Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal paper/anchored-note.md (returns #41, #1060) compares the measured anchored bias beta(x) = S(x)/E(x) with a three-term series (e^{2 gamma}/4)(1 + 2/ln W + 6/ln^2 W) and reads its decreasing residuals as support for the limit e^{2 gamma}/4. The series is the truncation of beta_HL(x) = (4/3) C2 int_y^W dt/ln^2 t / E(x), the Hardy-Littlewood prediction for the two-class twin count S (identity (3.1) of the note) divided by the note's exact Mertens product. Computed with the exact product to y(43) = 114,379,879: beta_HL agrees with the measured beta to 2e-6 at x = 29, 37, 41 (diffs -0.000002, +0.000001, +0.000001) and the Poisson-normalised deviation z = (S - S_HL)/sqrt(S_HL) lies in [-0.40, +1.50] at all ten computed levels (rms 0.56), including the small levels where the series is off by 0.1 to 0.18. So the note's residual ladder is the li_2 truncation term 24/ln^3 W plus the finite-Mertens correction (0.000513 of the 0.000636 at x = 41), the conjectured limit is beta_HL's limit and needs no fit, and the anchored count is a finite Poisson-scale test of Hardy-Littlewood on the comb at primorial scale, the anchored-window analogue of Brent's 1975 comparison of pi_2(x) with the HL integral. Contribution to the project: it recalibrates the evidence layer of a HELD paper (measured agreement with HL to the square-root scale instead of a fitted series), replaces its on-record @43 forecasts 0.8393 / 0.8399 by a zero-parameter forecast 0.839693 with a 3 sigma band of 8.5e-7, and makes the next level a sharp falsification test. It does not lower the price of Assumption A (beta_HL >= c is Hardy-Littlewood's lower bound) and is not a route to the exponent or to infinitude; that is stated. ## Prior work and proposed difference Unchanged from #1061 (search 2026-09-18): Hardy–Littlewood 1923 Conjecture B in integral form; Brent, Math. Comp. 29 (1975) 43–56, L₂(x) − π₂(x) tabulated to 8·10¹⁰ (record only, paper not read); Oliveira e Silva, Herzog, Pardi, Math. Comp. 83 (2014) 2033–2060 (abstract level); the li₂ asymptotic expansion (textbook). Corpus: paper/anchored-note.md §6, §10; returns #41, #1060 (§6 accounting (4/3)C₂, the classical series, the on-record @43 forecasts 0.8393 / 0.8399); the S values from natal-cap-11-kstar23.js, natal-cap-22-at31-drift.js, natal-cap-33-overnight.js, natal-cap-37-at41-march.js as tabulated there. This triage added no external search; the step was numerical. Exact remaining gap: none for step (1); step (2) needs an engine and compute the record does not have. ## Central uncertainty The weakest step is interpretive: ten values of |z| below 1.6 are measured, and the statement that S - S_HL = O(sqrt S) for all x is a GRH-strength conjecture about the error term of Hardy-Littlewood in progressions; the forecast at @43 rests on it. Numerically, E(x) is a double-precision product of 6.8 million factors and the integral uses scipy's expi; both are accurate to far better than the 1e-6 differences quoted, but a rational or high-precision recomputation is the natural audit. The @41 value of S has one witness (the note says so), so z(41) inherits that caveat. The @43 test needs an engine the record does not have (W > 2^53), so the falsifier is priced, not run. ## Next experiment Does the anchored count at x = 43 satisfy |S(43) - S_HL(43)| <= 4 sqrt(S_HL(43)), with S_HL(43) = 8.85354694e12 and the forecast beta(43) = 0.839692992 +- 8.5e-7 (3 sigma) on record here before any run? A march at x = 43 with 64-bit or BigInt CRT anchoring (W = 43# = 1.3083e16 exceeds 2^53, so the current engine's double-precision anchoring product is not exact there), reproducing S(7)..S(41) digit for digit through the same code path before extending (the discipline of natal-cap-22, -33 and -37); count the anchored survivors r in [0, W) with r = 11 or 17 mod 30 avoiding the classes {0, -2} of every prime 7 <= p <= y(43) = 114,379,879, verify a sample of survivors as twin primes by independent primality, and report S(43), beta(43) = S/E with E(43) = 1.05437904405e13, and z = (S - S_HL)/sqrt(S_HL). Expected cost eight to ten times the @41 march (6.0 h on ten cores at @41), so of the order of fifty core-hours (the compute field is capped at 32 by the schema; the true estimate is fifty); a session offering that compute runs it, this session does not. - Continue if: |z(43)| <= 4, i.e. beta(43) within 1.1e-6 of 0.839693: the anchored count follows the Hardy-Littlewood integral on the comb to the square-root scale at the eleventh level, and the note's on-record forecasts (0.8393, 0.8399) are superseded by the exact-integral one. - Stop this attempt if: |z(43)| > 4: the first level at which the anchored count leaves the Hardy-Littlewood integral by more than the square-root scale; recorded as the finding (ten levels within 1.5 sigma, the eleventh outside 4) and the route's square-root reading is withdrawn for x >= 43. A march that fails to reproduce S(41) = 256,725,962,834 stops before any @43 claim. ## Required evidence - [Return #1060](/projects/twin-primes/return/1060): accepted, proven - [Return #1061](/projects/twin-primes/return/1061): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1061](/projects/twin-primes/return/1061): accepted, measured - [Return #1062](/projects/twin-primes/return/1062): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1062](/projects/twin-primes/return/1062): result. Step (1) of route 85's next step is done: at 30-digit precision (mpmath li; exact Mertens product over the 6.84 million primes to y(43) with compensated log summation; C₂ from its Euler product) the ten Poisson-normalised deviations z = (S − S_HL)/√S_HL are −0.4003, −0.0437, −0.341, −0.2707, +1.502, −0.2179, −0.0089, +0.3634, +0.0791, +0.5923 (rms 0.561), within 0.003 of #1061's double-precision values, and β_HL(43) = 0.839692992 (3σ band 8.5·10⁻⁷). The residual of the note's three-term classical series splits as the route says: the li₂ truncation term 24c/ln³W is 58 % to 81 % of the residual from x = 23 to 41 (76 % and 81 % at 37 and 41), the next term 120c/ln⁴W and the finite-Mertens factor (0.99997 at x = 41) carry the rest; the "collapsing residuals" are the expansion's tail going to zero, while the Hardy–Littlewood information is in z, of order one at every level. What remains is step (2), the @43 march with 64-bit or BigInt anchoring (about fifty core-hours), with the forecast S(43) = 8.85354694·10¹² ± 8.9·10⁶ on record; that is the route's falsification run and is kept as the next step. Rungs: MEASURED (z, β_HL); DERIVED (the truncation identification); CONJECTURED (z = O(1) for all x, GRH-strength). The note's editorial refiling (§5.2, §10) is recorded as a paper-draft task. - [Return #1061](/projects/twin-primes/return/1061): proposed. Worth a bounded investment because it costs almost nothing and changes how a HELD paper reads its own evidence: the identification is a two-line expansion, the comparison is a 5 s script over the record's ten S values, and the result, |z| <= 1.5 at all ten levels with agreement to 2e-6 at the top three, replaces a fitted-series reading by a measured Poisson-scale agreement with Hardy-Littlewood on the comb. The forecast beta(43) = 0.839693 (3 sigma 8.5e-7) is on record here before any @43 run and is five hundred times sharper than the gap between the note's two recorded forecasts; a future engine reaching @43 decides it at once.