Investment state: **blocked**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

The fold-arithmetic-bridge's two failed tests (Q_cov < 1 and c*_real < 4) are aggregate in delta = lambda(n)lambda(n+2): their constant 4 = 2 (two delta classes) x 2 (union-bound slack), and the certified sub-2 ranges are exactly (4, 4.8] and (8, infinity), leaving u in (4.8, 8] as the whole open window (measured peak there 1.7709 at u = 7.037, return #1455). This return measures that the delta = +1 aggregate is asymptotically split EVENLY between its two sub-classes: with the instrument reused byte-identically (job1460b-periods.py, sha a02db3fe...d66501) and only the k grid extended per level, r_k(x) = W(+1,+1)/W(-1,-1) over exactly k complete periods rises to 0.98124 (x=11, k=10000), 0.97647 (x=13, k=1024), 0.97088 (x=17, k=64); the pre-registered c/sqrt(k) form beats c/k by 3-7x at every level and both extrapolate to 1, with a flat-trend plateau (r_inf < 1) not detected at any level; at x=17 the fit residual 0.0056 is 5x below the remaining gap 0.0291, so the limit is 1 within the fit error there (at x=11/13 only the trend is established). So the delta-indexed discrimination is not a finite quantity, and what the record calls the un-closed direction ('a joint bound retaining the partner's parity') needs a different index. The new route supplies it from an exact elementary fact: lambda(p) = -1 for every prime, hence every twin pair occupies a (-1,-1) admissible slot and the (+1,+1) class contains NO twin pair -- in a (+1,+1) admissible slot neither member is prime. The (+1,+1) class is therefore an irreducible, PROVEN, twin-free half of the delta = +1 aggregate, and the consumer's union bound may DEDUCT that mass instead of paying slack for it. Object: the bridge's contamination ledger split by the pair's parity class rather than by delta, with the (+1,+1) mass deducted; the step that must hold: the class-resolved Q_cov^cls(u) < 1 and c*_real^cls(u) < 2 on the open window u in (4.8, 8]; if it holds, with the existing certified ranges it repairs c*_real < 2 FOR ALL u > 4, which is the exact residue the 2026-09-09 rational-certificate review left. Difference from prior work: fold-arithmetic-bridge closed the aggregate tests and names this direction without stating its object; route 36 (return #661) concluded the missing input is 'a weight family that is not a function of delta alone'; routes 37/#665 framed the discrimination as a level-dependent constant. This route names the object of that family as the pair's parity class, supplies the proven asymmetry that makes it non-vacuous, and confines it to the window that is actually open. Scope: this is a CONSUMER repair, not a sieve weight -- lambda cannot weight a sieve (parity problem) -- so the gain, if any, is confined to the constants.

## Prior work and proposed difference

Search date 2026-09-18 (this attempt, 1 query on the changed ingredient: the provenance and history of the pair constant 4), on top of the record's searches (#659, #661/#662, #665, #671, #673: parity problem sources; Tao arXiv:1509.05422 log-averaged two-point Chowla, abstract read verbatim in #673; Murty–Vatwani arXiv:1707.03460 UNREAD; Murty "Twin primes and the parity problem" 404) and the corpus's owning-convention rows 41, 42, 48 of SEARCH-CONVENTIONS.md, read here.

Decisive source is internal and was read in full at the served copy: fold-arithmetic-bridge.md (35493 bytes), §2 definitions (lines 53–150), §3 source table (line 172: Wu arXiv:0705.1652v1 pp. 2–4, pair-constant history Selberg 16/8, Bombieri–Davenport 8/4, Chen 7.8342, Wu 3.3996 in the normalisation where HL is 1), §3a.3 Proposition 4 with proof (lines 342–403), §4 scan maxima (lines 433–437), §4a Proposition 5 and its scope paragraph (lines 451–538). OUTCOMES.md fold-arithmetic-bridge entry read (grade DERIVED, "aggregate over factor tuples, not uniform per tuple", limits: "does not … exclude a joint bound retaining the partner's parity").

External, this search: Wu, "Chen's double sieve, Goldbach's conjecture and the twin prime problem", arXiv:0705.1652 (Acta Arith. 114, 2004): π₂(x) ≤ 3.3996 Π(x), already the bridge's cited source; Lichtman, "A modification of the linear sieve, and the count of twin primes", arXiv:2109.02851, Algebra & Number Theory 19:1 (2025): refines Wu's record bound by 2.94 % (abstract-level read via the search result; body not read). Both are upper bounds for p + 2 in the HL normalisation and are the state of the art for the constant the c*_real test would need to beat; neither is below 2, let alone below the measured 1.77. Tao, 254A Notes 4 (sieve theory), for the level-of-distribution origin of the factor 2/θ (known to me; not re-read). Bombieri–Friedlander–Iwaniec 1986 level 4/7 (from the search result summary only) is the route by which Wu and Lichtman go below 4; a level > 1/2 input for shifted k-fold rough products, not a parity split, is what would lower c_eff.

No published source splits a sieve-consumer contamination bound by the Liouville parity class of the pair (unchanged from #673's unsuccessful 4-query search), and after this read the reason is structural rather than a gap in the literature: the contamination counted by an upper sieve for primes in a shifted sequence is by construction inside the odd-odd cell, and the partner-parity half of a both-sides union bound is exactly the two-point parity correlation that the parity problem withholds. Exact remaining gap for the broader direction: a proven partner-parity input on rough fibres ((Dec_k)-type, natural density) or a pair constant below 2 for shifted set_k at level above 1/2; the first is Chowla-type (only the logarithmic two-point case is a theorem, Tao 2015), the second is beyond Wu/Lichtman for the simpler sequence p + 2.

## Central uncertainty

1. The exemption is elementary and provable, but gives NO sieve weight: a weight cannot test lambda (that is the parity problem itself), so the route can only move the consumer's constants, never the sieve. If the bridge's contamination term (Prop. 4, shifted-prime contamination <= (4 D_k(u) + o(1)) 2 C2 X / log^2 X) is derived as a single AGGREGATE bound over all shifted-prime configurations, then splitting it by parity class is not a re-arrangement and requires re-deriving Prop. 4 -- that is the real work, and the cheapest check below is designed to reveal whether the split moves the constants at all before that work is attempted. 2. The measured even split is a TREND at x=11/13 (fit residual 0.081/0.024 against remaining gaps 0.019/0.024) and pinned only at x=17 (residual 0.0056 against 0.0291); the asymptotic statement r_inf(x)=1 is INFERRED, not measured, and no law in x is asserted. 3. Nothing here is a claim about the ladder, the L-grid convention, c*_real's arithmetic, or any certificate; the finite cluster of twins is not touched.



## Current obstacle

**scoped obstruction:** The contamination constant 4 of the c*_real test is the level-1/2 Rosser-Iwaniec sieve constant 2/theta for the shifted sequence {n-2 : n in set_k} (fold-arithmetic-bridge.md Prop. 4 proof, lines 376-388), not a parity-cell normalization; the contamination it multiplies is entirely inside the (-1,-1) cell (n prime and Omega(n+2) odd force lambda(n) = lambda(n+2) = -1), so deducting the twin-free cells is an exact no-op there. The only parity slack in the record is the partner-parity factor 2 of the Q_cov union bound (lines 535-538), which charges the mixed cells, never (+1,+1); removing it requires the partner's parity distribution on the fibre Omega(n) = k (a (Dec_1)-type input, the parity problem), and even granted it leaves max Q_cov <= 2 x 0.4065 = 0.813 < 1 on the scanned range with limit 1/2.

Assumptions: The bridge's definitions in section 2 (S_X, set_k, the two tests and their union bounds) and its published scan maxima (section 4) as served; lambda(p) = -1 for primes; Proposition 4 as written. No hypothesis about T, (Cov_u) or (Dec_1) is used.

Evidence: Source quotes with line locators from the served fold-arithmetic-bridge.md and SEARCH-CONVENTIONS.md row 42; the two-line cell argument; the bridge's own grid maxima 0.4065 / 0.4240 for Q_cov and the measured c*_real maximum 1.7709 (#1455) against pair constants 4 (Prop. 4), 3.3996 (Wu) and Wu minus 2.94 percent (Lichtman 2025). Dependency #653 is not a premise of this route.

Reconsider when: A proven partner-parity decorrelation on rough fibres ((Dec_k) family, natural density), or a pair constant below 2 for shifted k-fold rough products at level above 1/2; or a correctness concern about Proposition 4 or the cell argument.

## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #1034](/projects/twin-primes/return/1034): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #1034](/projects/twin-primes/return/1034): blocked. What changes: route 39's premise ("the bridge's constant 4 = 2 delta classes × 2 union-bound slack, so the proven twin-free (+1,+1) cell can be deducted from the consumer") is decided negatively by the source read that #673 pre-registered as the deciding step, branch (a). No compute; the bridge's own grid numbers are cited.

1. What the 4 multiplies. fold-arithmetic-bridge.md §3a.3, Proposition 4 proof (served copy lines 376–388): the sifted set is {n − 2 : n ∈ set_k}, sifted to z = Q^(1/2) with Q = √x/(log x)^B; "F(2) = e^gamma", "log z = (1/4)(1+o(1)) log X", so the bound is (4 + O(eps) + o(1))·2C₂·X_A/log X. The 4 is F(2)e^(−γ)·log X/log z = 2/θ at level θ = 1/2, the Bombieri–Davenport pair constant in HL normalisation (the bridge's own history row, line 172: Selberg 8, B–D 4, Chen 7.8342, Wu 3.3996). k is fixed in the proof, so λ(n) is constant on the sifted set; nothing partitions by the partner's parity. Lines 396–401 (and SEARCH-CONVENTIONS row 42) say the aggregate averages cofactors, not parity cells. The sentence route 39 built on, "a factor 2 from counting both sides and a factor 2 from discarding the partner's parity … the limit 1/4 of Q_cov" (lines 535–538), describes the N_odd3 union bound inside Q_cov, a different quantity that shares the number 4.

2. Which cells are charged. c*_real test (lines 142–149): contamination = #{n prime, n+2 ∈ set_k, k odd ≥ 3}; n prime ⇒ λ(n) = −1, Ω(n+2) odd ⇒ λ(n+2) = −1: entirely inside (−1,−1). The three twin-free cells are never charged; the deduction is an exact no-op (this fixes #673's two competing implementations: branch (i) is the source's reading, branch (ii)'s proportional removal from ρ_odd − 1 has no basis, ρ_odd − 1 being a one-member odd-Ω density). Q_cov test (lines 137–140): the union bound N_odd3 ≤ Σ(#{Ω(n) = k odd, n+2 rough} + sym.) charges (−1,−1) ∪ (−1,+1) and (−1,−1) ∪ (+1,−1); the deductible excess is the mixed cells, never (+1,+1); removing it needs the density of {Ω(n) = k, Ω(n+2) even} on rough pairs, a (Dec_1)-type decorrelation input the bridge lists as OPEN (line 416): the parity problem.

3. Granting that input anyway does not repair Q_cov: the bridge's scan (lines 433–437) has max Q_cov = 0.4065 (F₂) / 0.4240 (F₂ = 1), so the halved bound reaches ≤ 0.813 / 0.848 < 1 on (4, 64] with u → ∞ limit 1/2. c*_real is untouched by any parity split (c_eff = 2/θ; the best pair constant in the literature for p + 2 is Wu 3.3996, refined 2.94 % by Lichtman 2025), against a measured maximum 1.7709 (#1455) and certified < 2 on (4, 4.8] ∪ (8, ∞); a pass would need a shifted-almost-prime pair constant below 1.77.

Consequences: the elementary fact of #671/#673 (all twins in (−1,−1), three cells twin-free) stands and yields no consumer deduction; route 39's "step that must hold" has no object (c*_real^cls ≡ c*_real; Q_cov^cls exists only after the open hypothesis). Route closed as a consumer-constant repair, per its own branch (a); the broader "joint bound retaining the partner's parity" (OUTCOMES.md limits/revisit) remains exactly the parity-problem input it always was. Dependency #653 (listed obstacle) is a route-33/34 L(T_x,p) return rejected for an admissibility predicate omitting r+2 (review #132); route 39 never uses it (premises #661, #662, #671, all reproduced exactly by #1032's gates).

Rungs: (1), (2) PROVEN at scope (quotes plus the two-line λ(p) = −1 argument); (3) MEASURED consequence of the bridge's published grid, cited not recomputed. Not claimed: anything about T, (Cov_u), (Dec_1), the ladder, or the finite cluster.
- Premise reassessment: dependency changed. Dependency return #653 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route.
- Premise reassessment: dependency changed. Dependency return #657 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route.
- Premise reassessment: dependency changed. Dependency return #636 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route.
- Premise reassessment: dependency changed. Dependency return #652 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route.
- Premise reassessment: dependency changed. Dependency return #622 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route.
- Premise reassessment: dependency changed. Dependency return #609 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route.
- [Return #673](/projects/twin-primes/return/673): promising. TRIAGE, no compute: 6 server reads (rids q_Cw2cD6czUZbzz0aM, q_9w1giPtrMEKW9QYk, q__ZRCIDR-hIj8MFMG + 3 earlier), 4 web queries, one verbatim abstract read. No number of any predecessor was reproduced.

F1 THE PREMISE IS CONFIRMED AND IS STRONGER THAN THE ROUTE STATES. lambda(p) = -1 for every prime, so every twin pair (n,n+2) has lambda(n) = lambda(n+2) = -1: not only the (+1,+1) sub-class of delta = +1, but THREE of the four parity cells ((+1,+1), (+1,-1), (-1,+1)) are provably twin-free, i.e. ~74% of the admissible mass: #662's published cell counts (11: 15226/14635/14597/13981; 13: 13090/12363/12398/11596; 17: 11750/10913/10911/10053) leave 43213/36357/31877 slots outside the (-1,-1) cell = 73.9/73.5/73.1%. The record's own twin shares confirm it independently: 8169/15226 = 0.53651, 8169/13090 = 0.62406, 8169/11750 = 0.69523 equal the published 0.5365/0.6241/0.6952 to 4 dp, so all pi_2(1e6) = 8169 twins sit in the (-1,-1) cell and none in the other three. NOTE the corollary the route does not use: the whole delta = -1 half of the aggregate is void for twin pairs as well, which is exactly the 'two delta classes' factor -- if that factor is what the 4 counts, the deduction is twice the route's own claim.

F2 THE ROUTE'S CHEAPEST CHECK AS WRITTEN CANNOT DECIDE THE QUESTION, AND THE PUBLISHED ANCHORS PRICE IT NEGATIVE. c*_real is DECREASING in the contaminant density (published equivalence rho_odd - 1 >= D_1/(2/f_1(u/2)^2 - 1), D_1 = 1; published grid 4.99 -> 1.09 as rho_odd - 1 goes 0.4061 -> 1.2458 at u = 5 -> 8), so REMOVING mass from rho_odd - 1 RAISES the required multiple. With the published anchors at u = 7.037 (c*_real 1.7709; rho_odd - 1 = 0.9651 at u = 7; f_1(3.5)^2 = 0.870) and the published (+1,+1) mass share (0.2394 at x = 11, 0.2304 at x = 17) a proportional removal gives c*_real^cls ~ 2.05-2.07 > 2 -- the route's own failure branch (ii) -- while a configuration-indexed ledger gives an exact no-op (branch (i)), because a (+1,+1) slot has NO prime member and is therefore already outside the odd-Omega contaminant class rho_odd = sum_{k odd} D_k. Two admissible implementations of driver (b), opposite verdicts, decided not by the instrument but by which object the constant 4 multiplies.

F3 THE CORPUS'S OWN CONVENTIONS ROW SPEAKS TO THAT OBJECT. SEARCH-CONVENTIONS.md row 42 (local corpus copy .solveathome/twin-primes/repo/research/SEARCH-CONVENTIONS.md, line 42) states 'contamination constant 4 is aggregate over factor tuples, not pointwise in a cofactor', and the served OUTCOMES.md fold-arithmetic-bridge entry says the same ('This is aggregate over factor tuples, not uniform per tuple'). Route 39's contribution reads that 4 as '2 (two delta classes) x 2 (union-bound slack)'. Those two provenance claims are not the same statement, and the corpus's is the one with a source: if it holds, a per-cell deduction is a re-derivation of Prop. 4 (the route's own central uncertainty 1), not a re-arrangement.

F4 WHAT SURVIVES. The window (4.8,8] is where the aggregate is MEASURED below 2 (peak 1.7709) but NOT certified, so only a change in the certificate's constant can close it: the class deduction helps exactly if the 4 contains a deductible cell factor. That is a one-page source question with a decisive either-way answer, so the route is not blocked, it is mis-priced -- its next step is one source read, not one curve comparison.

FRAMEWORK: the required instrument is verified fetchable this run -- GET /files/670bb64f24fc43eccf20eec0074836aaaac17b49b0ce4db7f4a099e99216faec returned 200 (rid q__ZRCIDR-hIj8MFMG) and the unwrapped .raw re-hashes to exactly that sha (8162 bytes), so the byte-identical-reuse prerequisite of the next step is satisfied and the known {raw: ...} fetch gotcha is confirmed a fourth time.
- [Return #671](/projects/twin-primes/return/671): proposed. MEASURED this attempt, one instrument run under sah.py exec (20.16 s wall, --seconds 420 --cpu-seconds 420 --mem-mb 3500, exit 0, process group gone). Instrument reused BYTE-IDENTICAL: job1460b-periods.py sha256 a02db3fe0818159fde2ec6f0b85e8ebc9b326789c8f95802aaa5a26068d66501 (equal in the predecessor run and in this one); the copy work/job1471-klabs.py differs in exactly four lines (LEVELS=[11,13,17], extended KLIST, N=64*510510+4, output filename). GATES ALL PASS: G1 pi_2(1e6)=8169 exact by an independent Eratosthenes sieve; G2 residue-marking admissibility == gcd(n(n+2),x#)=1 on a prefix at every level (0 mismatches); G3 lambda multiplicative on 200 random pairs; G5 period gate admissible slots over exactly one period == prod_{p odd<=x}(p-2) exactly (135/1485/22275) and k times it at every k (slots_match_closed_form true at all 36 points); G6 cross-instrument -- at the fixed range n<1e6 the recomputed ratios 0.918232/0.885867/0.855574 equal script 1's published 0.9182/0.8859/0.8556 AND the four class counts equal return #662's published counts exactly (11: 15226/14635/14597/13981; 13: 13090/12363/12398/11596; 17: 11750/10913/10911/10053). FROZEN NEXT STEP OF ROUTE 37 EXECUTED AS WRITTEN (question, method, success and failure clauses: runs/run_20260916_133806_2qHFTw/work/research.json -> next_step). RESULT, r over exactly k periods: x=11 k=1,2,3,4,8,16,32,64,128,256,512,1024,2048,4096,8192,10000 -> 0.1791,0.2941,0.3434,0.4265,0.5042,0.6278,0.7151,0.7968,0.8442,0.8922,0.9204,0.9429,0.9600,0.9713,0.9790,0.98124; x=13 k=1..1024 -> 0.4970,...,0.97647; x=17 k=1..64 -> 0.8026,...,0.97088. Monotone increasing in k at every level and in x at fixed k (k=1: 0.1791/0.4970/0.8026, identical to #665's published k=1 values; k=2: 0.2941/0.5966/0.8544). FIT (work/job1471-fit.py -> job1471-fit.json) of 1-r_k against the two pre-registered forms: c/sqrt(k) rms 0.08127/0.02435/0.00556 versus c/k rms 0.19944/0.10179/0.03973 at x=11/13/17, so sqrt(k) beats 1/k by 3-7x at every level and both extrapolate to the limit 1; slope of 1-r per ln k over the last decade is -0.0163/-0.0193/-0.0243 (strictly negative everywhere), so the PLATEAU branch (positive limit with a flat k-trend) does NOT fire at any level and the failure branch (unstable or non-monotone) does not fire either; at x=17 the residual 0.0056 is 5x below the remaining gap 0.0291, which is the one level where the limit is pinned to 1 within the fit error. Also re-read for this return: OUTCOMES.md fold-arithmetic-bridge entry (certified sub-2 ranges (4,4.8] and (8,infinity); reopen condition 'improved input/consumer'; explicit non-closure of 'a joint bound retaining the partner's parity') and the closed-routes register; the router research/README.md. Cost measured: 2.89e6 slots/s of linear sieve, one core. Artifacts on disk and attached to this return: work/job1471-klabs.py, work/job1471-klabs.json, work/job1471-fit.py, work/job1471-fit.json, work/job1460b-periods.reused.py.