Base: b8bb28c4de357827be0bdad95a5e9b40d9c4e669 (author file at 3d5c666634766357e2c6484afb0e6fbd4c8f2540; review at §7 repair spec incorporated 2026-09-20). Date: 2026-09-20. Scope: one concrete pairing only. No survey, no further construction after this mechanism. Prior pairing work in this hunt: none (narrow grep over hunts/prime_pair_error/*.md for pairing / opposite-sign / mu(pm) returns only unrelated sign mentions in arc-split and residue notes). Earlier attempt produced no improved D_N bound; no finer analytic claim is made here. The sign identity mu(2m) = -mu(m) for odd m is standard, not a discovery.
1. Setup and exact statement
Inherit N >= 4, K = floor(sqrt(N)), Y = N/K, M = floor(N/2), U = floor(sqrt(M)) and
D_N = S_smooth(N,K) + Sig1(N) + Sig2(N), Sig2 = sum_{b=2..U} log b * M_b(N),
M_b(N) = sum_{a=U+1..V(b)} mu(a) w_N(ab), V(b) = floor(M/b),
from FACTORIZATION_NEXT_STEP (sections 2-3) and FINAL_ACCEPTANCE (sections 2.5-2.6). S_smooth and Sig1 are carried unchanged. The pairing acts inside each M_b only.
Theorem (exact p = 2 pairing with complete boundary). Fix N >= 4, b in [2, U], V = V(b), F = floor(V/2), and general kernel w (either branch). With
P_b = sum_{m odd, U < m <= F} mu(m) [w(mb) - w(2mb)] (paired differences), T_b = sum_{m odd, max(U,F) < m <= V} mu(m) w(mb) (upper-tail singles), H_b = sum_{t <= U, U < 2t <= V} mu(2t) w(2tb) (lower-head singles), Z_b = sum_{m = 2t, U < t <= F, t even} mu(m) w(mb) (identically zero),
M_b = P_b + T_b + H_b + Z_b holds exactly, and Z_b = 0 termwise.
Ranges: for N <= 11, Sig2 is empty (V(b) <= U for all b) and the identity is vacuous 0 = 0. For N >= 12 all sums are finite integer-indexed sums; the m <= Y versus m > Y split inside w is over integers against rational Y, hence exact.
Proof. Partition (U, V] into disjoint sets: odd m <= V/2 (A_odd_low), odd m > V/2 (A_odd_high), m = 2t with t <= U (head), m = 2t with U < t <= V/2 and t odd (mid-odd-half), m = 2t with U < t <= V/2 and t even (mid-even-half). Every even m <= V has half t = m/2 <= V/2, so the three even classes exhaust the evens; the two odd classes exhaust the odds. The map m |-> 2m is a bijection A_odd_low -> mid-odd-half with inverse n |-> n/2 (when F < U the low class is empty and the tail is all odds of (U, V]). For odd m: if m is squarefree, mu(2m) = -mu(m); if m is not squarefree (odd squared prime divides m, hence divides 2m), mu(m) = mu(2m) = 0, so mu(m) w(mb) + mu(2m) w(2mb) = mu(m)[w(mb) - w(2mb)] termwise in all cases. Summing the paired classes gives P_b; the remaining classes give T_b, H_b, Z_b. On mid-even-half, 4 | m gives mu(m) = 0 termwise, so Z_b = 0. Q.E.D.
Preserved structure. b-side prime powers (log 4, log 8, log 9, ...) are untouched (b is fixed by the pairing). a-side proper prime powers (exponent >= 2, not primes themselves) contribute exact zeros via mu = 0 on both sides of the identity (e.g. m = 9, 25, 49 in tails; m divisible by 4 in Z_b). p-divisibility (p = 2): m even in range is either head (partner below range), a paired partner (m = 2t with t odd in range), or a proved zero (4 | m). Floor jumps are inside w and never smoothed: the difference below keeps them exactly.
2. Paired weight difference (exact, under the proved ab > Y gate)
Section 4.2 of the prior note proves ab > Y for every Sig2 pair at every N >= 4 (small N checked by hand, gate asserted per-pair in the check script). Hence on Sig2, w(mb) = 1 - floor(N/(mb)) and the paired difference is exactly
Delta(m, b) = w(mb) - w(2mb) = floor(N/(2mb)) - floor(N/(mb)) <= 0,
i.e. minus the integer floor drop. No Taylor expansion, no smoothing cost.
3. Complete quantitative bound versus the same baseline
Decisive arithmetic hypotheses: the sign flip mu(2m) = -mu(m) for odd m (exact, elementary) and |mu| <= 1 for every estimate. No Mertens, PNT, or RH input. Majorants used on both sides: |w(m)| <= N/m (both branches: N/m - 1, floor(N/m) - 1) and, in the integer branch, |Delta(m,b)| <= N/(2mb) + 1 (the +1 is the exact floor-jump cost per pair: floor(X) - floor(X/2) < X/2 + 1).
Per-b paired majorant:
|M_b| <= P^maj_b + T^maj_b + H^maj_b, P^maj_b = sum_{odd U<m<=F} (N/(2mb) + 1), T^maj_b = sum_{odd max(U,F)<m<=V} N/(mb), H^maj_b = sum_{t<=U: U<2t<=V} N/(2tb).
Pre-pairing baseline with the same majorants: |M_b| <= B_b = sum_{U<a<=V} N/(ab). Weighted totals: B(N) = sum_b log b * B_b, Ptot(N) = sum_b log b * (P^maj+T^maj+H^maj); |Sig2| <= Ptot(N) versus |Sig2| <= B(N).
Declared majorants, audited asymptotics (review §4; finite-N ratios in results_mobius_pairing.json approach from above and are diagnostics, not the constants):
B(N) = (1/48) N log^3 N + O(N log^2 N), Ptot(N) = (1/192) N log^3 N + O(N log^2 N).
Measured Ptot/B = 0.60 at N = 10000, 0.69 at N = 1000, 0.87 at N = 100 against the asymptotic 1/4; exactly 1.0 at N <= 36 where every paired range is vacuous and the identity reduces to an odd/even split with zero gain. This is a factor-of-four asymptotic leading-constant reduction only. No exponent and no log power is improved.
Non-vacuous range is exactly b <= b* = floor(M / [2(U+1)]) (F > U iff b <= b*). Per-b Theta(N/b) behavior of the tail is asserted only on this restricted range; no uniform theta is asserted at empty endpoints (for b > b*, P^maj_b = 0 and T^maj_b + H^maj_b = B_b exactly, and that strip contributes only O(N log N)). The boundary totals T^maj + H^maj with the +1 floor costs are Theta(N log^2 N), one log power below the main term, so the boundaries do not repay the entire leading gain; the paired majorant itself remains Theta(N log^3 N) from below (summed paired main over b <= b* gives (1/192) N log^3 N - O(N log^2 N)).
Scope of the lower bound: it pins this chosen absolute-value majorant only. It does not bound the true signed sum Sig2 (whose mu(a) signs can cancel), and it does not constrain every p=2 technique (signed pairings, multi-prime mechanisms, or any estimate that does not pass through |mu| <= 1 with this floor majorant).
Combination with the principal term: D_N = S_smooth + Sig1 + sum_b log b (P_b+T_b+H_b) exactly. S_smooth = N(log Y + H_K - 2 - gamma) is preserved untouched in closed form. Sig1 is untouched and has no accepted bound (formal heuristic scale match only, per FINAL_ACCEPTANCE section 4). Therefore the joint target |S_smooth + Sig1 + Sig2| << N^{1/2+eps} is unmoved: the complete bound stays |S_smooth + Sig1| (unpriced) + O(N log^3 N). For the D_N target, no gain survives.
4. Verdict on this mechanism
Retire only this explicit majorant construction: the declared absolute-value majorant for this p = 2 Sig2 pairing is pinned at Theta(N log^3 N) from above and below (leading constants 1/48 baseline, 1/192 paired), so it cannot advance the D_N target. This is not a failed estimate (the identity is exact and the comparison apples-to-apples), and it is not a claim about any other pairing or method: the true signed Sig2, general signed pairings, and multi-prime mechanisms are unconstrained, and no bound is asserted false. Stop this construction here.
Relation to the conditional smooth zero-mode R_eta (review §5; SIGNED_MEAN_RENEWAL §8): R_eta is constructed conditionally on a hypothetical off-critical zero rho = beta + i gamma with beta > 1/2, and shows macroscopic scale-renewal properties alone cannot rule out such a mode. It is a C^1 continuum profile: it lacks genuine discrete prime-power jumps (the true R has jump log p at each p^k) and lacks any discrete mu convolution, so it cannot instantiate Sig2, which directly evaluates mu(a). Its failure to instantiate refutes no generic intermediate lemma of the hyperbola decomposition, and nothing here claims every conceivable RH proof must use mu. Conversely, the majorant obstruction above is unconditional and assumes no off-critical zero. The earlier empirical Type I ratios |S_smooth + Sig1|/N^{3/4} (0.17 at N = 400, 0.75 at N = 1000) are superseded by this paragraph for purposes of this memo.
5. Evidence
Scope distinction: the general-N claim (identity for all N >= 4) rests on the combinatorial proof in §1, not on machine checks. The exact tests below cover only the listed finite cutoffs. Ownership: mobius_pairing_check.py and results_mobius_pairing.json are author-worker artifacts (dispatch producing 3d5c666); mobius_pairing_independent_check.py and results_mobius_pairing_independent.json are independent review-worker artifacts; both run under the repo .venv and neither imports the other.
mobius_pairing_check.py (N <= 10000, predicted < 60 s: ~20k exact Fraction terms at the largest N): per-b exact rational identity M_b = P_b+T_b+H_b (zero Fraction defect), ab > Y gate asserted per pair, mu(2m) = -mu(m) asserted on odd squarefree m with mu(2m) = 0 on even m, square/nonsquare/proper-prime-power mu spot checks, missing-head/missing-tail/wrong-sign lesions (exact nonzero defects), before/after per-b prime-log coefficient match, bound totals B(N) vs Ptot(N) and unpaired-term fractions. Outputs results_mobius_pairing.json. Observed: all exact checks pass with zero defect; all three lesions detected; Ptot/B from 1.0 (N <= 36, vacuous) down to 0.60 at N = 10000; unpaired fraction from 1.0 down to 0.43; small-N ratios reported as diagnostics only, never as asymptotics. Exact commands and outputs recorded in section 6.
6. Run record
$ git status --short # clean; git rev-parse HEAD -> a626c916a0082e59b2c4c0eca83035e948160d3e
$ .venv/bin/python hunts/prime_pair_error/mobius_pairing_check.py
exit 0, elapsed_s 0.73 (estimate was < 60 s; numerical compute well under 2 min)
N=100: defect 0, gate_min 60, Ptot/B 0.872, unpaired 0.778
N=1000: defect 0, gate_min 426, Ptot/B 0.691, unpaired 0.514
N=10000: defect 0, gate_min 4200, Ptot/B 0.600, unpaired 0.425
lesions N=100: missing_head 11, missing_tail 5, wrong_sign 2 (all exact, nonzero)
lesions N=400: missing_head 39, missing_tail 17, wrong_sign 20 (all exact, nonzero)
mu sieve == sympy.mobius on 1..10000; divisor-sum identity to 200; mu spots
(4,8,9,25,27,36,49,121 -> 0; 6 -> 1; 30 -> -1) all pass.
$ .venv/bin/python # Sig2 cross-check vs results_factorization_diagnostic.json
all 11 table N (16..1000) agree to <= 1.5e-14 (float64 rounding).
$ .venv/bin/python -m pytest tests/test_prime_pair_residue.py -q -n0
27 passed in 0.96s.
$ git add hunts/prime_pair_error/MOBIUS_PAIRING.md \
hunts/prime_pair_error/mobius_pairing_check.py \
hunts/prime_pair_error/results_mobius_pairing.json
$ git commit -m "prime_pair_error: exact p=2 Mobius pairing in Sigma_2 ..."Author integration rerun (2026-09-20, base b8bb28c; JSONs backed up to /tmp and restored afterward, so evidence files are byte-identical and unstaged):
$ .venv/bin/python hunts/prime_pair_error/mobius_pairing_check.py
exit 0, elapsed 0.84s; zero Fraction defect; lesions 11/5/2 at N=100, 39/17/20
at N=400 (exact, nonzero); Ptot/B 0.60 at N=10000.
$ .venv/bin/python hunts/prime_pair_error/mobius_pairing_independent_check.py
exit 0, 0.46s; 15794 pairs over 20 cutoffs; global min gate margin 3 (at N=15).
$ .venv/bin/python -c "from fractions import Fraction; ..." # radical check
(sqrt(2)*sqrt(35/36)*5/6)^2 = 875/648 > 1 -> strict gate inequality exact.
$ grep -i "impossib|every.*pairing|all.*technique|cannot be" MOBIUS_PAIRING.md
one match: line 104 "does not constrain every p=2 technique" (scoping negation).
$ git status --short # only the two scoped docs modified; checkers/JSONs untouched.