name: global-proof description: Write a detailed branch-local theorem-level diagnostic draft from an accepted proof sketch before step-level proof work, with review-ready claim identity, block mapping, audits, hard-step diagnostics, and complete, partial, or sketch-blocked status. metadata: version: "1.0.0"

Global Proof

Use this internal skill after /proof-sketch-review accepts a sketch and before any /proof-step runs.

Contract

Required inputs:

• perspective_M/idea_N/setting.md
• accepted perspective_M/idea_N/proof_sketch.md
• accepted perspective_M/idea_N/proof_sketch_review.md

Optional inputs:

• perspective_M/idea_N/proof_tracker.md
• triggering perspective_M/idea_N/global_proof_review.md when rerunning after REVISE_GLOBAL
• prior same-branch sketch, global-proof-review, step, assembly, or final review failures when the controller provides them as diagnostic history

Output:

• perspective_M/idea_N/global_proof.md

Do not modify sketches, step artifacts, final proofs, trackers, worker logs, or accepted results. Do not write or overwrite latex_template/.

Responsibilities

• Draft a detailed theorem-level proof implied by the accepted sketch.
• Make the draft review-ready for a later global-proof reviewer by exposing claim identity, theorem-level blocks, dependency flow, assumption provenance, cited-tool obligations, quantitative-dependence obligations, and unresolved links.
• Check that the sketch steps cohere globally and plausibly compose to the formalized goal.
• Attempt theorem-critical derivations deeply enough to expose whether generated conditions, recurrences, invariants, stability, boundedness, membership, convergence, quantitative specializations, structural lower/sign/coercivity/nondegeneracy/support claims, scope upgrades, or theorem-closure blocks have a concrete closure mechanism and mechanism source under the declared theorem scope.
• Trace theorem-facing generated outputs from their producer step, theorem-level block, primitive source, accepted-derived source, cited tool with a valid diagnostic discharge path under the Source-To-Claim Adequacy Gate, direct derivation, standard fact or tool, current-notation wrapper, primitive-source derivation with checked source-convention compatibility and raw-assumption feasibility, or explicitly conditional target to every consumer block, closure block, specialization, and final theorem use.
• Expose missing links, hard steps, and likely step-local bottlenecks before step workers run.
• Expose candidate idea-level obstructions for /global-proof-review only when the global diagnostic identifies the required theorem-contract change: changed primitive assumptions, changed algorithm/model/procedure, changed theorem scope/mode/metric, changed exposed dependence, changed success criterion, adding a theorem-facing mechanism source not supported by the setting, or weakening the conclusion.
• Route sketch-level or roadmap-level blockers back to /proof-sketch.

global_proof.md is diagnostic guidance. It is not proof evidence, not a cited result, not an assumption source, and not a substitute for accepted step proofs.

Workflow

1. Read the exact setting and goal from setting.md.
2. Verify proof_sketch_review.md accepts the current proof_sketch.md; if not, stop as an input contract failure, do not write or overwrite global_proof.md, and ask the controller to rerun or hand off the correct upstream artifact or stage.
3. Read every sketch step, dependency, assumption, intended proof tool, output target, rate objective, and blocker.
4. If rerunning after global_proof_review.md returned REVISE_GLOBAL, read its blocking issues and required repair bundle. Repair only diagnostic-artifact defects under the same accepted sketch; do not change sketch steps, dependencies, assumptions, output targets, generated-output flows, generated-condition interfaces, accumulation behavior, closure mechanisms, mechanism witnesses, mechanism sources, boundary exclusions, theorem scope/mode/metric, exposed dependence, success criterion, or conclusion inside /global-proof.
5. Determine goal mode:

• exact-goal mode: setting.md states the exact theorem claim.
• target-spec mode: setting.md gives a theorem-ready target specification whose exact final bound, constants, or dependence were not fixed.

1. Apply the shared Assumption Provenance Contract and Generated Output Flow Gate from ../_shared/checklists/artifact-contracts.md. Check whether the accepted sketch proves every theorem-facing generated-object, event, local-validity, recurrence, boundedness, membership, structural condition, or invariant fact needed by an unconditional target from primitive conditions and accepted dependencies. Check whether every theorem-facing generated output consumed by the attempted theorem has a legal producer, consumers, final theorem use, and dependency path.
2. Apply the shared Explicit Rate Contract from ../_shared/checklists/artifact-contracts.md whenever the setting, goal, sketch, or review exposes quantitative dependence. Apply the shared Baseline Invariance Obligation whenever those artifacts expose a recovery, reduction, specialization, zero-defect, exact-limit, exact/noiseless, or baseline-case conclusion.
3. Apply the shared Theorem-Critical Mechanism Witness Gate and Residual-To-Target Adequacy Gate from ../_shared/checklists/artifact-contracts.md. For every theorem-critical generated condition, recurrence, invariant, stability, boundedness, membership, convergence, quantitative specialization, structural lower/sign/coercivity/nondegeneracy/support claim, scope upgrade, or theorem-closure block, name a derivation path from primitive conditions, accepted dependencies, cited tools, direct derivations, standard facts or tools, current-notation wrappers, primitive-source derivations, or explicitly conditional targets to the target condition or control, including the claim class and theorem role, closure mechanism, mechanism source, source-to-claim match, source-convention compatibility when a source supplies the mechanism, residual-to-target adequacy when a bridge transfers a produced, baseline, transformed, or surrogate object or control into the consumed target, key positive/control term or structural source, opposing defect terms, closure/dominance/absorption relation, accumulation behavior under the declared scope when repeated, iterated, recursive, limiting, or otherwise accumulated, boundary regimes handled or excluded, and producer-consumer provenance when generated outputs are consumed; otherwise record the missing witness or bridge as a blocker.
4. Apply the shared Exported Interface Feasibility Gate and Residual-To-Target Adequacy Gate from ../_shared/checklists/artifact-contracts.md. For every theorem-critical downstream-facing output target, generated condition, margin, threshold, simplified bound, basin or membership certificate, recurrence interface, cited-tool wrapper conclusion, direct derivation output, standard fact or tool output, current-notation wrapper output, primitive-source derivation output, or public bridge, expose the raw controls, exported interface, defect terms, controlled and uncontrolled defect classes, residual-to-target adequacy when a bridge transfers a produced, baseline, transformed, or surrogate object or control into the consumed target, dominance or transfer relation, margin or threshold source, source-convention compatibility when a source supplies the output, object-target compatibility when an entry, reference, population, no-error, or baseline object is consumed, branch or boundary handling when applicable, and consumers. If a direct derivation, standard fact or tool, current-notation wrapper, or primitive-source derivation supplies the output, also expose the exact setting convention, source-convention compatibility, object-target compatibility when applicable, and raw-assumption-to-output path. If this path or residual-to-target adequacy is missing under the accepted sketch, classify it as sketch/interface defect or candidate idea/theorem-contract defect.
5. Classify every theorem-critical hard obligation as step-local, sketch/interface defect, or idea/theorem-contract defect under the shared Step-Locality And Theorem-Contract Gate.
6. Write a detailed theorem-level proof draft that follows the sketch dependency order and explicitly marks unresolved links.
7. Build a theorem-level block map and a coverage map from each Step ID to the portion of the whole-proof draft it supports.
8. Build the required ## Scope And Closure Certificate for every theorem-critical generated condition, recurrence, invariant, stability, boundedness, membership, convergence, quantitative specialization, structural lower/sign/coercivity/nondegeneracy/support claim, scope upgrade, or theorem-closure block covered by the Theorem-Critical Mechanism Witness Gate, including the concrete recurrence/potential or mechanism-specific control relation at diagnostic granularity for every all-time or accumulated certificate row, mechanism source / boundary exclusion, obligation locality classification, noncircular closure status, and entry-state / first-update stress entries when applicable.
9. Build ## Generated Output Flow for theorem-facing generated outputs consumed by blocks, closure, specialization, or final theorem use.
10. Audit dependency closure, generated-output flow, assumption provenance, cited-tool applicability, quantitative dependence, exported-interface feasibility, scope-accumulation compatibility, early obstruction patterns, and same-setting repair plausibility.
11. Diagnose the smallest required repair level using the status rules below.
12. Write global_proof.md using ../_shared/templates/global-proof.md.

Global Draft Rules

• Treat the accepted proof_sketch.md as binding. Do not search for a new roadmap, change Step IDs, change step claims, change dependencies, change assumptions, or change output targets inside /global-proof.
• A REVISE_GLOBAL rerun from global_proof_review.md is a rewrite of the diagnostic only. If the required repair bundle cannot be satisfied without changing the accepted sketch or setting, write the obstruction explicitly and use SKETCH_BLOCKED rather than silently repairing the roadmap.
• In ## Attempted Theorem Claim, state the target theorem or strongest attempted claim faithful to setting.md. Include goal mode, theorem-facing assumptions, conclusion, scope, and any exposed quantitative mode or dependence.
• In exact-goal mode, do not weaken the claim unless the draft marks the mismatch as a blocker.
• In target-spec mode, do not guess unsupported rates, constants, convergence modes, or dependence. Draft only the strongest concrete claim that the accepted sketch appears able to support, and flag missing required dependence as a gap.
• Do not weaken or replace an inherited baseline invariance obligation inside the attempted claim. If the diagnostic can support only a conservative, conditional, stopped, finite-scope, or remainder-only surrogate, mark the baseline mismatch as a blocker or candidate idea-level obstruction rather than presenting it as the same theorem contract.
• In ## Whole-Proof Draft, write a substantive theorem-level diagnostic draft, not a summary of the sketch. Include named theorem-level blocks, the intended logical order, the main derivational transitions, and an assembly paragraph explaining how the blocks would combine to prove the target theorem or where composition fails.
• The draft should normally be long enough to let a future reviewer inspect the mathematical architecture without opening each future step proof. Avoid artificial length quotas, but do not compress nontrivial bridges, condition checks, or quantitative arguments into phrases such as "standard" or "routine".
• Every theorem-critical transition in ## Whole-Proof Draft must expose the key inequality, implication, recurrence, comparison, or cited-tool instantiation that the transition depends on. A transition is too shallow when the reader cannot see what mathematical interface would turn the inputs into the claimed output.
• For every theorem-critical generated condition, recurrence, invariant, stability, boundedness, membership, convergence, quantitative specialization, structural lower/sign/coercivity/nondegeneracy/support claim, scope upgrade, or theorem-closure block, name the derivation path from primitive conditions, accepted dependencies, cited tools, direct derivations, standard facts or tools, current-notation wrappers, primitive-source derivations, or explicitly conditional targets to the target condition or control. The path may be diagnostic and may point to future step-local proof units, but it must identify the claim class and theorem role, quantities and interfaces those units would control, key positive/control term or structural source, defect or forcing terms, closure/dominance/absorption relation, accumulation behavior when the target is repeated, iterated, recursive, limiting, or otherwise accumulated, the mechanism source that makes the interface nonvacuous, any source-convention compatibility and raw-assumption-to-claim or output feasibility needed by direct/standard/current-notation/primitive-source sources, producer-consumer provenance when generated outputs are consumed, and any allowed boundary regimes the target must handle or explicitly exclude.
• Map every accepted sketch Step ID to a named theorem-level block, direct assembly use, or explicit blocker in ## Sketch-Step Coverage.
• In ## Theorem-Level Block Map, include every named theorem-level block needed by the attempted claim, its source Step IDs or direct assembly role, dependency inputs, theorem-facing assumptions, and unresolved discharge obligations.
• Every selected high-risk roadmap item must appear as a named block, a named hard step, or an explicit blocker. High-risk classes include structural property claims, approximation or perturbation controls, recursive or invariant maintenance arguments, mode upgrades, explicit dependence in any relevant quantitative category, public simplification of technical quantitative statements, and hidden generated-condition assumptions.
• Every derived condition needed by the final theorem must appear as a named block, a sketch-step conclusion, or an explicit blocker. Do not let theorem-facing admissibility, good-event, local-validity, stability, recurrence, boundedness, membership, structural, or invariant packages serve as assumptions for an unconditional target unless a sketch step proves them from primitive conditions.
• Every theorem-facing generated output consumed by a theorem-level block, closure block, specialization, or final theorem use must appear in ## Generated Output Flow. Do not classify a missing producer path, missing consumer dependency, or unsupported generated-output export as a hard local step unless the accepted sketch already contains a legal flow and the diagnostic only needs to expose it.
• In ## Scope And Closure Certificate, certify every theorem-critical generated condition, recurrence, invariant, stability, boundedness, membership, convergence, quantitative specialization, structural lower/sign/coercivity/nondegeneracy/support claim, scope upgrade, or theorem-closure block covered by the Theorem-Critical Mechanism Witness Gate needed by the attempted theorem. For each target condition or control, record the declared theorem scope, local interface or recurrence, key positive/control term or structural source, defect or forcing terms, accumulation behavior / scope compatibility, closure mechanism, mechanism source / boundary exclusion, source-to-claim adequacy when the target needs a structural lower/sign/support/nondegeneracy claim, obligation locality classification, noncircular closure status when applicable, entry-state / first-update stress result when applicable, primitive controls, generated controls, boundary stress result, and failure mode if the mechanism is absent. Every all-time or accumulated row must expose the concrete recurrence/potential or mechanism-specific control relation at diagnostic granularity, including the accumulated defect or forcing term, sign status, one-step charge/absorption/potential-drop, preservation, projection, coupling, stopping, or conditioning relation, and why that relation has a finite budget or is valid under the declared scope.
• For every theorem-critical exported interface consumed downstream, the certificate or ## Whole-Proof Draft must expose the raw-controls-to-exported-interface path at diagnostic granularity. A hard step is not step-local when the accepted sketch only names a future proof, threshold, margin, wrapper, direct derivation, standard tool, current-notation proof, primitive-source derivation, or absorption but does not identify which available controls dominate which defect terms, which source supplies any remaining margin or threshold, any required object-target bridge and residual-to-target adequacy from produced entry/reference/baseline/transformed/surrogate object to consumed theorem target object, and any exact setting convention, source-convention compatibility, raw-assumption-to-output path, quantitative dominance or transfer relation, branch or boundary handling, and downstream interface needed by direct/standard/current-notation/primitive-source sources.
• When a baseline invariance obligation is present, record the baseline conclusion, the specialization or entry case, the mechanism source preserving the original conclusion, and any first-transition or stationary trace needed to show the original conclusion remains valid.
• For every theorem-critical hard obligation, record obligation locality using only step-local, sketch/interface defect, or idea/theorem-contract defect. step-local is valid only when the mechanism source already exists in primitive assumptions, accepted derived outputs, cited tools with a valid discharge path under the Source-To-Claim Adequacy Gate, explicitly conditional targets, or direct derivations, standard facts or tools, current-notation wrappers, or primitive-source derivations whose source-convention compatibility and raw-assumption-to-claim or output feasibility have already been checked at obstruction-level granularity, and the future proof step only instantiates or derives the claim under unchanged sketch claims, dependencies, assumptions, output targets, generated-output flows, generated-condition interfaces, accumulation behavior, closure mechanisms, mechanism witnesses, mechanism sources, boundary exclusions, theorem scope/mode/metric, exposed dependence, success criterion, and conclusion.
• For every hard step or unresolved local proof bottleneck, the certificate or ## Global Gaps And Hard Steps must explicitly carry the Theorem-Critical Mechanism Witness when the step is central to final theorem closure, recurrence, invariant maintenance, structural lower/sign/coercivity/nondegeneracy/support claims, scope upgrades, or quantitative specialization. Without that witness, use SKETCH_BLOCKED; do not use PARTIAL_BLOCKED.
• The closure mechanism must be one or more of: self-contraction, dissipative/restoring recurrence, telescoping, summable control, signed cancellation, monotone potential, reserve/ledger under declared scope, stopping/conditioning argument, projection/nonexpansive maintenance, algebraic coupling, structural lower/upper comparison, or explicitly conditional target. If none applies, record the absence as a blocker rather than deferring it. A closure category such as monotone potential, reserve/ledger, finite-budgeted, first-exit, stopped/conditioned, locally absorbed, projection, dissipative recurrence, algebraic coupling, or small-gain is not enough unless the certificate states the concrete mechanism-specific control relation and finite-budget or declared-scope validity justification.
• Valid mechanism sources include a primitive condition, accepted derived control, cited tool with a valid diagnostic discharge path under the Source-To-Claim Adequacy Gate, direct derivation, standard fact or tool, current-notation wrapper, primitive-source derivation with checked source-convention compatibility and raw-assumption feasibility, or explicitly conditional theorem target. Merely naming a closure category, direct derivation, standard tool, current-notation proof, or primitive-source derivation is not enough.
• For repeated, iterated, recursive, limiting, or otherwise accumulated targets, the certificate must classify the defect behavior as locally absorbed, contractive, telescoping, summable, signed-cancelled, monotone-potential controlled, finite-budgeted under the declared scope, source-excluded, stopped/conditioned, explicitly conditional, or unsupported. A barrier, reserve, ledger, invariant, stability, bootstrap, first-exit, Lyapunov, finite-budget, small-gain, projection, dissipative recurrence, algebraic coupling, or future-step label is not a concrete proof interface without this accumulation classification, mechanism source, named accumulated defect, sign status, budget/potential or mechanism-specific control relation, one-step charge/absorption/potential-drop, preservation, projection, coupling, stopping, or conditioning relation, and finite-budget or declared-scope validity justification.
• If an all-time or accumulated certificate row only names a closure category or says a future local proof will supply the controlling relation, use Status = SKETCH_BLOCKED and record the row as a sketch/interface defect or candidate idea/theorem-contract defect; do not use PARTIAL_BLOCKED.
• For all-time, uniform, limsup, invariant, stability, recurrence, support, basin, boundedness, and generated-condition claims, apply the Noncircular Closure Gate. The certificate must name the noncircular producer or mechanism source, the exit/defect/control relation, and the dependency path to each consumer; if the proposed source assumes the target closure, generated condition, or final theorem consequence, use SKETCH_BLOCKED.
• For recursive, iterative, descent, contraction, convergence, all-time, recurrence closure, invariant, basin/support, mode-conversion, and exact/noiseless specialization claims, apply the Entry-State / Activation Trace Gate. The certificate or ## Early Obstruction And Repair Plausibility must name an allowed entry, initial, stationary, null, degenerate, exact/noiseless, or boundary state, trace the first update, transition, or stationary behavior at diagnostic granularity, and state whether the mechanism source is active before the target conclusion is consumed. If the mechanism is inactive while the theorem-facing conclusion remains false, use SKETCH_BLOCKED and label candidate idea-level evidence for /global-proof-review only when repair would require the required theorem-contract change: changed primitive assumptions, changed algorithm/model/procedure, changed theorem scope/mode/metric, changed exposed dependence, changed success criterion, adding a theorem-facing mechanism source not supported by the setting, or weakening the conclusion.
• The mechanism source must be adequate for the claim class. Upper bounds, smallness, finite budgets, local boxes, admissibility inequalities, or generic regularity are not by themselves adequate sources for positive lower, coercive, nondegenerate, signed, or support claims unless paired with a primitive, earlier-derived, cited, direct derivation, standard fact or tool, current-notation wrapper, primitive-source derivation, or conditional source of the required lower/sign/support/conditioning content with the required feasibility checks.
• A statement such as "a later step closes the generated condition" is invalid unless the certificate names the actual closure mechanism, the quantities controlled, the defect accumulation behavior, and the mechanism source. If a recurrence, interface, accumulation, or defect term is needed, the certificate must show why it is compatible under the declared theorem scope or explain the failure.
• In ## Early Obstruction And Repair Plausibility, run obstruction-level checks only. Identify contract contradictions, missing theorem-critical mechanisms or mechanism sources, missing exported-interface feasibility, missing residual-to-target adequacy, unsupported mode or dependence upgrades, generated-condition provenance failures, limiting or boundary stress failures, entry-state trace failures, and whether repair appears possible under the same setting and goal. Do not attempt substantial local step proofs.
• If the strongest diagnosis is a candidate idea-level obstruction, keep Status = SKETCH_BLOCKED and Suggested Routing = /proof-sketch, because /global-proof has no /subagent-idea-generator route. Explicitly label the obstruction as candidate idea-level evidence for /global-proof-review to classify independently.
• If any theorem-critical hard obligation is classified as sketch/interface defect or idea/theorem-contract defect, use Status = SKETCH_BLOCKED and Suggested Routing = /proof-sketch. For idea/theorem-contract defect, explicitly label the finding as candidate idea-level evidence for /global-proof-review; do not emit IDEA_FAIL from global_proof.md.

Audit Rules

• In ## Dependency And Assumption Audit, classify theorem-facing conditions as primitive conditions, accepted derived conclusions from sketch dependencies, local conditional hypotheses, or missing derived-condition bridges. A final unconditional target may not assume missing derived conditions.
• In ## Citation And Tool Audit, for each cited theorem, standard fact, technical tool, direct derivation, current-notation wrapper, primitive-source derivation, or dependency use, restate the source or name, current-branch objects, assumptions, conclusion needed, source-convention compatibility when theorem-critical, object-target compatibility when an entry, reference, population, no-error, or baseline object is consumed, and where each assumption would have to be discharged. For theorem-critical cited results, also record source identity, version or stable locator when relevant, exact label or stable statement identifier when used, statement role, source-object mapping, hypothesis discharge, conclusion-interface match, source-convention compatibility, object-target compatibility when applicable, known non-output boundaries, and any bridge or wrapper obligation before treating the cited result as a step-local mechanism source. For theorem-critical direct derivations, standard facts or tools, current-notation wrappers, and primitive-source derivations, record the exact setting convention, source-convention compatibility, object-target compatibility when applicable, raw-assumption-to-claim or output path, quantitative dominance or transfer relation when needed, branch or boundary handling, and downstream interface before treating the output as step-local.
• In ## Quantitative Dependence Audit, when quantitative dependence is exposed or a baseline invariance obligation is present, record exposed variables, allowed hidden-constant dependence, fixed quantities, probability or stochastic mode, horizon or limiting mode, norm or metric mode, auxiliary tolerances, public specialization obligations, baseline-reduction obligations, and whether inherited baseline/recovery conclusions are preserved. Write None only when no quantitative dependence is exposed and no baseline invariance obligation is present.
• Track notation, quantifier order, constants, parameter dependence, stochastic or convergence mode, boundary cases, and interchanges of limits, expectations, suprema, derivatives, integrals, or conditionings when any are theorem-critical.
• In ## Scope And Closure Certificate, do not classify a transition as a hard local step unless the theorem-level interface is concrete enough for a step worker to prove under the unchanged sketch and the Theorem-Critical Mechanism Witness is explicit, source-convention compatible when a source supplies the mechanism, and scope-compatible. The interface must name the controlled quantities, accepted inputs, generated outputs, accumulation or recurrence shape when present, defect behavior, accumulation mode, closure mechanism category, mechanism source, key positive/control term or structural source, opposing defect terms, closure/dominance/absorption relation, boundary regimes handled or excluded, and entry-state / first-update stress result when the Entry-State / Activation Trace Gate applies.
• In ## Scope And Closure Certificate, do not classify an exported-interface gap as a hard local step. If proving the downstream-facing output requires a new bridge step, changed output target, changed dependency, new margin or threshold source, changed cited-tool wrapper conclusion, changed direct/standard/current-notation/primitive-source output conclusion, separation of defect classes, changed dominance/transfer relation, object-target compatibility or residual-to-target adequacy not exposed by the accepted sketch when an entry, reference, population, no-error, baseline, transformed, or surrogate object is consumed, or a same-target bridge for a transformed, weighted, preconditioned, whitened, reference-operator-modified, or otherwise surrogate produced object, source-convention compatibility not exposed by the accepted sketch, or raw-assumption/output feasibility not exposed by the accepted sketch, use SKETCH_BLOCKED.
• In ## Generated Output Flow, do not classify a missing producer-consumer path as a hard local step. A theorem-facing generated output must have a legal producer block/step/source, consumer block/step/final use, dependency path, and provenance class. If a new bridge, changed dependency, changed output target, or changed generated-condition interface is needed, use SKETCH_BLOCKED.
• In ## Early Obstruction And Repair Plausibility, distinguish a hard local proof obligation from a missing theorem-level mechanism. A hard obligation is step-local only when the accepted sketch already assigns the needed conclusion to a step, the theorem-level interface is source-adequate under unchanged scope and dependence, any cited tool has a valid discharge path under the Source-To-Claim Adequacy Gate, any required object-target compatibility has already been exposed, any transformed, weighted, preconditioned, whitened, reference-operator-modified, or surrogate produced object already has a same-target bridge with residual-to-target adequacy to the consumed interface, and any direct derivation, standard fact or tool, current-notation wrapper, or primitive-source derivation has checked source-convention compatibility and raw-assumption-to-claim or output feasibility.
• Before classifying a hard local step as PARTIAL_BLOCKED, run the Theorem-Critical Mechanism Witness Gate plus mechanism-source nonvacuity, source-convention compatibility, object-target compatibility and residual-to-target adequacy when an entry, reference, population, no-error, baseline, transformed, or surrogate object is consumed, and source-to-claim adequacy checks. The diagnostic must state why the claim is not false in allowed null, degenerate, exact/noiseless, or boundary regimes, and identify the primitive, earlier-derived, cited-tool, direct/standard/current-notation/primitive-source, or explicitly conditional source that prevents the mechanism from vanishing while the target conclusion would remain false or the theorem-critical obstruction would remain present. If the produced object is transformed, weighted, preconditioned, whitened, reference-operator-modified, or otherwise surrogate relative to the consumed target and no same-target bridge with residual-to-target adequacy is already exposed, use SKETCH_BLOCKED. If the witness is missing, scope-incompatible, lacks required object-target compatibility, lacks required residual-to-target adequacy, lacks required source-convention compatibility or raw-assumption feasibility, or the source is only an upper/size/budget/local-validity/small-gain control and no lower/sign/support/conditioning source is available under the unchanged sketch, use SKETCH_BLOCKED.
• For each hard Step ID, outline the likely local proof units needed later by /proof-step, using provisional unit roles such as lemma, proposition, claim, hypothesis, invariant, cited-result application, or assembly bridge.
• Do not hide difficult obligations behind phrases such as "clearly", "obviously", "standard", "similarly", or "by routine arguments"; name the exact missing local unit, cited-result condition, quantitative bridge, assumption-provenance bridge, or assembly link instead.
• Local units and cited-result applications in global_proof.md are diagnostic planning guidance only. They must be independently proved, checked, or discharged by /proof-step before they can be used as proof evidence.

Self-Check Before Writing

Before writing or finalizing global_proof.md, verify:

• Sketch faithfulness: the draft follows the accepted sketch without silently repairing claims, dependencies, assumptions, output targets, or rate objectives.
• Claim fidelity: the attempted claim does not strengthen assumptions, narrow the regime, change quantifiers, drop exposed dependence, or drift away from setting.md.
• Step coverage: every accepted sketch step has a coverage-map entry and a role in the whole-proof draft or an explicit blocker.
• Block coverage: every theorem-level block needed for the attempted claim is mapped to accepted sketch steps, direct assembly use, or an explicit blocker.
• Dependency flow: theorem-level blocks respect the accepted dependency order and identify any missing dependency closure.
• Assumption provenance: every theorem-facing generated-object, event, local-validity, recurrence, boundedness, membership, structural, or invariant condition is primitive, proved by a sketch step, diagnostic-only inside a conditional lemma, or named as a blocker.
• Scope and closure: every theorem-critical generated condition, recurrence, invariant, stability, boundedness, membership, convergence, quantitative specialization, structural lower/sign/coercivity/nondegeneracy/support claim, scope upgrade, or theorem-closure block covered by the Theorem-Critical Mechanism Witness Gate has a certificate with a named derivation path, a concrete interface or recurrence when relevant, concrete recurrence/potential or mechanism-specific control relation for every all-time or accumulated row, key positive/control term or structural source, a valid closure mechanism category, a valid mechanism source / boundary exclusion, source-to-claim adequacy, source-convention compatibility when a source supplies the mechanism, object-target compatibility and residual-to-target adequacy when an entry, reference, population, no-error, baseline, transformed, or surrogate object is consumed, any direct/standard/current-notation/primitive-source raw-assumption feasibility when relevant, entry-state / first-update stress result when applicable, and a boundary stress result.
• Theorem-critical mechanism witness: every hard step or theorem-level block central to closure, recurrence, invariant maintenance, structural lower/sign/coercivity/nondegeneracy/support claims, scope upgrades, or quantitative specialization has an explicit, scope-compatible witness, or is routed as SKETCH_BLOCKED.
• Generated-output flow: every theorem-facing generated output consumed by the attempted theorem, block, closure, or specialization has a legal producer, consumer, final use, dependency path, and provenance class, or is named as a blocker.
• Scope accumulation: every theorem-critical repeated, iterated, recursive, limiting, or otherwise accumulated condition or control has a certificate that classifies defect behavior and explains why it is compatible with the declared theorem scope by naming the accumulated defect, sign status, budget/potential or mechanism-specific control relation, one-step charge/absorption/potential-drop, preservation, projection, coupling, stopping, or conditioning relation, and finite-budget or declared-scope validity justification, or records the noncompatible defect as a blocker.
• Citation and source discipline: cited tools are not treated as discharged unless their diagnostic source identity, source-convention compatibility, object-target compatibility when applicable, and interface preflight are stated. Direct derivations, standard facts or tools, current-notation wrappers, and primitive-source derivations are not treated as step-local unless their diagnostic source-convention compatibility, object-target compatibility when applicable, and raw-assumption feasibility are stated. For theorem-critical cited results, this includes source identity, version or stable locator when relevant, exact label or stable statement identifier when used, statement role, source-object mapping, hypothesis discharge, conclusion-interface match, source-convention compatibility, object-target compatibility when an entry, reference, population, no-error, or baseline object is consumed, known non-output boundaries, and any required bridge or wrapper obligations. For theorem-critical direct derivations, standard facts or tools, current-notation wrappers, and primitive-source derivations, this includes exact setting convention, source-convention compatibility, object-target compatibility when applicable, raw-assumption-to-claim or output path, quantitative dominance or transfer relation when needed, branch or boundary handling, and downstream interface.
• Quantitative discipline: exposed quantitative dependence and public simplification obligations are preserved or marked as gaps.
• Baseline discipline: inherited recovery, reduction, specialization, zero-defect, exact-limit, exact/noiseless, or baseline-case conclusions are preserved or marked as blockers; weaker surrogates are not presented as target-preserving.
• Early obstruction discipline: candidate idea-level obstructions are not hidden as generic hard steps; same-setting repair plausibility is explicitly recorded for reviewer classification.
• Diagnostic boundary: the artifact says it is not proof evidence, not a cited result, not an assumption source, and not authority to change step claims.
• Status alignment: choose PARTIAL_BLOCKED only for hard local obligations classified as step-local under an unchanged sketch with an explicit, scope-compatible theorem-critical mechanism witness, a concrete raw-control-to-exported-interface path for every downstream-facing theorem-critical output, source-convention compatibility when a source supplies the mechanism or output, object-target compatibility and residual-to-target adequacy when an entry, reference, population, no-error, baseline, transformed, or surrogate object is consumed, any direct/standard/current-notation/primitive-source raw-assumption feasibility needed by the output, a concrete scope-accumulation-compatible proof interface, a legal generated-output flow when theorem-facing generated outputs are consumed, a source-adequate and nonvacuous mechanism source, an explicit null/boundary stress check, and an entry-state / first-update stress result when the Entry-State / Activation Trace Gate applies. If the produced object is transformed, weighted, preconditioned, whitened, reference-operator-modified, or otherwise surrogate relative to the consumed target, the unchanged sketch must already expose a same-target bridge with residual-to-target adequacy. Choose SKETCH_BLOCKED when a sketch row, coverage item, assumption bridge, exported-interface bridge, object-target bridge, residual-to-target bridge, generated-output flow, generated-condition interface, closure mechanism, accumulation behavior, mechanism witness, mechanism source, source-to-claim adequacy, source-convention compatibility, direct/standard/current-notation/primitive-source source feasibility, entry-state trace, boundary exclusion, quantitative bridge, proof roadmap, same-target bridge, or nonvacuity source must change.

Status Rules

• COMPLETE_DRAFT: the whole proof draft closes at the theorem-diagnostic level expected before local step proving, every sketch step has a coherent role, all required audits have no material unresolved issue, and Global Gaps And Hard Steps = None.
• PARTIAL_BLOCKED: the sketch still looks viable, but one or more hard obligations should be handled by step-local proof work under unchanged sketch claims, dependencies, assumptions, and output targets. Every unresolved theorem-critical hard obligation must be classified as step-local. The unresolved work must have an explicit, scope-compatible Theorem-Critical Mechanism Witness, a concrete raw-control-to-exported-interface path for every downstream-facing theorem-critical output, source-convention compatibility when a source supplies the mechanism or output, object-target compatibility and residual-to-target adequacy when an entry, reference, population, no-error, baseline, transformed, or surrogate object or control is consumed, any direct/standard/current-notation/primitive-source raw-assumption feasibility needed by that output, and a concrete, scope-accumulation-compatible proof interface in the ## Scope And Closure Certificate, legal generated-output flow in ## Generated Output Flow when theorem-facing generated outputs are consumed, including the local interface or recurrence, controlled quantities, key positive/control term or structural source, defect behavior and accumulation mode when present, required primitive or generated controls, a named closure mechanism category, a source-adequate and nonvacuous mechanism source, boundary regimes handled or excluded, an entry-state / first-update stress result when applicable, and a null/boundary stress result showing why the mechanism cannot vanish while the target conclusion would remain false or the theorem-critical obstruction would remain present. For all-time or accumulated obligations, the interface must also name the accumulated defect or forcing term, sign status, controlling budget/potential or mechanism-specific control relation, one-step charge/absorption/potential-drop, preservation, projection, coupling, stopping, or conditioning relation, and finite-budget or declared-scope validity justification. Category-only all-time or accumulated rows force SKETCH_BLOCKED, not PARTIAL_BLOCKED. Name the affected Step IDs and continue to /proof-step.
• SKETCH_BLOCKED: a step claim, dependency, assumption, missing step, missing assumption-provenance bridge, missing quantitative bridge, missing generated-output flow, missing generated-condition interface, missing scope-compatible closure mechanism, missing or incompatible accumulation behavior, missing/shallow/scope-incompatible mechanism witness, missing or changed mechanism source, source-to-claim inadequacy, missing boundary exclusion, coverage item, proof roadmap, sketch/interface defect, or candidate idea/theorem-contract defect must be resolved before local proving can be sound. Use Suggested Routing = /proof-sketch; candidate idea-level obstructions are routed this way only for controller compatibility and must be made explicit for /global-proof-review.

Do not use SKETCH_BLOCKED for a merely hard local proof obligation that can be isolated to an unchanged step with an explicit, scope-compatible Theorem-Critical Mechanism Witness, a concrete scope-accumulation-compatible interface, legal generated-output flow when needed, source-adequate mechanism source, source-convention compatibility when a source supplies the mechanism or output, object-target compatibility and residual-to-target adequacy when an entry, reference, population, no-error, baseline, transformed, or surrogate object or control is consumed, any direct/standard/current-notation/primitive-source raw-assumption feasibility needed by that source, and step-local classification. Do not use PARTIAL_BLOCKED if any unresolved theorem-critical obligation is classified as sketch/interface defect or idea/theorem-contract defect, or if resolving the obligation would require changing the target step claim, dependency, assumption, output target, generated-output flow, generated-condition interface, accumulation behavior, closure mechanism, mechanism witness, mechanism source, source-to-claim adequacy, source-convention compatibility, object-target compatibility, residual-to-target adequacy, direct/standard/current-notation/primitive-source source feasibility, boundary exclusion, adding a missing derived-condition bridge, adding a missing quantitative bridge, changing the algorithm/model/procedure, changing theorem scope/mode/metric, changing exposed dependence, changing success criterion, adding a theorem-facing mechanism source not supported by the setting, weakening the conclusion, or changing the proof roadmap.

Writing Rules

• Keep the draft faithful to the accepted sketch; do not silently repair the sketch.
• Include all template sections: Reviewed Sketch Identity, Attempted Theorem Claim, Whole-Proof Draft, Theorem-Level Block Map, Sketch-Step Coverage, Dependency And Assumption Audit, Citation And Tool Audit, Quantitative Dependence Audit, Scope And Closure Certificate, Generated Output Flow, Early Obstruction And Repair Plausibility, Global Gaps And Hard Steps, Diagnostic Boundary Note, and Suggested Routing.
• Mention every sketch step in ## Sketch-Step Coverage.
• If a cited theorem, technical tool, direct derivation, standard fact or tool, current-notation wrapper, or primitive-source derivation appears necessary, state what must be checked later; do not treat the source as already discharged or step-local unless it has a valid discharge path under the Source-To-Claim Adequacy Gate or checked source-convention compatibility, object-target compatibility when applicable, and raw-assumption feasibility path when applicable, including source identity, version or stable locator when relevant, exact label or stable statement identifier when used, statement role, source-object mapping, hypothesis discharge, conclusion-interface match, source-convention compatibility, object-target compatibility when an entry, reference, population, no-error, or baseline object is consumed, known non-output boundaries, and any bridge or wrapper obligation for theorem-critical cited results, and exact setting convention, source-convention compatibility, object-target compatibility when applicable, raw-assumption-to-claim or output path, quantitative dominance or transfer relation when needed, branch or boundary handling, and downstream interface for theorem-critical direct derivations, standard facts or tools, current-notation wrappers, and primitive-source derivations.
• If a gap implies a changed step claim, dependency, assumption, added step, missing Assumption Provenance Bridge, missing generated-output flow, missing generated-condition interface, missing scope-compatible closure mechanism, missing mechanism source, missing boundary exclusion, or missing quantitative bridge, route to /proof-sketch.
• If a gap implies the overall roadmap is untenable or the same-setting idea appears implausible, route to /proof-sketch and label the candidate idea-level obstruction for /global-proof-review.
• ## Diagnostic Boundary Note must state that global_proof.md is diagnostic only and cannot be consumed as proof evidence, a cited result, an assumption source, or authority to change step claims.
• ## Suggested Routing first line must be None for COMPLETE_DRAFT and PARTIAL_BLOCKED, and /proof-sketch for SKETCH_BLOCKED.