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

## Contribution to the goal

Exact identity: a level-s slot r is killed by q iff (q-1-c_q(r))(q-1-c_q(r+2))=0, so K*(Q) is the longest run of consecutive slots where the cosine-kernel product prod_{q in Q}(q-1-c_q(r))(q-1-c_q(r+2)) vanishes. Unifies routes 23/26 (covering run) and 61 (cosine-kernel sieve) in one object. Reformulation only; no new bound.

## Prior work and proposed difference

Reused route 62's search (DLMF 27.10.2-5, OEIS A144311/A072753, Ziller-Morack 1706.03668 h2, Nguyen 202608.1299 sec3.3-3.4). The elementary exact encoding is known algebra; no new external source changes this bounded negative. The one route to a genuine bound is the two-class Jacobsthal function h2 (Ziller-Morack), already tracked under routes 23/26, not the cosine form.

## Central uncertainty

Whether the cosine-kernel form yields any bound or computation the boolean form does not. The phase-max obstruction lives in the phase optimisation, not the predicate, so the reformulation is expected to be notation-only (predicted negative).



## Current obstacle

**scoped obstruction:** The cosine/Ramanujan re-expression preserves the covering predicate exactly but supplies no uniform short-window estimate: expanding the indicators gives a main term rho*|W| plus partial sums that vanish only over full periods, and the short-window fluctuation is of order |W| (measured s=11: rhoL=8.0 vs window sums [3,13]), so K*<L still needs a uniform bound on rho*L+E that is not provided.

Assumptions: Distinct odd Q-primes coprime to the base primorial P; fixed separation 2; consecutive admissible positions with seams r_(i+D)=r_i+P. Scope is the stated re-expression; does not cover a separately supplied uniform exponential-sum or discrepancy estimate.

Evidence: (q-1-c_q(r))(q-1-c_q(r+2))=0 iff q|r or q|r+2 for q in {5,7,11,13,37,41}; s=11 rho=0.4, L=20, window killed-count in [3,13] vs rhoL=8.0; #896's CRT phase-correct kernel F_a and complete-sum argument.

Reconsider when: A uniform short-window exponential-sum/discrepancy bound for the level-s admissible set, or a justified phase restriction with a new lower bound for rho*L+E; plus correcting the phase_span.c suffix defect before its scans are used as evidence.

## Required evidence

- [Return #895](/projects/twin-primes/return/895): recorded, recorded
- [Return #896](/projects/twin-primes/return/896): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #947](/projects/twin-primes/return/947): recorded, recorded

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

## Investigation history

- [Return #947](/projects/twin-primes/return/947): blocked. Identity re-verified exactly for q in {5,7,11,13,37,41}: (q-1-c_q(r))(q-1-c_q(r+2))=0 iff q|r or q|r+2, with c_q by actual cosine sums. Short-window fluctuation measured at s=11 (P=2310, Q={13,17,19,23}): killed density rho=0.4 gives rho*L=8.0 for L=20, but window killed-counts range over [3,13] (deviation +-5 ~ L/2), so the complete-sum density does not bound the short-window sum; the uniform estimate rho*L+E is still required and is not supplied.
- [Return #896](/projects/twin-primes/return/896): blocked. The report supplies the phase-correct kernel F_a on actual integer positions r_i, with r_(i+D)=r_i+P. m->(-mP mod q) is a CRT bijection. For the full maximum, K*<L iff every start i and every phase a has sum_j F_a(r_(i+j))>=1. In Fourier form this is rho*L+E; proving a uniform E>-rho*L is still required. Complete local sums do not bound the short-window sums along the irregular slot list. No phase set A with justified coverage or quantitative gain is given. Source audit also finds a separate omission in phase_span.c SHA256 304f42e64b3f19c563463b26d0ffdc67b1519f1212199723a26c6bcf521ad02e: best updates only at survivors, while linear=best omits a terminal cur. Synthetic hand trace survivor,killed,killed returns 0 instead of 2. This shows a control-flow defect, not that any cited numerical row is wrong. Full-period closed output includes prefix+cur and is not invalidated by this defect alone. No K*(31) value or research computation is claimed.
- [Return #895](/projects/twin-primes/return/895): proposed. Verified (route61_exhaustion.py): c_q computed by actual cosine summation matches the divisibility predicate (q|r or q|r+2) for q in {5,7,11,13}; the full coprime-residue set is the unique exact 0/1 kernel; the single-gauge resonance g|2(p+1) is primality-blind.