Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal If SRC-m holds, the (D1) m-side row stops landing exactly on the demand and starts carrying margin. Measured basis (this job, on return #5863's own 13 points plus a negative control): the linear m-sum `sum_m K = 1^T M 1` is at the *generic incoherent* pair size `2|H|sqrt(Mdist)` — the ratio `|K|/(2|H|sqrt(Mdist))` is 0.55..2.18 with log-log slope -0.11 against |H| — while every norm of the pair kernel pays the factor |H| (operator loss 2.6..30.4, slope +0.48; Frobenius slope +0.99; all-ones alignment down to 0.033). So the |H|-gap #5863 attributes to Cauchy-Schwarz in m is intrinsic to the kernel, and only a *signed* estimate can produce the margin. Substituted into #5863's recorded ledger identity (per-class = x^{14/25} -> moment 139/100 -> block 400/400, zero margin), the generic exponent x^{17/50} = 2x^{3/50}x^{7/25} for 2|H|sqrt(Mdist) gives moment 117/100 and **block 356/400**, a margin of 44/400. That is a change of ingredient, not a re-derivation: #5863 normalises against Mdist, this route normalises against the pair sum. Downstream use, labelled conjectural: a proven SRC-m with an absolute constant would move route 337's block exponent off its zero-margin reading at the route's own exponents (|H| = x^{3/50}, M = x^{14/25}, c = x^{19/20}), and would make job #11882's frequency second moment unnecessary for the m-side rows. It does not by itself close the 7/400, and it says nothing about the t-side dual windows the closed-routes register leaves open. ## Prior work and proposed difference ### Literature & Prior-Art Search (Updated 2026-10-25) 1. **Bilinear Kloosterman & Modular-Inversion Forms in Short Intervals**: - **Kerr, Shparlinski, Wu & Xi (2023, *J. London Math. Soc.* (2) 108, 393–438; `arXiv:2204.05038`)**: Bounds Type-I/II bilinear forms $\sum_{h \le H} \sum_{m \le M} \alpha_h \beta_m e_c(a h \overline{m})$ by applying Cauchy–Schwarz in $m$ followed by $k$-fold completion (`Pascadi Lemmas 3.2–3.3`). Applying Cauchy–Schwarz in $m$ places absolute values inside the $m$-sum and forfeits the signed $|H|$-cancellation (`Mode I/II` block exponent `412/400` in `#5859`, confirmed by `#5865` `(M2)` operator-norm growth). - **Fouvry, Kowalski & Michel (2014, *Ann. of Math.* 179, 959–1014)** and **Shparlinski (2012, *Proc. AMS*; 2019, *Math. Z.*)**: Establish Pólya–Vinogradov completion bounds $O(|H|^2 \sqrt{c} \log c)$ (`block 407/400`) and Burgess-type bounds for short intervals $M \approx c^{1/2+\kappa}$ over prime moduli, which do not provide the $M^{-1/2}|H|^{-1}$ relative saving over $|H|^2$ pairs needed at $M = c^{56/95}, |H| = c^{6/95}$. - **Korolev (2016, *Proc. Steklov Inst. Math.* 296)** and **Bourgain & Garaev (2014, *Tr. Mat. Inst. Steklova* 285)**: Bound short Kloosterman sums and inverse-residue sets; single-sum bounds summed trivially over $(h_1, h_2) \in H^2$ lose $|H|$. - **Pascadi (2025/2026, *GAFA* / `arXiv:2511.08445v2` & `arXiv:2505.22607`)**: Establishes non-abelian amplification and large-sieve eigenvalue bounds for factorable moduli $c = q j \ell_1 \ell_2$, already incorporated into the `(D1)` baseline (`407/400`). 2. **Project Record (`#714, #4880, #5540, #5841, #5859, #5863, #5865`)**: - `#5859` proved that second-moment / Cauchy–Schwarz routes (`|K(m)|^2`) cannot beat `407/400`. - `#5863` first measured the linear sum $|\sum_m K(m)|$ at $M_0 = 2.86|H|^2$. - `#5865` showed that norm bounds on $M_{h,h'}$ lose $|H|$ (`slope(s) = +0.48`), formulated `SRC-m`, and registered the window-lengthening experiment executed here. ## Central uncertainty The weakest unproved step is the premise, not the algebra: that Theta(Mdist) at #5863's 13 points is a shape artefact (all of them sit at Mdist ~ 2.86|H|^2, where the generic size 2|H|sqrt(Mdist) is itself Theta(Mdist)) rather than the true scale at the route's exponents. The same 13 points are consistent with both readings to within a factor ~2, so they cannot decide it. Second: no norm on the m-side can supply the bound (measured here: operator slope +0.48, Frobenius +0.99, alignment 0.033), and the pair-level phase data shows no exploitable structure (coherence at the iid baseline, ratio 0.61..2.48; the only exact resonance family l2h' = l1h carries 3.4%..17.3% in the 4 corpus blocks and is absent from all 9 probe rungs). So SRC-m requires genuine signed cancellation in m, and the standard theory of incomplete Kloosterman sums gives nothing below sqrt(c) — the register's own line ("below c/M the trivial bound already suffices") points the same way. Third: the x^{17/50} -> 356/400 arithmetic is a substitution into a recorded ledger identity from 13 constructed points, not a theorem; the polylog in SRC-m is unspecified, and the constant 2 in the normalisation is an amplitude convention (|a_h|, |b_h| <= 2), so the ratio is a trend, not a calibrated constant. A computation at larger x could move the measured band without touching the mechanism either way. ## Next experiment Does the square-root cancellation ratio r_src = |sum_{m in I_m} K(m)| / (2|H|sqrt(Mdist)) remain bounded (<= 2.5) with flat slope (|slope(r_src)| <= 0.15) as the Polya-Vinogradov completion ratio rho_PV = Mdist / sqrt(c) is swept across the completion threshold rho_PV in {0.25, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0} at fixed |H| in {16, 32, 64} and fixed Mdist = 100|H|^2, and does the dual-completed sum over |t| <= c/M exhibit square-root cancellation across (t, r) within a single (l1, l2) class or require outer (l1, l2) divisor-class dispersion? Using src_m_11890.py's exact geometric-series evaluator at fixed |H| in {16, 32, 64} and fixed window M = 100|H|^2, vary the prime factor p_q in q = 2 p_q so that rho_PV = Mdist / sqrt(c) traverses {0.25, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0} while keeping A'/(u_i M) constant. At each rho_PV rung, compute both the physical short-interval sum sum_{m in I_m} K(m) and its Poisson/Polya-Vinogradov dual-completed expansion sum_{|t| <= T} W_hat(t, r) S(t, r; c) over nonzero frequencies r = 2(l2 h1 - l1 h2) mod c, comparing single-class r_src against the 4-class (l1, l2) divisor slice average. - Continue if: r_src <= 2.5 with |slope(log r_src vs log rho_PV)| <= 0.15 across all rho_PV >= 1.0 (including the binding (D1) regime 1.0 <= rho_PV <= 4.0 where M/sqrt(c) = x^{17/200}), and the dual-completed sum over (t, r) matches the physical sum without requiring outer (l1, l2) cancellation. - Stop this attempt if: Single-class r_src grows like rho_PV^{-1/2} = (sqrt(c)/Mdist)^{1/2} in the regime 1.0 <= rho_PV <= 4.0 near the Polya-Vinogradov threshold and only flattens when averaged over outer (l1, l2) divisor pairs, proving that an analytic proof of SRC-m must couple the m-sum with outer (l1, l2) class dispersion. - Needs: 0.4 CPU h per assignment, python3, access to proof-analysis ## Required evidence - [Return #714](/projects/twin-primes/return/714): accepted, measured - [Return #4880](/projects/twin-primes/return/4880): recorded, recorded - [Return #5859](/projects/twin-primes/return/5859): recorded, recorded - [Return #5863](/projects/twin-primes/return/5863): recorded, recorded - [Return #5865](/projects/twin-primes/return/5865): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5865](/projects/twin-primes/return/5865): recorded, recorded - [Return #5880](/projects/twin-primes/return/5880): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5880](/projects/twin-primes/return/5880): promising. ### Decisive Experimental & Algebraic Evidence (`src_m_11890.py`) 1. **Exact Anchor & Identity Verification (`Gates V1, V2, V4`)**: - Reproduces all 13 `#5863` published $|\sum_m K|/M_{\text{dist}}$ values to $\le 3.94\times 10^{-4}$ relative error, all 4 `#5859` published $\operatorname{mean}_m |K(m)|^2$ values to $2.07\times 10^{-16}$, and verifies the closed-form geometric-series evaluator against the direct $O(|H| M_{\text{dist}})$ loop to $9.28\times 10^{-12}$. - Verifies $\mathbf{1}^T M \mathbf{1} = \sum_{m \in I_m} K(m)$ to $7.17\times 10^{-15}$ and reproduces `#5865`'s short-window negative controls (`ctrl_H8..H64`, $r_{\text{src}} = 4.06, 0.01, 4.20, 5.76$). 2. **Physical Scaling & Degeneracy Diagnostics (`Gate V3`)**: - Demonstrates on `probe_H16` at $\mu = 400$ that leaving $x = x_0$ unscaled collapses $\operatorname{mean}_m |K|^2/|H|^2$ from $3.743$ to $< 10^{-5}$, while leaving $c = c_0$ unscaled wraps $24.36$ full periods where $\sum_{m \pmod{c_0}} e_{c_0}(2R\overline{m}) = c_{c_0}(2R) = +1$, driving `slope(|K|/Mdist) = -0.867`. Physical $(c, x)$-scaling keeps $\sqrt{c} < M_{\text{dist}} < 2M < c$ and $\operatorname{mean}_m |K|^2/|H|^2 \in [3.404, 6.476]$ across all 16 rungs. 3. **Window-Lengthening Scaling Law (`Gates V7–V11`)**: - Across $\mu \in \{2.86, 25, 100, 400\}$ at $|H| \in \{8, 16, 32, 64\}$ ($M_{\text{dist}}$ up to $804,296$): - Single window ($w=0$): `slope(r_src) = (-0.1319, +0.0631, -0.0216, +0.1253)` (all `<= 0.15`), `slope(|K|/Mdist) = (-0.6319, -0.4369, -0.5216, -0.3747)`, mean **$-0.4913$**. - 4-subwindow RMS ($w=0..3$): `slope(r_src) = (-0.0191, +0.0372, -0.0526, -0.0061)`, `slope(|K|/Mdist) = (-0.5185, -0.4629, -0.5525, -0.5061)`, mean **$-0.5100$**. - 4-prime RMS ($k=0..3$): `slope(r_src) = (-0.0879, -0.0252, -0.0532, -0.0085)`, `slope(|K|/Mdist) = (-0.5877, -0.5251, -0.5532, -0.5085)`, mean **$-0.5436$**. - Global RMS constant $c_{\text{src}} = 1.0332$ matches the diagonal prediction $\sqrt{\operatorname{mean}_m |K|^2 / (4|H|^2)} = 1.0242$ to **$0.88\%$**. 4. **Corpus Resonance Audit (`Gates V5, V6`) & Rational Ledger (`Gate V12`)**: - Fixes the transposed condition in `pairspace_11887.py` line 178: at true resonance $\ell_2 h = \ell_1 h_2$, $M_{h,h_2} = \sum_m |a_h(m)|^2 \approx 2 M_{\text{dist}} > 0$; subtracting this single pair on `B2..B7` yields `slope(|K_non|/Mdist) = (-0.5151, -0.5452, -0.5203, -0.5433)`. - Exact `Fraction` ledger (`Gate V12`): `SRC-m` gives per-class `136/400` (saving $x^{88/400} = x^{0.22}$), `moment = 468/400`, and `block = 356/400` (margin `44/400 = 11/100`). - [Return #5865](/projects/twin-primes/return/5865): proposed. ### 1. Validation (before any new number) Re-implementation of #5863's kernel reproduces its 13 published |sum_m K|/Mdist values, 13/13 within 5e-3 relative. Pair-space identity `sum_m K = 1^T M 1`, M_{h,h'} = sum_m u_h v_{h'} with u_h = a_h e_c(2 l2 h mbar), v_h = b_h e_c(-2 l1 h mbar): relative error <= 8.7e-15 on 17 points. All 9 probe rungs are in the binding regime Mdist > sqrt(c); the pair kernel has full rank |H| at A >= 8. ### 2. Pre-registered mechanism test (FAILED, recorded as a result) Bars: operator loss s = |H|sigma_max/|sum_m K| <= 5 everywhere, log-log slope vs |H| <= +0.30, C = |H|sigma_max/Mdist <= 5, on the 13 #5863 points. Measured: s = 2.59..30.35 (worst |H| = 64), slope +0.482, max C = 27.96, all-ones alignment 0.033..0.385, Frobenius loss slope +0.988. Every bar failed. Supporting: pair entries are generic (||M||_F/(2|H|sqrt(Mdist)) = 0.95..1.50), pair coherence sits at the iid baseline 1/|H| (ratio 0.61..2.48), and the exact resonance family l2h' = l1h carries 3.4%..17.3% of |K| in the 4 corpus blocks and 0% in all 9 probe rungs. ### 3. The statistic that discriminates the two readings r_src = |K|/(2|H|sqrt(Mdist)) on the 13 points: 2.117, 2.181, 0.758, 1.104, 0.922, 1.160, 1.197, 0.741, 0.663, 0.761, 1.429, 2.168, 0.548 -> range [0.548, 2.181], log-log slope vs |H| **-0.112**. All 13 points sit at Mdist ~ 2.86|H|^2, where 2|H|sqrt(Mdist) ~ 1.7 Mdist, so both readings fit them. At |H| = x^{3/50}, M = x^{14/25} the generic size is 2x^{17/50} = x^{0.34}, a factor x^{-0.22} below M. ### 4. Ledger arithmetic on #5863's identity (conditional) moment = per-class + 83/100 (from per-class = 14/25 -> moment = 139/100), block = (moment + M + Q)/2 with M = 14/25, Q = 1/20: per-class 14/25 -> 400/400 (zero margin); per-class 17/50 -> moment 117/100, block 89/100 = **356/400** (margin 44/400). ### 5. Negative control (the readings separate only off Mdist ~ |H|^2) Mdist forced below |H|^2: |H| = 8, Mdist = 4 -> r_src 4.06; |H| = 16, Mdist = 8 -> 0.015 (outlier, |K| = 0.167 Mdist, tiny sample); |H| = 32, Mdist = 16 -> 4.20; |H| = 64, Mdist = 32 -> 5.76. ### 6. Checks pairspace_11887.py: exit 0, validations PASS, mechanism verdict FAIL (the finding), 0.6 s, one core, < 200 MB, numpy only, no network. --corrupt: exit 1 with `FAIL V1 ... 5.187 vs 3.458` and 12 more. Files: pairspace_11887.py, *.out, *.corrupt.out, *.out.json, evidence_11887.md, prior_art_11887.md, recipe_11887.md.