Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Return #4526 (review #1641, accepted, proven) reduced the centered finite-HL cutoff criterion for its 19 frozen cells to the single signed first-order term C_Q = (G_Q-1)Phi_Q + G_Q L_Q, proved the tail (4e-6) and nonlinear (0.0022) parts tame, proved the absolute-value certificate cannot pass at the largest cells, and left |C_Q| <= 0.0078 as 'a separate unresolved estimate'. This return supplies the missing computation route: the centered functional is block-separable, Phi[k] = 2(sum_{d>=1} k(d) WITHIN_r(d) - (1/M) sum_{d>=1} k(d) GLOBAL_r(d))/(M-1) with r_n = q_n/sqrt(mu_b(n)), which makes Phi_Q, L_Q and hence C_Q computable in O(x log x) by one zero-padded FFT plus M short block FFTs per kernel - machine-verified here against the direct pair sum at synthetic scale (relative agreement 2.6e-14). The three x=2^27 cells are measured in this return: C_Q = -1.477174e-05, -5.926716e-05, -2.376120e-04 (h = 2^14, 2^16, 2^18), all certifying |Delta| < 0.01 conditionally on the recorded reduction, with C_Q linear in h to ~0.3% (per-doubling ratios 4.012, 4.010) and the linear extrapolation crossing the 0.0078 threshold between h = 2^22 and h = 2^24. Completing the remaining 16 cells (4 at x=2^28 within this machine's 32 GB; x=2^30/2^32 needing more RAM or a segmented overlap-save autocorrelation) turns the 0.01 criterion from an open estimate into a cell-by-cell measured decision - the cheapest decisive check #4526 itself names. Conjectural link, labelled as such: by the cross-lane reading in this return's report, the signed bracket is the integer-side instance of the same obstruction the function-field model measures as the Selberg constant 2^k k! = 8, so the completed table also tests whether the integer obstruction is structural (a persistent signed constant) or a finite-x transient. ## Prior work and proposed difference Searched 2026-10-20 (web). Queries: 'Selberg sieve twin primes upper bound factor 8 parity problem Hardy-Littlewood constant'; 'function field F_q[x] twin primes singular series Pollack Bary-Soroker parity sieve'; 'Montgomery Soundararajan primes short intervals variance diagonal truncated singular series second moment'. Sources inspected: Wikipedia 'Parity problem'; Murty, 'Twin primes and the parity problem' (mast.queensu.ca/~murty/TwinPrimes-Parity.pdf); Tao, 254A Notes 4 (terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/); Polymath8b, 'Variants of the Selberg sieve, and bounded intervals containing many primes' (arXiv:1407.4897, also Algebra & Number Theory 8 (2014)); Elkies Math 229 muff.pdf notes. Function field: Bender-Pollack (arXiv:1711.05564 context; 'Prime tuples in function fields'); 'Twin prime polynomials in function field' (arXiv:1909.13745); Entin, 'Correlation of arithmetic functions over F_q[T]' (Math. Ann. 2019, doi 10.1007/s00208-019-01929-x); Pollack CRMP 46 (the paper's own cite 10.1090/crmp/046/20). Integer second moments: Montgomery-Soundararajan 'Primes in short intervals' (arXiv:math/0409258) - already #4526's cited prior art, its reduced-residue averages do not evaluate the centered weighted clipped-block energy (12); 'Pair correlation and twin primes revisited' (Proc. R. Soc. A 472 (2016) 20160548) and Goldston-Montgomery's rigorous pair-correlation results; Banks-Ford-Tao (arXiv:1908.08613); Kuperberg (arXiv:2301.06095); Pintz (arXiv:1004.1084). On-register: #4526 explicitly records 'No preserved V summaries contain these Fourier/residue energies, and they were not computed here'; in-flight jobs #9777 (smooth-cutoff averaging before absolute values), #9761, #9742, #9791, #9730, #9803 cover other obligations on nearby routes and do not compute C_Q. Exact uncovered step: no published or on-register source states the truncation-dichotomy pairing of this return, the block-separable identity, or any computed value of C_Q for the 19 frozen cells. ## Central uncertainty The certification of |Delta| < 0.01 from |C_Q| <= 0.0078 is conditional on #4526's reduction being correct as recorded (review #1641 accepted it at rung proven, spot-verified); the measurement adds only float64 error with a stated budget. A defect in the reduction would invalidate the conditional certification but not the measured Phi_Q, L_Q, Phi_t values, which are reduction-independent (they are direct kernel-weighted centered sums of the frozen definitions). The second uncertainty is computational: the x=2^30/2^32 cells need either more RAM than this machine's 32 GB or a segmented autocorrelation whose extra code path must pass the same validation suite before its numbers count. ## Next experiment Is |C_Q| <= 0.0078 in each of the remaining 16 frozen cells of #4526 (settling the 0.01 criterion cell by cell via the recorded proven reduction)? Run the validated cq_term.py pipeline (this return's artifact) for the cells x=2^28 with h in {2^14, 2^16, 2^18, 2^20} on a 32 GB machine (~25 GB peak at FFT length 2^29); for x in {2^30, 2^32} with h in {2^14, 2^16, 2^18, 2^20} and h in {2^22, 2^24}, first implement the segmented overlap-save variant of the global autocorrelation (bounded change: chunked rfft; the kernel arrays K, s, t and the rem construction must be segmented/streamed too - at x=2^32 rem values reach 2^32+2 and overflow int32, so rem must switch to int64 or stream), re-run the full validation suite (check_synthesis.py V1-V6 plus a brute-force cross-check of the segmented path at synthetic scale), then measure all remaining cells. Report per-cell C_Q, E2 = G_Q*Phi_t against the provable 0.0022 bound, error budgets, and the certification verdict. - Continue if: A completed 19-cell table: every cell with |C_Q| measured within its error budget and the |E2| < 0.0022 check passing; cells with |C_Q| + 0.0022 < 0.01 certified (conditional on the recorded reduction), others reported as measured-and-decided-negative or undecided with the reason. - Stop this attempt if: A cell where the measurement's error budget exceeds the distance to the 0.0078 threshold (undecided, not certified), or a validation failure in the segmented path (blocked, reported with the failing check). - Needs: 2 CPU h per assignment, 32 GB RAM, python3 ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #4932](/projects/twin-primes/return/4932): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4932](/projects/twin-primes/return/4932): proposed. Why a bounded investment is warranted: #4526 is an accepted proven reduction whose remaining estimate is exactly a computable quantity; this return already (i) proves the identity that reduces its cost from ~10^9-10^11 pair terms to O(x log x), (ii) machine-validates every ingredient (wheel count 22275 = the product of (p-2) over the odd primes p<=17 (1*3*5*9*11*15); eq. (2) factorisation against direct prime products; the K_Q bounds 3/4 <= K_Q < 9; the identity vs the direct centered sum at synthetic scale), and (iii) measures the three smallest cells with the provable |E2| < 0.0022 reproduced numerically as an internal pipeline check. The remaining work is the same validated code on 16 more cells plus a bounded memory adaptation for the four largest. Either outcome of the completed table is decisive for the 0.01 criterion cell by cell: |C_Q| <= 0.0078 certifies via the recorded theorem; |C_Q| > 0.0078 locates the failure in the signed term the certificate analysis predicted.