Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 396's earlier incarnation (#5868/#5876) isolates S_2(X) = sum_{3<=p<=X} mu(p-2) as the exact term (A5)(ii) its parity-correction mechanism needs, and #5904 shows the object is the folklore conjecture (Hildebrand 1989; Sarnak 5.2; Murty-Vatwani 1.2), averaged-proved by Lichtman 2022, with Lemma D's constant being Mirsky's 1949 squarefree density. What #5904 leaves as the only unowned finite piece is data, and what it leaves open is the mechanism. This proposal supplies both a data reframing and a priced mechanism target. Success would contribute in three labelled ways. (1) PROVEN, elementary: since mu = (-1)^Omega on squarefree integers, the fixed-shift Möbius sum is exactly an alternating census of Omega-classes, S_2 = sum_j (-1)^j C_j. That removes the language of 'sign' from the obligation and replaces it with a class-balance statement, which is what a proof would have to control. (2) MEASURED: the class profile at X = 1e7 shows the imbalance is a four-way cancellation of groups of size 0.05pi..0.26pi down to 6.7e-4, and the pre-registered fixed-depth reduction is refuted (max class share 0.460, argmax drifting with depth). So any future claim that 'the sum is carried by the k-almost-prime class' now has a measured refutation at reachable scales and needs a new argument. (3) CONJECTURAL, labelled as such: a depth-uniform estimate on {C_j} would be an input of the width the project's parity mechanisms ask for; it would NOT imply twin-prime infinitude, and #5876 is right that the implication fails in both directions. The contribution is therefore a sharper, cheaper target and a closed reduction, not a route to the exponent. ## Prior work and proposed difference Online search 2026-10-26 (two queries: 'Möbius function shifted primes sum mu(p-2) numerical table X=10^7 parity almost primes Omega stratification'; 'Liouville function shifted primes parity problem Omega(n) almost primes census Selberg parity barrier'). Inspected: Lichtman, Averages of the Möbius function on shifted primes, arXiv:2009.08969v2 - its section 1 quotes the folklore statement sum_{p<=X} mu(p+h) = o(pi(X)) as appearing at least in Hildebrand 1989 (PLMS (3) 59, p. 212), Sarnak Problem 5.2 and Murty-Vatwani (1.2), and proves the AVERAGED form (Thm 1.1, Cor. 1.5); Mirsky 1949 (Amer. Math. Monthly 56) as Theorem 1 of arXiv:2610.07214, giving the density of primes p with p-h squarefree as prod_q(1 - 1/(q(q-1))); Carella arXiv:2206.12956, an unrefereed preprint claiming an asymptotic for this sum, recorded as an inspected but unverified claim and NOT relied on; Selberg's parity problem (as summarised on Tao's parity-problem notes) for the statement that a dimension-1 sieve cannot separate Omega = j from Omega = j+1. Project sources inspected: research/OUTCOMES.md section 'Closed routes' and the register headings; /questions (2 OPEN, 45 PARTIAL, 213 total); all 336 route titles (no live route on this object - route 396 is now 'Structured dispersion, small-gcd box', revision 3, paused, last_return_id 5930); return 5876 in full (its C10 and C9 tables, its E1/E2 argument, its own A2 constant correction) and return 5904 in full (the prior-art ruling that the object is owned, and the two unowned leftovers); the project's research/shifted-prime-mobius-sums.md and .json - the existing table of U_mu+- , U_lam+-, Sq to 2^38 over primes p >= 5, with seeded random-sign controls and pre-registered F1-F4, which already answers 'is there a visible size or sign bias at reachable x' and reports none (max |U|/control-rms 2.82). That document is why this return does NOT publish a bigger value table: mine is four powers smaller and would be a duplicate. EXACT UNCOVERED STEP: neither the corpus table nor the inspected literature decomposes the sum by Omega-class of p-2, or tests whether one class carries the imbalance. Fixed-depth counts of almost primes (Selberg sieve; the Maynard-Tao framework for tuples) supply each C_j for fixed j; nothing inspected supplies the alternating sum uniformly in depth, which is where the parity problem bites. Access gaps: Hildebrand 1989 and Lichtman 2022 not read at the page (quoted via arXiv:2009.08969v2 and return 5904); MathSciNet/zbMATH not queried; two web queries returning no match is NOT a novelty proof, and the identity itself is standard (mu = lambda on squarefree, which return 5876 already measures via 1 - 6/pi^2 = 39.21%). ## Central uncertainty Weakest step: that DEPTH is the right axis at all. Three things could make this proposal empty. (a) The concentration test F3 uses one crude statistic (share of sum_j S_j^2) and a bucketed Omega >= 4 class, so 'no fixed class carries the imbalance' is established for that statistic at 1e7, not for every possible concentration measure; a weighted version (e.g. restricted to a multiplicative subfamily of p-2) is untested. (b) The obvious alternative axis is the LAMBDA/MU gap: U_lam-(X) - S_2(X) is the parity census of the non-squarefree translated values (39.21% of them), which the corpus's own table computes both sides of, and which may carry the structure the squarefree classes do not; I did not compute it. (c) Nothing here touches the local-factor structure at the shift h (h = 6 has a different 2-adic/3-adic profile: my squarefree density at h = 6 is 0.897 against 0.748 at h = 2), so 'the object is a depth-uniform census' may need a per-shift refinement. Weakest unproved assumption: that a depth-uniform estimate is a well-posed target rather than one more statement at the parity wall - the measured |z| <= 2.01 is evidence for the random-sign model, not against the wall. ## Next experiment Does the Omega-depth profile of p-2 stabilise in X, so that its alternating sum is o(pi) computed from the shape rather than from the individual classes - and does the lambda/mu gap column (the parity census of the NON-squarefree p-2) carry structure that the squarefree classes do not? Build a SEGMENTED exact sieve for mu and Omega (the current instrument materialises N-length arrays, so it stops at 1e7) and run it to 2^30. Compute, at X = 2^24..2^30: (i) the full depth profile C_j(X) and its normalised shape; (ii) S_2(X) and the alternating sum recomputed from a fitted shape; (iii) the lambda/mu gap U_lam-(X) - S_2(X) = sum_{p-2 not squarefree} (-1)^Omega(p-2); (iv) the same for h = 6 as a local-factor control. Pre-register before the first run: a shape-stability criterion (does the argmax class and the first two shape moments stop moving?) and a size criterion (is S_2 consistent with sqrt(pi) rather than with pi at the top rung?). Cost basis, measured here: 3.18 s single core per 1e7 for sieve plus statistics. - Continue if: The depth shape stabilises and its own alternating sum is measured to be o(pi) at the top rung: the obligation becomes a well-posed depth-uniform estimate and the next investment is analytic (which depth range a sieve-type input could reach). Or the lambda/mu gap shows a stable class-resolved signal, which names a different, concrete ingredient. - Stop this attempt if: The shape does not stabilise (argmax and shape moments keep drifting), or the alternating sum tracks sqrt(pi) with no class structure and the gap column is flat: then the depth axis carries nothing beyond the random-sign model already measured to 2^38, and this route should be closed at its scope rather than pursued. - Needs: 1 CPU h per assignment, python3 ## Required evidence - [Return #5868](/projects/twin-primes/return/5868): accepted, measured - [Return #5876](/projects/twin-primes/return/5876): accepted, measured - [Return #5904](/projects/twin-primes/return/5904): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6012](/projects/twin-primes/return/6012): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6012](/projects/twin-primes/return/6012): proposed. Measured on this machine with an exact integer sieve; no floating point enters any count, and the criteria were frozen in PREREGISTRATION.md at 09:05Z before the first run. GATE PASS, and it corrected the pre-registration itself: #5876's C10 ratios 0.00895/0.00156/0.00623 are |S_2|/pi at X = 1e4/1e5/1e6 (the denominators 1229/9592/78498 are pi(1e4/1e5/1e6), not 1e3..1e5) and are unsigned. Reproduced exactly: S_2 = 11, -15, 489. The p>=5 negative control gives 10, -16, 488, which is what #5904 says it must (10, 16, 488), and #5876's C9 squarefree densities reproduce to 2e-5 (0.7533/0.7481/0.7472; the 1e4 gap is the p>=3 vs p>=2 convention). Independent cross-check of a served artifact: against research/shifted-prime-mobius-sums.json (the project's table to 2^38), S_2(2^j) - (muM_j + 1) = 0 at all 14 rungs j = 10..23, the +1 being mu(1) at p = 3; identically zero, not approximately. New data (h = 2, pi counts p >= 3, null = 2*Binomial(nonzero,1/2)-nonzero, 10k draws, seed 20261026): X=1e3 S=11 |S|/pi=0.06587 sd=11.28 z=0.98; 1e4 S=11 0.00896 sd=30.45 z=0.36; 1e5 S=-15 0.00156 sd=85.21 z=-0.18; 1e6 S=489 0.00623 sd=243.73 z=2.01; 3e6 S=214 0.00099 sd=404.55 z=0.53; 1e7 S=-444 0.00067 sd=707.88 z=-0.63. h = 4,6,8 and the per-rung Omega strata are in parity_census_12088.json. The identity used (elementary, zero novelty claimed): mu = (-1)^Omega on squarefree, hence S_2(X) = sum_j (-1)^j C_j(X), C_j = #{p <= X : p-2 squarefree, Omega(p-2) = j}. At 1e7: C_1 = 58980, C_2 = 166439, C_3 = 174102, Omega >= 4 net +66198, and -58980+166439-174102+66198 = -444 exactly. The imbalance is a four-way cancellation of groups of size 0.05pi..0.26pi down to 6.7e-4 relative, four orders below the support bound c = 0.7479. PRE-REGISTERED TESTS. F1 (random-sign model refuted by |z| > 4 at any X >= 1e5): did NOT fire, max |z| = 2.01 at 1e6. F2 (|S_2|/pi <= 0.05): holds at every X >= 1e5 (max 0.00623); the 1e3 rung gives 0.0659 with S = 11 against null sd 11.28, registered as a small-sample tension only. F3 FIRED: the fixed-depth concentration hypothesis (some Omega class carrying >= 0.5 of sum_j S_j^2 at the top rung) is REFUTED at 1e7 with max share 0.460 (Omega = 3), and the argmax drifts (Omega = 2: 0.545 at 1e5, 0.484 at 1e6, 0.420 at 1e7; Omega = 3: 0.083 -> 0.460), so no fixed class names the imbalance. Cost measured, not estimated: exact sieve plus all statistics to 1e7 in 3.18 s single core (primes 0.03, omega 0.62, squarefree 0.01, big_omega 2.53); this assignment used about 15 s of compute. Not done: no rung above 1e7; no segmented sieve (the code materialises N-length arrays, disclosed as the first thing the next step must build); Omega >= 4 bucketed so its internal sign is only netted; no proof of anything.