{ "artifact_id": "qft.artifact.holography-quantum-gravity.wormholes-gravitational-path-integrals-and-ensembles.replica-wormhole-saddle-dominance-exchange", "owner_page_id": "qft.topic.black-hole-information.replica-wormholes-saddle-competition", "title": "Replica-Wormhole Fillings and the Purity Crossover", "revision": 3, "created_on": "2026-08-29", "updated_on": "2026-08-30", "creator": "OpenAI Codex, for QFT.org", "original_work": true, "reader_question": "Which gravitational replica connectivities compete in the microcanonical JT plus end-of-the-world-brane model, and how accurately does a dominant endpoint describe the n = 2 crossover?", "takeaway": "Identical replica boundary data admit disconnected, partial, and fully connected gravity fillings; the orbit-summed partial planar class leads at the endpoint crossing, while the exact annealed Haar quantity smooths the n = 2 endpoint minimum by almost log 2.", "scope": { "panel_A": "Qualitative Euclidean topology and permutation schematic for the n = 3 planar microcanonical PSSY bookkeeping.", "panel_B": "Quantitative exact annealed second Renyi quantity -log E P_2 for a sample-normalized Haar-random bipartite pure state with d_B = 64 and integer k from 1 through 4096; the quenched average E[-log P_2] is not computed.", "schematic_components": [ "the embedding and shapes of the dynamical-gravity regions in panel A", "the positions and sizes of the external replica tabs" ], "quantitative_components": [ "the n = 3 planar weights", "the n = 2 exact annealed Haar curve and endpoint exponents", "the crossing coordinate and entropy gap" ], "not_to_scale": true, "horizontal_axis_is_time": false }, "conventions": { "logarithm": "natural logarithm", "k": "dimension of the nongravitating reference system R", "d_B": "effective dimension exp(S_micro) of the declared microcanonical black-hole window; an integer in the Haar checkpoint", "tau": "the cyclic replica permutation (1 2 ... n)", "disconnected": "disconnected components of the dynamical-gravity filling; the nongravitating trace boundary data remain cyclically sewn", "replica_wormhole": "a Euclidean dynamical-gravity filling connecting distinct replicas; not a Lorentzian communication channel" }, "panels": [ { "id": "A", "title": "Same n = 3 boundary problem, several gravity fillings", "boundary_data": { "symbol": "B_3(R)", "fixed_across_classes": true, "description": "three copies with the same cyclic sewing on R and the same nongravitating boundary preparation", "warning": "The boundary sewing does not select a dynamical-gravity connectivity." }, "model": "microcanonical planar Penington-Shenker-Stanford-Yang JT plus EOW-brane model", "classes": [ { "id": "disconnected", "representative_permutation": "identity", "gravity_components": 3, "planar_multiplicity": 1, "moment_contribution": "1/k^2", "crossing_weight_for_k_equal_d_B": 1 }, { "id": "partial", "representative_connectivity": "2 + 1", "gravity_components": 2, "planar_multiplicity": 3, "moment_contribution": "3/(k d_B)", "crossing_weight_for_k_equal_d_B": 3, "qualification": "the drawing shows one representative; the coefficient three includes all noncrossing transposition placements" }, { "id": "fully-connected", "representative_permutation": "tau inverse", "gravity_components": 1, "planar_multiplicity": 1, "moment_contribution": "1/d_B^2", "crossing_weight_for_k_equal_d_B": 1, "topology_label": "replica-wormhole filling" } ], "mean_normalized_planar_relation": "P_hat_3,planar = 1/k^2 + 3/(k d_B) + 1/d_B^2", "crossing_relation": "at k = d_B, disconnected : partial : fully connected = 1 : 3 : 1", "scientific_point": "The orbit-summed partial planar class is larger than either endpoint at the endpoint crossing because it contains three transpositions; each individual transposition has the same weight as either endpoint there, so a two-endpoint approximation is not uniform." }, { "id": "B", "title": "Exact annealed n = 2 normalization check", "ensemble": "Haar-random normalized pure state on C^k tensor C^d_B", "frozen_parameters": { "d_B": 64, "k_min": 1, "k_max": 4096, "x_coordinate": "x = log k", "entropy_unit": "nats" }, "equations": { "exact_average_purity": "E P_2 = (k + d_B)/(k d_B + 1)", "exact_annealed_second_renyi": "S_2,ann = -log E P_2 = log(k d_B + 1) - log(k + d_B)", "quenched_average_second_renyi": "E[-log P_2] is not computed and is not equal to S_2,ann at finite nontrivial dimensions", "identity_contraction": "d_B/(k d_B + 1)", "swap_contraction": "k/(k d_B + 1)", "disconnected_endpoint_exponent": "Gamma_disc = log k", "connected_endpoint_exponent": "Gamma_wh = log d_B", "crossing": "k = d_B", "crossing_x": "log d_B", "dominant_endpoint_error": "delta_2,ann = log(1 + m/M) - log(1 + 1/(m M)), with m = min(k,d_B) and M = max(k,d_B)", "crossing_error": "delta_2,ann = log 2 - log(1 + d_B^(-2))" }, "crossing_values": { "k": 64, "x_log_k": 4.158883083, "endpoint_exponent": 4.158883083, "exact_S_2_ann": 3.465980014, "gap": 0.69290307 }, "samples": [ { "k": 1, "log_k": 0.0, "exact_S_2_ann": 0.0 }, { "k": 2, "log_k": 0.693147181, "exact_S_2_ann": 0.670157662 }, { "k": 4, "log_k": 1.386294361, "exact_S_2_ann": 1.32956838 }, { "k": 8, "log_k": 2.079441542, "exact_S_2_ann": 1.963609726 }, { "k": 16, "log_k": 2.772588722, "exact_S_2_ann": 2.550421257 }, { "k": 32, "log_k": 3.465735903, "exact_S_2_ann": 3.060758957 }, { "k": 64, "log_k": 4.158883083, "exact_S_2_ann": 3.465980014 }, { "k": 128, "log_k": 4.852030264, "exact_S_2_ann": 3.753540038 }, { "k": 256, "log_k": 5.545177444, "exact_S_2_ann": 3.935800565 }, { "k": 512, "log_k": 6.238324625, "exact_S_2_ann": 4.041130565 }, { "k": 1024, "log_k": 6.931471806, "exact_S_2_ann": 4.09827372 }, { "k": 2048, "log_k": 7.624618986, "exact_S_2_ann": 4.128119054 }, { "k": 4096, "log_k": 8.317766167, "exact_S_2_ann": 4.143382712 } ], "scientific_point": "The exact finite-dimensional annealed quantity is a smooth log-sum. It lies below the smaller endpoint exponent and differs from it by nearly log 2 at the crossing; it is not the quenched average second Renyi entropy." } ], "model_relations": { "microcanonical_planar_moment": "P_hat_n,planar = [E Tr rho_hat_R^n]_leading planar = k^(1-n) sum from j=1 to n of N(n,j) (k/d_B)^(j-1)", "narayana_number": "N(n,j) = binomial(n,j) binomial(n,j-1)/n", "endpoint_terms": [ "P_hat_n,planar disconnected = k^(1-n)", "P_hat_n,planar fully connected = d_B^(1-n)" ], "crossing_catalan_sum": "at k = d_B, P_hat_n,planar = C_n k^(1-n)", "large_dimension_match": "the exact Haar identity and swap contractions approach 1/k and 1/d_B, matching the two PSSY endpoint scalings without assigning bulk topology to the Haar contractions" }, "assumptions": [ "the same integer-replica boundary data are used for every displayed filling", "the declared topology policy admits the displayed planar gravity connectivities", "the microcanonical window is narrow enough that the window error is small", "k and d_B are large for the planar PSSY moment formula", "nonplanar handles and other omitted topologies are below the stated error budget", "the PSSY matrix-integral completion supplies the retained topology weights; no independent unique Euclidean thimble decomposition is asserted", "all compared actions use the same normalization and renormalization scheme", "the Haar plot is used as a finite-dimensional annealed random-state checkpoint rather than an exact generic gravity dual", "the quenched average E[-log P_2] is outside the plotted observable and is not inferred from the mean purity" ], "failure_cases": [ { "intervention": "exclude connected topologies", "outcome": "the fully connected term is absent and no replica-wormhole endpoint crossing exists" }, { "intervention": "set the connected coefficient to zero in a generic saddle model", "outcome": "the geometric connected solution contributes no term on that hypothetical contour; this is not an alternate PSSY contour derived from the matrix integral" }, { "intervention": "drop partial n = 3 connectivities at k = d_B", "outcome": "reject; the omitted class has weight three while either retained endpoint has weight one" }, { "intervention": "identify d_B with exp(S_0) without the microcanonical density and window", "outcome": "reject; the state-count and topology-control parameters are distinct" }, { "intervention": "interpret the horizontal coordinate as evaporation time", "outcome": "reject; it is log k in a fixed-parameter laboratory" }, { "intervention": "present the endpoint intersection as an exact sharp entropy transition", "outcome": "reject; the exact finite-dimensional annealed log-sum is smooth and differs by nearly log 2 at the crossing" }, { "intervention": "identify -log E P_2 with E[-log P_2]", "outcome": "reject; the former is annealed, the latter is quenched, and they differ whenever the purity fluctuates" }, { "intervention": "infer a fixed-theory unitary S-matrix", "outcome": "reject; neither panel contains microscopic time evolution or a fixed-theory amplitude" } ], "visual_encoding": { "repeated_numbered_tabs": "identical nongravitating replica boundary data in all panel-A cards", "separate_pale_regions": "disconnected dynamical-gravity components", "dashed_partial_region": "one representative partially connected filling; the text supplies its multiplicity", "continuous_pale_region": "fully connected replica-wormhole filling", "solid_heavy_curve": "exact annealed Haar quantity -log E P_2", "long_dashed_line": "disconnected endpoint exponent log k", "short_dashed_line": "connected endpoint exponent log d_B", "vertical_dashed_guide": "endpoint crossing k = d_B", "double_arrow": "exact annealed gap at the crossing", "canvas": "explicit white background; no scientific distinction is encoded by color alone" }, "accessibility": { "reading_order": [ "panel A common boundary-data banner", "panel A disconnected, partial, and fully connected cards from left to right", "panel A 1 to 3 to 1 crossing-weight summary", "panel B exact annealed equation, explicit not-quenched qualification, and frozen d_B value", "panel B axes and endpoint lines", "panel B exact annealed curve and crossing gap", "claim-ceiling footer" ], "non_color_encoding": "Connectivity is expressed by the number and continuity of outlined regions plus direct labels. Plot curves differ by direct labels, line weight, and dash pattern. The crossing and gap have separate guide and arrow marks.", "canvas": "explicit white background for light, dark, monochrome, and print contrast", "motion": "none" }, "claim_ceiling": { "licensed": [ "the n = 3 orbit-summed planar connectivity classes and their PSSY microcanonical weights", "the exact Haar-averaged n = 2 mean purity and its annealed logarithm for the frozen finite-dimensional fixture", "the common endpoint crossing k = d_B", "the failure of a two-endpoint approximation to be uniform at the crossover" ], "not_licensed": [ "topology selection from boundary sewing alone", "a unique gravitational contour or nonzero thimble coefficient", "a universal finite-n to n = 1 analytic continuation", "the quenched average second Renyi entropy E[-log P_2]", "an exact finite-G_N gravity entropy curve", "a time-dependent Page curve or microscopic evaporation dynamics", "decoding, a unitary S-matrix, or factorization in one fixed theory" ] }, "caption": "The same n = 3 cyclic boundary data admit disconnected, partially connected, and fully connected dynamical-gravity fillings. In the microcanonical planar model their orbit-summed contributions at k = d_B occur in the ratio 1:3:1; the middle class contains three transpositions, so it cannot be discarded near the endpoint crossing. The lower panel fixes d_B = 64 and compares the exact annealed Haar quantity S_2,ann = -log E P_2 with the endpoint exponents log k and log d_B; the exact annealed sum lies below their lower envelope by almost log 2 at k = d_B. It does not plot the quenched average E[-log P_2]. The horizontal axis is log k, not time. Euclidean topology schematic and quantitative finite-dimensional checkpoint; no microscopic-unitarity or fixed-theory claim.", "alt_text": "The same three-replica boundary data admit disconnected, partially connected, and fully connected gravitational fillings, whose orbit-summed crossing weights are one to three to one. A second panel plots the exact annealed Haar second Renyi quantity below the two endpoint exponents, with a nearly log-two gap at their crossing; the quenched average is not shown.", "adjacent_semantic_table": [ "one fixed cyclic boundary problem B_3(R) does not select a gravity topology", "the disconnected, partial, and fully connected planar weights are k^-2, 3/(k d_B), and d_B^-2", "the exact Haar checkpoint uses S_2,ann = -log E P_2 = log(k d_B + 1) - log(k + d_B) at d_B = 64", "the quenched average E[-log P_2] is not computed", "the crossing gap is log 2 - log(1 + d_B^-2)", "neither panel claims Lorentzian communication, microscopic unitarity, or fixed-theory factorization" ], "independent_checks": [ "all three panel-A cards repeat the same three exterior replica tabs", "only the connectivity of the dynamical-gravity region changes between panel-A cards", "N(3,1), N(3,2), and N(3,3) equal 1, 3, and 1", "at k = d_B the n = 3 planar class weights reduce to 1:3:1", "the exact Haar identity and swap contractions sum to (k + d_B)/(k d_B + 1) and have ratio k/d_B", "for d_B = k = 64 the endpoint exponent is 4.158883083 and exact S_2,ann is 3.465980014", "the plotted crossing gap 0.692903070 equals log 2 - log(1 + 64^-2) to the displayed precision", "the exact annealed curve begins at S_2,ann = 0 for k = 1 and approaches log 64 from below", "the horizontal coordinate is log k and no label denotes time" ], "sources": [ { "citation": "Penington, Shenker, Stanford, and Yang, Replica Wormholes and the Black Hole Interior, sections 2.1-2.5 and appendices D-E (2022)", "url": "https://doi.org/10.1007/JHEP03(2022)205", "role": "JT plus EOW-brane state, purity topologies, microcanonical planar resolvent, noncrossing permutation sum, and ensemble interpretation" }, { "citation": "Lubkin, Entropy of an n-System from Its Correlation with a k-Reservoir (1978)", "url": "https://doi.org/10.1063/1.523763", "role": "finite-dimensional Haar purity checkpoint" }, { "citation": "Zyczkowski and Sommers, Induced Measures in the Space of Mixed Quantum States, equation 44 (2001)", "url": "https://doi.org/10.1088/0305-4470/34/35/335", "role": "exact finite-dimensional Haar purity formula" }, { "citation": "Page, Average Entropy of a Subsystem (1993)", "url": "https://doi.org/10.1103/PhysRevLett.71.1291", "role": "random-subsystem entropy context" }, { "citation": "de Boer, Hollander, and Rolph, Page Curves and Replica Wormholes from Random Dynamics, section 1 (2024)", "url": "https://doi.org/10.1007/JHEP07(2024)023", "role": "annealed-versus-quenched entropy distinction in a replica-wormhole random-dynamics setting" }, { "citation": "QFT.org, Semiclassical Gravitational Replicas", "url": "/holography-quantum-gravity/wormholes-gravitational-path-integrals-and-ensembles/semiclassical-gravitational-replicas/", "role": "site convention separating boundary sewing, gravity fillings, replica symmetry, and continuation" } ] }