{
  "artifact_id": "qft.artifact.supersymmetry-duality.breaking.pseudomoduli-soft-decoupling-map",
  "schema_version": "qft-semantic-figure/pseudomoduli-soft-decoupling-map-v1",
  "fixture_id": "L08",
  "fixture_revision": 1,
  "fixture_sha256": "e40d92a3bf44fe48bfee0631fb25eb3209b94287b1fd1b139351514670f38e08",
  "title": "Pseudomodulus stability, nonlinear-EFT gate, and soft decoupling",
  "reader_question": "How do the exact O'Raifeartaigh scalar spectrum, the validity gate for a nonlinear goldstino EFT, and the stability plane of a softly split two-chiral model distinguish local control, supersymmetry restoration, and large-soft decoupling?",
  "takeaway": "The O'Raifeartaigh nominal valley is tachyon-free only for 0<y<1 and develops light modes at large z; a nilpotent goldstino description additionally needs F nonzero and energy parametrically below both heavy thresholds and an order-sqrt(f) nonlinear scale; the soft two-chiral model restores its supersymmetric spectrum only at the zero-soft origin, whereas a large positive scalar-mass ray requires matching and proves no phase theorem.",
  "alt_text": "A vertically stacked monochrome quantitative figure. Panel A plots the exact minimum O'Raifeartaigh scalar mass squared divided by m squared against z=h times absolute X over m. The solid y=one-half curve stays positive and approaches zero from above, the dashed y=one curve is exactly massless, and the dot-dashed y=three-halves curve stays negative and approaches zero from below. A shaded large-z band warns that light modes require reorganizing the EFT. A compact box states that the one-loop coefficient C(y) is positive for zero<y<one and begins four y squared over three plus two y fourth over fifteen. A separate warning says the benchmark supplies no lower basin or lifetime. The middle gate requires F nonzero and E parametrically below the smaller of the heavy threshold and an order-square-root-f nonlinear scale; it fails as F goes to zero or a removed partner becomes light. Panel B plots beta=absolute b over absolute m squared against r=m_s squared over absolute m squared. The exact line beta=one+r separates a stable lower wedge from a tachyonic upper region. A dashed trajectory reaches the supersymmetric origin, while a dotted fixed-beta ray moves toward large positive r and scalar decoupling. The latter is labeled as requiring matching and a cutoff and as proving no phase continuity theorem.",
  "caption_semantics": "Exact data for two displayed benchmark models. In panel A, mu_min=1+(z^2-y)/2-sqrt((z^2-y)^2+4z^2)/2 is positive for y=1/2, identically zero for y=1, and negative for y=3/2 at every finite z; all three approach zero in magnitude at large z, where the light spectrum must be reorganized. The DRbar-type one-loop anchor C(y)>0 applies only on the controlled 0<y<1 branch and is a local-curvature result: no lower basin, bounce, or lifetime is supplied. The nonlinear goldstino gate additionally requires a nonzero auxiliary branch and E parametrically below both the first omitted mass and an order-sqrt(f) nonlinear scale. In panel B, nu_plus/minus=1+r plus/minus beta, so beta=1+r is exactly the massless boundary. The zero-soft trajectory restores the displayed supersymmetric degeneracy at (0,0); the large-r trajectory makes the scalars heavy relative to the fermion but requires interaction-dependent threshold matching and establishes no phase theorem.",
  "canonical_embed": {
    "page_id": "qft.topic.susy-holomorphy.pseudomoduli-quantum-lifting-metastability",
    "anchor": "pseudomoduli-soft-decoupling-map",
    "svg_route": "/figures/supersymmetry-duality/pseudomoduli-soft-decoupling-map.svg",
    "semantic_route": "/figures/supersymmetry-duality/pseudomoduli-soft-decoupling-map.json"
  },
  "owner_pages": [
    {
      "page_id": "qft.topic.susy-holomorphy.pseudomoduli-quantum-lifting-metastability",
      "path": "src/content/docs/supersymmetry-duality/susy-breaking-controlled-deformations/pseudomoduli-quantum-lifting-metastability.md",
      "route": "/supersymmetry-duality/susy-breaking-controlled-deformations/pseudomoduli-quantum-lifting-metastability/",
      "integration_role": "canonical figure embed at the declared anchor"
    },
    {
      "page_id": "qft.topic.susy-breaking.nonlinear-goldstino-constrained-eft",
      "path": "src/content/docs/supersymmetry-duality/susy-breaking-controlled-deformations/nonlinear-goldstino-constrained-eft.md",
      "route": "/supersymmetry-duality/susy-breaking-controlled-deformations/nonlinear-goldstino-constrained-eft/",
      "integration_role": "contextual link to the canonical figure anchor"
    },
    {
      "page_id": "qft.topic.susy-holomorphy.soft-breaking-controlled-decoupling",
      "path": "src/content/docs/supersymmetry-duality/susy-breaking-controlled-deformations/soft-breaking-controlled-decoupling.md",
      "route": "/supersymmetry-duality/susy-breaking-controlled-deformations/soft-breaking-controlled-decoupling/",
      "integration_role": "contextual link to the canonical figure anchor"
    }
  ],
  "exactness_and_scope": {
    "panel_a_mass_curves": "exact eigenvalues of the displayed tree-level quadratic mass matrices at the frozen y and z values",
    "panel_a_curvature": "one-loop DRbar-type local curvature under the declared perturbative and higher-operator hierarchy",
    "nonlinear_gate": "parametric domain statement with no asserted order-one or 4 pi coefficient",
    "panel_b": "exact tree-level quadratic masses and boundary in the displayed two-chiral benchmark",
    "excluded_claims": [
      "no lower basin or runaway is supplied",
      "no bounce or lifetime is supplied",
      "no large-soft phase-continuity theorem is asserted",
      "no universal numerical nonlinear cutoff coefficient is asserted"
    ]
  },
  "conventions": {
    "spacetime_and_supersymmetry": "four-dimensional rigid N=1 supersymmetry with the repository (+---) metric convention",
    "panel_a_units": "all plotted scalar eigenvalues are divided by m^2; z=h|X|/m and y=hf/m^2 with f,h,m positive",
    "panel_b_units": "r=m_s^2/|m|^2 and beta=|b|/|m|^2; beta is nonnegative and both scalar eigenvalues are divided by |m|^2",
    "curve_interpolation": "the SVG joins adjacent frozen L08 samples by straight line segments; the displayed analytic formulas, not the interpolation, are authoritative between samples",
    "renormalization": "panel A tree-level mass curves are scheme independent; the C(y) one-loop curvature anchor is stated in a mass-independent DRbar-type scheme at a scale near the heavy masses; panel B tree-level masses are defined at one common renormalization scale, while large-soft threshold coefficients require an explicit matching scheme",
    "global_claim_boundary": "the fixture establishes local spectra, signs, and declared limits only; it supplies no lower vacuum, runaway endpoint, bounce, lifetime, or theorem of phase continuity"
  },
  "formulas": {
    "panel_a": {
      "superpotential": "W=f X+(h/2) X phi_1^2+m phi_1 phi_2",
      "nominal_valley": "phi_1=phi_2=0 with V_0=f^2 and F_X=-f",
      "definitions": "y=hf/m^2 and z=h|X|/m",
      "scalar_pair_formula": "mu_{B,s,t}=1+(z^2+s y)/2+(t/2) sqrt((z^2+s y)^2+4 z^2), s,t in {+1,-1}",
      "minimum_scalar_formula": "mu_min(z;y)=1+(z^2-y)/2-(1/2) sqrt((z^2-y)^2+4 z^2)",
      "pair_determinant_formula": "mu_{B,s,+} mu_{B,s,-}=1+s y",
      "fermion_determinant": "det M_F=-m^2, so one fermion and the corresponding scalar modes become light at large finite z"
    },
    "coleman_weinberg_anchor": {
      "formula": "C(y)=(2/y)[(1+y)^2 log(1+y)-(1-y)^2 log(1-y)-2y]",
      "curvature": "m_X^2=(h^2 m^2/(64 pi^2)) C(y)",
      "positivity_argument": "with F(y) equal to the bracket, F(0)=F'(0)=0 and F''(y)=2 log((1+y)/(1-y))>0 for 0<y<1, so C(y)>0",
      "weak_splitting_series": "C(y)=(4/3)y^2+(2/15)y^4+O(y^6)",
      "sample_values": [
        {
          "y": 0.1,
          "C": 0.0133467049214497,
          "sign": "positive"
        },
        {
          "y": 0.25,
          "C": 0.0838637174606358,
          "sign": "positive"
        },
        {
          "y": 0.5,
          "C": 0.342333153533425,
          "sign": "positive"
        },
        {
          "y": 0.75,
          "C": 0.801244661659267,
          "sign": "positive"
        },
        {
          "y": 0.9,
          "C": 1.20026306669397,
          "sign": "positive"
        }
      ]
    },
    "nonlinear_eft_gate": {
      "order_parameter": "F is nonzero on the auxiliary branch used to solve the nilpotent constraint",
      "energy_domain": "E<<min(m_heavy,Lambda_NL) with Lambda_NL=O(sqrt(f))",
      "normalization_scope": "f has mass dimension two; the order-one and 4 pi factors in Lambda_NL depend on the channel and UV completion, so the figure asserts no numerical coefficient",
      "failure_conditions": [
        "F tends to zero, so the nilpotent auxiliary branch is singular",
        "a purportedly removed superpartner becomes light",
        "E is not parametrically below a heavy threshold or the nonlinear strong-coupling scale"
      ],
      "claim_boundary": "passing this gate licenses a nonlinear low-energy field organization only; it is not evidence for a lower vacuum, a metastable basin, or a lifetime"
    },
    "panel_b": {
      "superpotential": "W=m Phi_+ Phi_-",
      "soft_potential": "V_soft=m_s^2(|phi_+|^2+|phi_-|^2)+(b phi_+ phi_-+h.c.)",
      "definitions": "r=m_s^2/|m|^2 and beta=|b|/|m|^2",
      "scalar_eigenvalues": "nu_+=1+r+beta and nu_-=1+r-beta",
      "fermion_mass_squared": "m_F^2/|m|^2=1",
      "scalar_determinant": "nu_+ nu_-=(1+r)^2-beta^2"
    }
  },
  "plotted_geometry": {
    "panel_a_axes": {
      "horizontal": {
        "quantity": "z=h|X|/m",
        "minimum": 0,
        "maximum": 6,
        "ticks": [
          0,
          1,
          2,
          3,
          4,
          5,
          6
        ]
      },
      "vertical": {
        "quantity": "mu_min=m_{B,min}^2/m^2",
        "minimum": -0.6,
        "maximum": 0.6,
        "ticks": [
          -0.5,
          0,
          0.5
        ]
      }
    },
    "panel_a_large_z_band": {
      "z_minimum": 4.5,
      "z_maximum": 6,
      "polygon_vertices": [
        {
          "z": 4.5,
          "mu": -0.6
        },
        {
          "z": 6,
          "mu": -0.6
        },
        {
          "z": 6,
          "mu": 0.6
        },
        {
          "z": 4.5,
          "mu": 0.6
        }
      ],
      "meaning": "the light eigenvalues visibly approach zero; this shaded band is an explanatory warning, not a sharp threshold"
    },
    "panel_a_curve_points": [
      {
        "curve_id": "stable_y_0p5",
        "y": 0.5,
        "line_style": "solid",
        "status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity",
        "z": 0,
        "mu_min": 0.5,
        "eigenvalue_sign": "positive",
        "plotted_formula": "mu_min(z;y)=1+(z^2-y)/2-(1/2) sqrt((z^2-y)^2+4 z^2)"
      },
      {
        "curve_id": "stable_y_0p5",
        "y": 0.5,
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        "z": 0.25,
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        "plotted_formula": "mu_min(z;y)=1+(z^2-y)/2-(1/2) sqrt((z^2-y)^2+4 z^2)"
      },
      {
        "curve_id": "stable_y_0p5",
        "y": 0.5,
        "line_style": "solid",
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        "z": 0.5,
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        "plotted_formula": "mu_min(z;y)=1+(z^2-y)/2-(1/2) sqrt((z^2-y)^2+4 z^2)"
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      {
        "curve_id": "stable_y_0p5",
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        "z": 0.75,
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      {
        "curve_id": "stable_y_0p5",
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        "curve_id": "tachyonic_y_1p5",
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        "line_style": "dot dash",
        "status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity",
        "z": 4,
        "mu_min": -0.0302475808395979,
        "eigenvalue_sign": "negative",
        "plotted_formula": "mu_min(z;y)=1+(z^2-y)/2-(1/2) sqrt((z^2-y)^2+4 z^2)"
      },
      {
        "curve_id": "tachyonic_y_1p5",
        "y": 1.5,
        "line_style": "dot dash",
        "status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity",
        "z": 5,
        "mu_min": -0.0195927891221341,
        "eigenvalue_sign": "negative",
        "plotted_formula": "mu_min(z;y)=1+(z^2-y)/2-(1/2) sqrt((z^2-y)^2+4 z^2)"
      },
      {
        "curve_id": "tachyonic_y_1p5",
        "y": 1.5,
        "line_style": "dot dash",
        "status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity",
        "z": 6,
        "mu_min": -0.0136934928288817,
        "eigenvalue_sign": "negative",
        "plotted_formula": "mu_min(z;y)=1+(z^2-y)/2-(1/2) sqrt((z^2-y)^2+4 z^2)"
      }
    ],
    "panel_b_axes": {
      "horizontal": {
        "quantity": "r=m_s^2/|m|^2",
        "minimum": -1,
        "maximum": 3,
        "ticks": [
          -1,
          0,
          1,
          2,
          3
        ]
      },
      "vertical": {
        "quantity": "beta=|b|/|m|^2",
        "minimum": 0,
        "maximum": 4.2,
        "ticks": [
          0,
          1,
          2,
          3,
          4
        ]
      }
    },
    "panel_b_region_polygons": [
      {
        "region_id": "stable",
        "inequality": "0<=beta<1+r with r>-1",
        "status": "both complex scalar eigenvalues are positive",
        "polygon_vertices": [
          {
            "r": -1,
            "beta": 0
          },
          {
            "r": 3,
            "beta": 0
          },
          {
            "r": 3,
            "beta": 4
          }
        ],
        "sample": {
          "r": 0,
          "beta": 0.4,
          "nu_minus": 0.6,
          "eigenvalue_sign": "positive"
        }
      },
      {
        "region_id": "tachyonic",
        "inequality": "beta>1+r, including every beta>0 when r<=-1",
        "status": "nu_- is negative",
        "polygon_vertices": [
          {
            "r": -1,
            "beta": 0
          },
          {
            "r": -1,
            "beta": 4.2
          },
          {
            "r": 3,
            "beta": 4.2
          },
          {
            "r": 3,
            "beta": 4
          }
        ],
        "sample": {
          "r": 0,
          "beta": 1.5,
          "nu_minus": -0.5,
          "eigenvalue_sign": "negative"
        }
      }
    ],
    "panel_b_boundary_points": [
      {
        "curve_id": "massless_boundary",
        "line_style": "solid",
        "status": "nu_-=0: one complex scalar is massless",
        "r": -1,
        "beta": 0,
        "nu_minus": 0,
        "eigenvalue_sign": "zero"
      },
      {
        "curve_id": "massless_boundary",
        "line_style": "solid",
        "status": "nu_-=0: one complex scalar is massless",
        "r": 0,
        "beta": 1,
        "nu_minus": 0,
        "eigenvalue_sign": "zero"
      },
      {
        "curve_id": "massless_boundary",
        "line_style": "solid",
        "status": "nu_-=0: one complex scalar is massless",
        "r": 1,
        "beta": 2,
        "nu_minus": 0,
        "eigenvalue_sign": "zero"
      },
      {
        "curve_id": "massless_boundary",
        "line_style": "solid",
        "status": "nu_-=0: one complex scalar is massless",
        "r": 2,
        "beta": 3,
        "nu_minus": 0,
        "eigenvalue_sign": "zero"
      },
      {
        "curve_id": "massless_boundary",
        "line_style": "solid",
        "status": "nu_-=0: one complex scalar is massless",
        "r": 3,
        "beta": 4,
        "nu_minus": 0,
        "eigenvalue_sign": "zero"
      }
    ],
    "panel_b_restoration_points": [
      {
        "trajectory_id": "supersymmetry_restoration",
        "line_style": "long dash with arrow toward t=0",
        "status": "stable along the displayed path",
        "t": 1,
        "r": 0.8,
        "beta": 0.6,
        "nu_minus": 1.2,
        "nu_plus": 2.4,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      },
      {
        "trajectory_id": "supersymmetry_restoration",
        "line_style": "long dash with arrow toward t=0",
        "status": "stable along the displayed path",
        "t": 0.75,
        "r": 0.6,
        "beta": 0.45,
        "nu_minus": 1.15,
        "nu_plus": 2.05,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      },
      {
        "trajectory_id": "supersymmetry_restoration",
        "line_style": "long dash with arrow toward t=0",
        "status": "stable along the displayed path",
        "t": 0.5,
        "r": 0.4,
        "beta": 0.3,
        "nu_minus": 1.1,
        "nu_plus": 1.7,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      },
      {
        "trajectory_id": "supersymmetry_restoration",
        "line_style": "long dash with arrow toward t=0",
        "status": "stable along the displayed path",
        "t": 0.25,
        "r": 0.2,
        "beta": 0.15,
        "nu_minus": 1.05,
        "nu_plus": 1.35,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      },
      {
        "trajectory_id": "supersymmetry_restoration",
        "line_style": "long dash with arrow toward t=0",
        "status": "stable along the displayed path",
        "t": 0,
        "r": 0,
        "beta": 0,
        "nu_minus": 1,
        "nu_plus": 1,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      }
    ],
    "panel_b_large_soft_points": [
      {
        "trajectory_id": "large_soft_decoupling",
        "line_style": "dot dash with arrow toward increasing r",
        "status": "stable local quadratic spectrum; matching still required",
        "r": 0.8,
        "beta": 0.4,
        "nu_minus": 1.4,
        "nu_plus": 2.2,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      },
      {
        "trajectory_id": "large_soft_decoupling",
        "line_style": "dot dash with arrow toward increasing r",
        "status": "stable local quadratic spectrum; matching still required",
        "r": 1.5,
        "beta": 0.4,
        "nu_minus": 2.1,
        "nu_plus": 2.9,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      },
      {
        "trajectory_id": "large_soft_decoupling",
        "line_style": "dot dash with arrow toward increasing r",
        "status": "stable local quadratic spectrum; matching still required",
        "r": 2.25,
        "beta": 0.4,
        "nu_minus": 2.85,
        "nu_plus": 3.65,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      },
      {
        "trajectory_id": "large_soft_decoupling",
        "line_style": "dot dash with arrow toward increasing r",
        "status": "stable local quadratic spectrum; matching still required",
        "r": 3,
        "beta": 0.4,
        "nu_minus": 3.6,
        "nu_plus": 4.4,
        "eigenvalue_signs": [
          "positive",
          "positive"
        ],
        "normalized_fermion_mass_squared": 1
      }
    ],
    "plotted_numeric_geometry_count": 58,
    "interpolation_policy": "the SVG joins adjacent frozen L08 samples by straight line segments; the displayed analytic formulas, not the interpolation, are authoritative between samples"
  },
  "eigenvalue_sign_table": [
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=0",
      "smallest_normalized_scalar_eigenvalue": 0.5,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=0.25",
      "smallest_normalized_scalar_eigenvalue": 0.449057943352042,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=0.5",
      "smallest_normalized_scalar_eigenvalue": 0.359611796797792,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=0.75",
      "smallest_normalized_scalar_eigenvalue": 0.28059924065848,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=1",
      "smallest_normalized_scalar_eigenvalue": 0.219223593595585,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=1.5",
      "smallest_normalized_scalar_eigenvalue": 0.138444501318775,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=2",
      "smallest_normalized_scalar_eigenvalue": 0.0924635468163375,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=3",
      "smallest_normalized_scalar_eigenvalue": 0.0478369883287968,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=4",
      "smallest_normalized_scalar_eigenvalue": 0.0286182287437953,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=5",
      "smallest_normalized_scalar_eigenvalue": 0.0188813776007581,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=0.5, z=6",
      "smallest_normalized_scalar_eigenvalue": 0.0133380774482674,
      "sign": "positive",
      "local_status": "stable: mu_min is positive at every finite z and approaches zero from above as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=0",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=0.25",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=0.5",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=0.75",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=1",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=1.5",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=2",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=3",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=4",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=5",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1, z=6",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "massless boundary: mu_min is exactly zero for every finite z"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=0",
      "smallest_normalized_scalar_eigenvalue": -0.5,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=0.25",
      "smallest_normalized_scalar_eigenvalue": -0.47973722886787,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=0.5",
      "smallest_normalized_scalar_eigenvalue": -0.425390529679106,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=0.75",
      "smallest_normalized_scalar_eigenvalue": -0.353185731130307,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=1",
      "smallest_normalized_scalar_eigenvalue": -0.280776406404415,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=1.5",
      "smallest_normalized_scalar_eigenvalue": -0.171164609606623,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=2",
      "smallest_normalized_scalar_eigenvalue": -0.108495283014151,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=3",
      "smallest_normalized_scalar_eigenvalue": -0.0523431780746364,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=4",
      "smallest_normalized_scalar_eigenvalue": -0.0302475808395979,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=5",
      "smallest_normalized_scalar_eigenvalue": -0.0195927891221341,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "O'Raifeartaigh nominal valley",
      "locus": "y=1.5, z=6",
      "smallest_normalized_scalar_eigenvalue": -0.0136934928288817,
      "sign": "negative",
      "local_status": "tachyonic: mu_min is negative at every finite z and approaches zero from below as z tends to infinity"
    },
    {
      "model": "soft two-chiral benchmark",
      "locus": "r=-1, beta=0",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "nu_-=0: one complex scalar is massless"
    },
    {
      "model": "soft two-chiral benchmark",
      "locus": "r=0, beta=1",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "nu_-=0: one complex scalar is massless"
    },
    {
      "model": "soft two-chiral benchmark",
      "locus": "r=1, beta=2",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "nu_-=0: one complex scalar is massless"
    },
    {
      "model": "soft two-chiral benchmark",
      "locus": "r=2, beta=3",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "nu_-=0: one complex scalar is massless"
    },
    {
      "model": "soft two-chiral benchmark",
      "locus": "r=3, beta=4",
      "smallest_normalized_scalar_eigenvalue": 0,
      "sign": "zero",
      "local_status": "nu_-=0: one complex scalar is massless"
    },
    {
      "model": "soft two-chiral benchmark",
      "locus": "stable sample r=0, beta=0.4",
      "smallest_normalized_scalar_eigenvalue": 0.6,
      "sign": "positive",
      "local_status": "both complex scalar eigenvalues are positive"
    },
    {
      "model": "soft two-chiral benchmark",
      "locus": "tachyonic sample r=0, beta=1.5",
      "smallest_normalized_scalar_eigenvalue": -0.5,
      "sign": "negative",
      "local_status": "nu_- is negative"
    }
  ],
  "scheme_local_and_global_status": {
    "panel_a": {
      "renormalization_prescription": "tree spectrum exact at quadratic order; C(y) is the one-loop DRbar-type curvature coefficient with renormalized parameters evaluated near the heavy scale",
      "hierarchy": "h^2/(16 pi^2) is small, 0<y<1 for the controlled local minimum, higher-dimensional Kahler contributions are smaller than the one-loop curvature, and z is not taken beyond the regime in which the displayed field content is the useful Wilsonian organization",
      "expansion_order": "mass curves use the exact tree-level quadratic spectrum; C(y) is one loop; the displayed weak-splitting series is through y^4 with remainder O(y^6)",
      "stability_domain": "0<y<1 is tachyon-free at every finite z; y=1 has one exactly massless scalar; y>1 has one tachyonic scalar at every finite z",
      "local": "for 0<y<1 the exact tree spectrum is tachyon-free and the displayed one-loop curvature at X=0 is positive under the declared hierarchy",
      "global": "not determined by this fixture",
      "metastability": "not established: the benchmark supplies no lower basin, runaway endpoint, bounce, or lifetime"
    },
    "panel_b": {
      "renormalization_prescription": "the tree-level mass parameters are compared at one declared renormalization scale; any finite large-soft threshold coefficient depends on the matching prescription and interactions",
      "local": "the line beta=1+r exactly separates a positive quadratic Hessian from a tachyon in the displayed free mass benchmark",
      "global": "the quadratic benchmark has no interactions from which to infer nontrivial threshold corrections, remote vacua, or phase structure",
      "restoration": "the origin (r,beta)=(0,0) restores the displayed supersymmetric mass degeneracy",
      "decoupling": "large positive r at fixed beta makes the scalars heavy but leaves a nonsupersymmetric light fermion EFT after matching"
    }
  },
  "thresholds_and_limits": {
    "panel_a_large_z": {
      "z_minimum": 4.5,
      "z_maximum": 6,
      "polygon_vertices": [
        {
          "z": 4.5,
          "mu": -0.6
        },
        {
          "z": 6,
          "mu": -0.6
        },
        {
          "z": 6,
          "mu": 0.6
        },
        {
          "z": 4.5,
          "mu": 0.6
        }
      ],
      "meaning": "the light eigenvalues visibly approach zero; this shaded band is an explanatory warning, not a sharp threshold"
    },
    "nonlinear_eft_gate": {
      "order_parameter": "F is nonzero on the auxiliary branch used to solve the nilpotent constraint",
      "energy_domain": "E<<min(m_heavy,Lambda_NL) with Lambda_NL=O(sqrt(f))",
      "normalization_scope": "f has mass dimension two; the order-one and 4 pi factors in Lambda_NL depend on the channel and UV completion, so the figure asserts no numerical coefficient",
      "failure_conditions": [
        "F tends to zero, so the nilpotent auxiliary branch is singular",
        "a purportedly removed superpartner becomes light",
        "E is not parametrically below a heavy threshold or the nonlinear strong-coupling scale"
      ],
      "claim_boundary": "passing this gate licenses a nonlinear low-energy field organization only; it is not evidence for a lower vacuum, a metastable basin, or a lifetime"
    },
    "soft_massless_boundary": {
      "formula": "beta=1+r",
      "status": "nu_-=0: one complex scalar is massless",
      "points": [
        {
          "r": -1,
          "beta": 0,
          "nu_minus": 0,
          "eigenvalue_sign": "zero"
        },
        {
          "r": 0,
          "beta": 1,
          "nu_minus": 0,
          "eigenvalue_sign": "zero"
        },
        {
          "r": 1,
          "beta": 2,
          "nu_minus": 0,
          "eigenvalue_sign": "zero"
        },
        {
          "r": 2,
          "beta": 3,
          "nu_minus": 0,
          "eigenvalue_sign": "zero"
        },
        {
          "r": 3,
          "beta": 4,
          "nu_minus": 0,
          "eigenvalue_sign": "zero"
        }
      ]
    },
    "supersymmetry_restoration": {
      "parameter": "t in [0,1]",
      "formula": "r=0.8 t and beta=0.6 t, with the arrow directed toward t=0",
      "limit": "as t tends to zero, nu_+ and nu_- both tend to 1 and become degenerate with the fermion; the supersymmetric spectrum is restored",
      "points": [
        {
          "t": 1,
          "r": 0.8,
          "beta": 0.6,
          "nu_minus": 1.2,
          "nu_plus": 2.4,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        },
        {
          "t": 0.75,
          "r": 0.6,
          "beta": 0.45,
          "nu_minus": 1.15,
          "nu_plus": 2.05,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        },
        {
          "t": 0.5,
          "r": 0.4,
          "beta": 0.3,
          "nu_minus": 1.1,
          "nu_plus": 1.7,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        },
        {
          "t": 0.25,
          "r": 0.2,
          "beta": 0.15,
          "nu_minus": 1.05,
          "nu_plus": 1.35,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        },
        {
          "t": 0,
          "r": 0,
          "beta": 0,
          "nu_minus": 1,
          "nu_plus": 1,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        }
      ]
    },
    "large_soft_decoupling": {
      "parameter": "r",
      "formula": "beta=0.4 fixed with the arrow directed toward increasing r",
      "limit": "as r tends to infinity, nu_+/r and nu_-/r tend to 1 while the normalized fermion mass squared stays 1; the two scalars become heavy relative to the fermion",
      "points": [
        {
          "r": 0.8,
          "beta": 0.4,
          "nu_minus": 1.4,
          "nu_plus": 2.2,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        },
        {
          "r": 1.5,
          "beta": 0.4,
          "nu_minus": 2.1,
          "nu_plus": 2.9,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        },
        {
          "r": 2.25,
          "beta": 0.4,
          "nu_minus": 2.85,
          "nu_plus": 3.65,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        },
        {
          "r": 3,
          "beta": 0.4,
          "nu_minus": 3.6,
          "nu_plus": 4.4,
          "eigenvalue_signs": [
            "positive",
            "positive"
          ]
        }
      ],
      "matching_warning": "integrating out the heavy scalars requires an interacting-theory matching calculation; relevant threshold corrections can scale as a coupling times M_soft^2/(16 pi^2) and are absorbed into low-energy relevant parameters rather than guaranteed small",
      "cutoff_warning": "the low-energy cutoff lies below the first omitted threshold and any UV completion scale",
      "phase_warning": "the large-soft ray is a decoupling trajectory, not a theorem that vacuum structure or phase connectivity is preserved"
    }
  },
  "semantic_tables": {
    "panel_a_columns": [
      "curve_id",
      "y",
      "z",
      "mu_min",
      "eigenvalue_sign",
      "line_style",
      "status"
    ],
    "panel_b_boundary_columns": [
      "r",
      "beta",
      "nu_minus",
      "eigenvalue_sign",
      "status"
    ],
    "trajectory_columns": [
      "trajectory_id",
      "r",
      "beta",
      "nu_minus",
      "nu_plus",
      "eigenvalue_signs",
      "normalized_fermion_mass_squared",
      "status"
    ]
  },
  "independent_checks": {
    "o_raifeartaigh": {
      "frozen_curve_points_checked": 33,
      "pair_determinants_checked": 66,
      "global_minimum_identifications_checked": 33,
      "determinant_identity": "mu_{B,s,+} mu_{B,s,-}=1+s y",
      "y_half_sign": "positive for every frozen finite-z point",
      "y_one_sign": "zero for every frozen finite-z point",
      "y_three_halves_sign": "negative for every frozen finite-z point"
    },
    "coleman_weinberg": {
      "positivity_samples_checked": 5,
      "analytic_positivity_chain": "F(0)=F'(0)=0 and F''(y)=2 log((1+y)/(1-y))>0 for 0<y<1",
      "independently_recovered_series_coefficients": {
        "y_squared": 1.333333333333333,
        "y_fourth": 0.1333333333333333,
        "odd_terms_through_y_fifth": 0
      }
    },
    "soft_plane": {
      "boundary_points_checked": 5,
      "restoration_points_checked": 5,
      "large_soft_points_checked": 4,
      "eigenvalue_identity": "nu_+=1+r+beta and nu_-=1+r-beta",
      "determinant_identity": "nu_+ nu_-=(1+r)^2-beta^2",
      "restoration_endpoint": "(r,beta)=(0,0), nu_+=nu_-=m_F^2/|m|^2=1",
      "large_soft_limit": "at fixed beta=0.4, nu_+/r and nu_-/r tend to 1 while the normalized fermion mass squared remains 1"
    },
    "nonlinear_gate": {
      "dimension_check": "[F]=[f]=mass^2, so sqrt(f), m_heavy, Lambda_NL, and E all have mass dimension one",
      "singular_limits_checked": [
        "F->0 invalidates the nilpotent auxiliary branch",
        "E/m_heavy not << 1 invalidates partner removal",
        "E^2/f not << 1 invalidates the nonlinear expansion"
      ]
    }
  },
  "accessibility_encoding": {
    "color_independence": "black, white, and gray only; stable, massless, and tachyonic cases use solid, long-dashed, and dot-dashed lines plus direct status words; trajectories use distinct dash patterns and arrow directions",
    "unstable_warning": "tachyonic regions are adjacent to the explicit nu_-<0 label and the exact massless boundary",
    "long_description": "A vertically stacked monochrome quantitative figure. Panel A plots the exact minimum O'Raifeartaigh scalar mass squared divided by m squared against z=h times absolute X over m. The solid y=one-half curve stays positive and approaches zero from above, the dashed y=one curve is exactly massless, and the dot-dashed y=three-halves curve stays negative and approaches zero from below. A shaded large-z band warns that light modes require reorganizing the EFT. A compact box states that the one-loop coefficient C(y) is positive for zero<y<one and begins four y squared over three plus two y fourth over fifteen. A separate warning says the benchmark supplies no lower basin or lifetime. The middle gate requires F nonzero and E parametrically below the smaller of the heavy threshold and an order-square-root-f nonlinear scale; it fails as F goes to zero or a removed partner becomes light. Panel B plots beta=absolute b over absolute m squared against r=m_s squared over absolute m squared. The exact line beta=one+r separates a stable lower wedge from a tachyonic upper region. A dashed trajectory reaches the supersymmetric origin, while a dotted fixed-beta ray moves toward large positive r and scalar decoupling. The latter is labeled as requiring matching and a cutoff and as proving no phase continuity theorem.",
    "semantic_equivalent": "/figures/supersymmetry-duality/pseudomoduli-soft-decoupling-map.json"
  },
  "primary_sources": [
    {
      "citation": "Coleman, S., and E. Weinberg, Physical Review D 7 (1973) 1888-1910",
      "locator": "one-loop effective-potential construction, pp. 1888-1910",
      "url": "https://doi.org/10.1103/PhysRevD.7.1888",
      "use": "one-loop curvature framework and renormalization dependence"
    },
    {
      "citation": "Shih, D., Journal of High Energy Physics 2008 no. 02 (2008) 091",
      "locator": "Appendix A.1, arXiv PDF pp. 13–14, Eqs. (A.1)–(A.3)",
      "url": "https://doi.org/10.1088/1126-6708/2008/02/091",
      "use": "exact one-loop curvature of the displayed O'Raifeartaigh benchmark"
    },
    {
      "citation": "Komargodski, Z., and N. Seiberg, Journal of High Energy Physics 2009 no. 09 (2009) 066",
      "locator": "sections 2 and 5",
      "url": "https://arxiv.org/abs/0907.2441",
      "use": "nilpotent goldstino branch and constrained low-energy multiplets"
    },
    {
      "citation": "Graesser, M. L., R. Kitano, and M. Kurachi, Journal of High Energy Physics 2009 no. 10 (2009) 077",
      "locator": "§ 4.1, arXiv PDF pp. 16–17, Eqs. (63), (65)–(66)",
      "url": "https://arxiv.org/abs/0907.2988",
      "use": "goldstino amplitude growth and the order-sqrt(f) nonlinear validity scale"
    },
    {
      "citation": "Girardello, L., and M. T. Grisaru, Nuclear Physics B 194 (1982) 65-76",
      "locator": "classification and ultraviolet behavior of soft operators",
      "url": "https://doi.org/10.1016/0550-3213(82)90512-0",
      "use": "soft scalar masses and bilinears"
    },
    {
      "citation": "Appelquist, T., and J. Carazzone, Physical Review D 11 (1975) 2856-2861",
      "locator": "decoupling theorem and low-energy parameter matching, pp. 2856-2861",
      "url": "https://doi.org/10.1103/PhysRevD.11.2856",
      "use": "large-mass decoupling with matching rather than automatic small threshold coefficients"
    }
  ]
}