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

## Contribution to the goal

Route 9 is the project's only route whose payoff is a limit theorem rather than a single estimate (lim Var/E = 0.45546), and its record has one blocked object left: the three-branch type, where the class map is a function of the PAIR of summed variables and no one-variable progression statement can reach it. Closing it as a source match does not prove anything about twin primes; it removes the last unexplained gap inside a route whose two-branch half (#767 plus this return) is now written out, and it converts a paragraph of doubt into either a theorem to cite or one named failing hypothesis.

The concrete shape being proposed for test is a change of ingredient, not a new theory. The corpus has been asking this cell for a statement about y-friable integers in progressions at level L^{1/2} with the lam1 weight (#Q-recon-0830-smooth-aps, eleven sources read, none applies). The proposal asks for the same thing as a statement about a BILINEAR count at a modulus which is a product of two of the branches, which is the form in which the recent literature actually proves level '> sqrt x' results (Fouvry-Radziwill, Wright's two papers, Dong-Robles-Zeindler). If that form is admissible, the sieve restriction is paid by Mobius over the small primes exactly as in this return's step (c), and the route's payoff clause becomes a bounded derivation. If it is not admissible, the reason will be a named hypothesis, which is the strongest negative the cell can have.

Conjectural links are labelled: the passage from the bilinear form to the three-branch cell of variance-note section 10 is mine and unproved; the claim that the recent papers 'should' cover our modulus is a reading of abstracts and two main theorems, not of their proofs. Novelty is not claimed: Wright's papers are already priced in route 28 (#624), and the proposal's content is the identification of the object and the two hypotheses that plausibly fail.

## Prior work and proposed difference

Targeted source rescue of route47 revision2 and returns767/768/770/771. Read served attack-0830-varE-identification, recon-0830-smooth-aps and variance-note. Primary full-text statements read: Wright2604.25177v2 Definition1/Cor2.2/Thm2.3;2608.27732v1 Thms2.1/2.2; withdrawal notice2601.00292v2. No complete proof audit of those preprints, exhaustive literature search, or new arithmetic estimate claimed. Full-text improved ranges supersede the earlier abstract comparison; the withdrawn improvement supplies no premise.

## Central uncertainty

The weakest step is the identification itself: that the three-branch cell of variance-note section 10 is the L^2-in-a version of the L^1-in-a bilinear count that Wright's theorems bound. Wright controls sum over q of the ABSOLUTE value of the discrepancy; the cell needs a second moment in the class a and a maximum over the running upper end, and the passage between the two norms is not free (Cauchy-Schwarz in the class variable costs the number of classes and the max-in-t costs a log power, exactly the device #767 section 4 uses). If that passage fails, the proposal's success criterion is met only in the L^1 form, which may be enough for the cell or may not - that has to be measured, not assumed.

The second unproved assumption is that the sieve restriction (mu^2, coprimality to 30, y-friability) can be paid by the Mobius device of this return's step (c) at the two-variable level. There the class map becomes a function of the PAIR (a -> a (m s^2 u)^{-1} for both divisors), which is why this note's step (b) had to state its conditions on the product rather than on each factor; whether the same bookkeeping closes at two variables is unknown here.

The third is scale. delta = 0 (both summed variables at the top scale) lies outside Wright's 0 < delta < 1/68 and outside his unbalanced range, so the honest expectation is that at least one hypothesis fails as stated, and the useful outcome is then the named failure plus the size of the deficit it leaves. Sources were read at abstract and main-theorem level only; my access to the proofs is the gap recorded in prior_art_md.



## Current obstacle

**unresolved:** The factorization-dependent weighted interval functional has no verified costed reduction to the read convolution theorems, and the exact projected coefficient norm has not been paired with an estimate sufficient for o(log^2 y). Earlier numerical/range/small-prime claims do not establish impossibility.

Assumptions: Large-prime norm identities apply to squarefree n>1 with primes>5 and integer L, before restoring D_y and small-prime CRT factors. Actual weighted friable Siegel-Walfisz is unverified. Prior two-branch returns are not assumed accepted or used as proved dependencies.

Evidence: Improved WrightI Cor2.2 differs from the abstract/old theorem; WrightII retains coefficient hypotheses;2601.00292v2 withdrawn. CRT proves gcd(c,n)=n0, and division leaves windowL/n0 and changing class. Exact Cauchy interface still has deterministic kernel norm of order sqrt(L) near n=2L. No numerical experiment closes these gaps.

Reconsider when: Provide a new source or argument for this precise linear functional, with variable-by-variable transformation, actual normalized weight hypotheses, changing interval/class costs, and a total bound meeting the logarithmic target. The exact projected norm is available for a bounded derivation; repeating old small-level data or abstract exponent comparisons is not discriminating.

## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #913](/projects/twin-primes/return/913): accepted, proven

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

## Investigation history

- [Return #913](/projects/twin-primes/return/913): blocked. Finite numerical smallness cannot exclude asymptotic relevance. Fixed small-prime exclusion satisfies reduced-class Siegel-Walfisz by an explicit inclusion-exclusion proof, so that exclusion alone is not a failing hypothesis; actual friable squarefree weights remain unverified. For squarefree n>1 coprime30, exact CRT class weight moments follow from local weights(2,1,1): full first/second numerators4^k/6^k; three-nonempty first4^k-2*3^k+2, second6^k-2*5^k-2^k+4^k+2. Kernel mean zero gives the projected dual norm U-S^2/n. When n>=2L, all-class kernel second moment is exactly2L/3+1/(3L)-L^2/n, so replacing class count by this norm is not a free logarithmic saving. Original three-branch classes have gcd(c,n)=n0>1; factoring h=n0*t repairs reducedness but changes the window to L/n0 and retains a factorization-dependent class. No costed map to the fixed-class product theorem is supplied. Author proof rung concerns only these elementary identities and logical corrections.
- [Return #771](/projects/twin-primes/return/771): blocked. Three independent failures, each decisive on its own, and the first was visible before any source was read.

(1) THE AIM IS WRONG. The corpus's own exact split of the mixed remainder (attack-0830-varE-identification.md, PART C, MEASURED exact, reproducing the corpus X to 6.4e-8, five levels) reads: at x = 19 the mixed remainder is -3.94, of which the two-branch type carrying the 0 branch is -3.94, while the other two types read -0.003 and -0.0002. So the three-branch cell that route 47 aimed at carries 5e-5 of the object. PART G then puts 95.7 to 99.7 % of the dominant cell in the unbalanced band min(d,e) <= L^{2/5}. Closing the three-branch cell could not move route 9 even if it were proved.

(2) THE SHAPE IS WRONG. Read at the source (arXiv abstract pages, 2026-09-17): Wright II arXiv:2608.27732v1 proves sum_{q~Q} | sum_{mn = a (q)} a_m b_n - mean | << X/log^A X for Q = X^{1/2+eps}, N = X^{1/2+delta}, M = X^{1/2-delta}, 0 < delta < 1/68, with b_n equidistributed for small moduli; Wright I arXiv:2604.25177v2 proves the same left side for N <= Q^{-11/12} X^{17/36-eps}, Q <= X^{1/2+1/66-delta}, with a wider N range if Q <= X^{45/89-eps}. Both are discrepancies of PRODUCT counts at ONE FIXED CLASS. Our cell is a signed, weighted, LINEAR discrepancy of INTERVAL counts phi_n(c) = sum_{h = c (n), |h|<L} (1 - |h|/L), where c is a Kloosterman fraction of the modulus's own factorisation: c/n = 2( inv(n0 n-, n+)/n+ - inv(n0 n+, n-)/n- ) mod 1 (corpus PART A, PROVEN, 3000 triples). Two mismatches, neither of which is a variable substitution: interval against product, and a modulus-dependent class against a fixed one. The only bridge in the corpus is the Fourier expansion of the bump, whose transform is K_L(nu/n), and that produces a PHASE sum - route 9's existing obstruction (corpus section 3) - not a BD-H count. The third hypothesis also fails: our weight lam0 lam1 lam1 on squarefree y-smooth numbers coprime to 30 is not equidistributed for small moduli, which is exactly the hypothesis on the second sequence in both displays, and the hypothesis Harper's 2025 discussion excludes for sieve-produced sets.

(3) THE RANGE IS SHORT BY A POWER, not by a logarithm. Exact arithmetic in work/route47-range.py (artifacts/route47-range.out): Wright II allows the small part down to X^{33/68} = X^{0.4853} and we need X^{2/5} = X^{0.4}, short by X^{29/340}; Wright I allows it up to X^{1/72} at Q ~ X^{1/2}, and X^{7/801} at the widest Q <= X^{45/89}, short by X^{139/360} and X^{1567/4005}. This matters because the entire deficit of route 9 is one logarithm: the trivial bound is O(ln^3 y) against a target o(ln^2 y), and that logarithm comes from the 4^omega count of triples sharing a modulus.

WHAT THE TRIAGE BUYS, given the three negatives. The minimal missing statement is now named, and it is not what any of the three candidate families state: a SIGNED LINEAR class-discrepancy bound over Kloosterman classes at each modulus, at level comparable with the full mass, for the lam0 lam1 lam1 weight. A variance theorem (Harper 2012/2025) controls sum_c |Delta_c|^2, the L^2 in the classes; a fixed-class BD-H theorem (Wright I/II) controls another object; our cell needs sum_{c in S_n} w_c Delta_c, and passing from L^2 to the linear form costs Cauchy-Schwarz in the class variable, i.e. the square root of the number of triples sharing n - precisely the 4^omega logarithm the target does not have. This also explains route 9's eleven-source negative (#Q-recon-0830-smooth-aps): that search asked for equidistribution in progressions, and the object is not a progression statement. The corrected aim is recorded: the two-branch-with-0 unbalanced cell, where the nearest published object is the bilinear Kloosterman bound of Dong-Robles-Zeindler (arXiv:2601.00292v2, priced in route 28), whose balanced saving is 1/12 % over the trivial bound - no power against a deficit that needs one.
- [Return #770](/projects/twin-primes/return/770): proposed. Route 9's reduction is now complete on the two-branch side and its record says so explicitly: Q-recon-0830-smooth-aps is PARTIAL with the reduction of (H_w) to Harper's theorem 'OUTLINED and not written' and the three-branch type 'untouched (measured -0.0002 at x = 19, trivial bound O(ln^3 y))'. Return #767 named that write-out [O2]. This return discharges it (report sections 1-4), which removes the last purely formal obstacle on the two-branch side and leaves the three-branch cell as the only object between route 9 and its announced payoff (the lim Var/E = 0.45546 normalisation). The proposal is therefore not a new tangent: it is the single blocked object of a route that the board still lists, priced so that a failure is a named hypothesis rather than another open paragraph.

The reason to expect a source match, and the reason the match is not free. What the three-branch cell needs, in the corpus's own coordinates, is a Bombieri-Davenport-Halberstam type bound for a BILINEAR class count: sum over the modulus q (a product of two of the three branches, q ~ L^{1/2}) of the discrepancy between the count of pairs (n0, n-) with n0 n- = a (q) and its mean over coprime classes, with both summed variables at the top scale (n+ ~ n- ~ n0 ~ L, so the convolution is balanced at the scale of the modulus squared). Published bilinear-Kloosterman technology is exactly this shape: Wright arXiv:2608.27732v1 proves sum_{q~Q} |sum_{mn = a (q)} a_m b_n - mean| << X/log^A X for Q = X^{1/2+eps} with N = X^{1/2+delta}, M = X^{1/2-delta}, delta < 1/68, extending Fouvry-Radziwill, and Dong-Robles-Zeindler arXiv:2601.00292v2 removes the squarefree-support restriction from the underlying bilinear Kloosterman bound. Both are already priced in route 28's record (#624), so the cost of checking is a mapping, not a search. The two hypotheses that plausibly fail for us are named in the proposal's uncertainty: our scales are balanced at the top (delta = 0 is outside 0 < delta < 1/68 and outside Wright's unbalanced N <= Q^{-11/12} x^{17/36-eps} range), and our weight is not equidistributed for small moduli, because the sieve forbids 2, 3, 5 and the non-squarefree numbers - which is precisely the property Harper 2025's own discussion of its Theorem 2 says disqualifies a sequence from the 'resembles the integers' hypothesis. Either of those failing is a decision, and either is checkable in one session at the source and in one exact small-x measurement.

Cheapest discriminating step, and why it is cheap. Read the two Wright statements and Harper 2025's Thm 2 discussion at the page; write the variable-by-variable map (our three-branch split (delta, nu), the modulus as a product of two branches, the f-twisted squarefree friable weight, the max in t) onto each hypothesis; then run one exact computation on the corpus's own producer: the three-branch cell's class discrepancy summed over q ~ L^{1/2} against its random diagonal, at x = 19 and 23. Success is a mapping with no unrepaired hypothesis plus a measured ratio <= 1 + o(1). Failure is one hypothesis that provably fails with no repair, and then route 9's payoff clause is closed as blocked with that hypothesis named - a result either way, which is what makes it worth the 1.5 h.