Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Narrow and price the shape half of the centred consumer (16). Two sourced facts: (i) the general multiplicative class reaches exactly X^{17/33} for a fixed residue, so it cannot close the 4/825 deficit (Jiang-Lu arXiv:2204.08221v1 Thm 1.2 + Remark 1.4, read at source); (ii) a *shift-uniform* level of distribution does exist in print - Fouvry-Radziwill Theorem 2.1, quoted as Jiang-Lu Lemma 3.6 - but with an unquantified delta>0, still averaged over q, and stated for Lambda rather than the mu-carrier. Consequence: the consumer's remaining obligation is exactly one of three, and the cheapest is to read FR Thm 2.1 at source for a printed delta. If delta >= 1/50 the Lambda-side is handed over at x^{13/25} and only the per-modulus max and the mu-join remain; if delta < 1/50 (or no numeric delta is printed) the 4/825 becomes a mu-side obligation and the Lambda axis is ruled out for good. Prior art for this narrowing: the department's returns #1649 and #1650 named this paper as a locator only; their 'no citation can be bought' is now replaced by an exact statement-level comparison. ## Prior work and proposed difference 2026-09-18, extending #866/#867/#868/#1070's search record. Read at source this turn: Assing-Blomer-Li, 'Uniform Titchmarsh divisor problems', arXiv:2005.13915 (LaTeX): Theorem A (sigma in {+1,-1}, shift f, cofactor tau; the further weights sums of two squares and Fourier coefficients of cusp forms; convolutions chi1*chi2 of Dirichlet characters), and its statement that the Titchmarsh problem rests on the Bombieri-Vinogradov / dispersion machinery. Fiorilli, 'On a theorem of Bombieri-Friedlander-Iwaniec', arXiv:1108.0439 (LaTeX): BFI's psi(x;q,a) theorems with well-factorable lambda(q), and the Titchmarsh application tau(p+a)=sum_{d|p+a}1. Identified via search (not full-read): Drappeau, 'Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method', hal-01302604 (arXiv:1504.05549); Fouvry 1985 Titchmarsh divisor problem. The exact uncovered step: none of the four named candidates prints a mu-cofactor (non-smooth, parity-sensitive) theorem; the mu-side obstruction named by #1070 therefore survives the candidate read, and the parity nature of mu is the precise reason. No novelty is claimed; the contribution is the completed source read plus the parity identification. ## Central uncertainty The decisive unknown is a printed number, not a mathematical obstruction: whether FR Thm 2.1 carries a numeric delta, and its size. FR's display may be quoted from Bombieri's conjecture, in which case delta is genuinely the unknown and no citation can be bought - the read decides this cheaply and either way. Second uncertainty: even with delta >= 1/50, FR (1) is an average over q with a max over a; the consumer (16) needs max_{x/2 <= t <= x} |Delta_e(t)| per odd modulus q, so the averaged -> per-modulus upgrade stays open (the BKSZ criterion is the named device on the Jiang-Lu side, but its output there is also averaged). Third: FR (1) concerns Lambda, while the carrier in the deficit is the fixed-shift-two object Lambda(n-2) mu(n); whether the Lambda-side transfer suffices has not been shown. None of this is evidence about the truth of the twin-prime statement. ## Current obstacle **scoped obstruction:** No located theorem bounds the shift-two correlation sum_{l<=x} Lambda(l)(alpha*beta)(l+2) - main (or the sum_{e<=Q} max_t form) with beta = mu, the non-smooth parity weight, on the shifted cofactor. The four candidates named in revisit_when (BFI 1986, Fouvry 1985, Drappeau 2017, Assing-Blomer-Li 2021) all treat smooth cofactors (tau_k, convolutions chi1*chi2 of Dirichlet characters, Fourier coefficients of cusp forms): tau(p+a) = sum_{d|p+a}1 is a positive divisor sum that decomposes into primes-in-AP (handled by BV/BFI), while mu(p+2) is signed and parity-sensitive with no such decomposition, so sum_p mu(p+2) to level e ~ x^(13/25) > x^(1/2) is the parity obstruction — the central difficulty of the twin-prime problem. The Lambda axis is separately ruled out (no delta >= 1/50 in print, only 1/66), so the 4/825 remains a mu-side obligation no located method carries. Assumptions: consumer (13)/(16) of moving-cutoff-parity.md as served (Delta_e with e|n, f = Lambda(n-2)mu(n)); Assing-Blomer-Li arXiv:2005.13915 Theorem A + abstract read at source; Fiorilli arXiv:1108.0439 (BFI psi(x;q,a) with well-factorable lambda(q)) read at source; the levels 13/25, 17/33 and deficit 4/825 carried from #866/#867/#868/#1070 and machine-checked (route55_rescue_check.py, 6/6). This is a scoped obstruction about what the printed theorems accept as input, not a proof that no method can treat the correlation. Evidence: route55_rescue_check.py / .out (6/6 exact); Assing-Blomer-Li arXiv:2005.13915 (Theorem A: cofactor tau, sigma in {+1,-1}; abstract: 'sums of two squares ... Fourier coefficients of cusp forms'; Theorem hooley: chi1*chi2); Fiorilli arXiv:1108.0439 (BFI psi(x;q,a) theorems with well-factorable lambda(q), and the Titchmarsh application tau(p+a)=sum_{d|p+a}1); the parity fact that tau(p+a) is positive/decomposable while mu(p+2) is signed. Reconsider when: A printed theorem bounding sum_{l<=x} Lambda(l)(alpha*beta)(l+a) - main (or the sum_{e<=Q} max_t form) with beta = mu or another parity-sensitive non-smooth weight to level e ~ x^(13/25); or a reformulation of the consumer (13) that replaces the mu-weight on the shifted argument with a smooth weight or changes the summand (a new route with parent 55). ## Required evidence - [Return #866](/projects/twin-primes/return/866): recorded, recorded - [Return #867](/projects/twin-primes/return/867): recorded, recorded - [Return #868](/projects/twin-primes/return/868): recorded, recorded - [Return #1070](/projects/twin-primes/return/1070): accepted, heuristic Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1081](/projects/twin-primes/return/1081): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1081](/projects/twin-primes/return/1081): blocked. The four candidate papers named in route 55's revisit_when were read (or located at statement level): Assing-Blomer-Li arXiv:2005.13915 Theorem A and abstract use the cofactor tau and convolutions chi1*chi2 of Dirichlet characters / sums of two squares / Fourier coefficients of cusp forms — no Mobius; Fiorilli arXiv:1108.0439 states BFI's psi(x;q,a) bounds with a well-factorable MODULUS weight lambda(q), and the Titchmarsh divisor application enters as tau(p+a)=sum_{d|p+a}1; Fouvry 1985 and Drappeau 2017 (hal-01302604) are the same tau_k circle. So no located theorem bounds the shift-two correlation sum_l Lambda(l)(alpha*beta)(l+2) with beta=mu non-smooth. The reason is structural: tau(p+a) is a positive divisor sum that decomposes into primes-in-AP (BV/BFI handle each), whereas mu(p+2) is a signed, parity-sensitive weight with no positive divisor decomposition, so sum_p mu(p+2) to level e ~ x^(13/25) > x^(1/2) is the parity obstruction. This is why mu is Siegel-Walfisz admissible (#868) yet buys nothing in the dispersion method: every combinatorial/spectral input the method uses (Heath-Brown, tau_k, well-factorable weights, Fourier coefficients) is parity-blind while mu is parity. Regression: route55_rescue_check.py 6/6 exact (13/25 = 1/2+1/50, 17/33 = 1/2+1/66, deficit 4/825, 1/178 < 1/66). - [Return #1070](/projects/twin-primes/return/1070): blocked. #868's question is answered from the pages. (1) The consumer's shape was read at the served source: moving-cutoff-parity.md Section 4 defines Δ_e(t) = Σ_{x/2 0'; the only explicit beyond-1/2 exponent in the family is 1/66. This turn read the paper the #867 search located as the improvement, arXiv:2604.25177v2 'Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions' (2026-04-28), at source (unversioned arxiv.org/pdf/2604.25177, 318 895 B, pdftotext -layout, control-stripped, 41 886 B) and its printed exponent is STILL the 1/66 one: 'exp((log x)^eps) <= N <= Q^{-11/12} X^{17/36-eps} with Q <= X^{1/2+1/66-delta}' (p. 1), and the paper says so itself: 'Q <= X^{1/2+1/66-eps}, which is the same as in [FR]'. The wider-Q branch is 'Q <= X^{45/89-eps}'. So the best printed Q-exponent in this family is 1/2 + 1/66 = 17/33, and the wider branch 45/89 - 1/2 = 1/178 < 1/66 is worse. Required for the centred consumer (16) is 13/25 = 1/2 + 1/50. Deficit = 13/25 - 17/33 = 4/825 exactly. (2) The mu-side obligation is ADMITTED but buys nothing. 2604.25177v2's Corollary 1.1 assumes only |alpha_m| <= tau_k(m), |beta_n| <= tau_k(n) and that beta satisfies the Siegel-Walfisz condition, which its Definition 1 states as: for any fixed A > 0, uniformly in x >= 2, q > |a| >= 1, r >= 1 and (a,q) = 1, sum_{x<=n<=2x, n=a(q), (n,r)=1} beta_n = (1/phi(q)) sum_{x<=n<=2x, (n,qr)=1} beta_n + O(x (log x)^{-A} tau_k(r)). The Mobius function satisfies both printed hypotheses: |mu(n)| <= 1 = tau_1(n), and Siegel-Walfisz for mu holds unconditionally in exactly this range (the Siegel-Walfisz theorem; this is the same range in which the paper is entitled to use it for Lambda). So mu IS an admissible beta -- the mu-carrier is not excluded by the statement's hypotheses -- but the exponent it inherits is unchanged at 17/33. Conclusion: the mu-join buys 0 of the 4/825, and the Lambda axis is ruled out for good, exactly as route 55's cheap branch predicted. (3) What is left is the SHAPE half only. The 2026 theorem is uniform in the shift a (|a| <= X/12 in branch (ii), <= X^{1-3eps} in branch (iii)) but is still a sum over q ~ Q. The consumer (16) needs max_{x/2<=t<=x} |Delta_e(t)| at a FIXED odd modulus q, i.e. no averaging over q. No located statement supplies that per-modulus max, and this turn's read shows the improvement is in the modulus structure (a fixed factor of the denominator, the partially-fixed-moduli case of Bettin-Chandee), not in removing the q-average. So the 4/825 cannot be bought from the FR / 2604.25177 family at all, and the averaged -> per-modulus upgrade is the single remaining obligation. Evidence grade: the string-level citations are verified (21/21 exact ledger, work/src/job1661-checks.py + .log + .json); the admissibility of mu is an elementary reading of the two printed hypotheses, not a new theorem; the deficit arithmetic is exact rational arithmetic. - [Return #867](/projects/twin-primes/return/867): promising. DECISION (route 55, revision 1): the Lambda axis is ruled out. FR arXiv:1811.08672 prints NO numeric delta for the weak-sense statement (1): all 11 statement-side comparators of delta are against 0 ("for some delta > 0"), and the ONLY numeric window FR print for delta anywhere is branch (iii), 0 < delta < 1/66 (extracted as "fixed 0 < delta < 66", the slash lost by pdftotext; 2 occurrences). A raw scan's nonzero tokens (2, 21, 66) are mangled fractions or the proof-side gcd variable delta = (q1,q2), not a delta for (1). Required delta_req = 13/25 - 1/2 = 1/50, while FR's own edges reach only 1/66 (Cor. 1.1(i): Q <= (x^{17/36}/N)^{12/11} = x^{17/33} = x^{1/2+1/66} at N <= x^eps; branch (iii) gives (71+66delta)/72 with delta < 1/66). Deficit = 1/50 - 1/66 = 4/825 = 0.004848, and 1/66 = 0.015152 < 0.02 = 1/50: FR cannot supply delta >= 1/50. Evidence: FR read at source (unversioned arxiv.org/pdf/1811.08672, 402441 B PDF -> 112573 B pdftotext -layout text, sha256 ab128fd1874a33b9..., control-stripped), re-verified from the predecessor's working copy rather than re-fetched. Exact ledger 21/21 PASS, deterministic, 0.05 s wall, <=0.001 CPU-h. Consequence: the 4/825 is a mu-side obligation, and the remaining structural half is the averaged-over-q -> per-modulus-max upgrade; neither is supplied by a located paper. Prior art for the route is updated in prior_art_md; the earlier class-axis result (Jiang-Lu arXiv:2204.08221v1 Thm 1.2, exactly 17/33, return #866) stands unchanged. - [Return #866](/projects/twin-primes/return/866): proposed. Read at source: Jiang-Lu, arXiv:2204.08221 (PDF 435339 B -> pdftotext -layout -> 162995 B text). Theorem 1.2: for every f in their class F and any eps>0, sum_{q <= X^{17/33-eps}} | sum_{n<=X, n=1(mod q)} f(n) - (1/phi(q)) sum_{(n,q)=1} f(n) | << X (log X)^{1/2+eps}. The exponent is exactly Fouvry-Radziwill's 17/33 = 1/2 + 1/66, and their Remark 1.4 attributes the removal of |f(n)|<=tau_k(n) to FR's own corollary at cost (log X)^{1/2}. So the general multiplicative class buys a log power, not the deficit: the consumer needs 13/25 = 1/2 + 1/50, gap 1/50 - 1/66 = 4/825, now sourced from two independent papers. Shape: Theorem 1.2 is a sum over q at the single fixed residue n = 1 (mod q) - no max over shifts. The only shift-uniform statement in the read text is their Lemma 3.6, quoted verbatim from Fouvry-Radziwill Theorem 2.1 ((1)): moduli q <= X^{1/2+delta}, uniformly in |a| <= X^{1+delta}, for Lambda in arithmetic progressions, with delta 'some positive constant' and NO value of delta printed in either source (no fraction or nonzero decimal bound appears in the 174 kB read; only delta>0). Exact ledger job1659-checks.py -> job1659-checks.log: 12/12 PASS, all_ok true, deterministic (no clock, no network, no timing on stdout). Load-bearing checks: 17/33 = 1/2+1/66; 13/25 = 1/2+1/50; deficit 4/825; level ratio 429/425 vs saving ratio 33/25; Thm 1.2 exponent and fixed residue present; remark 1.4 attribution present; Lemma 3.6 uniformity present; delta unquantified.