{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.competing-orders-primary-vestigial-nematicity",
  "title": "Primary and vestigial routes to electronic nematicity",
  "classification": "schematic symmetry-and-inference diagram",
  "reader_question": "How can a primary q = 0 fermion bilinear and a vestigial composite of fluctuating finite-wavevector stripes produce the same B1g nematic symmetry, and what evidence can distinguish the routes?",
  "dominant_point": "The same B1g symmetry can arise from a primary q = 0 bilinear or from vestigial finite-Q stripe fluctuations, so common symmetry evidence establishes nematicity but route-sensitive momentum and scaling evidence is needed to identify the microscopic contribution.",
  "takeaway": "Primary and vestigial fields can transform in the same B1g representation and therefore mix linearly; C4-to-C2 symmetry, domains, strain response, or coincident onset establishes nematicity but does not by itself identify the microscopic route, which requires route-sensitive q = 0 and finite-Q diagnostics.",
  "status": {
    "schematic": true,
    "not_to_scale": true,
    "contains_model_data": false,
    "contains_material_specific_band_structure": false,
    "quantitative_peak_widths_or_amplitudes": false
  },
  "conventions": [
    "The square-lattice x and y axes lie along the bonds. In this convention x squared minus y squared transforms as B1g; a rotated crystallographic convention can interchange B1g and B2g labels.",
    "The primary operator mathcal N_B1g is extensive, while n_B1g = mathcal N_B1g / V is its intensive density. Only intensive quantities are combined in the displayed mixed response Phi_B1g.",
    "Q_x = (Q, 0) and Q_y = (0, Q) are finite ordering wavevectors exchanged by a ninety-degree rotation. For real-space spin or charge densities, each channel appears in momentum space as a conjugate pair at plus and minus Q_x or plus and minus Q_y, so the diagram contains four broad peaks.",
    "M_x and M_y are intensive magnetic amplitudes, and rho_x and rho_y are intensive charge amplitudes. Their squared magnitudes are used for the translation-invariant composites phi_M and phi_rho.",
    "The drawing chooses the domain with the x-channel variance larger than the y-channel variance. Its degenerate partner is obtained by exchanging x and y, which reverses every B1g order parameter.",
    "External B1g strain epsilon_B1g is treated as a conjugate symmetry-breaking field. Spontaneous order requires the thermodynamic limit before the zero-strain limit."
  ],
  "panels": [
    {
      "id": "A",
      "title": "Primary q = 0 fermionic route",
      "purpose": "Represent a B1g Fermi-contour distortion generated by an extensive one-particle bilinear and identify its intensive order-parameter density.",
      "object_ids": [
        "momentum_axes_primary",
        "symmetric_reference_contour",
        "primary_distorted_contour",
        "primary_operator_card"
      ]
    },
    {
      "id": "B",
      "title": "Vestigial finite-Q stripe route",
      "purpose": "Represent a parent-disordered regime with zero first moments but unequal broad fluctuation strengths in the C4-related conjugate peak pairs at plus and minus Q_x and plus and minus Q_y.",
      "object_ids": [
        "momentum_axes_stripe",
        "broad_Qx_fluctuation_cloud",
        "broad_Qy_fluctuation_cloud",
        "vestigial_composite_card"
      ]
    },
    {
      "id": "C",
      "title": "Same broken symmetry, different microscopic evidence",
      "purpose": "Show symmetry-allowed mixing into one measured B1g response, distinguish route-sensitive diagnostics, and state the zero-strain thermodynamic order check.",
      "object_ids": [
        "primary_source_chip",
        "vestigial_source_chip",
        "mixing_arrows",
        "shared_B1g_response",
        "primary_sensitive_diagnostic",
        "vestigial_sensitive_diagnostic",
        "shared_order_check"
      ]
    }
  ],
  "objects": [
    {
      "id": "momentum_axes_primary",
      "panel_id": "A",
      "kind": "momentum-space axes",
      "content": "Bond-aligned k_x and k_y axes centered on Gamma; no numerical momentum scale is supplied."
    },
    {
      "id": "symmetric_reference_contour",
      "panel_id": "A",
      "kind": "reference Fermi contour",
      "content": "A directly labeled dashed circle used only as a visual C4-symmetric reference, not as an asserted normal-state band contour."
    },
    {
      "id": "primary_distorted_contour",
      "panel_id": "A",
      "kind": "schematic nematic Fermi contour",
      "content": "A directly labeled solid horizontal ellipse representing one B1g domain of a q = 0 primary fermionic distortion. Its aspect ratio is layout-only."
    },
    {
      "id": "primary_operator_card",
      "panel_id": "A",
      "kind": "operator definition",
      "content": "Defines the extensive operator mathcal N_B1g as the momentum sum of the B1g form factor times occupation and defines the intensive density n_B1g = mathcal N_B1g / V."
    },
    {
      "id": "momentum_axes_stripe",
      "panel_id": "B",
      "kind": "wavevector-space axes",
      "content": "q_x and q_y axes centered at zero, with broad schematic conjugate fluctuation clouds at plus and minus Q_x and plus and minus Q_y."
    },
    {
      "id": "broad_Qx_fluctuation_cloud",
      "panel_id": "B",
      "kind": "finite-wavevector correlation channel",
      "content": "Solid-outlined, diagonally hatched, broad conjugate clouds around both plus Q_x and minus Q_x represent the larger parent variance in the chosen nematic domain; the two clouds form one real-field conjugate pair and are not Bragg peaks or a quantitative structure factor."
    },
    {
      "id": "broad_Qy_fluctuation_cloud",
      "panel_id": "B",
      "kind": "finite-wavevector correlation channel",
      "content": "Dashed-outlined, stippled, broad conjugate clouds around both plus Q_y and minus Q_y represent the smaller parent variance in the chosen nematic domain; the two clouds form the other real-field conjugate pair, are not absent, and are not a quantitative structure factor."
    },
    {
      "id": "vestigial_composite_card",
      "panel_id": "B",
      "kind": "composite-order definition",
      "content": "States that both parent first moments vanish while phi_X = expectation of |X_x|^2 minus |X_y|^2 is nonzero, for magnetic X = M or charge X = rho."
    },
    {
      "id": "primary_source_chip",
      "panel_id": "C",
      "kind": "mixing input",
      "content": "The intensive primary density n_B1g."
    },
    {
      "id": "vestigial_source_chip",
      "panel_id": "C",
      "kind": "mixing inputs",
      "content": "The intensive magnetic and charge composites phi_M and phi_rho."
    },
    {
      "id": "mixing_arrows",
      "panel_id": "C",
      "kind": "symmetry-allowed relation",
      "content": "Dashed arrows from the primary and composite source chips to the shared B1g response. They denote allowed linear mixing among fields in the same irrep, not a causal direction or sequence."
    },
    {
      "id": "shared_B1g_response",
      "panel_id": "C",
      "kind": "mixed observable channel",
      "content": "A generic intensive B1g response Phi_B1g = a_n n_B1g + a_M phi_M + a_rho phi_rho, all of whose terms reverse sign under the chosen C4 operation and characterize C4-to-C2 symmetry breaking."
    },
    {
      "id": "primary_sensitive_diagnostic",
      "panel_id": "C",
      "kind": "mechanism-sensitive evidence",
      "content": "Form-factor-resolved q = 0 Fermi-contour deformation together with selective forward-channel tuning, analyzed without assuming that a finite-Q parent generated it."
    },
    {
      "id": "vestigial_sensitive_diagnostic",
      "panel_id": "C",
      "kind": "mechanism-sensitive evidence",
      "content": "Broad unequal Q_x and Q_y parent correlations, composite scaling, and a controlled two-step onset in which nematic order can survive while the finite-Q parent remains disordered."
    },
    {
      "id": "shared_order_check",
      "panel_id": "C",
      "kind": "thermodynamic order criterion",
      "content": "Demonstrate opposite zero-strain domains and take V to infinity before epsilon_B1g tends to zero. This establishes spontaneous nematic order but is not a microscopic mechanism test."
    }
  ],
  "relations": [
    {
      "from": "primary_operator_card",
      "to": "primary_distorted_contour",
      "type": "can_generate",
      "meaning": "A primary q = 0 B1g fermion bilinear can deform a Fermi contour with x-y anisotropy while preserving translation symmetry."
    },
    {
      "from": "broad_Qx_fluctuation_cloud",
      "to": "broad_Qy_fluctuation_cloud",
      "type": "C4_related_conjugate_pairs_with_unequal_variance",
      "meaning": "A ninety-degree rotation exchanges the full conjugate pair at plus and minus Q_x with the full conjugate pair at plus and minus Q_y; unequal pair variances in one domain produce a B1g composite, and the partner domain reverses the inequality."
    },
    {
      "from": "broad_Qx_fluctuation_cloud",
      "to": "vestigial_composite_card",
      "type": "contributes_to",
      "meaning": "The total parent variance represented by the conjugate peaks at plus and minus Q_x enters phi_X with a positive sign in the declared convention."
    },
    {
      "from": "broad_Qy_fluctuation_cloud",
      "to": "vestigial_composite_card",
      "type": "contributes_to",
      "meaning": "The total parent variance represented by the conjugate peaks at plus and minus Q_y enters phi_X with a negative sign in the declared convention."
    },
    {
      "from": "primary_source_chip",
      "to": "shared_B1g_response",
      "type": "mixes_linearly_by_symmetry",
      "visual": "dashed arrow",
      "meaning": "The primary density may contribute to a measured B1g response because it belongs to the same one-dimensional irrep; the arrow does not assert that it is the microscopic cause."
    },
    {
      "from": "vestigial_source_chip",
      "to": "shared_B1g_response",
      "type": "mixes_linearly_by_symmetry",
      "visual": "dashed arrow",
      "meaning": "Magnetic and charge composites may contribute to the same measured B1g response; the arrow does not assert that either composite is the microscopic cause."
    },
    {
      "from": "shared_B1g_response",
      "to": "shared_order_check",
      "type": "requires_for_spontaneous_order",
      "meaning": "A nonzero response at imposed strain is not by itself spontaneous order; the system-size limit must precede removal of the conjugate strain."
    },
    {
      "from": "primary_sensitive_diagnostic",
      "to": "shared_B1g_response",
      "type": "helps_discriminate_contribution",
      "meaning": "A q = 0 form-factor-resolved deformation and selective forward-channel tuning provide evidence about a primary contribution beyond the common broken symmetry."
    },
    {
      "from": "vestigial_sensitive_diagnostic",
      "to": "shared_B1g_response",
      "type": "helps_discriminate_contribution",
      "meaning": "Finite-Q parent correlations, composite scaling, and separated parent and nematic onsets provide evidence about a vestigial contribution beyond the common broken symmetry."
    }
  ],
  "arrow_semantics": {
    "dashed_arrows": "Symmetry-allowed linear mixing among intensive B1g fields. They do not encode time order, renormalization-group flow, derivation, dominance, or causality.",
    "solid_axis_arrows": "Positive momentum or wavevector directions only.",
    "no_causal_arrow": "The figure deliberately contains no arrow claiming that a shared nematic observation identifies either microscopic route."
  },
  "equations": [
    {
      "id": "primary_extensive_operator",
      "equation": "mathcal N_B1g = sum_(k,sigma) (cos k_x - cos k_y) n_(k,sigma)",
      "meaning": "An extensive q = 0 fermion bilinear in the bond-aligned B1g channel."
    },
    {
      "id": "primary_density",
      "equation": "n_B1g = mathcal N_B1g / V",
      "meaning": "The intensive primary order-parameter density used in the mixing relation."
    },
    {
      "id": "finite_wavevectors",
      "equation": "Q_x = (Q, 0), Q_y = (0, Q); displayed peaks are at plus or minus Q_x and plus or minus Q_y",
      "meaning": "C4 exchanges the x and y channels, while reality of the underlying spin or charge density supplies the conjugate momentum partner within each channel."
    },
    {
      "id": "vestigial_magnetic_composite",
      "equation": "phi_M = expectation(|M_x|^2 - |M_y|^2)",
      "conditions": "expectation(M_x) = expectation(M_y) = 0 in the parent-disordered vestigial regime",
      "meaning": "A uniform spin-rotation-invariant and translation-invariant B1g composite of magnetic stripe fluctuations."
    },
    {
      "id": "vestigial_charge_composite",
      "equation": "phi_rho = expectation(|rho_x|^2 - |rho_y|^2)",
      "conditions": "expectation(rho_x) = expectation(rho_y) = 0 in the parent-disordered vestigial regime",
      "meaning": "A uniform translation-invariant B1g composite of charge-stripe fluctuations."
    },
    {
      "id": "mixed_B1g_response",
      "equation": "Phi_B1g = a_n n_B1g + a_M phi_M + a_rho phi_rho",
      "meaning": "The most elementary linear mixing of normalized intensive fields in the same one-dimensional irrep; coefficients depend on operator definitions and microscopic couplings."
    },
    {
      "id": "spontaneous_zero_strain_limit",
      "equation": "lim_(epsilon_B1g -> 0+) lim_(V -> infinity) expectation(Phi_B1g)_epsilon != 0",
      "meaning": "The thermodynamic limit precedes removal of the conjugate strain when testing spontaneous nematic order."
    }
  ],
  "diagnostics": {
    "shared_but_not_mechanism_specific": [
      "B1g transformation and C4-to-C2 symmetry breaking",
      "opposite nematic domains after the conjugate strain is removed in the correct order of limits",
      "a strong B1g susceptibility, elastoresistive response, or elastic softening after probe-coupling and ensemble effects are controlled",
      "a nematic onset temperature, even if it nearly coincides with another anomaly"
    ],
    "primary_sensitive": [
      "a form-factor-resolved q = 0 deformation of the one-particle spectrum or Fermi contour",
      "selective tuning of the forward-scattering or Pomeranchuk channel without a corresponding finite-Q parent enhancement",
      "a quantitative susceptibility model whose normalized eigenvector has a stable primary-density component"
    ],
    "vestigial_sensitive": [
      "broad or growing correlations at both members of each conjugate pair, plus and minus Q_x and plus and minus Q_y, with a controlled variance anisotropy between the x and y channels",
      "composite susceptibility or scaling tied to the parent correlators",
      "a controlled regime with nonzero nematic order but vanishing parent first moments and no thermodynamic finite-Q parent order",
      "selective tuning of magnetic or charge parent correlations that tracks the nematic response",
      "a separated two-step onset when such a separation is allowed by dimensionality, elasticity, disorder, and model coefficients"
    ]
  },
  "zero_strain_thermodynamic_limit": {
    "required_order": "First take V to infinity at a fixed infinitesimal sign of epsilon_B1g; only then take epsilon_B1g to zero from that sign.",
    "controls": [
      "calibrate residual B1g strain and distinguish fixed-strain from fixed-stress conditions",
      "show the opposite domain under the opposite infinitesimal field or otherwise establish domain degeneracy",
      "control finite-size tunneling and domain averaging before interpreting a zero first moment",
      "separate a rounded conjugate-field response from spontaneous symmetry breaking"
    ],
    "interpretation": "Passing this check establishes spontaneous nematic order in the declared ensemble but does not determine whether the dominant microscopic field is primary, magnetic-vestigial, charge-vestigial, orbital, or lattice assisted."
  },
  "assumptions": [
    "The high-symmetry reference state is tetragonal and admits a bond-aligned B1g channel.",
    "All fields in the displayed mixing equation have been normalized as intensive quantities.",
    "The finite-Q stripe amplitudes occur in a C4-related pair and the shown vestigial regime is parent-disordered after thermodynamic scaling, not merely a finite-volume symmetric average of an ordered parent.",
    "The linear mixing equation is a symmetry-level local description; additional fields, gradients, frequency dependence, nonlinearities, and lattice degrees of freedom may be required in a material or microscopic model.",
    "The diagnostic statements are evidentiary tests whose interpretation requires consistent temperature, strain or stress ensemble, momentum resolution, probe matrix elements, domains, finite-size control, and covariance."
  ],
  "limits": [
    {
      "condition": "rotated crystallographic axes or a different unit-cell convention",
      "statement": "The physical x-y anisotropy is unchanged, but the B1g and B2g labels may interchange; axes and unit cell must be stated."
    },
    {
      "condition": "nonzero external or residual B1g strain",
      "statement": "The strain acts as a conjugate field, selects a domain, and rounds an Ising singularity; a finite response at that strain is not proof of spontaneous order."
    },
    {
      "condition": "finite volume or domain-averaged sampling",
      "statement": "Vanishing parent or nematic first moments do not by themselves prove disorder; correlation-length, structure-factor, Binder, or equivalent thermodynamic scaling is required."
    },
    {
      "condition": "strong mixing among same-irrep fields",
      "statement": "The eigenmode at an instability can be a hybrid. Describing it as purely primary or purely vestigial can be basis dependent without a microscopic and probe-coupling analysis."
    },
    {
      "condition": "incommensurate charge or spin parents with disorder",
      "statement": "Random fields, pinning, glassiness, and dimensionality can modify or eliminate parent long-range order while leaving a discrete vestigial channel; the displayed clean schematic does not classify those regimes."
    }
  ],
  "nonclaims": [
    "The dashed reference circle and solid ellipse are not calculated Fermi surfaces and do not specify a carrier density, dispersion, material, or distortion magnitude.",
    "The four broad clouds at plus and minus Q_x and plus and minus Q_y contain no measured or simulated structure-factor data; their widths, shapes, hatch or stipple density, contrast, and relative areas have no quantitative meaning.",
    "The chosen x-dominant domain is not universal; the y-dominant partner is symmetry related and equally allowed in zero field.",
    "Zero parent first moments in a finite symmetric system do not by themselves establish a vestigial phase or exclude parent long-range order.",
    "The figure does not assert that primary and vestigial descriptions are mutually exclusive; same-irrep fields generally hybridize.",
    "The mixing coefficients a_n, a_M, and a_rho are not universal, are normalization dependent, and are not inferred from the drawing.",
    "A dashed mixing arrow does not establish causality, ancestry, temporal order, or the dominant microscopic mechanism.",
    "A common B1g symmetry, domain pattern, strain response, elastic anomaly, or coincident onset does not distinguish primary from vestigial order.",
    "A separated two-step transition is not guaranteed; dimension, elastic coupling, disorder, commensurability, and interaction coefficients can merge, split, or change the order of transitions.",
    "The figure does not exclude orbital, phononic, structural, pair-density-wave, or other same-symmetry contributions to a measured nematic response.",
    "No universal critical exponents, onset hierarchy, spectral line shape, or fluctuation amplitude are shown or inferred.",
    "Strain is not treated as a neutral probe; it is a conjugate field whose ensemble and limiting procedure must be stated."
  ],
  "visual_encodings": {
    "canvas": "explicit white background for light, dark, monochrome, and print contexts",
    "reading_order": "three vertically stacked regions A, B, and C for a phone-first reading path; panels A and B each place the diagram first, the full-width equation and explanation card second, and a compact scope note last",
    "primary_reference": "directly labeled dashed gray circle",
    "primary_distortion": "directly labeled solid black ellipse",
    "Qx_cloud_pair": "two directly labeled broad peaks at plus and minus Q_x, each with a solid black outline and diagonal hatching",
    "Qy_cloud_pair": "two directly labeled broad peaks at plus and minus Q_y, each with a dashed gray outline and stippling",
    "mixing": "two dashed gray arrows, explicitly labeled as symmetry-allowed mixing rather than genealogy",
    "shared_response": "gray filled black-outlined box containing the common B1g symmetry and normalized mixing equation",
    "diagnostics": "stacked full-width text cards; distinctions do not depend on color",
    "color_dependency": "none; direct labels, solid versus dashed outlines, hatch versus stipple patterns, position, and text redundantly encode every distinction"
  },
  "sources": [
    {
      "citation": "Vadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin, Quantum Theory of a Nematic Fermi Fluid, Physical Review B 64 (2001) 195109",
      "doi": "10.1103/PhysRevB.64.195109",
      "url": "https://doi.org/10.1103/PhysRevB.64.195109",
      "use": "primary q = 0 electronic nematic order as a fermionic quadrupolar or Pomeranchuk distortion and its Fermi-surface interpretation"
    },
    {
      "citation": "Steven A. Kivelson, Eduardo Fradkin, and Victor J. Emery, Electronic Liquid-Crystal Phases of a Doped Mott Insulator, Nature 393 (1998) 550-553",
      "doi": "10.1038/31177",
      "url": "https://doi.org/10.1038/31177",
      "use": "electronic nematic order with orientational symmetry breaking but without long-range positional stripe order"
    },
    {
      "citation": "Rafael M. Fernandes, Andrey V. Chubukov, Johannes Knolle, Ilya Eremin, and Joerg Schmalian, Preemptive Nematic Order, Pseudogap, and Orbital Order in the Iron Pnictides, Physical Review B 85 (2012) 024534",
      "doi": "10.1103/PhysRevB.85.024534",
      "url": "https://doi.org/10.1103/PhysRevB.85.024534",
      "use": "preemptive Ising-nematic order as a composite of C4-related stripe-magnetic amplitudes with vanishing magnetic first moments"
    },
    {
      "citation": "Laimei Nie, Akash V. Maharaj, Eduardo Fradkin, and Steven A. Kivelson, Vestigial Nematicity from Spin and/or Charge Order in the Cuprates, Physical Review B 96 (2017) 085142",
      "doi": "10.1103/PhysRevB.96.085142",
      "url": "https://doi.org/10.1103/PhysRevB.96.085142",
      "use": "spin- and charge-stripe doublets, their vestigial nematic composites, same-symmetry coupling, and model-dependent two-stage ordering"
    }
  ],
  "independent_checks": [
    {
      "check": "extensivity and normalization",
      "result": "mathcal N_B1g is explicitly extensive, n_B1g = mathcal N_B1g / V is intensive, M and rho are declared intensive amplitudes, and only intensive quantities enter Phi_B1g."
    },
    {
      "check": "point-group transformation",
      "result": "cos k_x minus cos k_y, n_B1g, phi_M, and phi_rho all reverse sign when x and y are exchanged; the associated order breaks C4 to C2 in the declared bond-aligned convention."
    },
    {
      "check": "translation and spin symmetry of the composite",
      "result": "The squared-magnitude differences are uniform and spin-rotation invariant even when the finite-Q parent first moments vanish, so they can order vestigially without static stripe magnetism or charge order."
    },
    {
      "check": "momentum distinction",
      "result": "Panel A labels the primary bilinear q = 0, while panel B places four parent-correlation peaks at the two conjugate pairs plus and minus Q_x and plus and minus Q_y; the diagram never identifies the two momentum structures."
    },
    {
      "check": "domain convention",
      "result": "The visual chooses phi greater than zero, states that x and y exchange in the partner domain, and assigns no universal meaning to that sign."
    },
    {
      "check": "thermodynamic limit",
      "result": "The displayed formula takes V to infinity before epsilon_B1g tends to zero, consistent with a spontaneous discrete-symmetry-breaking test."
    },
    {
      "check": "arrow semantics",
      "result": "Every dashed arrow terminates on the shared B1g response and is explicitly labeled mixing, not genealogy; no other relation is drawn as a causal arrow."
    },
    {
      "check": "accessibility and non-color redundancy",
      "result": "Direct labels, solid versus dashed outlines, diagonal hatching versus stippling, box position, and explicit text preserve every distinction in grayscale and without color perception."
    },
    {
      "check": "claim-source correspondence",
      "result": "The primary Fermi-fluid route, electronic liquid-crystal interpretation, magnetic vestigial route, and coupled spin/charge vestigial route are each tied to a distinct primary scientific source."
    }
  ],
  "caption": "Schematic primary and vestigial routes to the same bond-aligned B1g nematic symmetry. A q = 0 fermion bilinear can distort a Fermi contour, while unequal broad fluctuations in the two C4-related conjugate peak pairs, at plus and minus Q_x and at plus and minus Q_y, can yield a composite nematic order even when both magnetic or charge parent first moments vanish. Because these intensive fields share an irrep, they may mix linearly: zero-strain domains and the thermodynamic order of limits establish spontaneous C4-to-C2 order, but route-sensitive q = 0 and finite-Q correlations are needed to test the microscopic interpretation. Shapes, widths, hatch or stipple density, contrast, and mixing coefficients are schematic, not to scale, and contain no material data.",
  "alt_text": "Three vertically stacked schematic regions compare two routes to B1g nematicity, with each of the first two regions placing its diagram above a full-width equation card. The top shows a dashed circular reference Fermi contour and a solid horizontal elliptical contour produced by an extensive q-equals-zero B1g occupation bilinear, with its intensive density defined by division by volume. The middle shows four broad peaks: a solid-outlined, diagonally hatched conjugate pair at plus and minus Q_x with the larger variance in the chosen domain, and a dashed-outlined, stippled conjugate pair at plus and minus Q_y with the smaller variance. Both magnetic or charge parent first moments vanish, but the difference of their x- and y-channel squared amplitudes is nonzero. The bottom sends primary density and vestigial composite labels through dashed arrows marked as mixing rather than causal direction into one shared C4-to-C2 B1g response, then lists primary-sensitive and vestigial-sensitive diagnostics. A final box requires the thermodynamic limit before zero strain and warns that symmetry, strain response, or coincident onset alone cannot identify the route.",
  "rights": {
    "basis": "Original QFT.org schematic created from independently checked symmetry relations and equations; no third-party figure, geometry, tracing, material data, or simulation output was copied or restyled.",
    "source_role": "The cited papers support the scientific concepts and transformation laws only; they are not visual sources.",
    "creator": "OpenAI Codex, for QFT.org",
    "date": "2026-08-31"
  }
}