# Completed signed transition research round > Public mirror edition: repository-internal working instructions were removed. See MIRROR.md. **The twin-prime margin remains OPEN.** This contract was executed from `cb51292`; the returned reports were reviewed in [transition-round-audit.md](transition-round-audit.md) and reconciled in [transition-round-review.md](transition-round-review.md). Read those conclusions before reusing this specification. The workflow below records the bounded assignments that ran; another execution needs a specified new estimate, changed hypothesis or correctness concern. For the current prepared assignments, use [RESEARCH-EXECUTION.md](RESEARCH-EXECUTION.md). Repository: `~/Files/Git/primeoire`. Mathematical baseline: `029208295099b60d7a8b837d19eb361cfc12e6b1`. Use the current checkout containing these briefs, after reconciling any later changes. Do not reset to the baseline and lose the briefs. Record the actual commit and working-tree state in each report. ## 1. Mission and ownership The immediate question is whether the **signed** transition correlation can be estimated beyond its present absolute budget, with all other terms priced. A useful round can also correct an argument or identify the precise additional theorem needed. It need not obtain a proof of twins to return useful, completed work. | Owner | Read and execute | Deliverable | |---|---|---| | Reviewer | [Energy review specification](transition-energy-review-spec.md) | Independent check or correction of the sharp energy and failed norm-transfer argument | | Source researcher | [Source matching specification](transition-source-match-spec.md) | At most three detailed theorem-interface assessments, with actual coefficients and all losses | | Joint-budget researcher | [Joint budget specification](transition-joint-budget-spec.md) | Analysis of mixed terms, possible cancellation between pieces, and a complete sufficient inequality | | Core handler | Sections 2–5 below, plus the three returns | A bounded attempt on the central signed kernel, dependency review, global payoff and integrated record | Launch the three workers in parallel if capacity permits; with fewer slots, start the reviewer first and queue the other assignments. Keep the central signed estimate and final mathematical judgment with the handler. Workers must not expand into additional agents or unrelated campaigns. Model settings do not alter the evidence requirements. ## 2. Mandatory context and mathematical contract ### Exact coefficients Let x=2^j tend to infinity, J_x=(x/2,x] intersected with the integers, and fix 0W}\Lambda(r),\qquad C_i(m)=\sum_{d\mid m,\ D_i=z_i. Let h_i=1-rho_i, \[ \widetilde C_i(m)=\sum_{d\mid m}\mu(d)h_i(d)\beta_{W_i}(m/d), \qquad T_i=C_i-\widetilde C_i. \] The four signed sums, always with the right factor at n-2, are \[ \begin{split} R_{00}&=\sum_{n\in J_x}\widetilde C_L(n)\widetilde C_R(n-2),\\ R_{10}&=\sum_{n\in J_x}T_L(n)\widetilde C_R(n-2),\qquad R_{01}=\sum_{n\in J_x}\widetilde C_L(n)T_R(n-2),\\ R_{11}&=\sum_{n\in J_x}T_L(n)T_R(n-2). \end{split} \] Their sum is the full raw corner R_cor=sum C_L(n)C_R(n-2). The exact central kernel is \[ R_{11}=\sum_{\substack{D_L0, the sharp and transition squared norms are also bounded below by a positive eta-dependent multiple of x log^2 x. The recorded range is eta2)(1-1/(p-1)^2)>0. The target remains fixed c,K>0 and an unbounded set of dyadic x with C_2x+E_dagger(x)>=c x/log^K x. A sufficient route through (E2) must keep positive slack, choose fixed H>K, and put **all** estimates on the same scales. The sufficiently strong rescaled-average and cumulative consumers in the handoff are also admissible. Separate unbounded good sets need not intersect. Even R_cor=o(x) would leave E_out to estimate. ## 3. Execution sequence and dependency handling 1. **Preflight.** Record commit, local modifications, assigned paths and the existing compute allocation. Read the four specs. Check for already completed reports before dispatch; reuse completed work unless a precise defect or changed hypothesis justifies a repeat. 2. **Dispatch.** Give each worker its spec, this document and the actual launch commit. Require it to read the owning arguments, not just the summary. Ask for an immediate message on a decisive defect or useful lemma, followed by the complete written return. 3. **Work locally.** The handler attempts a matched signed estimate for (E1). The exact small-prime cutoff difference in coefficient-energy (16) is one candidate starting point. Freeze x-dependent cutoffs when applying it; retain exceptional divisibility and all separation costs. A different method is allowed if its coefficient interface is stated. 4. **Use returns conditionally.** Source matching and joint-budget work may proceed while the audit runs. Mark dependence on reviewed norm claims explicitly. If a defect is found, suspend only the affected inference, notify the other workers, and repair or remove it before accepting a downstream result. Independent work can continue. 5. **Integrate once the arguments are available.** Read the decisive proofs, resolve discrepancies by derivation or counterexample, and price every accepted result in (E2). Worker agreement is not proof. Have the relevant worker check any substantive repair before closing the round, within the same assigned scope. The handler must finish at least one bounded analytic attempt: state the proposed lemma, derive its strongest justified bound, and compare it with the required normalization. If it stops at a missing input, write that input with exact quantifiers and identify the step that needs it. Do not turn the round into an indefinite search for a complete proof. ## 4. Files, computation and authorization Each worker owns only its report and any specifically justified companion validator. Planned report paths and question ids are in the individual specs; they are not completed findings. The handler owns these planned outputs (to be created during execution): ```yaml central_note: research/transition-signed-estimate.md central_question: Q-transition-signed-estimate integration_note: research/transition-round-review.md integration_question: Q-transition-round-review ``` Each result note needs the repository ledger block with its own question id, status matching the result, todo C and an honest parity scope. The handler acknowledges new ids in TODO when those notes exist. Only the handler updates shared records: OUTCOMES, TODO, QUESTIONS, SCRIPTS, SEARCH-CONVENTIONS, IMPORT-MAP, the owning baseline notes and current-state entry points. Workers put proposed outcome entries and precise corrections in their reports. They must not concurrently stage, commit, reset, or edit shared files. Preserve unrelated local work. Isolated worktrees are acceptable if the handler needs them; record the commit used and integrate reviewed changes deliberately. This round is primarily analytic. Numerical work needs a named falsifier, control and stopping condition before execution. Reuse retained inputs where applicable; label proxy cutoffs honestly. The existing six-hour compute allowance is aggregate for the campaign, not six hours per worker or a fresh allocation from this document. Account for prior consumption and sum concurrent job allocations before scheduling substantial compute. Unknown remaining allowance is not permission to start a large run. Use `node research/qc/embed.js` for retained script output. Routine exact checks are evidence for finite identities, not an asymptotic correlation rate. Do not repeat the empty fixed-corner census or increase enumeration merely to collect more twins. Existing internal research and local-commit authorization applies. No push, publication or external circulation is authorized; the repository moratorium remains in force. ## 5. Required returns, acceptance and closeout Every report must contain: 1. Question, baseline, scope and calibration; state the open twin margin. 2. Exact result or conditional statement, including cutoffs, quantifiers, constants, scale set and normalization. 3. Load-bearing derivation and source locators, versions and access status; separate imported results from proof steps actually checked. 4. Complete cost table, with each term marked derived, imported, conditional or open; include its contribution to (E2). 5. Falsifier/control and whether it ran; no required computation when the decisive test is a mathematical counterexample or source mismatch. 6. Reusable findings, precise failed steps and remaining obligations. 7. A proposed OUTCOMES entry, affected dependencies and one justified next move, or a reason no further repeat is useful. A completed specification is not an executed assignment. A completed audit is not a new signed bound. A theorem match must include the parameter substitution and final cost; a conditional lemma must retain its unproved premise. A weak saving is an intermediate estimate if it does not reach the consumer. A scoped failure must not become a universal closure or a claim that no theorem exists. The handler's integration note must include a finding-by-finding disposition, a dependency table for the baseline conclusions, the best justified combined budget, and an explicit verdict on whether any region, exact cut or sufficient twin margin changed. Record successes and failures together by research question in OUTCOMES; Git owns revisions and authorship. Add result-ledger ids to TODO C when their notes exist; never mark an arithmetic question solved merely because its report is done. Reconcile affected live statements in CLAUDE, README Status, G2-STATE section 0, TWIN-REDUCTION, the research router, AGENT-START and RESEARCH-HANDOFF. Regenerate indexes and run the relevant validators and the repository gates after integration: ```sh node research/qc.js --index --strict node research/qc/selftest.js node research/audit-numbers.js git diff --check ``` Fix failures, commit the reviewed internal work locally, and report the commit, changed mathematical claims, unresolved target and next precise input. If the round produces no signed improvement, say so and preserve the audit and failed-step evidence. The handler should not ask for a new round merely to repeat an unchanged failed interface.