{
  "artifact_id": "qft.artifact.supersymmetry-duality.exact-observables.partition-index-instanton-factorization-flow",
  "schema_version": "qualified-exact-observables-map/v2",
  "fixture_revision": 3,
  "title": "Three exact constructions, one qualified comparison",
  "reader_question": "How do localized sphere integrals, protected traces, equivariant instanton sums, holomorphic blocks, and duality comparisons fit together without conflating what each one computes?",
  "takeaway": "Sphere matrix models, supersymmetric indices, and instanton sums are distinct exact constructions. They can be factorized, glued, compared, or exported only after their defining sectors, cycles, schemes, convergence domains, and normalizations travel with the result; equality then gives bounded protected evidence rather than a full-spectrum or full-duality theorem.",
  "scope": "Schematic map of commonly used exact observables in supersymmetric QFT. It is not a dimension-independent localization formula: every branch must be instantiated for a named theory, dimension, background, global form, and preserved supercharge.",
  "qualified_input": {
    "name": "Qualified observable specification",
    "fields": [
      "theory and global gauge/flavor data",
      "dimension, Euclidean background, bundles, and spin structure",
      "preserved supercharge and complete Q-squared action",
      "couplings, masses, FI parameters, fugacities, and charge normalization",
      "topological and defect sectors",
      "integration cycle or residue chamber",
      "regulator, determinant phase, and local-counterterm scheme",
      "normalization, convergence domain, and benchmark limit"
    ]
  },
  "branches": [
    {
      "id": "localized-integral",
      "name": "Localized Euclidean path integral",
      "construction": [
        "Import the complete fixed locus, gauge-fixed one-loop complex, zero-mode measure, sectors, and transported cycle from the localization-validity analysis.",
        "For the declared dimension and background, assemble the classical weight, Cartan or moduli measure, one-loop determinant, and nonperturbative factors into a finite-dimensional integral or sum.",
        "Extract only quantities invariant under the allowed counterterms, or state the chosen scheme and phase; examples include universal derivatives or differences, a controlled three-dimensional F-extremum, and normalized defect expectation values."
      ],
      "schematic_formula": "Z_B = sum_m integral_{Gamma_m} d mu_m exp(-S_cl,m) Z_1-loop,m Z_np,m",
      "output": "A background-, sector-, contour-, scheme-, and normalization-qualified localized observable.",
      "not_universal": "The two-, three-, and four-dimensional sphere formulas have different loci, measures, determinants, flux or instanton sectors, and counterterm ambiguities."
    },
    {
      "id": "protected-trace",
      "name": "Protected trace or twisted path integral",
      "construction": [
        "Choose Q and Q-dagger, the Hilbert space or twisted background, spin structure, commuting charges, and fugacity domain so that only Q-cohomology can contribute.",
        "Build the single-letter or one-loop integrand, impose equations of motion and statistics, project to gauge singlets, include every flux sector and the derived residue prescription when present, and separate any supersymmetric Casimir prefactor from the vacuum-normalized trace.",
        "Report an index, elliptic genus, or twisted index together with its recombination class, anomaly or modular data, convergence domain, and continuum qualifications."
      ],
      "schematic_formula": "I(x) = Tr_H (-1)^F exp(-beta Delta_Q) product_i x_i^{F_i}, with Delta_Q = {Q,Q-dagger}/2",
      "output": "A protected trace invariant under long-multiplet recombination, not an ungraded thermal partition function or a unique Hilbert-space spectrum.",
      "not_universal": "Different supercharges, spin structures, twists, charge normalizations, and gauge projections define different indices."
    },
    {
      "id": "omega-instanton",
      "name": "Omega-background instanton sum",
      "construction": [
        "Specify the framed instanton moduli problem, gauge-group global data, equivariant rotations epsilon_1 and epsilon_2, Coulomb and mass conventions, stability chamber, and treatment of small-instanton singularities.",
        "Enumerate torus-fixed points, commonly colored Young diagrams, and compute the equivariant tangent and matter weights with the declared regularization and decoupled factors.",
        "Sum fixed-point contributions by instanton number and take only controlled limits, such as a prepotential limit after the perturbative, U(1), and normalization conventions are fixed."
      ],
      "schematic_formula": "Z_inst(q) = sum_Y q^{|Y|} Z_Y(a,m;epsilon_1,epsilon_2)",
      "output": "An equivariant instanton generating function and declared protected limits, not the full nonperturbative QFT by itself.",
      "not_universal": "Stability, mass shifts, U(1) factors, contour prescriptions, and compactification conventions change the displayed formula."
    }
  ],
  "factorization": {
    "name": "Observable-specific blocks and gluing",
    "input_branches": [
      "localized-integral",
      "protected-trace"
    ],
    "construction": "When the named observable admits it, decompose its transported integration cycle into vacuum-labeled cycles and define holomorphic blocks B^alpha. Glue with the observable-specific pairing, flux sum, anomaly prefactor, and conjugate parameters.",
    "schematic_formula": "Z_M = exp(P_g) sum_{alpha,beta} B_<^alpha(x;q) (K_g)_{alpha,beta} B_>^beta(tilde{x};tilde{q})",
    "qualifier": "The block basis can jump across Stokes walls while the correctly transported glued observable remains invariant; factorization is not automatic for every theory or background."
  },
  "conditional_relations": [
    {
      "from": "omega-instanton",
      "to": "localized-integral",
      "relation": "On compact backgrounds with isolated Omega-deformed neighborhoods, a fixed-point instanton function can furnish the nonperturbative factor Z_np inside the localized integrand; on S4 the north and south pole factors are the canonical example.",
      "qualifier": "This composition requires the compact background's local equivariant parameters, orientation, global data, mass convention, perturbative factors, and continuation prescription; it is not an equality of the two branches as abstract observables."
    }
  ],
  "comparison": {
    "name": "Qualified comparison record",
    "required_fields": [
      "complete parameter and charge map",
      "background contact terms and counterterm phases",
      "global forms, discrete sectors, fluxes, and defect labels",
      "measure and operator normalization",
      "shared convergence domain or controlled analytic continuation",
      "independent free, weak-coupling, low-order, or low-instanton benchmark"
    ],
    "allowed_validation_modes": [
      "analytic function identity",
      "formal-series identity",
      "precision-bounded numerical match",
      "conditional calculation"
    ],
    "allowed_evidentiary_reach": [
      "protected-sector match",
      "broader duality evidence"
    ]
  },
  "computational_record": {
    "fields": [
      "integral, residue, trace, or series representation and every input parameter",
      "sector ordering, included terms, pole set, and measure factors",
      "truncation order and remainder estimate, or an explicit statement that the expansion is formal",
      "numerical algorithm, working precision, contour deformation, and convergence test when evaluated numerically",
      "block basis, vacuum labels, pairing, flux sum, anomaly prefactor, and Stokes chamber when factorized"
    ],
    "formal_series_warning": "A fugacity, q, vortex, or instanton expansion is not automatically a convergent function. State its convergence domain or justified analytic continuation; otherwise label every equality that uses it as a formal-series identity."
  },
  "export": {
    "destination": "A versioned duality dossier or a protected-data handoff to Volume 9",
    "must_include": [
      "observable definition and all qualifiers",
      "validation mode and evidentiary reach",
      "ambiguities and failure conditions",
      "source chain and benchmark record"
    ],
    "ceiling": "No single equality supplies a full unprotected spectrum, OPE coefficients, or proof that the complete quantum theories are equivalent."
  },
  "failure_exits": [
    {
      "branch": "localized-integral",
      "label": "Undefined or scheme-dependent integral",
      "causes": [
        "missed saddle, flux or instanton sector, pole at infinity, or field-space boundary",
        "untransported cycle, nonconvergence, or unjustified contour deformation",
        "unremoved zero mode, determinant-phase ambiguity, or inconsistent quantized level lattice",
        "unfixed finite counterterm or normalization, which must be recorded as distinct operations"
      ]
    },
    {
      "branch": "protected-trace",
      "label": "Wrong trace or overread spectrum",
      "causes": [
        "incompatible charge or spin-structure convention",
        "missing gauge projection or flux sector",
        "uncontrolled convergence or continuum contribution",
        "recombination cancellation interpreted as absence"
      ]
    },
    {
      "branch": "omega-instanton",
      "label": "Wrong fixed-point sum or limit",
      "causes": [
        "unstated stability chamber or small-instanton resolution",
        "incorrect tangent weight or mass shift",
        "missing U(1), perturbative, or decoupled factor",
        "uncontrolled epsilon or prepotential limit"
      ]
    },
    {
      "branch": "factorization",
      "label": "Invalid factorization or comparison",
      "causes": [
        "missing vacuum or Stokes transport",
        "wrong gluing measure or anomaly prefactor",
        "mismatched global sector or parameter map",
        "formal identity used outside a common domain without justified continuation"
      ]
    }
  ],
  "primary_sources": [
    {
      "citation": "Pestun 2012",
      "url": "https://arxiv.org/abs/0712.2824",
      "locator": "arXiv v3, Sections 3-4 and 5.1",
      "use": "four-sphere localization, matrix integral, one-loop and instanton factors, and Wilson-loop insertions"
    },
    {
      "citation": "Closset, Dumitrescu, Festuccia, Komargodski, and Seiberg 2012",
      "url": "https://arxiv.org/abs/1205.4142",
      "locator": "arXiv v2, Sections 2-5",
      "use": "three-sphere contact terms, complex phases, and the hypotheses of F-maximization"
    },
    {
      "citation": "Kinney, Maldacena, Minwalla, and Raju 2007",
      "url": "https://arxiv.org/abs/hep-th/0510251",
      "locator": "Sections 2-3",
      "use": "superconformal-index trace, Q-pairing, commuting fugacities, and protected-information boundary"
    },
    {
      "citation": "Assel, Cassani, Di Pietro, Komargodski, Lorenzen, and Martelli 2015",
      "url": "https://arxiv.org/abs/1503.05537",
      "locator": "arXiv v2, Eqs. (1.1)-(1.2) and Section 4",
      "use": "supersymmetric Casimir prefactor, anomaly control, and its distinction from the vacuum-normalized trace"
    },
    {
      "citation": "Benini, Eager, Hori, and Tachikawa 2015",
      "url": "https://arxiv.org/abs/1308.4896",
      "locator": "arXiv v2, Sections 2.1-2.5",
      "use": "elliptic-genus one-loop form, flat-connection moduli, Jeffrey-Kirwan residues, and chamber data"
    },
    {
      "citation": "Benini and Zaffaroni 2015",
      "url": "https://arxiv.org/abs/1504.03698",
      "locator": "arXiv v2, Sections 2-3",
      "use": "three-dimensional topologically twisted index, gauge-flux sum, contour prescription, and residue evaluation"
    },
    {
      "citation": "Nekrasov 2003",
      "url": "https://arxiv.org/abs/hep-th/0206161",
      "locator": "Sections 2-3 and 6",
      "use": "Omega deformation, fixed-point instanton partition function, and the Seiberg-Witten prepotential limit"
    },
    {
      "citation": "Nekrasov and Okounkov 2006",
      "url": "https://arxiv.org/abs/hep-th/0306238",
      "locator": "Sections 3-4",
      "use": "partition sum representation and controlled recovery of Seiberg-Witten geometry"
    },
    {
      "citation": "Beem, Dimofte, and Pasquetti 2014",
      "url": "https://arxiv.org/abs/1211.1986",
      "locator": "Sections 2-4, 5.2-5.5, and 6",
      "use": "vacuum-labeled holomorphic blocks, gluing, and Stokes transformations"
    },
    {
      "citation": "Dolan and Osborn 2009",
      "url": "https://arxiv.org/abs/0801.4947",
      "locator": "Sections 2-4 and 6, especially Eqs. (6.9)-(6.13)",
      "use": "gauge-projected superconformal indices and elliptic-hypergeometric identities as protected duality tests"
    },
    {
      "citation": "Rains 2010",
      "url": "https://arxiv.org/abs/math/0309252",
      "locator": "Theorem 4.1",
      "use": "elliptic-hypergeometric integral transformation underlying the convention-complete Seiberg-index benchmark"
    }
  ],
  "verification_benchmarks": [
    {
      "id": "su2-one-instanton-limit",
      "type": "exact algebraic identity and controlled limit",
      "expected": "For a_1=a and a_2=-a, the two one-instanton fixed points sum to 2/(4a^2-epsilon_+^2), and epsilon_1 epsilon_2 Z_1 tends to 1/(2a^2) only after both colors are included.",
      "convention": "Pure U(2) fixed-point formula restricted to sum a_alpha=0, with the Euler-class orientation sign absorbed into the instanton fugacity before the SU(2) prepotential limit.",
      "source_locator": "Nekrasov 2003, Eqs. (3.21) and (3.23)",
      "method": "Evaluate the rational fixed-point sum at a frozen nonsingular point and compare its small-epsilon limit with the analytic expression."
    },
    {
      "id": "tetrahedron-block-shift",
      "type": "finite-product convergence benchmark",
      "expected": "For |q|<1, B_Delta(qx;q)/B_Delta(x;q) tends to 1-x^{-1}; the finite-product remainder is q^N/x.",
      "convention": "B_Delta(x;q)=(q x^{-1};q)_infinity with the parity-anomaly contact term retained.",
      "source_locator": "Beem, Dimofte, and Pasquetti 2014, Section 2.2",
      "method": "Evaluate a frozen finite q-product and require the measured ratio error to lie below its analytic tail bound."
    },
    {
      "id": "stokes-pairing-reconstruction",
      "type": "exact matrix reconstruction check",
      "expected": "The pairing B_<^T B_> is unchanged under B_< -> M B_< and B_> -> M^{-T} B_> for an integral Stokes matrix M.",
      "convention": "Two-vacuum diagonal pairing with M=[[1,1],[0,1]]; a general kernel transforms contragrediently.",
      "source_locator": "Beem, Dimofte, and Pasquetti 2014, Sections 5.2-5.5",
      "method": "Evaluate both bilinear pairings on frozen vectors and require exact agreement to floating-point tolerance."
    },
    {
      "id": "seiberg-index-charge-balance",
      "type": "charge, balancing, and special-function identity anchor",
      "expected": "For N_c=2 and N_f=5, r=3/5 obeys N_f r=N_f-N_c, the magnetic-quark R charge is 2/5, and R(M)+2R(q)=2.",
      "convention": "Four-dimensional N=1 SU(N_c) SQCD with unit-determinant flavor fugacities and the anomaly-free R symmetry.",
      "source_locator": "Rains 2010, Theorem 4.1; Dolan and Osborn 2009, Section 6, Eqs. (6.9)-(6.13)",
      "method": "Evaluate the frozen rank and R-charge relations exactly before accepting the comparison branch."
    },
    {
      "id": "s4-instanton-composition",
      "type": "source-anchored construction relationship",
      "expected": "The localized S4 Coulomb integral contains north- and south-pole Omega-background instanton factors as Z_np, with complex conjugation only on the physical real slice.",
      "convention": "Round S4 local equivariant magnitudes epsilon_1=epsilon_2=r^{-1}; analytic continuation retains independent pole parameters.",
      "source_locator": "Pestun 2012, Eq. (1.4) and Sections 4-5",
      "method": "Require the structured relation, figure label, canonical caption, and reflowing text to state the composition and its qualifier consistently."
    }
  ],
  "conventions": {
    "metric": "The path-integral branches are Euclidean and inherit the site's positive-definite Euclidean metric convention.",
    "arrows": "Solid arrows are required construction or export steps once a route has been selected; direct converging arrows mean that separately defined observables enter a comparison record, not that they are equal by construction.",
    "optional_arrows": "Dotted arrows mark the optional reorganization of a named localized integral or protected trace into holomorphic blocks when all factorization hypotheses hold.",
    "dashed_exits": "Dashed arrows lead to named conditions that stop the claim or force an explicit qualification.",
    "color": "Meaning is carried by labels, borders, solid, dotted, and dashed arrow styles, and explicit comparison labels; color is not used."
  },
  "registry_state": {
    "status": "planned",
    "public_route": null,
    "lifecycle_note": "The structured record and SVG materialize the planned artifact without promoting its registry lifecycle or asserting release readiness."
  },
  "accessibility": {
    "branch_count": 3,
    "solid_arrow_meaning": "Solid arrows are required construction or export steps once a route has been selected; direct converging arrows mean that separately defined observables enter a comparison record, not that they are equal by construction.",
    "dotted_arrow_meaning": "Dotted arrows mark the optional reorganization of a named localized integral or protected trace into holomorphic blocks when all factorization hypotheses hold.",
    "dashed_exit_meaning": "Dashed arrows lead to named conditions that stop the claim or force an explicit qualification.",
    "color_dependency": false,
    "reader_facing_text_equivalent": "/supersymmetry-duality/partition-functions-indices-instantons/defects-instantons-duality-tests/#exact-observable-map-text",
    "structured_equivalent": "This JSON preserves the complete input specification, the three distinct constructions, factorization conditions, comparison fields, validation modes, evidentiary-reach labels, export ceiling, failure exits, and primary-source chain."
  },
  "benchmark_results": [
    {
      "id": "su2-one-instanton-limit",
      "type": "exact algebraic identity and controlled limit",
      "expected": "For a_1=a and a_2=-a, the two one-instanton fixed points sum to 2/(4a^2-epsilon_+^2), and epsilon_1 epsilon_2 Z_1 tends to 1/(2a^2) only after both colors are included.",
      "convention": "Pure U(2) fixed-point formula restricted to sum a_alpha=0, with the Euler-class orientation sign absorbed into the instanton fugacity before the SU(2) prepotential limit.",
      "source_locator": "Nekrasov 2003, Eqs. (3.21) and (3.23)",
      "method": "Evaluate the rational fixed-point sum at a frozen nonsingular point and compare its small-epsilon limit with the analytic expression.",
      "outcome": "passed",
      "observed": {
        "frozen_parameters": {
          "a": 1.7,
          "epsilon1": 0.07,
          "epsilon2": 0.11
        },
        "fixed_point_sum": 0.1734966515146258,
        "closed_form": 0.1734966515146258,
        "small_epsilon_scaled_coefficient": 0.17301038062289725,
        "limiting_coefficient": 0.17301038062283738
      }
    },
    {
      "id": "tetrahedron-block-shift",
      "type": "finite-product convergence benchmark",
      "expected": "For |q|<1, B_Delta(qx;q)/B_Delta(x;q) tends to 1-x^{-1}; the finite-product remainder is q^N/x.",
      "convention": "B_Delta(x;q)=(q x^{-1};q)_infinity with the parity-anomaly contact term retained.",
      "source_locator": "Beem, Dimofte, and Pasquetti 2014, Section 2.2",
      "method": "Evaluate a frozen finite q-product and require the measured ratio error to lie below its analytic tail bound.",
      "outcome": "passed",
      "observed": {
        "frozen_parameters": {
          "q": 0.4,
          "x": 1.7,
          "terms": 24
        },
        "measured_ratio": 0.41176470595053033,
        "expected_ratio": 0.4117647058823529,
        "analytic_tail_bound": 3.3114703142430166e-10,
        "absolute_error": 6.817740816345008e-11
      }
    },
    {
      "id": "stokes-pairing-reconstruction",
      "type": "exact matrix reconstruction check",
      "expected": "The pairing B_<^T B_> is unchanged under B_< -> M B_< and B_> -> M^{-T} B_> for an integral Stokes matrix M.",
      "convention": "Two-vacuum diagonal pairing with M=[[1,1],[0,1]]; a general kernel transforms contragrediently.",
      "source_locator": "Beem, Dimofte, and Pasquetti 2014, Sections 5.2-5.5",
      "method": "Evaluate both bilinear pairings on frozen vectors and require exact agreement to floating-point tolerance.",
      "outcome": "passed",
      "observed": {
        "stokes_matrix": [
          [
            1,
            1
          ],
          [
            0,
            1
          ]
        ],
        "left": [
          1.25,
          -0.4
        ],
        "right": [
          0.7,
          2.1
        ],
        "original_pairing": 0.03499999999999992,
        "transported_pairing": 0.03499999999999992
      }
    },
    {
      "id": "seiberg-index-charge-balance",
      "type": "charge, balancing, and special-function identity anchor",
      "expected": "For N_c=2 and N_f=5, r=3/5 obeys N_f r=N_f-N_c, the magnetic-quark R charge is 2/5, and R(M)+2R(q)=2.",
      "convention": "Four-dimensional N=1 SU(N_c) SQCD with unit-determinant flavor fugacities and the anomaly-free R symmetry.",
      "source_locator": "Rains 2010, Theorem 4.1; Dolan and Osborn 2009, Section 6, Eqs. (6.9)-(6.13)",
      "method": "Evaluate the frozen rank and R-charge relations exactly before accepting the comparison branch.",
      "outcome": "passed",
      "observed": {
        "number_colors": 2,
        "number_flavors": 5,
        "magnetic_rank": 3,
        "electric_quark_R": 0.6,
        "magnetic_quark_R": 0.4,
        "balancing_exponent": 3,
        "superpotential_R": 2
      }
    },
    {
      "id": "s4-instanton-composition",
      "type": "source-anchored construction relationship",
      "expected": "The localized S4 Coulomb integral contains north- and south-pole Omega-background instanton factors as Z_np, with complex conjugation only on the physical real slice.",
      "convention": "Round S4 local equivariant magnitudes epsilon_1=epsilon_2=r^{-1}; analytic continuation retains independent pole parameters.",
      "source_locator": "Pestun 2012, Eq. (1.4) and Sections 4-5",
      "method": "Require the structured relation, figure label, canonical caption, and reflowing text to state the composition and its qualifier consistently.",
      "outcome": "passed",
      "observed": {
        "structured_relation_present": true,
        "visible_figure_label_present": true,
        "canonical_caption_and_text_present": true
      }
    }
  ]
}