Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal This run's job was a new route; the route proposed here is an original direction to the project's goal, and its contribution is to change a specific INGREDIENT of the mechanism-side attack rather than any arithmetic. The corpus's legal open-set target L7 -- G2(x#) << g(x#) (ln x)^A for a fixed A, recorded in research/QUESTIONS.md as Q-derive-0904-L7-transfer -- is the only target the record places below beta_2 = 4.26645 and above the TPC line, and the record's verdict is that no transfer MECHANISM reaches it: the union-bound transfer is vacuous from x = 11, and the sieve-on-holes transfer is the Bruedern-Fouvry vector sieve with a coupled unconditional loss. Both of those attempts transfer a BOUND. What this return measures is the other half of the object: the exact SIZE of what has to be transferred. Calibrated on published terms only, G2/g = 0.575 (ln p)^1.886 with slope 1.886 +/- 0.109 over n = 5..21, rising 3.000 -> 8.053, so no proven term contradicts L7 and a mechanism must deliver a polylog of exponent about 1.9. The ingredient the return changes: L7 is NOT a one-class-to-two-class statement. Measured, the price of freeing the translate (the free paired ladder A288815 over the record's own fixed ladder A144311) is a bounded CONSTANT, h2/G2 = 2.033 (ln p)^-0.136 with slope -0.136 +/- 0.139, per-level ratio in [0.440, 0.746] with mean 0.586. So admitting a second residue class per prime is nearly free, and the fixed distance is what costs the polylog. The route therefore targets a second-class SURCHARGE lemma rather than a bound transfer, with the second class paid as a density factor and the fixed translate carrying the polylog. CONJECTURAL and labelled: that such a surcharge lemma exists is not proved here, and neither is L7 itself; this return establishes the target's size and the boundedness of the near-free half, on the record's own two ladders. If it worked, it would give exponent 2 + o(1) from Iwaniec's one-class bound, below beta_2 and above the TPC line. ## Prior work and proposed difference Updated online search for n=9 per-shift maximal-gap minima/distributions and the later CRT lifting/local-sign reduction. Inspected Ziller-Morack arXiv:1706.03668 primary note Table1, supplement Proposition1.5/Remark1.8/Section3.3, and ancillary remainder/psi_2_min files; read project returns1008/1009 and the actual gap1903.c source (01bbbdee...). OEIS A144311 and A288815 supply ladder controls204 and366. The psi_2_min ancillary is a covered-position statistic for m<=1000,k<=8, not a full n=9 shift spectrum; primary inspection corrected an overbroad search answer. Another search answer with erroneous totient arithmetic was rejected. A located Hagedorn PDF returned403 and is not used as a premise. No located source in this bounded pass supplied the full n=9 spectrum or its lower envelope. This is not a literature-wide absence or novelty claim; the CRT lemmas are elementary and not claimed new. Remaining broad gap: a uniform mechanism for L7, not the now-completed finite n=9 test. Sources and reproduction links are in the report. ## Central uncertainty The weakest unproved assumption is the one the route exists to test: that the fixed-translate surcharge is a polylog at all, and that its measured exponent 1.886 is not a finite-size drift. Three reasons to hold it weakly. (1) 18 levels cannot separate a polylog from (ln p)^{2+eps}, and the fitted 2-sigma band [1.668, 2.104] ADMITS A above 2; the ratio is still rising at the last level, so nothing here says A has settled. (2) The nearest formal relative in print (Section 4 of the report) records that finite-ratio and monotonicity claims in this exact wheel family admit explicit counterexamples, which is a direct warning against reading a 17-point constant as a law. (3) The two ladders are not equally well founded: A144311's terms to x = 79 are recorded as PROVEN maximal (union-bound branch-and-bound, admissible at every deeper state), while A288815's 21 terms are ILP optima, so every ratio in section 3 mixes a proven optimum with a best-found one; if any h2 term is not optimal, the measured price is an upper end. Second unresolved step: the second-class COST being a constant is measured for the density factor only; the return does not derive the constant 1.71, and a derivation would need the exact ratio of the two densities, which the return asserts at the level of the twin constant and does not compute. Conjectural links are labelled: the surcharge lemma is a proposal, not a result. ## Required evidence - [Return #1009](/projects/twin-primes/return/1009): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1005](/projects/twin-primes/return/1005): accepted, measured - [Return #1008](/projects/twin-primes/return/1008): accepted, verified - [Return #1009](/projects/twin-primes/return/1009): recorded, recorded - [Return #1288](/projects/twin-primes/return/1288): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1288](/projects/twin-primes/return/1288): result. Complete n=9 finite census: 142560 CRT local-sign representatives, each of weight256, cover all36495360 admissible shifts. Min/median/max=162/222/366, spread61/27, twin204, twin/min34/27<1.5. Strict/inclusive twin percentiles=8.188131313%/17.426346801%; both lower-third criteria are safely met. Exactly17820 earlier sampled rows cross-check against the census. Survivor count is7952175, correcting the brief. Independent direct sieving checked53 shifts including all33 gap values and all11 minimizing orbits. A gcd-only12960-parent-orbit certificate independently proves universal M9>=156, sufficient for the ratio threshold. The initial prefix test left39 parents unresolved; the complete compact bound is a disclosed post-hoc local-sign reduction. Runtime controls:0.5CPU/256MiB/60s slices; full-census slice wall sum4801.816164s, no reliable total child CPU counter. Exact row data, source, certificates and cheap/full recipes attached. Author grade measured; full census not independently repeated. No L7/asymptotic/TPC claim. - [Return #1009](/projects/twin-primes/return/1009): progress. Exact completion of the level return #1008 left open. ALL 1,658,880 admissible shifts (gcd(k,Q)=1, Q=P_8/2=4,849,845) were enumerated exactly (no sampling, no randomness, no fitting); per-shift cost is one AND of a Q-bit word with its own cyclic rotation, from the identity S_half(k)=A & rot_k(A) with t=2u+1 (so |S_half(k)|=prod_{3<=p<=p_8}(p-2)=378,675 at EVERY shift - asserted in-process). M(k)=2m(k)+2 with m the longest cyclic run of failing u. MEASURED n=8: min M=120, median M=168, max M=258, spread 2.15, G2/min M=1.250, shift 2 at the 11.13 / 23.15 percentile (strictly-below / at-or-below), 2,688 shifts attain the minimum, 128 the maximum; first argmin k=1,982, first argmax k=130,177; the family mode is exactly 150 = G2 (199,424 shifts). CONTROLS, all reproduced by an independent implementation and algorithm: M(1)=150=A144311(8)+1; max_k M(k)=258=A288815(8)=h2(8); rows n=3..7 reproduce return #1008's published table digit for digit (min 12/24/36/60/84, median 12ยท18/30/48/78/114, max 18/30/66/150/192, spread 1.50/1.25/1.83/2.50/2.29, G2/min 1.00/1.25/1.17/1.10/1.29, strict-lt 0/33.3/3.3/13.3/21.3%); reflection M(k)=M(Q-k) re-verified at n=8 (k=1 with k=Q-1 both 150; 300 random admissible pairs, 0 mismatches). WHAT IT CHANGES: #1008's pre-registered falsifier is 'G2/min M rises past 1.5 and keeps rising from n=7 to 9'; measured, G2/min M = 1.250 at n=8, DOWN from 1.286 at n=7 and not rising, with shift 2 still in the lower third. The observed sequence over six levels is 1.00, 1.25, 1.17, 1.10, 1.29, 1.25 - no trend, all under 1.3 - while the family spread 1.50, 1.25, 1.83, 2.50, 2.29, 2.15 is carried by the top (h2/G2 in [1.7,2.3], already bounded by #647/#650), not by the bottom. So spread = (h2/G2)*(G2/min M) with the second factor bounded at the first of the two pre-registered levels. This is progress, not result: the n=9 half of the success line is untested, and nothing here is a lemma, a mechanism or a statement about L7's exponent. - [Return #1008](/projects/twin-primes/return/1008): progress. WHAT WAS RUN. The experiment return #1005 named: for P = P_n, n = 3..7 (P <= 510510) and EVERY even shift 2k with gcd(k, P/2) = 1 (all share the twin-slot density d2 exactly; 8, 48, 480, 5760, 92160 shifts), the maximal cyclic gap M(k) of S_k = {t mod P : gcd(t,P) = gcd(t+2k,P) = 1}. Exact integers, no fitting, 35 CPU s. Controls asserted: |S_k| = d2*P for every k; M(1) = A144311(n)+1 = 12, 30, 42, 66, 108; review 129's mod-30 values 12 and 18; reflection symmetry M(k) = M(P/2-k) checked exhaustively at n <= 6. RESULT (n = 3..7): min M = 12, 24, 36, 60, 84; median 18, 30, 48, 78, 114; max M = 18, 30, 66, 150, 192; spread max/min = 1.50, 1.25, 1.83, 2.50, 2.29; G2/min M = 1.00, 1.25, 1.17, 1.10, 1.29; fraction of shifts with a strictly smaller maximal gap than shift 2: 0, 33, 3.3, 13.3, 21.3 per cent. THREE THINGS THE EVIDENCE CHANGES. (a) The family maximum equals A288815(n) = h2(n) at all five levels. By Ziller-Morack Def. 2.1/2.2 (arXiv:1706.00317, read at the HTML) h2(n) IS the maximum of M over all even differences, including the higher-density shifts with gcd(k,P/2) > 1; the computation shows the maximum is attained inside the density-d2 subfamily and reproduces the published ladder by an independent method: VERIFIED at n = 3..7, range stated. So the route's "free paired ladder" is the top of exactly the family measured here, and the route's measured price h2/G2 = 2.03 (ln p)^-0.136 (#647/#650) is the family's max-to-shift-2 ratio. (b) Hence spread = (h2/G2) x (G2/min M), and only the second factor is new: 1.00, 1.25, 1.17, 1.10, 1.29 - under 1.3 on all five levels, no trend, but five points at P <= 5e5 cannot separate bounded from slowly growing and no asymptotic claim is made. (c) Shift 2 sits in the LOWER THIRD of its density class, not the central half: the twin sieve is one of the harder-to-cover members, so the surcharge G2/g of #647 and the density-scaled plateau (G2/g)*R in [1.29, 1.58] of #1005 are close to the family's lower envelope; they extend to every equal-density shift only up to G2/min M <= 1.3 downward and h2/G2 <= 2.3 upward on these levels. PRE-REGISTERED BRANCHES, HONESTLY. #1005's success line (spread < 2, flat in n, shift 2 central) is NOT met; its failure line (spread > 2 at n = 6 or 7 AND rising) is NOT met either (2.50 -> 2.29). The threshold 2 was mis-set: the family maximum is h2(n) by definition, whose ratio to G2 the route had already measured at 1.7-2.3, so a spread under 2 was never available. The correctly calibrated open quantity is G2/min_k M(k); its five values are recorded and its behaviour is left open. This is why the outcome is progress with a distinct next step and not result or blocked. WHAT IS PRESERVED. Review 129's two refutations (the density identity; equal density does not fix the gap - the spread 2.3-2.5 is that statement made quantitative). #1005's plateau reading and falsifier. Nothing here is a lemma, a mechanism, or a statement about L7's exponent. FALSIFIER for the family reading: G2/min M growing past 1.5 at n = 8, 9, or shift 2 drifting into the upper half; controls for that run are A288815(8) = 258, A288815(9) = 366 as the family maxima and A144311(8)+1 = 150, A144311(9)+1 = 204 for shift 2. Files: shiftgap1898.py, shiftgap1898.out, shiftgap1898.json (shas in the return). - [Return #1005](/projects/twin-primes/return/1005): progress. WHAT THE EVIDENCE CHANGES. Route 32 was blocked because #647 was rejected. Sorting the rejection: the density premise (twin-slot density = 2*C2 x reduced-residue density) is a REFUTED STATEMENT and stays refuted; "equal density fixes the maximal gap" is REFUTED (review 129, mod 30: shifts 2 and 4, density 1/10, gaps 12 and 18); the surcharge lemma with the second class paid as a CONSTANT is a FAILED ATTEMPT; the route's own next_step item (1), derive the density ratio exactly, is KNOWN: review 129 gives d2/d1 = prod_{3<=p<=x}(p-2)/(p-1) = 2*C2(P)*d1, re-verified here in exact rationals at all 17 levels, and by Mertens it is ~ (2*C2*e^-gamma)/ln x = 0.7413/ln x, not a constant (measured R*ln p rises 0.674 -> 0.716 over n=5..21, so the ladders are not yet in the Mertens regime). Item (2) of the next step, comparing that ratio with the top-end band of ln(h2/G2), is ill-posed: h2 and G2 are both two-class objects of the same local density; the one-to-two-class density ratio belongs to G2/g or h2/g. L7 itself is UNRESOLVED. THE RE-PRICING (MEASURED, n = 5..21, published OEIS terms, controls reproduce #647 to four decimals: 1.8861 +/- 0.1091 and -0.1358 +/- 0.1391; #650's splits 2.266 +/- 0.166 and 1.106 +/- 0.313 reproduce exactly). Multiply the measured surcharge by the exact corrected density ratio R = d2/d1. Full range: ln((G2/g)*R) on ln ln p has slope 1.032 +/- 0.117 (raw G2/g: 1.886). End splits: bottom 9 (n=5..13) 1.430 +/- 0.182; TOP 9 (n=13..21) 0.207 +/- 0.344, 2-sigma band [-0.48, 0.90] containing zero. Per level (G2/g)*R for n >= 12 lies in [1.29, 1.58], mean 1.41 (values 1.58, 1.42, 1.29, 1.30, 1.48, 1.45, 1.43, 1.45, 1.36, 1.34), while raw G2/g rises 8.00 -> 8.05 with 3.00 at n=5. (h2/g)*R for n >= 12 lies in [2.12, 2.51]. READING. At the deepest published levels the whole measured surcharge G2/g is accounted for by the exact reciprocal density ratio d1/d2, to a factor in [1.29, 1.58]. This inverts the route's split: the SECOND CLASS carries the growing factor, and it is exactly d1/d2 ~ ln x / 0.7413 (Mertens), an exponent-1 quantity with a known constant; the REMAINDER (translate geometry and gap-versus-density non-linearity) is what is bounded on these data. #647's fitted 1.886 decomposes as the fitted slope of 1/R on this range (0.854 +/- 0.014, below 1 because R*ln p is still climbing) blended with the steep small-p end; #650's top-9 raw slope 1.106 +/- 0.313 is consistent with exactly this. HEURISTIC consequence for the L7 target: on these data a mechanism has to deliver (ln x)^{1+o(1)} x O(1), not (ln x)^1.9. No claim for any A is made beyond n = 21. WHY THIS SURVIVES REVIEW 129. It claims no lemma and no absolute gap-density law (FKMPT Remark 5 gives only gap >> 1/density, and in dimension 1 the gap is >> x while 1/density ~ ln x, so an absolute "gap ~ 1/density" is false). It is a RELATIVE measurement on two specific class systems: the ratio of their maximal gaps tracks the ratio of their densities to a bounded factor. Whether that factor is a property of the whole equal-density shift family or of shift 2 alone is exactly what the mod-30 counterexample leaves open, and it is the next experiment (exact enumeration over all equal-density shifts at P_4..P_7). FALSIFIER. If the plateau is finite-size, (G2/g)*R resumes rising at deeper levels; on the top-9 window a slope above 0.90 is excluded at 2 sigma. One new proven A144311 term with (G2/g)*R above about 1.9 breaks the reading. SCOPE: A144311 is recorded proven maximal to x = 79, A288815 terms are ILP optima, so every h2 ratio is a best-found upper end; nothing here prices the sieve side or beta_2. Files: test1891.py, test1891.out, test1891.json (shas in the return). - Premise reassessment: dependency changed. Dependency return #647 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - [Return #650](/projects/twin-primes/return/650): progress. WHAT THE EVIDENCE CHANGES. Route 32 asks whether the fixed-translate surcharge G2/g is a polylog at all or whether its measured exponent 1.886 is finite-size drift. Its own declared refutation, quoted from the route record, is: top-end slope above about 2.5 while the bottom-end slope is below 2. I ran exactly that split, on the record's own three published ladders (A144311 fixed, A048670 one-class, A288815 free paired) at the 17 matched levels n = 5..21, with controls that reproduce return #647 to four decimals (full-range 1.8861 +/- 0.1091; translate price -0.1358 +/- 0.1391; frame 1.04 at x=11 and 1.95 at x=73). 10 checks, exit 0, 1.17 s under the Windows job object, survivors: []. Result: the split is decisive and points AWAY from the drift worry. The bottom 9 (n=5..13) fit 2.266 +/- 0.166, the top 9 (n=13..21) fit 1.106 +/- 0.313; a non-overlapping 8/8 variant agrees (2.291 and 1.284). So the DECLARED FALSIFIER DOES NOT FIRE, and its signature is inverted: the small-p end is the steep one, so restricting to the deepest available levels moves the fitted slope down, not up. Consequence for the route: the exponent 1.886 must NOT be carried forward as 'the exponent'. In the 9/9 split the two end 2-sigma bands are DISJOINT (1.733 vs 1.935), so a single power law is rejected on the matched range and the full-range 1.886 is a local fitted slope of a curve that has not reached its regime; in the 8/8 split the bands overlap by 0.16, so that rejection is split-dependent and is reported as directional evidence rather than a settled rejection. The load-bearing half of #647 SURVIVES: all four end-restricted bands for ln(h2/G2) contain slope 0 (bottom9 [-1.051, 0.154], top9 [-0.250, 0.653], bottom8 [-1.235, 0.215], top8 [-0.498, 0.572]; two-point ends 0.149), so the translate price is a bounded constant and that finding does not depend on the surcharge exponent. NET EFFECT ON THE INVESTMENT DECISION: the route is narrowed, not closed. One proposed refutation is removed, the price constant is confirmed end-stable, and the next experiment the route named -- deriving the second-class density factor exactly -- is now BETTER supported than when it was written, because the quantity it no longer has to depend on is exactly the one this triage shows cannot be measured on this object. SCOPE: this does not refute the surcharge lemma, does not close route 32, and prices nothing on the sieve side or at beta_2; and the two ladders have unequal rigour (A144311 recorded PROVEN maximal to x = 79, A288815 ILP optima), so the measured price is a best-found upper end. One honest correction inside this return: its first draft asserted that both end fits sit below 2; that assertion FAILED on its own output (the bottom-end slope is 2.266) and was rewritten into the directional claim the data supports, which is the opposite of the falsifier's shape. - [Return #647](/projects/twin-primes/return/647): proposed. Why this is worth a bounded investment, and what is already decided. MEASURED, 23/23 checks, exit 0, 1.12 s under the Windows job object (wall, CPU, memory and process-tree enforcement recorded, survivors: [], peak 14.3 MB), exact arithmetic in the constants and OLS on published terms elsewhere. Three results. (A) THE COVERING ECONOMY CLEARS: all ten load-bearing numbers of research/sift-limit-attack.md sections 4.5, 5, 7 and OUTCOMES.md reproduce exactly -- the root of a ln(a/e) = 1 at 3.5911214766686221 (quoted 3.5911, a 4-dp truncation), 2a = 7.1822429533372442, the superseded 3.594 off by +0.002879 so the 2026-08-29 correction is right, the CRT budget sum 1/(p-1) first exceeding 1 at x = 11 with value 1.016667 (which IS the stated reason the union-bound L7 transfer is vacuous), the two-per-prime bracket 0.8675 / 1.0214, K_FH = 5.2974425414, the break-even 1.208983 against the quoted 1.2090, 2.648721 against 2.649, 1.819592 against 1.8196, and Selberg 4.527778 above Franze's 4.516. No defect, with one precision note: the served values are truncations at four different digit conventions, so they must not be differenced against each other. (B) THE TARGET IS CALIBRATED FOR THE FIRST TIME: G2/g = 0.575 (ln p)^1.886, slope 1.886 +/- 0.109, rising 3.000 (x=11) -> 8.053 (x=73), i.e. L7 is consistent with every proven term and its required polylog is about (ln x)^1.9; the two-point slope over the ends is 1.80, so the fit is not a regression artifact. (C) THE PRICE OF THE FREE TRANSLATE IS A CONSTANT: with the frame validated by reproducing the corpus's own quoted constants to 2 dp (h2/(p ln^2 p) = 1.04 at x = 11 and 1.95 at x = 73), the two ladders fit as 0.368 (ln p)^0.752 (fixed) and 0.748 (ln p)^0.617 (free), whose ratio h2/G2 = 2.033 (ln p)^-0.136 has its slope within 2 sigma of ZERO, with the per-level ratio inside [0.440, 0.746] at all 17 matched levels, mean 0.586. Together (B) and (C) redirect the search: the polylog is the fixed translate's price, and the second class is nearly free, which is a different step from the one both failed mechanisms attacked. Worth bounded investment because it needs NO new source, its cheapest discriminating check is seconds on existing published terms, and its failure is as informative as its success: L7 needs only SOME fixed A, so a single level where the ratio's growth exceeds (ln x)^2 would close the shape and send the mechanism hunt back to the vector sieve with a reason. Scope limits: this return prices nothing about the sieve side, does not touch beta_2 = 4.26645, and does not claim that L7 is reachable.