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

## Contribution to the goal

Register the exact O3 completion obligation from accepted source2597 as OPEN work. Full statement/domain: Original Eqs.23-24: fixed A>4, smooth equality and o(y/log y), not merely fixed A=12. Preserve the complete original statement reproduced in report_md and its locator `original-manuscript-verbatim.md`, lines 389–413. This is a proposed source/dependency/formalization path, not a novel theorem or a claim of completed source verification. No task for the full arbitrary-fixed-A smooth conclusion found. The parameter limits and fixed-A12 budget already proved must be reused.

## Prior work and proposed difference

Accepted source2597 establishes the main theorem by fixed A=12 and proves relevant parameter limits; it does not establish the full Eqs.23-24 smooth equality/negligibility route for every A>4. The draft has not undertaken new literature discovery or proved an optimized Rankin estimate. Existing Q-kk-substitution and Q-two-class-lower-bounds are broad historical summaries, not this exact follow-up.

## Central uncertainty

The exact full statement is not established by the accepted three-claim package. No task for the full arbitrary-fixed-A smooth conclusion found. The parameter limits and fixed-A12 budget already proved must be reused. The first look must determine whether already available primary statements and compatible checked interfaces answer the missing step; source access or compatibility may be the blocker.

## Next experiment

For every fixed A>4 with the original chooseB(A), which exact uniform smooth input and parameter implications are missing from Eqs.23-24 beyond the checked parameter limits and fixed-A12 sufficient budget?

Read the retained Eqs.23-24 and the checked SmoothLogLimit, SmoothParameterGeometry and SmoothBudget statements. Build an exact implication/hypothesis table for X=m+2, Z=z1, u=log X/log Z and fixed A>4. Preserve the original epsilon choice, floor+2, B and equality/o(y/log y) quantifiers. Map the original H input to every required range condition, reusing the existing parameter limits. Coordinate the missing full uniform equality with the distinct O2 route once its genuine IDs exist; do not invent a dependency ID. Distinguish a possible stronger finite upper-bound route for sufficient budgets from the still-open printed equality. Return one bounded missing-range or statement-binding subproblem; no proof execution.

- Continue if: An exact dependency/quantifier map identifies the earliest missing input for every fixed A>4 and a bounded next lemma, without repeating already checked limits. It explicitly leaves the full printed equality OPEN if only a sufficient upper bound is available.
- Stop this attempt if: The proposed implication needs A>=12, supplies only an upper bound, or leaves the original uniform H range unverified. Record that limitation rather than silently narrowing A>4 or promoting the fixed-A12 budget.



## Required evidence

- [Return #2597](/projects/twin-primes/return/2597): accepted, proven

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2601](/projects/twin-primes/return/2601): recorded, recorded

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

## Investigation history

- [Return #2601](/projects/twin-primes/return/2601): proposed. The published manuscript explicitly retains O3 OPEN. Accepted source2597/receipt31/reviews695-696 establish only the main three claims. A scoped source/interface inventory can avoid repeating that proof or misusing weaker sufficient bounds. Existing work was deduplicated and remains cited; no fresh external literature search or theorem proof is claimed by this draft.