{
  "artifact_id": "qft.artifact.holography-quantum-gravity.wormholes-gravitational-path-integrals-and-ensembles.integer-replica-gluing-quotient-map",
  "owner_page_id": "qft.topic.black-hole-information.semiclassical-gravitational-replicas",
  "title": "Integer Replica Sewing and the Conditional Gravitational Quotient",
  "kind": "boundary-sewing-and-local-quotient schematic",
  "source_revision": 1,
  "source_date": "2026-08-29",
  "schematic": true,
  "not_to_scale": true,
  "reader_question": "How does integer-n density-matrix sewing define the replica boundary condition, and when may a regular gravitational saddle be replaced by a quotient with a conical locus?",
  "takeaway": "Cyclic sewing fixes the boundary problem at integer n; only an admitted regular saddle that actually preserves Z_n has a one-fundamental-domain quotient with opening angle 2 pi/n, while the full cover remains smooth.",
  "scope": {
    "construction": "Euclidean gravitational replica boundary-value problem for a regulated boundary region R",
    "replica_domain": "integer n >= 2",
    "drawn_example": "n = 3",
    "dimension": "arbitrary spacetime dimension admitting a codimension-two replica fixed locus; the lower panels show only its two-dimensional transverse section",
    "quantitative_status": "the identifications, group action, angular ranges, deficit, and Einstein-gravity tension relation are exact in their stated scope; positions, lengths, and wedge embedding are schematic",
    "excluded_claims": [
      "analytic continuation away from integer n",
      "existence or dominance of a particular gravitational saddle",
      "selection of a bulk topology",
      "replica-wormhole topology competition",
      "an ensemble or fixed-theory interpretation"
    ]
  },
  "conventions": {
    "banks": "R_k^+ and R_k^- denote the chosen upper and lower banks of the cut on copy k; reversing both bank labels reverses the displayed cycle orientation but not the replica geometry",
    "copy_index": "k is understood modulo n, so R_(n+1) is R_1",
    "chosen_region_sewing": "R_k^+ is identified with R_(k+1)^-",
    "complement_sewing": "the two banks of the complement are identified within the same copy: Rbar_k^+ is identified with Rbar_k^-",
    "boundary_branch_locus": "partial R labels the boundary endpoint or entangling locus of the cut and is not automatically identical to the bulk fixed locus Sigma_n",
    "euclidean_identifications": "arrowheads in panel A record the cyclic permutation only; they do not denote Lorentzian propagation or Euclidean time evolution",
    "replica_action": "on a regular Z_n-symmetric saddle, the generator acts in the transverse normal plane as theta -> theta + 2 pi/n and obeys g^n = 1",
    "smooth_cover_angle": "theta has period 2 pi on the unquotiented smooth cover M_n",
    "quotient_angle": "the quotient fundamental domain has intrinsic opening angle 2 pi/n",
    "deficit": "delta_n = 2 pi (1 - 1/n)",
    "einstein_source": "in Einstein gravity the auxiliary quotient defect may be represented by T_n = (n - 1)/(4 n G_N), for which delta_n = 8 pi G_N T_n",
    "defect_location": "the conical defect belongs to the quotient description; the regular unquotiented cover is smooth at Sigma_n"
  },
  "panels": [
    {
      "id": "A",
      "title": "Integer-replica boundary sewing",
      "example_n": 3,
      "objects": [
        "copy 1 with upper and lower banks of R and its complement",
        "copy 2 with upper and lower banks of R and its complement",
        "copy 3 with upper and lower banks of R and its complement",
        "boundary branch locus partial R"
      ],
      "relations": [
        {
          "type": "cyclic-identification",
          "equation": "R_k^+ ~ R_(k+1)^- with R_(n+1) = R_1",
          "drawn_cycle": "1 -> 2 -> 3 -> 1",
          "permutation": "one n-cycle"
        },
        {
          "type": "same-copy-identification",
          "equation": "Rbar_k^+ ~ Rbar_k^-",
          "permutation": "identity on the complement"
        }
      ],
      "warning": "The boundary sewing defines a boundary condition. It does not choose a bulk topology, contour coefficient, saddle, or dominant contribution."
    },
    {
      "id": "B",
      "title": "Smooth cover M_n",
      "condition": "an admitted regular gravitational saddle preserves the Z_n permutation symmetry",
      "local_geometry": "a smooth two-dimensional transverse disk around the codimension-two fixed locus Sigma_n",
      "angular_period": "theta ~ theta + 2 pi",
      "group_action": "g: theta -> theta + 2 pi/n, with g^n = 1",
      "sector_guides": "the n radial guides mark fundamental domains of the group action; they are not physical seams or curvature singularities",
      "regularity": "M_n is smooth at Sigma_n in the displayed branch"
    },
    {
      "id": "C",
      "title": "Conditional quotient Mhat_n",
      "condition": "the admitted saddle preserves Z_n",
      "definition": "Mhat_n = M_n/Z_n",
      "local_geometry": "one fundamental wedge with its radial sides identified",
      "opening_angle": "2 pi/n",
      "deficit_angle": "delta_n = 2 pi (1 - 1/n)",
      "einstein_gravity_representation": {
        "role": "auxiliary source in the quotient description, not a pre-existing matter brane in the smooth cover",
        "tension": "T_n = (n - 1)/(4 n G_N)",
        "check": "delta_n = 8 pi G_N T_n"
      }
    }
  ],
  "failure_branch": {
    "id": "replica-symmetry-breaking",
    "intervention": "include an admitted saddle that does not preserve the boundary Z_n permutation",
    "outcome": "analyze the full saddle M_n directly; no single-fundamental-domain quotient with one conical locus is licensed",
    "claim_boundary": "replica boundary conditions do not force replica symmetry"
  },
  "controls": [
    {
      "case": "n = 2",
      "opening_angle": "pi",
      "deficit_angle": "pi",
      "einstein_tension": "1/(8 G_N)"
    },
    {
      "case": "formal n = 1 normalization check",
      "opening_angle": "2 pi",
      "deficit_angle": "0",
      "einstein_tension": "0",
      "qualification": "this algebraic limit check does not establish or choose an analytic continuation"
    },
    {
      "case": "wrong-space defect control",
      "seeded_error": "place the conical deficit on the regular unquotiented cover",
      "required_result": "reject"
    },
    {
      "case": "broken-symmetry quotient control",
      "seeded_error": "replace a replica-symmetry-breaking saddle by one quotient wedge",
      "required_result": "reject"
    }
  ],
  "claim_ceiling": {
    "licensed": "the integer-replica boundary sewing and, for a declared regular Z_n-symmetric saddle, the local relation between its smooth cover and conical quotient",
    "not_licensed": [
      "a unique noninteger continuation or entropy derivative at n = 1",
      "existence, stability, contour membership, or dominance of the displayed saddle",
      "a replica wormhole or any other topology-changing saddle",
      "a cosmic brane as an independent source in the smooth cover",
      "the Einstein tension formula in a generic higher-derivative theory",
      "an ensemble interpretation or a failure of fixed-theory factorization"
    ]
  },
  "visual_encoding": {
    "numbered_copy_boxes": "the three density-matrix copies in the drawn example",
    "solid_arrowed_connectors": "the cyclic permutation on R; arrowheads carry no causal meaning",
    "double_strokes": "same-copy identifications on the complement",
    "dashed_radial_guides": "group fundamental domains in the smooth cover, not physical cuts",
    "filled_center_point": "transverse intersection with the codimension-two fixed locus Sigma_n",
    "matched_edge_ticks": "identified radial sides of the quotient wedge",
    "solid_conditional_arrow": "quotienting is allowed only after the symmetry condition",
    "dashed_stop_bar": "replica-symmetry-breaking exit",
    "canvas": "explicit white background; no physical distinction is encoded by color alone"
  },
  "accessibility": {
    "reading_order": [
      "panel A three copies and the same-copy complement identifications",
      "panel A cyclic R identifications and exact general equations",
      "panel B smooth full transverse disk and Z_n action",
      "conditional symmetry arrow",
      "panel C quotient wedge, opening angle, deficit, and qualified Einstein source",
      "replica-symmetry-breaking failure branch",
      "integer-only and claim-ceiling footer"
    ],
    "non_color_encoding": "Copies are numbered; cyclic and same-copy identifications use different line structures; group-domain guides are dashed; quotient edges carry matching ticks; the success and failure branches use solid arrowheads and dashed stop bars; every relation is directly labeled.",
    "canvas": "explicit white background for light, dark, monochrome, and print contrast",
    "motion": "none"
  },
  "caption": "For n = 3, the upper bank of R on copy k is identified with the lower bank on copy k + 1, while the complement closes within each copy. If an admitted regular gravitational saddle preserves the resulting Z_n permutation, its smooth transverse disk may be quotiented to a fundamental wedge of opening angle 2 pi/n and deficit 2 pi(1 - 1/n); in Einstein gravity the same quotient can be represented by the stated auxiliary cosmic-brane tension. Replica-symmetry-breaking saddles have no such single-wedge description. Euclidean schematic; not to scale.",
  "alt_text": "Three density-matrix copies cyclically identify the cut region but close its complement within each copy. A regular Z_3-symmetric saddle has a smooth full transverse disk around a codimension-two fixed surface; quotienting gives one identified 2 pi/3 wedge with deficit 4 pi/3. A separate branch shows that a symmetry-breaking saddle cannot be represented by this quotient.",
  "independent_checks": [
    "the displayed R identifications form the single cycle 1 -> 2 -> 3 -> 1 and return to the starting copy after three crossings",
    "the complement uses the identity permutation and closes separately on every copy",
    "g^n = 1 for the transverse action theta -> theta + 2 pi/n",
    "the unquotiented cover has angular period 2 pi and no conical singularity at Sigma_n",
    "the quotient opening angle is 2 pi/n and its deficit is 2 pi(1 - 1/n)",
    "T_n = (n - 1)/(4 n G_N) reproduces the quotient deficit through delta_n = 8 pi G_N T_n in Einstein gravity",
    "n = 2 gives opening pi, deficit pi, and T_2 = 1/(8 G_N)",
    "the formal n = 1 control gives zero deficit and zero tension without licensing an analytic continuation",
    "the boundary branch locus partial R and bulk fixed locus Sigma_n remain separately labeled",
    "no arrow, surface, or label asserts Lorentzian propagation, a replica-wormhole topology, saddle dominance, or ensemble averaging"
  ],
  "sources": [
    {
      "citation": "Lewkowycz and Maldacena, Generalized Gravitational Entropy (2013), sections 2-3",
      "url": "https://doi.org/10.1007/JHEP08(2013)090",
      "role": "regular replica-symmetric cover, quotient geometry, fixed locus, and analytic-continuation qualifications"
    },
    {
      "citation": "Dong, The Gravity Dual of Renyi Entropy (2016)",
      "url": "https://doi.org/10.1038/ncomms12472",
      "role": "Einstein-gravity cosmic-brane tension and quotient deficit relation"
    },
    {
      "citation": "QFT.org, Replica Constructions on Fixed and Semiclassical Backgrounds",
      "url": "/curved-spacetime/generalized-entropy-quantum-extremal-surfaces/replica-constructions-in-gravitational-backgrounds/",
      "role": "site convention for integer replica sewing and the separation between fixed-background and gravitational replicas"
    }
  ]
}