{ "schema_version": "1.0.0", "artifact_id": "qft.artifact.supersymmetry-duality.n1-gauge.phases-exact-results-evidence-ledger", "artifact_type": "semantic evidence tables", "title": "N=1 gauge dynamics: exact results and evidence limits", "evidence_cutoff": "2026-08-24", "scope": "Simply connected four-dimensional N=1 gauge theories under the hypotheses stated row by row; compactified rows separately fix spin structure, holonomy, and order of limits.", "interpretation_rule": "A row authorizes only the claim in its claimed-result or infrared-interpretation field, at the strength stated in its status field, and only until its explicit failure test is triggered.", "canonical_source": { "page_id": "qft.topic.susy-gauge-dynamics.infrared-phases-conformal-window", "file": "src/content/docs/supersymmetry-duality/four-dimensional-n1-gauge-dynamics/infrared-phases-conformal-window.md", "route": "/supersymmetry-duality/four-dimensional-n1-gauge-dynamics/infrared-phases-conformal-window/", "anchor": "sqcd-regimes-exact-results-and-evidence-limits", "markdown_sha256": "891b9b5d127c59439c0f24d6bdcb1da523e2efee954e7ffa7782fb92686d0339" }, "tables": [ { "id": "phases-exact-results-evidence-table", "caption": "Rank domains, protected records, infrared interpretations, and failure tests for simply connected four-dimensional N=1 gauge theories.", "column_count": 11, "row_count": 8, "headers": [ { "index": 0, "field_key": "domain_or_model", "label": "Domain or model" }, { "index": 1, "field_key": "theory_hypotheses_and_global_form", "label": "Theory hypotheses and global form" }, { "index": 2, "field_key": "light_variables_and_invariant_relations", "label": "Light variables and invariant relations" }, { "index": 3, "field_key": "exact_or_protected_record", "label": "Exact or protected record" }, { "index": 4, "field_key": "infrared_interpretation", "label": "Infrared interpretation" }, { "index": 5, "field_key": "logical_status_and_control", "label": "Logical status and control" }, { "index": 6, "field_key": "inputs_shared_with_other_checks", "label": "Inputs shared with other checks" }, { "index": 7, "field_key": "what_remains_uncontrolled", "label": "What remains uncontrolled" }, { "index": 8, "field_key": "small_rank_or_global_form_exception", "label": "Small-rank or global-form exception" }, { "index": 9, "field_key": "primary_source_and_evidence_date", "label": "Primary source and evidence date" }, { "index": 10, "field_key": "explicit_failure_test", "label": "Explicit failure test" } ], "rows": [ { "row_index": 0, "row_label": "UV SQCD theory card, Nf > 0", "fields": { "domain_or_model": { "column_index": 0, "header": "Domain or model", "field_key": "domain_or_model", "html": "UV SQCD theory card, Nf > 0", "text": "UV SQCD theory card, Nf > 0", "links": [] }, "theory_hypotheses_and_global_form": { "column_index": 1, "header": "Theory hypotheses and global form", "field_key": "theory_hypotheses_and_global_form", "html": "Simply connected SU(Nc); massless fundamental–antifundamental pairs; zero tree superpotential; the holomorphic scale convention and finite normalization are fixed before coefficients are compared.", "text": "Simply connected SU(Nc); massless fundamental–antifundamental pairs; zero tree superpotential; the holomorphic scale convention and finite normalization are fixed before coefficients are compared.", "links": [] }, "light_variables_and_invariant_relations": { "column_index": 2, "header": "Light variables and invariant relations", "field_key": "light_variables_and_invariant_relations", "html": "Q, Q-tilde; mesons and, when rank permits, baryons; classical rank and Plücker relations.", "text": "Q, Q-tilde; mesons and, when rank permits, baryons; classical rank and Plücker relations.", "links": [] }, "exact_or_protected_record": { "column_index": 3, "header": "Exact or protected record", "field_key": "exact_or_protected_record", "html": "be = 3Nc − Nf; local and mixed anomaly coefficients; faithful connected non-R symmetry after its finite kernel is removed.", "text": "be = 3Nc − Nf; local and mixed anomaly coefficients; faithful connected non-R symmetry after its finite kernel is removed.", "links": [] }, "infrared_interpretation": { "column_index": 4, "header": "Infrared interpretation", "field_key": "infrared_interpretation", "html": "A complete ultraviolet input card from which exact deformations and candidate infrared descriptions can be tested.", "text": "A complete ultraviolet input card from which exact deformations and candidate infrared descriptions can be tested.", "links": [] }, "logical_status_and_control": { "column_index": 5, "header": "Logical status and control", "field_key": "logical_status_and_control", "html": "Exact kinematics, perturbative anomaly data, and a one-loop coefficient; no infrared phase follows by itself.", "text": "Exact kinematics, perturbative anomaly data, and a one-loop coefficient; no infrared phase follows by itself.", "links": [] }, "inputs_shared_with_other_checks": { "column_index": 6, "header": "Inputs shared with other checks", "field_key": "inputs_shared_with_other_checks", "html": "The same charges enter holomorphy, anomaly matching, superconformal R assignments, and both electric and magnetic beta tests.", "text": "The same charges enter holomorphy, anomaly matching, superconformal R assignments, and both electric and magnetic beta tests.", "links": [] }, "what_remains_uncontrolled": { "column_index": 7, "header": "What remains uncontrolled", "field_key": "what_remains_uncontrolled", "html": "Kähler geometry, spectrum, vacuum existence, duality, and confinement.", "text": "Kähler geometry, spectrum, vacuum existence, duality, and confinement.", "links": [] }, "small_rank_or_global_form_exception": { "column_index": 8, "header": "Small-rank or global-form exception", "field_key": "small_rank_or_global_form_exception", "html": "For Nf=2, flavor cubic anomalies vanish but mod-two Witten anomalies can remain; SU(2) color has enhanced SU(2Nf) flavor.", "text": "For Nf=2, flavor cubic anomalies vanish but mod-two Witten anomalies can remain; SU(2) color has enhanced SU(2Nf) flavor.", "links": [] }, "primary_source_and_evidence_date": { "column_index": 9, "header": "Primary source and evidence date", "field_key": "primary_source_and_evidence_date", "html": "Intriligator–Seiberg 1996, §§ 2–3, arXiv PDF pp. 4–12; checked 2026-08-24.", "text": "Intriligator–Seiberg 1996, §§ 2–3, arXiv PDF pp. 4–12; checked 2026-08-24.", "links": [ { "url": "https://arxiv.org/pdf/hep-th/9509066", "label": "Intriligator–Seiberg 1996, §§ 2–3, arXiv PDF pp. 4–12" } ] }, "explicit_failure_test": { "column_index": 10, "header": "Explicit failure test", "field_key": "explicit_failure_test", "html": "Fail if a claimed symmetry acts trivially modulo a gauge center, a mixed gauge anomaly is nonzero, or a classical relation has the wrong rank.", "text": "Fail if a claimed symmetry acts trivially modulo a gauge center, a mixed gauge anomaly is nonzero, or a classical relation has the wrong rank.", "links": [] } } }, { "row_index": 1, "row_label": "Nf = 0", "fields": { "domain_or_model": { "column_index": 0, "header": "Domain or model", "field_key": "domain_or_model", "html": "Nf = 0", "text": "Nf = 0", "links": [] }, "theory_hypotheses_and_global_form": { "column_index": 1, "header": "Theory hypotheses and global form", "field_key": "theory_hypotheses_and_global_form", "html": "Pure simply connected SU(Nc) SYM; exact rigid supersymmetry; declared local-operator and scale normalization.", "text": "Pure simply connected SU(Nc) SYM; exact rigid supersymmetry; declared local-operator and scale normalization.", "links": [] }, "light_variables_and_invariant_relations": { "column_index": 2, "header": "Light variables and invariant relations", "field_key": "light_variables_and_invariant_relations", "html": "Glueball operator S and Nc chiral branches; the electric center one-form symmetry is present.", "text": "Glueball operator S and Nc chiral branches; the electric center one-form symmetry is present.", "links": [] }, "exact_or_protected_record": { "column_index": 3, "header": "Exact or protected record", "field_key": "exact_or_protected_record", "html": "Index and holomorphic decoupling give Nc branches; the anomaly permits the corresponding condensate phases.", "text": "Index and holomorphic decoupling give Nc branches; the anomaly permits the corresponding condensate phases.", "links": [] }, "infrared_interpretation": { "column_index": 4, "header": "Infrared interpretation", "field_key": "infrared_interpretation", "html": "Gapped confining vacua with flux tubes are the standard physical picture.", "text": "Gapped confining vacua with flux tubes are the standard physical picture.", "links": [] }, "logical_status_and_control": { "column_index": 5, "header": "Logical status and control", "field_key": "logical_status_and_control", "html": "Branch count and chiral relations are protected; nonzero normalization uses dynamics; the gap and flux tubes are dynamical expectations.", "text": "Branch count and chiral relations are protected; nonzero normalization uses dynamics; the gap and flux tubes are dynamical expectations.", "links": [] }, "inputs_shared_with_other_checks": { "column_index": 6, "header": "Inputs shared with other checks", "field_key": "inputs_shared_with_other_checks", "html": "The anomaly, index, decoupling path, and compactified monopoles constrain overlapping parts of the result.", "text": "The anomaly, index, decoupling path, and compactified monopoles constrain overlapping parts of the result.", "links": [] }, "what_remains_uncontrolled": { "column_index": 7, "header": "What remains uncontrolled", "field_key": "what_remains_uncontrolled", "html": "The mass spectrum, string tensions, and existence of every proposed BPS wall.", "text": "The mass spectrum, string tensions, and existence of every proposed BPS wall.", "links": [] }, "small_rank_or_global_form_exception": { "column_index": 8, "header": "Small-rank or global-form exception", "field_key": "small_rank_or_global_form_exception", "html": "For general simple groups the count is h∨, while center order need not equal h∨; centerless groups still have chiral branches.", "text": "For general simple groups the count is h∨, while center order need not equal h∨; centerless groups still have chiral branches.", "links": [] }, "primary_source_and_evidence_date": { "column_index": 9, "header": "Primary source and evidence date", "field_key": "primary_source_and_evidence_date", "html": "Witten 2000, §§ 3.1 and 4.2; Kac–Smilga 1999, § 1; Intriligator–Seiberg 1996, § 4.1, arXiv PDF pp. 12–15; checked 2026-08-24.", "text": "Witten 2000, §§ 3.1 and 4.2; Kac–Smilga 1999, § 1; Intriligator–Seiberg 1996, § 4.1, arXiv PDF pp. 12–15; checked 2026-08-24.", "links": [ { "url": "https://arxiv.org/abs/hep-th/0006010", "label": "Witten 2000, §§ 3.1 and 4.2" }, { "url": "https://arxiv.org/abs/hep-th/9902029", "label": "Kac–Smilga 1999, § 1" }, { "url": "https://arxiv.org/pdf/hep-th/9509066", "label": "Intriligator–Seiberg 1996, § 4.1, arXiv PDF pp. 12–15" } ] }, "explicit_failure_test": { "column_index": 10, "header": "Explicit failure test", "field_key": "explicit_failure_test", "html": "Fail any claim of an exact gap or wall spectrum that was inferred only from the holomorphic superpotential.", "text": "Fail any claim of an exact gap or wall spectrum that was inferred only from the holomorphic superpotential.", "links": [] } } }, { "row_index": 2, "row_label": "0 < Nf < Nc", "fields": { "domain_or_model": { "column_index": 0, "header": "Domain or model", "field_key": "domain_or_model", "html": "0 < Nf < Nc", "text": "0 < Nf < Nc", "links": [] }, "theory_hypotheses_and_global_form": { "column_index": 1, "header": "Theory hypotheses and global form", "field_key": "theory_hypotheses_and_global_form", "html": "Massless SU(Nc) SQCD in the finite scheme fixed by the one-instanton anchor.", "text": "Massless SU(Nc) SQCD in the finite scheme fixed by the one-instanton anchor.", "links": [] }, "light_variables_and_invariant_relations": { "column_index": 2, "header": "Light variables and invariant relations", "field_key": "light_variables_and_invariant_relations", "html": "Meson M on a generic Higgs branch; baryons are absent; det M ≠ 0 in the local formula.", "text": "Meson M on a generic Higgs branch; baryons are absent; det M ≠ 0 in the local formula.", "links": [] }, "exact_or_protected_record": { "column_index": 3, "header": "Exact or protected record", "field_key": "exact_or_protected_record", "html": "WADS=(Nc−Nf)(Λ^(3Nc−Nf)/det M)^(1/(Nc−Nf)), with local branch choice.", "text": "WADS=(Nc−Nf)(Λ^(3Nc−Nf)/det M)^(1/(Nc−Nf)), with local branch choice.", "links": [] }, "infrared_interpretation": { "column_index": 4, "header": "Infrared interpretation", "field_key": "infrared_interpretation", "html": "No stationary point at finite M; the massless theory runs toward large field. Generic masses yield isolated vacua.", "text": "No stationary point at finite M; the massless theory runs toward large field. Generic masses yield isolated vacua.", "links": [] }, "logical_status_and_control": { "column_index": 5, "header": "Logical status and control", "field_key": "logical_status_and_control", "html": "Exact Wilsonian F-term under its hypotheses; the weakly coupled large-field runaway is controlled, but its metric is not holomorphically fixed.", "text": "Exact Wilsonian F-term under its hypotheses; the weakly coupled large-field runaway is controlled, but its metric is not holomorphically fixed.", "links": [] }, "inputs_shared_with_other_checks": { "column_index": 6, "header": "Inputs shared with other checks", "field_key": "inputs_shared_with_other_checks", "html": "Symmetries, zero modes at Nf=Nc−1, holomorphic recursion, and independently normalized pure-SYM decoupling.", "text": "Symmetries, zero modes at Nf=Nc−1, holomorphic recursion, and independently normalized pure-SYM decoupling.", "links": [] }, "what_remains_uncontrolled": { "column_index": 7, "header": "What remains uncontrolled", "field_key": "what_remains_uncontrolled", "html": "The Kähler potential at strong field-space loci and nonchiral observables.", "text": "The Kähler potential at strong field-space loci and nonchiral observables.", "links": [] }, "small_rank_or_global_form_exception": { "column_index": 8, "header": "Small-rank or global-form exception", "field_key": "small_rank_or_global_form_exception", "html": "The Nc=2 operator basis is Pfaffian; det M=0 is outside the local ADS coordinate patch.", "text": "The Nc=2 operator basis is Pfaffian; det M=0 is outside the local ADS coordinate patch.", "links": [] }, "primary_source_and_evidence_date": { "column_index": 9, "header": "Primary source and evidence date", "field_key": "primary_source_and_evidence_date", "html": "Intriligator–Seiberg 1996, § 4.1, arXiv PDF pp. 12–15; checked 2026-08-24.", "text": "Intriligator–Seiberg 1996, § 4.1, arXiv PDF pp. 12–15; checked 2026-08-24.", "links": [ { "url": "https://arxiv.org/pdf/hep-th/9509066", "label": "Intriligator–Seiberg 1996, § 4.1, arXiv PDF pp. 12–15" } ] }, "explicit_failure_test": { "column_index": 10, "header": "Explicit failure test", "field_key": "explicit_failure_test", "html": "Fail if the expression has the wrong dimension or R-charge, does not reproduce the Nf=Nc−1 instanton, or violates one-flavor scale matching.", "text": "Fail if the expression has the wrong dimension or R-charge, does not reproduce the Nf=Nc−1 instanton, or violates one-flavor scale matching.", "links": [] } } }, { "row_index": 3, "row_label": "Nf = Nc", "fields": { "domain_or_model": { "column_index": 0, "header": "Domain or model", "field_key": "domain_or_model", "html": "Nf = Nc", "text": "Nf = Nc", "links": [] }, "theory_hypotheses_and_global_form": { "column_index": 1, "header": "Theory hypotheses and global form", "field_key": "theory_hypotheses_and_global_form", "html": "Massless simply connected SU(Nc) SQCD; the same composite and scale normalization as in adjacent ranks.", "text": "Massless simply connected SU(Nc) SQCD; the same composite and scale normalization as in adjacent ranks.", "links": [] }, "light_variables_and_invariant_relations": { "column_index": 2, "header": "Light variables and invariant relations", "field_key": "light_variables_and_invariant_relations", "html": "M, B, B-tilde with det M − B B-tilde = Λ^(2Nc).", "text": "M, B, B-tilde with det M − B B-tilde = Λ^(2Nc).", "links": [] }, "exact_or_protected_record": { "column_index": 3, "header": "Exact or protected record", "field_key": "exact_or_protected_record", "html": "The quantum-modified chiral constraint; no physical superpotential is generated on the constraint surface.", "text": "The quantum-modified chiral constraint; no physical superpotential is generated on the constraint surface.", "links": [] }, "infrared_interpretation": { "column_index": 4, "header": "Infrared interpretation", "field_key": "infrared_interpretation", "html": "A smooth supersymmetric moduli space whose classical origin is removed; symmetry realization depends on the branch.", "text": "A smooth supersymmetric moduli space whose classical origin is removed; symmetry realization depends on the branch.", "links": [] }, "logical_status_and_control": { "column_index": 5, "header": "Logical status and control", "field_key": "logical_status_and_control", "html": "Exact chiral relation. Smoothness follows by differentiating the constraint; the Kähler metric is not determined.", "text": "Exact chiral relation. Smoothness follows by differentiating the constraint; the Kähler metric is not determined.", "links": [] }, "inputs_shared_with_other_checks": { "column_index": 6, "header": "Inputs shared with other checks", "field_key": "inputs_shared_with_other_checks", "html": "Holomorphy, anomalies, mass deformation to ADS, and decoupling from the s-confining rank.", "text": "Holomorphy, anomalies, mass deformation to ADS, and decoupling from the s-confining rank.", "links": [] }, "what_remains_uncontrolled": { "column_index": 7, "header": "What remains uncontrolled", "field_key": "what_remains_uncontrolled", "html": "Metric distances, masses, and scattering along the moduli space.", "text": "Metric distances, masses, and scattering along the moduli space.", "links": [] }, "small_rank_or_global_form_exception": { "column_index": 8, "header": "Small-rank or global-form exception", "field_key": "small_rank_or_global_form_exception", "html": "For SU(2), Nf=2, use Pf V = Λ^4 and the enhanced SU(4) flavor symmetry.", "text": "For SU(2), Nf=2, use Pf V = Λ^4 and the enhanced SU(4) flavor symmetry.", "links": [] }, "primary_source_and_evidence_date": { "column_index": 9, "header": "Primary source and evidence date", "field_key": "primary_source_and_evidence_date", "html": "Seiberg 1994, § 4, arXiv PDF pp. 7–10; checked 2026-08-24.", "text": "Seiberg 1994, § 4, arXiv PDF pp. 7–10; checked 2026-08-24.", "links": [ { "url": "https://arxiv.org/pdf/hep-th/9402044", "label": "Seiberg 1994, § 4, arXiv PDF pp. 7–10" } ] }, "explicit_failure_test": { "column_index": 10, "header": "Explicit failure test", "field_key": "explicit_failure_test", "html": "Fail if the gradient of the constraint vanishes on the surface, if the origin is retained, or if a mass deformation misses the ADS scale power.", "text": "Fail if the gradient of the constraint vanishes on the surface, if the origin is retained, or if a mass deformation misses the ADS scale power.", "links": [] } } }, { "row_index": 4, "row_label": "Nf = Nc + 1", "fields": { "domain_or_model": { "column_index": 0, "header": "Domain or model", "field_key": "domain_or_model", "html": "Nf = Nc + 1", "text": "Nf = Nc + 1", "links": [] }, "theory_hypotheses_and_global_form": { "column_index": 1, "header": "Theory hypotheses and global form", "field_key": "theory_hypotheses_and_global_form", "html": "Massless simply connected SU(Nc) SQCD; all composites retained.", "text": "Massless simply connected SU(Nc) SQCD; all composites retained.", "links": [] }, "light_variables_and_invariant_relations": { "column_index": 2, "header": "Light variables and invariant relations", "field_key": "light_variables_and_invariant_relations", "html": "M, B, B-tilde and W=(B M B-tilde − det M)/Λ^(2Nc−1).", "text": "M, B, B-tilde and W=(B M B-tilde − det M)/Λ^(2Nc−1).", "links": [] }, "exact_or_protected_record": { "column_index": 3, "header": "Exact or protected record", "field_key": "exact_or_protected_record", "html": "Polynomial confining superpotential, its F-term relations, anomaly matching, and mass flow to the quantum-modified rank.", "text": "Polynomial confining superpotential, its F-term relations, anomaly matching, and mass flow to the quantum-modified rank.", "links": [] }, "infrared_interpretation": { "column_index": 4, "header": "Infrared interpretation", "field_key": "infrared_interpretation", "html": "A regular composite effective description, including a massless symmetry-preserving origin; the vacuum variety still has singular strata.", "text": "A regular composite effective description, including a massless symmetry-preserving origin; the vacuum variety still has singular strata.", "links": [] }, "logical_status_and_control": { "column_index": 5, "header": "Logical status and control", "field_key": "logical_status_and_control", "html": "Exact Wilsonian F-term plus the s-confining infrared interpretation. The canonically normalized cubic is marginally irrelevant at the free endpoint.", "text": "Exact Wilsonian F-term plus the s-confining infrared interpretation. The canonically normalized cubic is marginally irrelevant at the free endpoint.", "links": [] }, "inputs_shared_with_other_checks": { "column_index": 6, "header": "Inputs shared with other checks", "field_key": "inputs_shared_with_other_checks", "html": "Composite anomalies, deformations, and the same normalization chain used for lower ranks.", "text": "Composite anomalies, deformations, and the same normalization chain used for lower ranks.", "links": [] }, "what_remains_uncontrolled": { "column_index": 7, "header": "What remains uncontrolled", "field_key": "what_remains_uncontrolled", "html": "The exact Kähler potential and normalization of nonchiral correlators.", "text": "The exact Kähler potential and normalization of nonchiral correlators.", "links": [] }, "small_rank_or_global_form_exception": { "column_index": 8, "header": "Small-rank or global-form exception", "field_key": "small_rank_or_global_form_exception", "html": "For SU(2), Nf=3, use W=−Pf V/Λ^3 with enhanced SU(6) flavor.", "text": "For SU(2), Nf=3, use W=−Pf V/Λ^3 with enhanced SU(6) flavor.", "links": [] }, "primary_source_and_evidence_date": { "column_index": 9, "header": "Primary source and evidence date", "field_key": "primary_source_and_evidence_date", "html": "Seiberg 1994, § 5, arXiv PDF pp. 10–15; checked 2026-08-24.", "text": "Seiberg 1994, § 5, arXiv PDF pp. 10–15; checked 2026-08-24.", "links": [ { "url": "https://arxiv.org/pdf/hep-th/9402044", "label": "Seiberg 1994, § 5, arXiv PDF pp. 10–15" } ] }, "explicit_failure_test": { "column_index": 10, "header": "Explicit failure test", "field_key": "explicit_failure_test", "html": "Fail if the composite anomalies do not match, the F-terms miss the classical relations, or one-flavor decoupling misses the quantum constraint.", "text": "Fail if the composite anomalies do not match, the F-terms miss the classical relations, or one-flavor decoupling misses the quantum constraint.", "links": [] } } }, { "row_index": 5, "row_label": "Nc + 2 ≤ Nf < 3Nc/2", "fields": { "domain_or_model": { "column_index": 0, "header": "Domain or model", "field_key": "domain_or_model", "html": "Nc + 2 ≤ Nf < 3Nc/2", "text": "Nc + 2 ≤ Nf < 3Nc/2", "links": [] }, "theory_hypotheses_and_global_form": { "column_index": 1, "header": "Theory hypotheses and global form", "field_key": "theory_hypotheses_and_global_form", "html": "Massless SQCD plus Seiberg’s magnetic dictionary; the integer interval must be nonempty.", "text": "Massless SQCD plus Seiberg’s magnetic dictionary; the integer interval must be nonempty.", "links": [] }, "light_variables_and_invariant_relations": { "column_index": 2, "header": "Light variables and invariant relations", "field_key": "light_variables_and_invariant_relations", "html": "SU(Nf−Nc) magnetic gauge field, q, q-tilde, elementary dimension-one Mm, and W=y Mm q q-tilde.", "text": "SU(Nf−Nc) magnetic gauge field, q, q-tilde, elementary dimension-one Mm, and W=y Mm q q-tilde.", "links": [] }, "exact_or_protected_record": { "column_index": 3, "header": "Exact or protected record", "field_key": "exact_or_protected_record", "html": "bm=2Nf−3Nc<0; anomaly, baryon, moduli, and deformation matches.", "text": "bm=2Nf−3Nc<0; anomaly, baryon, moduli, and deformation matches.", "links": [] }, "infrared_interpretation": { "column_index": 4, "header": "Infrared interpretation", "field_key": "infrared_interpretation", "html": "Free magnetic infrared phase with a surviving weakly coupled gauge field.", "text": "Free magnetic infrared phase with a surviving weakly coupled gauge field.", "links": [] }, "logical_status_and_control": { "column_index": 5, "header": "Logical status and control", "field_key": "logical_status_and_control", "html": "Infrared freedom is perturbative once the magnetic description is accepted; the identification with the electric theory is duality-supported.", "text": "Infrared freedom is perturbative once the magnetic description is accepted; the identification with the electric theory is duality-supported.", "links": [] }, "inputs_shared_with_other_checks": { "column_index": 6, "header": "Inputs shared with other checks", "field_key": "inputs_shared_with_other_checks", "html": "The same duality dictionary supplies the magnetic rank, R-charges, anomalies, and deformation map.", "text": "The same duality dictionary supplies the magnetic rank, R-charges, anomalies, and deformation map.", "links": [] }, "what_remains_uncontrolled": { "column_index": 7, "header": "What remains uncontrolled", "field_key": "what_remains_uncontrolled", "html": "A first-principles proof of equivalence and generic unprotected electric observables.", "text": "A first-principles proof of equivalence and generic unprotected electric observables.", "links": [] }, "small_rank_or_global_form_exception": { "column_index": 8, "header": "Small-rank or global-form exception", "field_key": "small_rank_or_global_form_exception", "html": "The interval is empty for Nc=3; SU(2) color uses enhanced flavor; the lower endpoint is excluded.", "text": "The interval is empty for Nc=3; SU(2) color uses enhanced flavor; the lower endpoint is excluded.", "links": [] }, "primary_source_and_evidence_date": { "column_index": 9, "header": "Primary source and evidence date", "field_key": "primary_source_and_evidence_date", "html": "Seiberg 1995, §§ 2–4, arXiv PDF pp. 4–14; checked 2026-08-24.", "text": "Seiberg 1995, §§ 2–4, arXiv PDF pp. 4–14; checked 2026-08-24.", "links": [ { "url": "https://arxiv.org/pdf/hep-th/9411149", "label": "Seiberg 1995, §§ 2–4, arXiv PDF pp. 4–14" } ] }, "explicit_failure_test": { "column_index": 10, "header": "Explicit failure test", "field_key": "explicit_failure_test", "html": "Fail if magnetic anomalies, baryon charges, rank-changing mass flows, or the sign of bm disagree.", "text": "Fail if magnetic anomalies, baryon charges, rank-changing mass flows, or the sign of bm disagree.", "links": [] } } }, { "row_index": 6, "row_label": "3Nc/2 < Nf < 3Nc", "fields": { "domain_or_model": { "column_index": 0, "header": "Domain or model", "field_key": "domain_or_model", "html": "3Nc/2 < Nf < 3Nc", "text": "3Nc/2 < Nf < 3Nc", "links": [] }, "theory_hypotheses_and_global_form": { "column_index": 1, "header": "Theory hypotheses and global form", "field_key": "theory_hypotheses_and_global_form", "html": "Massless SQCD, the proposed electric–magnetic equivalence, and an interacting superconformal endpoint.", "text": "Massless SQCD, the proposed electric–magnetic equivalence, and an interacting superconformal endpoint.", "links": [] }, "light_variables_and_invariant_relations": { "column_index": 2, "header": "Light variables and invariant relations", "field_key": "light_variables_and_invariant_relations", "html": "Electric and magnetic variables; protected chiral operators obey the common superconformal R assignment.", "text": "Electric and magnetic variables; protected chiral operators obey the common superconformal R assignment.", "links": [] }, "exact_or_protected_record": { "column_index": 3, "header": "Exact or protected record", "field_key": "exact_or_protected_record", "html": "Δ(M)=3(1−Nc/Nf)>1; anomaly and deformation matches; exact R selection can use a-maximization.", "text": "Δ(M)=3(1−Nc/Nf)>1; anomaly and deformation matches; exact R selection can use a-maximization.", "links": [] }, "infrared_interpretation": { "column_index": 4, "header": "Infrared interpretation", "field_key": "infrared_interpretation", "html": "Interacting non-Abelian Coulomb phase described by either duality frame.", "text": "Interacting non-Abelian Coulomb phase described by either duality frame.", "links": [] }, "logical_status_and_control": { "column_index": 5, "header": "Logical status and control", "field_key": "logical_status_and_control", "html": "Duality-supported throughout; perturbatively controlled only parametrically near either edge in a large-rank fractional-edge limit.", "text": "Duality-supported throughout; perturbatively controlled only parametrically near either edge in a large-rank fractional-edge limit.", "links": [] }, "inputs_shared_with_other_checks": { "column_index": 6, "header": "Inputs shared with other checks", "field_key": "inputs_shared_with_other_checks", "html": "The unitarity and beta-function boundary tests share the rank, R-charge, and magnetic dictionary.", "text": "The unitarity and beta-function boundary tests share the rank, R-charge, and magnetic dictionary.", "links": [] }, "what_remains_uncontrolled": { "column_index": 7, "header": "What remains uncontrolled", "field_key": "what_remains_uncontrolled", "html": "Generic unprotected dimensions and OPE data away from the edges, and a mathematical construction of the fixed point.", "text": "Generic unprotected dimensions and OPE data away from the edges, and a mathematical construction of the fixed point.", "links": [] }, "small_rank_or_global_form_exception": { "column_index": 8, "header": "Small-rank or global-form exception", "field_key": "small_rank_or_global_form_exception", "html": "At fixed small ranks the nearest integer need not be weakly coupled; accidental symmetries require a revised R analysis.", "text": "At fixed small ranks the nearest integer need not be weakly coupled; accidental symmetries require a revised R analysis.", "links": [] }, "primary_source_and_evidence_date": { "column_index": 9, "header": "Primary source and evidence date", "field_key": "primary_source_and_evidence_date", "html": "Seiberg 1995, §§ 2–4, arXiv PDF pp. 4–14; Intriligator–Wecht 2003, §§ 1–2, arXiv PDF pp. 1–8; protected evidence through 2003, checked 2026-08-24.", "text": "Seiberg 1995, §§ 2–4, arXiv PDF pp. 4–14; Intriligator–Wecht 2003, §§ 1–2, arXiv PDF pp. 1–8; protected evidence through 2003, checked 2026-08-24.", "links": [ { "url": "https://arxiv.org/pdf/hep-th/9411149", "label": "Seiberg 1995, §§ 2–4, arXiv PDF pp. 4–14" }, { "url": "https://arxiv.org/pdf/hep-th/0304128", "label": "Intriligator–Wecht 2003, §§ 1–2, arXiv PDF pp. 1–8" } ] }, "explicit_failure_test": { "column_index": 10, "header": "Explicit failure test", "field_key": "explicit_failure_test", "html": "Fail a proposed dictionary if an exact anomaly, chiral dimension, deformation, or protected index is inconsistent in the two frames.", "text": "Fail a proposed dictionary if an exact anomaly, chiral dimension, deformation, or protected index is inconsistent in the two frames.", "links": [] } } }, { "row_index": 7, "row_label": "3–2 model, weak-λ hierarchy", "fields": { "domain_or_model": { "column_index": 0, "header": "Domain or model", "field_key": "domain_or_model", "html": "3–2 model, weak-λ hierarchy", "text": "3–2 model, weak-λ hierarchy", "links": [] }, "theory_hypotheses_and_global_form": { "column_index": 1, "header": "Theory hypotheses and global form", "field_key": "theory_hypotheses_and_global_form", "html": "SU(3)×SU(2) with one standard chiral generation and generic renormalizable tree coupling; g2/λ ≫ 1 in the displayed calculable regime.", "text": "SU(3)×SU(2) with one standard chiral generation and generic renormalizable tree coupling; g2/λ ≫ 1 in the displayed calculable regime.", "links": [] }, "light_variables_and_invariant_relations": { "column_index": 2, "header": "Light variables and invariant relations", "field_key": "light_variables_and_invariant_relations", "html": "Gauge invariants on the D-flat manifold and the exact SU(3) ADS term.", "text": "Gauge invariants on the D-flat manifold and the exact SU(3) ADS term.", "links": [] }, "exact_or_protected_record": { "column_index": 3, "header": "Exact or protected record", "field_key": "exact_or_protected_record", "html": "The F-equations are incompatible; minimization gives v∼Λ3 λ^(−1/7) and positive energy scaling V∼Λ3^4 λ^(10/7).", "text": "The F-equations are incompatible; minimization gives v∼Λ3 λ^(−1/7) and positive energy scaling V∼Λ3^4 λ^(10/7).", "links": [] }, "infrared_interpretation": { "column_index": 4, "header": "Infrared interpretation", "field_key": "infrared_interpretation", "html": "A stable calculable vacuum with spontaneous supersymmetry and R-symmetry breaking in the declared hierarchy.", "text": "A stable calculable vacuum with spontaneous supersymmetry and R-symmetry breaking in the declared hierarchy.", "links": [] }, "logical_status_and_control": { "column_index": 5, "header": "Logical status and control", "field_key": "logical_status_and_control", "html": "Exact F-term obstruction plus controlled semiclassical model calculation; not a general theorem about chiral gauge theories.", "text": "Exact F-term obstruction plus controlled semiclassical model calculation; not a general theorem about chiral gauge theories.", "links": [] }, "inputs_shared_with_other_checks": { "column_index": 6, "header": "Inputs shared with other checks", "field_key": "inputs_shared_with_other_checks", "html": "The same charges enter gauge consistency, faithful U(1)X/Z6, anomalies, the tree term, and the minimization.", "text": "The same charges enter gauge consistency, faithful U(1)X/Z6, anomalies, the tree term, and the minimization.", "links": [] }, "what_remains_uncontrolled": { "column_index": 7, "header": "What remains uncontrolled", "field_key": "what_remains_uncontrolled", "html": "Other coupling regimes, arbitrary Kähler corrections near strong field, and a general chiral-theory classification.", "text": "Other coupling regimes, arbitrary Kähler corrections near strong field, and a general chiral-theory classification.", "links": [] }, "small_rank_or_global_form_exception": { "column_index": 8, "header": "Small-rank or global-form exception", "field_key": "small_rank_or_global_form_exception", "html": "An omitted runaway, singular branch, or a regime with g2/λ not large invalidates the displayed minimization.", "text": "An omitted runaway, singular branch, or a regime with g2/λ not large invalidates the displayed minimization.", "links": [] }, "primary_source_and_evidence_date": { "column_index": 9, "header": "Primary source and evidence date", "field_key": "primary_source_and_evidence_date", "html": "Shacham 2012, § 2, arXiv PDF pp. 3–8; checked 2026-08-24.", "text": "Shacham 2012, § 2, arXiv PDF pp. 3–8; checked 2026-08-24.", "links": [ { "url": "https://arxiv.org/pdf/1202.6031", "label": "Shacham 2012, § 2, arXiv PDF pp. 3–8" } ] }, "explicit_failure_test": { "column_index": 10, "header": "Explicit failure test", "field_key": "explicit_failure_test", "html": "Fail if a simultaneous F-flat solution exists on the complete physical domain, the Hessian has a tachyon, or the minimum leaves the controlled hierarchy.", "text": "Fail if a simultaneous F-flat solution exists on the complete physical domain, the Hessian has a tachyon, or the minimum leaves the controlled hierarchy.", "links": [] } } } ], "source_html_sha256": "a076b89a82d53204a399972fba59ee70d4de4583774f1b5399a67292cc77d7de" }, { "id": "compactification-evidence-table", "caption": "What compactified pure SYM calculations establish, and when their inference fails.", "column_count": 8, "row_count": 6, "headers": [ { "index": 0, "field_key": "regime", "label": "Regime" }, { "index": 1, "field_key": "spin_structure_holonomy_and_scale_order", "label": "Spin structure, holonomy, and scale order" }, { "index": 2, "field_key": "relevant_saddles_or_variables", "label": "Relevant saddles or variables" }, { "index": 3, "field_key": "claimed_result", "label": "Claimed result" }, { "index": 4, "field_key": "status", "label": "Status" }, { "index": 5, "field_key": "source_scope", "label": "Source scope" }, { "index": 6, "field_key": "what_is_not_transported", "label": "What is not transported" }, { "index": 7, "field_key": "failure_mode", "label": "Failure mode" } ], "rows": [ { "row_index": 0, "row_label": "Finite small spatial circle", "fields": { "regime": { "column_index": 0, "header": "Regime", "field_key": "regime", "html": "Finite small spatial circle", "text": "Finite small spatial circle", "links": [] }, "spin_structure_holonomy_and_scale_order": { "column_index": 1, "header": "Spin structure, holonomy, and scale order", "field_key": "spin_structure_holonomy_and_scale_order", "html": "Periodic gaugino; center-symmetric holonomy selected dynamically; N L Λ ≪ 1 for SU(N).", "text": "Periodic gaugino; center-symmetric holonomy selected dynamically; N L Λ ≪ 1 for SU(N).", "links": [] }, "relevant_saddles_or_variables": { "column_index": 2, "header": "Relevant saddles or variables", "field_key": "relevant_saddles_or_variables", "html": "N fundamental monopole events, including the affine event; their product reassembles one instanton.", "text": "N fundamental monopole events, including the affine event; their product reassembles one instanton.", "links": [] }, "claimed_result": { "column_index": 3, "header": "Claimed result", "field_key": "claimed_result", "html": "The monopole superpotential, discrete vacua, condensate phases, and an abelian mass gap are semiclassically calculable.", "text": "The monopole superpotential, discrete vacua, condensate phases, and an abelian mass gap are semiclassically calculable.", "links": [] }, "status": { "column_index": 4, "header": "Status", "field_key": "status", "html": "Controlled semiclassical expansion for the named protected and long-distance observables.", "text": "Controlled semiclassical expansion for the named protected and long-distance observables.", "links": [] }, "source_scope": { "column_index": 5, "header": "Source scope", "field_key": "source_scope", "html": "Davies–Hollowood–Khoze 2003, §§ 4–5.", "text": "Davies–Hollowood–Khoze 2003, §§ 4–5.", "links": [ { "url": "https://arxiv.org/abs/hep-th/0006011", "label": "Davies–Hollowood–Khoze 2003, §§ 4–5" } ] }, "what_is_not_transported": { "column_index": 6, "header": "What is not transported", "field_key": "what_is_not_transported", "html": "A proof of smooth decompactification or all four-dimensional spectral data.", "text": "A proof of smooth decompactification or all four-dimensional spectral data.", "links": [] }, "failure_mode": { "column_index": 7, "header": "Failure mode", "field_key": "failure_mode", "html": "Fail when the lightest W-boson scale is no longer parametrically above strong dynamics.", "text": "Fail when the lightest W-boson scale is no longer parametrically above strong dynamics.", "links": [] } } }, { "row_index": 1, "row_label": "Strict three-dimensional limit", "fields": { "regime": { "column_index": 0, "header": "Regime", "field_key": "regime", "html": "Strict three-dimensional limit", "text": "Strict three-dimensional limit", "links": [] }, "spin_structure_holonomy_and_scale_order": { "column_index": 1, "header": "Spin structure, holonomy, and scale order", "field_key": "spin_structure_holonomy_and_scale_order", "html": "L→0 with g3²=g4²/L fixed, so the four-dimensional instanton factor tends to zero.", "text": "L→0 with g3²=g4²/L fixed, so the four-dimensional instanton factor tends to zero.", "links": [] }, "relevant_saddles_or_variables": { "column_index": 2, "header": "Relevant saddles or variables", "field_key": "relevant_saddles_or_variables", "html": "The affine monopole term disappears; only the non-affine Toda terms remain.", "text": "The affine monopole term disappears; only the non-affine Toda terms remain.", "links": [] }, "claimed_result": { "column_index": 3, "header": "Claimed result", "field_key": "claimed_result", "html": "The Coulomb coordinate runs away rather than producing the finite-circle four-dimensional vacua.", "text": "The Coulomb coordinate runs away rather than producing the finite-circle four-dimensional vacua.", "links": [] }, "status": { "column_index": 4, "header": "Status", "field_key": "status", "html": "Controlled but different limit.", "text": "Controlled but different limit.", "links": [] }, "source_scope": { "column_index": 5, "header": "Source scope", "field_key": "source_scope", "html": "Davies–Hollowood–Khoze 2003, § 4, Eq. (4.18).", "text": "Davies–Hollowood–Khoze 2003, § 4, Eq. (4.18).", "links": [ { "url": "https://arxiv.org/abs/hep-th/0006011", "label": "Davies–Hollowood–Khoze 2003, § 4, Eq. (4.18)" } ] }, "what_is_not_transported": { "column_index": 6, "header": "What is not transported", "field_key": "what_is_not_transported", "html": "The finite-L vacuum count.", "text": "The finite-L vacuum count.", "links": [] }, "failure_mode": { "column_index": 7, "header": "Failure mode", "field_key": "failure_mode", "html": "Fail if “small circle” is silently identified with strict three dimensions.", "text": "Fail if “small circle” is silently identified with strict three dimensions.", "links": [] } } }, { "row_index": 2, "row_label": "Decompactification", "fields": { "regime": { "column_index": 0, "header": "Regime", "field_key": "regime", "html": "Decompactification", "text": "Decompactification", "links": [] }, "spin_structure_holonomy_and_scale_order": { "column_index": 1, "header": "Spin structure, holonomy, and scale order", "field_key": "spin_structure_holonomy_and_scale_order", "html": "Increase L at fixed finite N and periodic spin structure.", "text": "Increase L at fixed finite N and periodic spin structure.", "links": [] }, "relevant_saddles_or_variables": { "column_index": 2, "header": "Relevant saddles or variables", "field_key": "relevant_saddles_or_variables", "html": "The abelian saddle expansion eventually loses parametric control.", "text": "The abelian saddle expansion eventually loses parametric control.", "links": [] }, "claimed_result": { "column_index": 3, "header": "Claimed result", "field_key": "claimed_result", "html": "Protected holomorphic quantities may agree if no singularity or phase transition intervenes.", "text": "Protected holomorphic quantities may agree if no singularity or phase transition intervenes.", "links": [] }, "status": { "column_index": 4, "header": "Status", "field_key": "status", "html": "Continuity assumption, observable by observable.", "text": "Continuity assumption, observable by observable.", "links": [] }, "source_scope": { "column_index": 5, "header": "Source scope", "field_key": "source_scope", "html": "Hollowood–Khoze–Lee–Mattis 1999, abstract and discussion.", "text": "Hollowood–Khoze–Lee–Mattis 1999, abstract and discussion.", "links": [ { "url": "https://arxiv.org/abs/hep-th/9904116", "label": "Hollowood–Khoze–Lee–Mattis 1999, abstract and discussion" } ] }, "what_is_not_transported": { "column_index": 6, "header": "What is not transported", "field_key": "what_is_not_transported", "html": "Unprotected masses, string tensions, and wall existence.", "text": "Unprotected masses, string tensions, and wall existence.", "links": [] }, "failure_mode": { "column_index": 7, "header": "Failure mode", "field_key": "failure_mode", "html": "Fail at a phase transition, singular branch, or nonuniform limit.", "text": "Fail at a phase transition, singular branch, or nonuniform limit.", "links": [] } } }, { "row_index": 3, "row_label": "Thermal circle", "fields": { "regime": { "column_index": 0, "header": "Regime", "field_key": "regime", "html": "Thermal circle", "text": "Thermal circle", "links": [] }, "spin_structure_holonomy_and_scale_order": { "column_index": 1, "header": "Spin structure, holonomy, and scale order", "field_key": "spin_structure_holonomy_and_scale_order", "html": "Antiperiodic gaugino; thermal ensemble.", "text": "Antiperiodic gaugino; thermal ensemble.", "links": [] }, "relevant_saddles_or_variables": { "column_index": 2, "header": "Relevant saddles or variables", "field_key": "relevant_saddles_or_variables", "html": "Thermal holonomy potential and thermal excitations, not supersymmetric monopole balance.", "text": "Thermal holonomy potential and thermal excitations, not supersymmetric monopole balance.", "links": [] }, "claimed_result": { "column_index": 3, "header": "Claimed result", "field_key": "claimed_result", "html": "A thermal center transition may occur.", "text": "A thermal center transition may occur.", "links": [] }, "status": { "column_index": 4, "header": "Status", "field_key": "status", "html": "Different physical theory; no supersymmetric-continuity inference.", "text": "Different physical theory; no supersymmetric-continuity inference.", "links": [] }, "source_scope": { "column_index": 5, "header": "Source scope", "field_key": "source_scope", "html": "Spin-structure distinction is exact.", "text": "Spin-structure distinction is exact.", "links": [] }, "what_is_not_transported": { "column_index": 6, "header": "What is not transported", "field_key": "what_is_not_transported", "html": "Periodic-circle vacuum claims.", "text": "Periodic-circle vacuum claims.", "links": [] }, "failure_mode": { "column_index": 7, "header": "Failure mode", "field_key": "failure_mode", "html": "Fail any argument that cites a thermal result for the spatial periodic theory without a new continuation.", "text": "Fail any argument that cites a thermal result for the spatial periodic theory without a new continuation.", "links": [] } } }, { "row_index": 4, "row_label": "Large N before small L", "fields": { "regime": { "column_index": 0, "header": "Regime", "field_key": "regime", "html": "Large N before small L", "text": "Large N before small L", "links": [] }, "spin_structure_holonomy_and_scale_order": { "column_index": 1, "header": "Spin structure, holonomy, and scale order", "field_key": "spin_structure_holonomy_and_scale_order", "html": "The W-boson spacing scales as 1/(N L).", "text": "The W-boson spacing scales as 1/(N L).", "links": [] }, "relevant_saddles_or_variables": { "column_index": 2, "header": "Relevant saddles or variables", "field_key": "relevant_saddles_or_variables", "html": "An increasingly dense tower of off-diagonal modes.", "text": "An increasingly dense tower of off-diagonal modes.", "links": [] }, "claimed_result": { "column_index": 3, "header": "Claimed result", "field_key": "claimed_result", "html": "The fixed-N abelian semiclassical hierarchy need not survive.", "text": "The fixed-N abelian semiclassical hierarchy need not survive.", "links": [] }, "status": { "column_index": 4, "header": "Status", "field_key": "status", "html": "Order-of-limits warning.", "text": "Order-of-limits warning.", "links": [] }, "source_scope": { "column_index": 5, "header": "Source scope", "field_key": "source_scope", "html": "The scale relation is perturbative and exact at the center-symmetric background.", "text": "The scale relation is perturbative and exact at the center-symmetric background.", "links": [] }, "what_is_not_transported": { "column_index": 6, "header": "What is not transported", "field_key": "what_is_not_transported", "html": "Uniform large-N continuity.", "text": "Uniform large-N continuity.", "links": [] }, "failure_mode": { "column_index": 7, "header": "Failure mode", "field_key": "failure_mode", "html": "Fail when N L Λ is not small even though L Λ is.", "text": "Fail when N L Λ is not small even though L Λ is.", "links": [] } } }, { "row_index": 5, "row_label": "SU(2) lattice evidence", "fields": { "regime": { "column_index": 0, "header": "Regime", "field_key": "regime", "html": "SU(2) lattice evidence", "text": "SU(2) lattice evidence", "links": [] }, "spin_structure_holonomy_and_scale_order": { "column_index": 1, "header": "Spin structure, holonomy, and scale order", "field_key": "spin_structure_holonomy_and_scale_order", "html": "Periodic adjoint Majorana fermion in the explored finite-mass and finite-cutoff window.", "text": "Periodic adjoint Majorana fermion in the explored finite-mass and finite-cutoff window.", "links": [] }, "relevant_saddles_or_variables": { "column_index": 2, "header": "Relevant saddles or variables", "field_key": "relevant_saddles_or_variables", "html": "Lattice order parameters and spectrum proxies.", "text": "Lattice order parameters and spectrum proxies.", "links": [] }, "claimed_result": { "column_index": 3, "header": "Claimed result", "field_key": "claimed_result", "html": "Center stability and no observed intervening transition in the sampled regime.", "text": "Center stability and no observed intervening transition in the sampled regime.", "links": [] }, "status": { "column_index": 4, "header": "Status", "field_key": "status", "html": "Numerical evidence, not a theorem of continuum all-N SYM.", "text": "Numerical evidence, not a theorem of continuum all-N SYM.", "links": [] }, "source_scope": { "column_index": 5, "header": "Source scope", "field_key": "source_scope", "html": "Bergner–Piemonte–Ünsal 2018, §§ 4–5.", "text": "Bergner–Piemonte–Ünsal 2018, §§ 4–5.", "links": [ { "url": "https://doi.org/10.1007/JHEP11(2018)092", "label": "Bergner–Piemonte–Ünsal 2018, §§ 4–5" } ] }, "what_is_not_transported": { "column_index": 6, "header": "What is not transported", "field_key": "what_is_not_transported", "html": "Other ranks, the continuum limit, and masses outside the simulation window.", "text": "Other ranks, the continuum limit, and masses outside the simulation window.", "links": [] }, "failure_mode": { "column_index": 7, "header": "Failure mode", "field_key": "failure_mode", "html": "Fail if cutoff, volume, mass, or rank extrapolations are presented as directly simulated facts.", "text": "Fail if cutoff, volume, mass, or rank extrapolations are presented as directly simulated facts.", "links": [] } } } ], "source_html_sha256": "25d6ac342dcda89a4a3c8b19207671a020bd85d577ba0df9a4c4657ac3b32e50" } ], "semantic_coverage": { "table_count": 2, "row_count": 14, "field_count": 136, "every_canonical_header_row_and_cell_preserved": true, "raw_cell_html_preserved": true, "normalized_cell_text_preserved": true, "links_preserved": true }, "cross_artifact_checks": { "phase_fixture_scope": "Nc=2,3,4,5 integer intervals, endpoints, beta signs, unitarity, and the shared-input caveat", "quantum_fixture_scope": "operator dimensions and R/B charges, threshold exponents, branch counts, one s-confining anomaly, and SU(2) Pfaffians", "compactification_scope": "finite circle, strict 3D, decompactification, thermal, large-N-first, and SU(2) lattice rows remain distinct", "dynamical_breaking_scope": "the 3–2 model is a controlled example in its hierarchy, not a theorem about all chiral theories" }, "cited_links": [ { "url": "https://arxiv.org/pdf/hep-th/9509066", "label": "Intriligator–Seiberg 1996, § 4.1, arXiv PDF pp. 12–15" }, { "url": "https://arxiv.org/abs/hep-th/0006010", "label": "Witten 2000, §§ 3.1 and 4.2" }, { "url": "https://arxiv.org/abs/hep-th/9902029", "label": "Kac–Smilga 1999, § 1" }, { "url": "https://arxiv.org/pdf/hep-th/9402044", "label": "Seiberg 1994, § 5, arXiv PDF pp. 10–15" }, { "url": "https://arxiv.org/pdf/hep-th/9411149", "label": "Seiberg 1995, §§ 2–4, arXiv PDF pp. 4–14" }, { "url": "https://arxiv.org/pdf/hep-th/0304128", "label": "Intriligator–Wecht 2003, §§ 1–2, arXiv PDF pp. 1–8" }, { "url": "https://arxiv.org/pdf/1202.6031", "label": "Shacham 2012, § 2, arXiv PDF pp. 3–8" }, { "url": "https://arxiv.org/abs/hep-th/0006011", "label": "Davies–Hollowood–Khoze 2003, § 4, Eq. (4.18)" }, { "url": "https://arxiv.org/abs/hep-th/9904116", "label": "Hollowood–Khoze–Lee–Mattis 1999, abstract and discussion" }, { "url": "https://doi.org/10.1007/JHEP11(2018)092", "label": "Bergner–Piemonte–Ünsal 2018, §§ 4–5" } ], "governance_snapshot": { "registry_status": "prototype", "registry_public_route": null, "integration_state": "complete" } }