# Research specification: mixed terms and the combined signed budget **The twin-prime margin remains OPEN.** Execute under the [core handler's contract](TRANSITION-RESEARCH-EXECUTION.md). Your question is whether separately bounding R_00,R_10,R_01,R_11 loses cancellation that can be preserved in a **justified estimate**, and how any useful bound enters the complete reduction. The handler owns the main R_11 attempt; coordinate without duplicating it. ## 1. Inputs and owned output Mathematical baseline: `029208295099b60d7a8b837d19eb361cfc12e6b1`. Read the common contract, [sharp-corner-transition.md](sharp-corner-transition.md) section 5, and [corner-coefficient-energy.md](corner-coefficient-energy.md) sections 3–4. For the consumer read [endpoint-target-audit.md](endpoint-target-audit.md), [RESEARCH-HANDOFF.md](RESEARCH-HANDOFF.md) section 3, and [cross-campaign-synthesis.md](cross-campaign-synthesis.md) for the common-scale issue. Check the relevant OUTCOMES and QUESTIONS entries. Planned outputs, to be created during execution: ```yaml report: research/transition-joint-budget.md question: Q-transition-joint-budget optional_validator: research/transition-joint-budget-validation.js ``` Own only this report and, for a new named falsifier, the optional validator. Shared records and final integration belong to the handler. Work conditionally if the reviewer disputes an input; state the dependency. ## 2. Exact task and minimum analytic attempt The definitions and equation (E2) in the common contract fix the object: \[ R_{\rm cor}=R_{00}+R_{10}+R_{01}+R_{11},\qquad E_\dagger=E_{\rm out}+R_{\rm cor}+O_H(x/\log^H x). \] First check the equality and all supports. The auxiliary smoothing does not alter E_dagger or the controlled cuts. The four individual norm-based budgets are already known; reproducing them is a consistency check, not completion of this assignment. Then undertake at least one concrete analytic attempt on a mixed term or a specified combination of the four pieces. State what structure the estimate retains before Cauchy or a triangle inequality. Possible starting points are the logarithmic cutoff integral, cancellation between nearby cutoff differences, or grouping a mixed term with the transition pair. Choose a mechanism and derive its actual cost; merely listing these possibilities is not a return. For every proposed grouping: - Write the exact coefficient or kernel after grouping, with both shifted endpoints and all divisor/cofactor branches present. - Identify the signed quantity that the method bounds and the first operation, if any, that discards cancellation. - Price cutoff variation, parameter integration, smoothing error, approximation terms and all logarithmic or power losses. - Compare the resulting inequality with the existing joint budget. If it does not improve it, isolate the first limiting term. One-point Gram positivity does not imply positive cross-correlation at shift 2. Conversely, large separate norms do not show that the four signed pieces cannot cancel. Any claimed obstruction must name the particular proposed inequality and prove failure at its stated scope. ## 3. State an actual sufficient consumer Provide a table for R_00,R_10,R_01,R_11 and E_out, or for a disjoint grouping of them. Mark each bound as derived, imported, conditional or open. Give its sign, scale set and contribution after normalization by x. Do not silently drop an open entry or count a term twice. One admissible conditional specification is the following. Write B_mix=R_00+R_10+R_01. On a **common unbounded set** of dyadic x, suppose the deductions actually supply \[ R_{11}\ge-F_{11}(x),\quad B_{\rm mix}\ge-F_{\rm mix}(x), \quad E_{\rm out}\ge-F_{\rm out}(x), \] with specified error functions, and \[ F_{11}+F_{\rm mix}+F_{\rm out} \le C_2x-2c x/\log^Kx \] for fixed c,K>0. Choosing fixed H>K in the remaining error would then give the sufficient positive margin for all sufficiently large x in that set. This is a **template of a conditional implication**, not a bound supplied by the assignment. Determine whether your proposed mechanism provides any of its actual inputs. A joint lower estimate for the entire sum is also admissible and may retain more cancellation. A sufficiently strong rescaled-average or cumulative consumer is equally acceptable. If using one, give the precise sampling, exceptional-set and uniformity argument. The earlier dyadic transfer for the multiplicative subfamily is not automatically a transfer for these full coefficients. No necessity of separate componentwise o(x) estimates is assumed. ## 4. Required return and stopping rule Return the checked decomposition, the concrete attempted joint or mixed estimate, its complete cost table, and the strongest justified combined inequality. Distinguish an actual improvement from a conditional specification and from a reformulation. A conditional statement must name the exact unproved lemma and the line where it is needed. For a failed attempt, state whether the loss came from a norm bound, boundary term, coefficient separation, scale alignment or another identified step. Preserve any reusable identity. Do not convert an unestimated E_out into a claimed impossibility for the corner method. Use exact finite checks only for a named gap in the algebra; do not infer a sign or asymptotic rate from a finite matrix. Return proposed OUTCOMES and next-step text to the handler. Stop after the specified attempt and complete accounting are written, or after a decisive defect has been identified and its dependency consequences supplied. Proving the entire outside remainder is not required to complete this bounded assignment; leaving it explicitly open is required when it is unproved.