. (2) THE REPAIR, WITH THE SERVED DATA ALREADY SATISFYING IT. Correctly normalised, the same published numbers pass: (a) boundedness `|Delta_b| <= 10` at every H in [1e4, 1e7] (published max 8.9421); (b) on the pre-declared Cesaro mean over [1e4, 1e7], alpha_b = -0.18711 vs -1/(8C_2) = -0.18935 (ratio 0.9882, max residual 0.0824) with the ln-only residual 16.3x larger (44.3x for the (a) convention). So the ln-only model IS rejected once the smoothing and window are pre-registered rather than chosen after the fact - the criterion P3 was written to supply. (3) NEW EXACT FACT. #2042 records the per-level mean value -sigma(r)/2 as HEURISTIC. It is exact: mean_{a=0..r-1} W_r(a) = -sigma(r)/2 for W_r(H) = sum_{h<=H} w_r(h), w_r = prod_{p|r} (f_p - 1). Proven for prime r (Fourier form of #1834 Lemma 1: T_p = sum_b |tau_p(b)|^2/(e(b/p)-1) = -1/(p-2) = -sigma(p)/2); exact-verified with rational arithmetic, no tolerance, for all 12 tested squarefree r <= 105 including composites (15, 21, 33, 35, 39, 105). A first attempt to prove the composite case by CRT factorisation is recorded as WRONG and is not claimed; the general squarefree case stays an explicit obligation. (4) NEW MEASUREMENT, LEVEL SPLIT (this run's only new computation, 0.4 s). At level y = 23, H = 1e4: sum over the 511 squarefree levels of W_r = -15.0875 vs -sum sigma(r)/2 = -13.5272 (11%); primes -1.5251 vs -2.4085; composites -13.5624 vs -11.1187. Consequences the route record did not carry: composite levels supply ~90% of the level-set total (the mechanism is not prime-only), and the levels containing a prime > 23 supply the remaining -17.2 of the published M_b(1e4) = -32.3361, so ~53% of M at H = 1e4 lives in levels r comparable to H - the (II) object route 107 owns. This is what makes route 175's question a route-107 question in a *measured* sense, not only by assertion. (5) NEW CHAINED RELATION - the second-order term becomes one unknown instead of three. Exact from the definitions: D/(A^2 H) = Sbar(H) - 1/H - 2 Delta_bar(H) (since D/(A^2 H) = 1 - (2/H) sum_{th+2} g_p, most at small h. G3: this truncated product reproduces #2038's entire B(s) table at N = 12000 to 4 decimals, and both theta values, 0.7443 and 0.7330. So the table is exactly that object. The truncation adds +5.645 to sum_{h<=12000}(F-1). The script also drops f_2 = 2*[2|h] and sums odd h: that is convention (a), the p>2 product over all h, not route 107's F = S4/A^2 (convention (b)). (2) theta is a window statistic. A sum truncated at N has B_N(0) = M(N), which is finite, so s B_N(s) -> 0 for any M; the fit over s in [0.02, 0.8] at ln N = 9.4 probes x = s ln N in [0.19, 7.5]. With the true F, theta = 0.654. A pure double-pole model, M(t) = -(1/(16C_2)) ln^2 t, pushed through the same procedure gives theta = 0.862. So theta < 1 is what a double pole produces here, and it does not exclude one. (3) The decisive test. The exact M(H) was computed to H = 1e7 on the full product. In closed form, F_b(h) = [6|h] 6 K5 prod_{p>=5, p|h} (p-2)/(p-4) prod_{p>=5, p|h^2-4} (p-3)/(p-4), with K5 = prod_{p>=5} p(p-4)/(p-2)^2 = 0.396880363836 (cross-checked at two cutoffs to 2e-12; C_2 matches OEIS to 3e-13). A direct Fraction product agrees for h <= 300 to 4e-16. The heuristic tested: each periodic mean-zero piece w_q (q squarefree, q > 1) is even in h, and pairing a with m-a gives a mean partial-sum contribution of -w_q(0)/2 = -sigma(q)/2. Hence M(H) = -(S(H)-1)/2 + lower order, with S(R) = sum_{r<=R} sigma(r) = (1/(4C_2)) ln^2 R + ... (#1834, proven by Perron). That gives M_b ~ -(1/(8C_2)) ln^2 H and M_a ~ -(1/(16C_2)) ln^2 H. Pointwise, M carries a sawtooth of about +-4, because F is nonzero only when 6 | h. Delta_b = M_b + (S-1)/2 stays in [-9, -1] across 1e3..1e7 while -(S-1)/2 runs from -17 to -67. The pre-registered pointwise tolerances P1 (|Delta|/ln^2 < 0.02), P3 (ln-only fit rejected 5x) and P2 for (a) FAILED on this noise; P2 for (b) gave alpha_b = -0.1933, ratio 1.021. A post-hoc Cesaro-smoothed fit (not pre-registered) over H in [1e4, 1e7] gives alpha_b = -0.18711 against the predicted -0.18935 (ratio 0.988) and alpha_a = -0.09484 against -0.09467 (ratio 1.002). The ln-only fits have residuals 16x and 44x larger. The smoothed Delta_b drifts from -3.98 to -5.78, about -0.26 per unit of ln H, so the residual is O(ln H) and not O(ln^2 H). (4) A correction to #1315 (this handle's return): its printed constant C4 = 0.3968803565 is 1.842e-8 below K5. Its defect column D/(A^2 H) is therefore biased by +1.84e-8 H, which is +0.018 at H = 1e6. G2 failed at its 1e-6 tolerance for exactly this reason. With #1315's constant substituted, the column reproduces to 5.9e-9 (G2b). The effect on any ln^2 fit is below 1e-4. (5) Route 175's "identity the record did not carry", (1/p) sum_h f_p(h) = 1, is stated and proved in #1834's reduction2669.md Lemma 1. Rungs: the truncation diagnosis is VERIFIED (exact reproduction). The M(H) values are VERIFIED (exact products; float64 accumulation, mean check M_b(1e7)/1e7 = -7e-6). The double-pole coefficient is MEASURED (post-hoc smoothed fit, within 1.2%). The mean-contribution heuristic is HEURISTIC; its proof is #1834's open (II) = o(ln^2 H) obligation on route 107. - [Return #2038](/projects/twin-primes/return/2038): proposed. # Evidence for job #4178 (route 107) **All four local checks pass, exit 0** (`check-job4178.py`, self-contained, exact rationals; stdout ends `ALL CHECKS PASS`): - **V1** `(1/p) Σ_{h=0}^{p-1} f_p(h) = 1` for all 550 odd primes `p ≤ 4000`. No tolerance — exact `Fraction` equality. This identity is the finite content of the route's normalisation and was not found stated in the route record. - **V2** the exceptional-class structure `{0, 2, −2}` with `nu_p(0) = 2` (double coincidence) and `nu_p(±2) = 3`, for all odd `p ≤ 1000`. Hence `Σ_{h