{ "schema_version": "1.0.0", "artifact_id": "qft.artifact.supersymmetry-duality.duality.dictionary-operations-evidence-map", "artifact_class": "typed duality-operation graph with failed-square diagnostics", "title": "Duality operations require a closed background-dependent square", "evidence_date": "2026-08-24", "reader_question": "When may a proposed duality be transported through gauging or through a deformation and renormalization-group flow?", "dominant_point": "A transported daughter claim is licensed only when the source and target theories are matched as background-dependent objects and the operation square closes with compatible anomalies, bundles, counterterms, scale hierarchy, operators, and retained or decoupled sectors.", "scope": { "spacetime_dimension": "abstract relativistic QFT framework; dimensions and supersymmetry are supplied by a family-specific theory card", "metric_convention": "inherit the QFT.org (+---) convention when a Lorentzian family-specific example is inserted", "operation_classes": [ "gauging a declared ordinary or generalized global symmetry", "flowing after a mapped relevant or marginal deformation" ], "claim_form": "Z_A[A;lambda_A] = exp(i C[A]) Z_B[f(A);lambda_B] for every allowed background and bundle in the stated regime, with the local functional C[A] recorded", "excluded_inference": "graph connectivity, agreement at A=0, or one successful protected check does not by itself establish equivalence or a transported daughter claim" }, "theory_records": [ { "id": "source_theory", "role": "source", "display_name": "T_A", "parameters": [ "lambda_A", "scales", "discrete theta data" ], "local_operators": [ "local operator basis O_i", "deformation operators and sources" ], "extended_operators": [ "genuine lines L_alpha", "surfaces and topological defects when present" ], "backgrounds_global_data": [ "background A and its allowed bundles", "global form and genuine-operator lattice", "ordinary and higher-form anomalies", "local counterterms and contact terms" ], "regime": "declared source regime R_A, including energy range and scale hierarchy", "evidence_date": "2026-08-24", "status": "abstract complete theory-card schema, not a family-specific duality assertion", "uncertainty": "field content, couplings, global form, and exact scope must be instantiated for each proposed dual pair" }, { "id": "target_theory", "role": "target", "display_name": "T_B", "parameters": [ "lambda_B = D_parameter(lambda_A)", "mapped scales", "mapped discrete theta data" ], "local_operators": [ "D_local(O_i)", "mapped deformation operators and sources" ], "extended_operators": [ "D_extended(L_alpha)", "mapped surfaces and topological defects" ], "backgrounds_global_data": [ "f(A) and the image of every allowed bundle", "mapped global form and genuine-operator lattice", "matched anomaly class", "specified local counterterm C[A] and contact terms" ], "regime": "mapped target regime R_B with the same claimed infrared or exact scope", "evidence_date": "2026-08-24", "status": "abstract complete theory-card schema, not a family-specific duality assertion", "uncertainty": "accidental symmetries, emergent sectors, and noninvertible parts of the dictionary require case-by-case checks" }, { "id": "source_daughter", "role": "source daughter", "display_name": "O_A(T_A)", "parameters": [ "operation couplings", "gauging measure or mapped deformation strength", "ordered scales" ], "local_operators": [ "gauge-invariant descendants", "operators generated or removed by the flow" ], "extended_operators": [ "post-gauging genuine lines and defects", "surviving topological sectors" ], "backgrounds_global_data": [ "bundle sum", "gauge volume", "discrete theta and local terms", "residual symmetries" ], "regime": "domain in which O_A exists and its scale ordering is controlled", "evidence_date": "2026-08-24", "status": "well-defined only after all operation inputs are declared", "uncertainty": "the daughter need not lie in the same phase if hierarchy or sector assumptions fail" }, { "id": "target_daughter", "role": "target daughter", "display_name": "O_B(T_B)", "parameters": [ "dictionary image of operation couplings", "mapped deformation strength", "mapped ordered scales" ], "local_operators": [ "dictionary image of source descendants", "target-generated operators" ], "extended_operators": [ "mapped post-operation lines and defects", "matched topological sectors" ], "backgrounds_global_data": [ "corresponding bundle sum", "gauge volume", "counterterms", "residual symmetries" ], "regime": "common domain in which O_B exists and the daughter comparison is meaningful", "evidence_date": "2026-08-24", "status": "conditionally comparable to the source daughter only if the square closes", "uncertainty": "a mismatch may signal a wrong map, missing sector, invalid hierarchy, or genuinely failed duality claim" } ], "nodes": [ { "id": "source_theory", "kind": "theory", "label": "complete source theory T_A" }, { "id": "target_theory", "kind": "theory", "label": "complete target theory T_B" }, { "id": "source_daughter", "kind": "theory", "label": "source daughter O_A(T_A)" }, { "id": "target_daughter", "kind": "theory", "label": "target daughter O_B(T_B)" }, { "id": "full_background_claim", "kind": "claim record", "label": "background-dependent parent claim" }, { "id": "operation_square", "kind": "logical gate", "label": "complete operation square" }, { "id": "zero_background_claim", "kind": "insufficient claim", "label": "equality established only at A=0" }, { "id": "symmetry_breaking_deformation", "kind": "failure input", "label": "deformation that breaks the symmetry G" }, { "id": "anomaly_check", "kind": "evidence", "label": "anomaly and contact-term check" }, { "id": "protected_check", "kind": "evidence", "label": "protected-observable or partition-function check" }, { "id": "working_conjecture", "kind": "status record", "label": "family-specific working conjecture" }, { "id": "dated_claim_record", "kind": "status record", "label": "dated claim, uncertainty, and falsifier record" }, { "id": "family_handoff", "kind": "handoff", "label": "family-specific duality analysis" } ], "edge_type_legend": { "claim": "double-headed theory relation whose exact scope and status must be stated; never inferred from connectivity alone", "map": "typed component of the dictionary, with explicit source and target payloads", "operation": "one-way construction or flow applied to a complete theory record", "check": "one-way dotted open-tip evidence edge that tests but cannot visually assert equivalence", "implication": "one-way logical consequence valid only under its listed hypotheses", "conjecture": "status-limited proposal awaiting independent tests or a stronger argument", "failed-square": "crossed dashed edge recording why a transported inference is blocked or undefined", "handoff": "one-way transfer of a qualified record to a family-specific page or later review" }, "visual_encodings": { "claim_or_dictionary": "black double-headed double line with a direct status label", "operation": "black solid single-headed line", "check": "black densely dotted single line with an open triangular tip and the words tests, never asserts equivalence", "failed_square": "black dashed box with an explicit multiplication-cross symbol and failure status", "handoff": "plain one-way record transfer in the structured equivalent only", "color_dependence": "none; fill shade, line style, symbols, and words are redundant" }, "edges": [ { "id": "claim-parent-candidate", "type": "claim", "source": "source_theory", "target": "target_theory", "label": "candidate duality in a declared regime", "hypotheses": [ "the complete typed dictionary is supplied", "the claimed equality is background-dependent" ], "reversible": "only if an inverse dictionary and inverse claim are separately established", "status": "candidate framework edge; no family-specific equivalence is asserted by this artifact", "uncertainty": "the claim can range from exact reformulation to infrared or protected-sector equivalence", "evidence_date": "2026-08-24", "falsifier": "a mismatch in any observable or global datum inside the declared scope", "visible_in_figure": "yes, aggregated with the typed dictionary" }, { "id": "map-parameters", "type": "map", "source": "source_theory", "target": "target_theory", "label": "parameter and scale map", "component": "parameters", "source_payload": "lambda_A, scales, and discrete theta data", "target_payload": "lambda_B, mapped scales, and mapped discrete theta data", "hypotheses": [ "normalizations and scheme-dependent local terms are declared" ], "reversible": "not assumed", "status": "required dictionary component", "uncertainty": "threshold and contact-term conventions can modify the written relation", "evidence_date": "2026-08-24", "falsifier": "inconsistent dimensions, periodicities, decoupling limits, or threshold matching", "visible_in_figure": "yes, as one item in the dictionary band" }, { "id": "map-local-operators", "type": "map", "source": "source_theory", "target": "target_theory", "label": "local-operator and deformation map", "component": "local operators", "source_payload": "O_i and their sources", "target_payload": "D_local(O_i), including mapped deformations", "hypotheses": [ "operator mixing, descendants, and accidental symmetries are treated in the claimed regime" ], "reversible": "not assumed", "status": "required dictionary component", "uncertainty": "some ultraviolet operators can become redundant, free, or emergent in the infrared", "evidence_date": "2026-08-24", "falsifier": "incompatible charges, chiral relations, correlators, or response to a mapped deformation", "visible_in_figure": "yes, as one item in the dictionary band" }, { "id": "map-extended-operators", "type": "map", "source": "source_theory", "target": "target_theory", "label": "line, surface, and defect map", "component": "extended operators", "source_payload": "genuine lines L_alpha and other defects", "target_payload": "D_extended(L_alpha) and mapped defect fusion or screening data", "hypotheses": [ "global form and the genuine-operator lattice are specified on both sides" ], "reversible": "not assumed", "status": "required dictionary component whenever the claim includes extended operators", "uncertainty": "a local-operator duality can have a narrower scope than an equivalence of genuine defects", "evidence_date": "2026-08-24", "falsifier": "different screening, braiding, fusion, or higher-form symmetry data within the claimed scope", "visible_in_figure": "yes, inside the complete theory cards" }, { "id": "map-backgrounds-global-data", "type": "map", "source": "source_theory", "target": "target_theory", "label": "background, bundle, anomaly, and counterterm map", "component": "backgrounds and global data", "source_payload": "A, allowed bundles, anomaly class, and local terms", "target_payload": "f(A), image bundles, matched anomaly class, and C[A]", "hypotheses": [ "all topological sectors relevant to the operation are included" ], "reversible": "not assumed", "status": "mandatory before gauging is transported", "uncertainty": "equality may hold only after an explicitly recorded invertible sector or local counterterm is added", "evidence_date": "2026-08-24", "falsifier": "a nontrivial-bundle, anomaly, discrete-theta, or contact-term mismatch", "visible_in_figure": "yes, inside the complete theory cards and closure gate" }, { "id": "map-regime", "type": "map", "source": "source_theory", "target": "target_theory", "label": "claim-regime map", "component": "regime", "source_payload": "R_A with energy range and scale ordering", "target_payload": "R_B with the corresponding range and ordering", "hypotheses": [ "the same limit is taken on both paths through the square" ], "reversible": "not assumed", "status": "required scope component", "uncertainty": "dangerously irrelevant couplings or phase boundaries can obstruct the naive order of limits", "evidence_date": "2026-08-24", "falsifier": "the two paths land in different phases or require incompatible scale orderings", "visible_in_figure": "yes, inside the complete theory cards" }, { "id": "operation-source", "type": "operation", "source": "source_theory", "target": "source_daughter", "label": "apply O_A by gauging G or flowing with a mapped deformation", "hypotheses": [ "O_A exists", "its measure, bundles, local terms, hierarchy, and sector treatment are declared" ], "reversible": "no", "status": "defined operation, not evidence for a duality", "uncertainty": "strong-coupling or ordering effects may change the daughter description", "evidence_date": "2026-08-24", "falsifier": "the declared operation is anomalous, ill-defined, or outside its controlled regime", "visible_in_figure": "yes, solid downward arrow" }, { "id": "operation-target", "type": "operation", "source": "target_theory", "target": "target_daughter", "label": "apply the corresponding O_B", "hypotheses": [ "O_B exists", "the operation data are the image of O_A under the dictionary" ], "reversible": "no", "status": "defined operation, not evidence for a duality", "uncertainty": "the target operation can require an extra topological or decoupled sector", "evidence_date": "2026-08-24", "falsifier": "O_B is undefined or its bundle sum, counterterms, or hierarchy do not match O_A", "visible_in_figure": "yes, solid downward arrow" }, { "id": "implication-background-to-square", "type": "implication", "source": "full_background_claim", "target": "operation_square", "label": "a complete parent claim permits the closure test", "hypotheses": [ "the equality holds on every background and bundle included by the operation" ], "reversible": "no", "status": "necessary input, not sufficient by itself", "uncertainty": "operation-specific measures and sectors still have to be checked", "evidence_date": "2026-08-24", "falsifier": "the parent relation is known only at zero background or only in a smaller sector", "visible_in_figure": "yes, represented by the central gate" }, { "id": "implication-source-to-background-claim", "type": "implication", "source": "source_theory", "target": "full_background_claim", "label": "upgrade the candidate claim to a background-dependent statement", "hypotheses": [ "target theory and every dictionary component are supplied" ], "reversible": "no", "status": "required strengthening before gauging", "uncertainty": "an A=0 identity need not possess this extension", "evidence_date": "2026-08-24", "falsifier": "no consistent map exists on a nontrivial allowed background", "visible_in_figure": "yes, encoded by the complete top-row cards" }, { "id": "claim-daughters-conditional", "type": "claim", "source": "source_daughter", "target": "target_daughter", "label": "conditional daughter duality", "hypotheses": [ "the complete parent claim holds", "both operations exist", "anomalies, bundles, counterterms, hierarchy, operator maps, and sectors agree" ], "reversible": "only to the extent justified by the daughter dictionary", "status": "licensed only after the operation square closes", "uncertainty": "its scope cannot exceed the parent claim or the controlled domain of the operations", "evidence_date": "2026-08-24", "falsifier": "any path-dependent daughter datum inside the declared scope", "visible_in_figure": "yes, conditional double-headed arrow" }, { "id": "check-anomaly", "type": "check", "source": "anomaly_check", "target": "full_background_claim", "label": "compare anomaly and contact-term responses", "hypotheses": [ "the same symmetry normalization and allowed local counterterms are used" ], "reversible": "no", "status": "necessary check when the claim includes that symmetry; not a proof of equivalence", "uncertainty": "matching anomalies is compatible with many inequivalent dynamics", "evidence_date": "2026-08-24", "falsifier": "an uncancelled anomaly or non-removable contact-term mismatch", "visible_in_figure": "yes, aggregated into the evidence-check record" }, { "id": "check-protected-observable", "type": "check", "source": "protected_check", "target": "full_background_claim", "label": "compare a protected observable or partition function", "hypotheses": [ "normalization, background, contour, and decoupled factors are matched" ], "reversible": "no", "status": "positive evidence with a ceiling set by the observable's protected sector", "uncertainty": "agreement need not determine unprotected correlators or the full defect category", "evidence_date": "2026-08-24", "falsifier": "a mismatch after allowed local terms and decoupled factors are handled consistently", "visible_in_figure": "yes, aggregated into the evidence-check record" }, { "id": "conjecture-to-claim", "type": "conjecture", "source": "working_conjecture", "target": "full_background_claim", "label": "proposed family-specific completion of the dictionary", "hypotheses": [ "current checks pass within their stated limits" ], "reversible": "no", "status": "conjectural until a stronger argument or complete independent evidence record is supplied", "uncertainty": "unprotected and global-data sectors may remain unresolved", "evidence_date": "2026-08-24", "falsifier": "one reproducible mismatch in the declared claim domain", "visible_in_figure": "no; preserved in the structured equivalent" }, { "id": "failed-zero-background-gauging", "type": "failed-square", "source": "zero_background_claim", "target": "source_daughter", "label": "A=0 equality does not license a gauged daughter claim", "hypotheses": [ "nontrivial bundles, anomaly response, and counterterms were not compared" ], "reversible": "no", "status": "inference blocked; this does not claim that either theory is intrinsically ungaugeable", "uncertainty": "a valid background-dependent extension may exist but has not been established", "evidence_date": "2026-08-24", "falsifier": "supply and verify the full background-dependent 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