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the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "ChenYang", "url": "https://arxiv.org/pdf/1503.02547v4", "version": "arXiv:1503.02547v4, 2018-07-09", "pins": [ "p.2, Conjectures 1.1-1.2", "Sections 2.2-2.3, quantum 6j identities and state-sum definition" ], "checked": "Odd levels, specified root, 2pi/r versus 4pi/r absolute-growth factors, and the schematic sum topology.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "TuraevVirelizier", "url": "https://arxiv.org/pdf/1006.3501v5", "version": "arXiv:1006.3501v5, 2013-06-24", "pins": [ "p.2, modular and unitary specializations of the center formula", "p.34, Theorems 11.1-11.2" ], "checked": "Spherical fusion category of nonzero dimension; state sum equals RT of its center. Modulus square needs matching unitary modular data.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "MurakamiMurakami", "url": "https://arxiv.org/pdf/math/9905075v2", "version": "arXiv:math/9905075v2, archive header 1999-06-10", "pins": [ "p.1, reduced normalization explanation", "p.15, Theorem 4.9", "Section 5, volume-conjecture formulation" ], "checked": "The normalized colored-Jones specialization agrees with the normalized Kashaev invariant for N>=2. This is distinct from proving a general volume conjecture.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "KashaevVolume", "url": "https://arxiv.org/pdf/q-alg/9601025v2", "version": "arXiv:q-alg/9601025v2, 1996-03-25", "pins": [ "p.3, equations (2.2) and (2.5)" ], "checked": "Figure-eight product sum. The original phase ambiguity noted after (2.5) is removed by the reduced Jones convention and the later Murakami-Murakami identification.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "MurakamiIntro", "url": "https://arxiv.org/pdf/1002.0126v1", "version": "arXiv:1002.0126v1, archive header 2010-01-31", "pins": [ "Section 2, braid/quantum-trace construction", "pp.18-22, Section 3.2, equations (3.1) onward" ], "checked": "Figure-eight positive-product argument, credit to Ekholm, Lobachevsky normalization, and two regular ideal tetrahedra.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "FaddeevKashaev", "url": "https://arxiv.org/pdf/hep-th/9310070v1", "version": "arXiv:hep-th/9310070v1, 1993-10-13", "pins": [ "Section 1, equations (1.1)-(1.5)", "Sections 2-3, quantum pentagon and classical limit" ], "checked": "Quantum pentagon has a classical Rogers five-term limit. The chapter proves its own elementary q-Pochhammer expansion separately.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "DimofteGaroufalidis", "url": "https://arxiv.org/pdf/1202.6268v2", "version": "arXiv:1202.6268v2, 2012-10-25", "pins": [ "p.6, Conjecture 1.8", "pp.30-32, Section 5.1, equations (5.5)-(5.8)", "pp.32-33, Sections 5.2-5.3" ], "checked": "Product shift and branched leading dilogarithm; logarithmic gluing equation; gluing one-loop torsion is conjecturally adjoint torsion in this source. The chapter does not import a general global state-integral asymptotic.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "WittenContinuation", "url": "https://arxiv.org/pdf/1001.2933v4", "version": "arXiv:1001.2933v4, 2010-08-28", "pins": [ "pp.8-12, Sections 2.1-2.3, especially (2.15)-(2.16)", "Sections 2.5-2.6, continued cycle and compact/complex gauge groups" ], "checked": "Relative-cycle/thimble coefficients, Stokes changes, and need to choose a cycle. This is framework material, not a rigorous existence theorem for every infinite-dimensional path integral.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "GukovApoly", "url": "https://arxiv.org/pdf/hep-th/0306165v1", "version": "arXiv:hep-th/0306165v1, 2003-06-17", "pins": [ "Section 1.1, complex Chern-Simons action", "Section 3, torus boundary and A-polynomial discussion" ], "checked": "Background complex-flat-connection and boundary-quantization interpretation; exact phase conventions are defined in the chapter and derived in Section16.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "DimofteQuantumRS", "url": "https://arxiv.org/pdf/1102.4847v3", "version": "arXiv:1102.4847v3, 2011-08-22", "pins": [ "Section 2.1, compact and analytically continued action", "Sections 2.2-2.4, equation (2.17) and logarithmic quantization", "Section 6.2, wavefunctions and changes of polarization" ], "checked": "A-polynomial as classical boundary relation and its quantization. The chapter does not identify every formal operator with a colored-Jones recursion.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "WongYangDehn", "url": "https://ems.press/content/serial-article-files/51093", "version": "Quantum Topology 16 (2025), 419-491; DOI10.4171/QT/240", "pins": [ "pp.420-421, Theorems 1.1-1.2", "p.482, RT/TV scalar conversion in proof of Theorem1.2" ], "checked": "All closed oriented hyperbolic rational figure-eight fillings, inverse-square-root adjoint torsion in RT, and the separate absolute-growth limits.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "KRV", "url": "https://arxiv.org/pdf/1707.08785v3", "version": "arXiv:1707.08785v3, 2019-08-30", "pins": [ "pp.7-9, equations (2.11)-(2.15)", "p.20, equation (6.1) and Theorem6.1" ], "checked": "First moment/reference metric and unit-volume continuation in a neighborhood of strict extended Seiberg bounds. No small-b uniform bound is imported.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "GKRVBootstrap", "url": "https://arxiv.org/pdf/2005.11530v2", "version": "arXiv:2005.11530v2", "pins": [ "p.7, equation (1.16) and Theorem1.1", "p.8, paragraph explaining extra discrete terms beyond the pair-sum domain" ], "checked": "Initial bootstrap charges are twice the chapter physics charges; pairing (0,z) corresponds to chapter labels (1,3). The theorem does not cover the screening locus by direct substitution.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "TeschnerRevisited", "url": "https://arxiv.org/html/hep-th/0104158v3", "version": "arXiv:hep-th/0104158v3, 2001-11-09", "pins": [ "Sections 2 and 6, conformal weights and Virasoro modules", "Sections 3.4 and 7.2, screening and spectral pole continuation", "Section18.1, screened chiral operators" ], "checked": "Virasoro/block background and screening interpretation. Infinite spectral continuation is retained as the separate explicit input of Section13.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "TeschnerVartanov", "url": "https://arxiv.org/pdf/1202.4698v3", "version": "arXiv:1202.4698v3, 2013-04-30", "pins": [ "pp.1-4, modular double and principal labels", "pp.10-11, equations (4.5)-(4.6),(4.10)-(4.11)", "p.23, equations (A.13)-(A.17) and double-Gamma quotient" ], "checked": "Continuous fusion normalization, principal spectrum, sine shifts and reflection. The reviewed Section15 supplies its own uniform strip estimate, not an unqualified use of the source fixed-argument asymptotic.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "Costantino", "url": "https://arxiv.org/pdf/math/0611399v4", "version": "arXiv:math/0611399v4, 2007-10-30", "pins": [ "p.4, Definition2.4 (D-block)", "pp.7-8, Definitions3.1-3.2", "p.10, Theorem3.9 and equation (4)" ], "checked": "Hyperbolic-type label regime, pole-removed evaluation ev_n([n]{6j}), and Vol(D(u)) for the doubled block. This is separated from Chen-Murakami actual finite values at q=xi_r^2.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "ChenMurakami", "url": "https://arxiv.org/pdf/1706.04887v3", "version": "arXiv:1706.04887v3, 2021-04-02", "pins": [ "p.1, Theorem1 and equations (1)-(2)", "pp.1-2, Theorem2 and equation (3)", "p.3, Racah-Wigner normalization and q^(1/2) quantum integer" ], "checked": "Theorem1 is the general leading logarithmic rate with convergence, admissibility and selected vertex-sum condition. Theorem2 is the refined sqrt(2)*pi prefactor only for odd numerator/denominator eta and exact shifted integer labels. Official arXiv HTML independently disambiguates sqrt(2)*pi.", "scope_limit": "The two theorem statements and their normalizations were inspected. Their full analytic proofs and any arbitrary-shape quantum limit were not verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "Roberts", "url": "https://arxiv.org/pdf/math-ph/9812013", "version": "Geometry & Topology 3 (1999), 21-66", "pins": [ "pp.22-23, Theorem1 and exterior dihedral convention" ], "checked": "Classical admissible large-label asymptotic for nondegenerate Euclidean tetrahedra, oscillatory Regge phase and volume amplitude. No passage to a shrinking hyperbolic root regime is inferred.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "PortiWeiss", "url": "https://arxiv.org/pdf/math/0510432v2", "version": "arXiv:math/0510432v2, 2006-11-14", "pins": [ "p.1, Definition1.2 and Theorem1.3" ], "checked": "Closed orientable Euclidean cone three-manifold, input cone angles <=pi, not almost product. Generic doubled convex polyhedra need not satisfy these hypotheses.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "RivinDegeneration", "url": "https://arxiv.org/pdf/math/0512065v1", "version": "arXiv:math/0512065v1, archive header 2005-12-02; published Geom. Dedicata131 (2008),73-85", "pins": [ "pp.2-3, continuity/vanishing discussion and Theorem1", "p.4, Schlaefli derivative estimate and Corollary1" ], "checked": "Continuity and control toward degeneracy; logarithmic derivative growth does not imply an analytic power series at every boundary point.", "scope_limit": "The statement, formula and stated hypotheses were inspected; the complete external proof was not independently verified.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "HodgsonThesis", "url": "https://researchers.ms.unimelb.edu.au/~craigdh@unimelb/public.html", "version": "Author publication list accessed2026-11-01", "pins": [ "Publication list entry for the 1986 Princeton thesis" ], "checked": "Author, title, institution, year and vii+169-page metadata only.", "scope_limit": "No thesis theorem or proof is cited or claimed inspected.", "access": "Primary material retrieved with the web tool on 2026-11-01; no PDF-byte hash claimed." }, { "key": "WittenJones", "url": "https://doi.org/10.1007/BF01217730", "version": "Comm. Math. Phys.121(1989),351-399", "pins": [ "Bibliography and original formal Chern-Simons/Jones attribution" ], "checked": "Historical attribution retained. Direct Project Euclid PDF access met an interstitial; no full-text inspection is claimed. The finite first-variation calculation is derived locally, and action conventions are also checked in Gukov and Dimofte primary papers.", "scope_limit": "No theorem from the inaccessible article is used as an unverified analytic step.", "access": "Primary PDF access unsuccessful; mathematical dependence uses locally derived identities and separately accessed primary descriptions." } ], "claims": [ { "id": "s10.recoupling", "labels": [ "sec:quantumintro", "sec:introrecoupling", "eq:introrecoupling" ], "statement": "Two multiplicity-free sl2 coupling bases have six labels and the four specified admissible triples.", "status": "external_statement_checked", "source_keys": [ "TaylorWoodward" ], "hypotheses": [ "Chosen trivalent bases; nonnegative half-integer spins." ], "dependencies": [], "derivation": "The two decompositions share the same invariant multiplicity space; dimension/sign factors are part of normalization.", "scope_limit": "Label tetrahedron is initially combinatorial, not a metric realization.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.quantum_integer", "labels": [ "eq:introqinteger" ], "statement": "The symmetric quantum integer tends to n and the compact fusion cutoff/admissibility conditions are stated in the same algebraic parameter.", "status": "external_statement_checked", "source_keys": [ "TaylorWoodward" ], "hypotheses": [ "q_alg=exp(pi*i/r), r>2 for the finite semisimple quotient." ], "dependencies": [], "derivation": "The limit follows from the quotient of first derivatives at q_alg=1.", "scope_limit": "", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.state_sum", "labels": [ "eq:introstatesum" ], "statement": "Pentagon, orthogonality and normalization identities underpin a tetrahedral state sum.", "status": "external_statement_checked", "source_keys": [ "ChenYang", "TuraevVirelizier" ], "hypotheses": [ "Prescribed spherical fusion data, triangulation and matching normalization." ], "dependencies": [], "derivation": "", "scope_limit": "Displayed sum is schematic and does not define a measure on hyperbolic metrics.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.invariant_dictionary", "labels": [ "sec:introinvariants" ], "statement": "Colored Jones, Kashaev, RT, TV, continuous recoupling and Liouville correlation have different inputs and normalization requirements.", "status": "external_statement_checked", "source_keys": [ "MurakamiIntro", "MurakamiMurakami", "ChenYang", "TeschnerVartanov", "KRV" ], "hypotheses": [], "dependencies": [], "derivation": "", "scope_limit": "A local 6j weight is not a manifold invariant; analytic kernels require their specified domains.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.jones_kashaev", "labels": [], "statement": "Form the reduced colored Jones polynomial at generic q, then evaluate at exp(2pi*i/N) to obtain the normalized Kashaev invariant.", "status": "external_statement_checked", "source_keys": [ "MurakamiMurakami" ], "hypotheses": [ "N>=2; unknot value1." ], "dependencies": [], "derivation": "", "scope_limit": "The top color is outside the retained compact level-N-2 category because its quantum dimension vanishes.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.knot_volume", "labels": [ "eq:introknotvolume" ], "statement": "The hyperbolic-knot volume conjecture uses changing color and root and ordinary three-dimensional volume.", "status": "external_conjecture", "source_keys": [ "MurakamiMurakami" ], "hypotheses": [ "Reduced Jones normalization; hyperbolic knot in S3." ], "dependencies": [], "derivation": "", "scope_limit": "No general proof is asserted; the figure-eight proof below is a special case.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.closed_growth", "labels": [ "eq:introclosedvolume" ], "statement": "Chen-Yang absolute-growth limits have factors4pi/r and2pi/r; Wong-Yang prove the stated hyperbolic figure-eight filling cases.", "status": "external_statement_checked", "source_keys": [ "ChenYang", "WongYangDehn" ], "hypotheses": [ "r odd, root exp(2pi*i/r); closed oriented hyperbolic manifold; source-specific RT/TV convention." ], "dependencies": [], "derivation": "", "scope_limit": "General statement remains designated a conjecture; source theorem proof is not independently re-audited.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.center_relation", "labels": [], "statement": "TV of a spherical fusion category is RT of its center; the modulus-square specialization requires matching unitary modular data.", "status": "external_statement_checked", "source_keys": [ "TuraevVirelizier" ], "hypotheses": [ "Nonzero categorical dimension; closed oriented manifold." ], "dependencies": [], "derivation": "", "scope_limit": "No automatic equality between differently normalized SU2 and SO3 invariants is used.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.root_color_scope", "labels": [], "statement": "q_J=q_alg^2 is a common conversion, while [N] at q_alg=exp(pi*i/N) is zero and spin(N-1)/2 is excluded.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [], "dependencies": [ "s10.quantum_integer", "s10.jones_kashaev" ], "derivation": "Direct sine evaluation and comparison with the retained maximum spin(N-2)/2.", "scope_limit": "Identical numerical roots after notation conversion do not identify observable categories.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.figure_product", "labels": [ "sec:introfigureeight", "eq:introfigureproduct" ], "statement": "The reduced figure-eight root evaluation is the positive product sum.", "status": "external_statement_checked", "source_keys": [ "KashaevVolume", "MurakamiIntro", "MurakamiMurakami" ], "hypotheses": [ "N>=2; the phase convention is fixed by reduced Jones." ], "dependencies": [], "derivation": "", "scope_limit": "", "computational_coverage": { "status": "finite_fixture_support", "implemented_result_groups": [ "Q1_figure_eight" ], "result_pointers": [ "/groups/Q1_figure_eight" ], "evidence_reference": "computational_companion.reviewed_author_result", "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.figure_maximum", "labels": [], "statement": "m=floor(5N/6), with possible adjacent tie, is a global product maximum and P_m<=J_N<=N*P_m.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [], "dependencies": [ "s10.figure_product" ], "derivation": "Successive ratio is4sin^2(pi*k/N); P_(N-1)=N^2 follows by evaluating (z^N-1)/(z-1) at1, so the final local maximum exceeds P_0=1.", "scope_limit": "", "computational_coverage": { "status": "finite_fixture_support", "implemented_result_groups": [ "Q1_figure_eight" ], "result_pointers": [ "/groups/Q1_figure_eight" ], "evidence_reference": "computational_companion.reviewed_author_result", "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.figure_integral", "labels": [ "eq:introfigureintegral" ], "statement": "The logarithmic product converges to the integral of log(4sin^2(pi*x)) from0 to5/6.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [], "dependencies": [ "s10.figure_maximum" ], "derivation": "Separate2log x near0; continuous remainder uses Riemann sums and log part uses Stirling. The log N upper/lower gap vanishes after division by N.", "scope_limit": "", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.figure_volume", "labels": [ "eq:introfigurevolume" ], "statement": "The limit is4Lambda(pi/6)=6Lambda(pi/3)=2v3, the figure-eight complement volume.", "status": "external_statement_checked", "source_keys": [ "MurakamiIntro" ], "hypotheses": [], "dependencies": [ "s10.figure_integral" ], "derivation": "Reflection and the duplication identity for Lambda give the constants; the complete complement decomposes into two regular ideal tetrahedra.", "scope_limit": "The decimal is a rounded illustrative value, not a certified volume enclosure.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.qpochhammer", "labels": [ "sec:introdilog", "eq:introqpochhammer" ], "statement": "The compact-disk q-Pochhammer logarithm equals -Li2(z)/epsilon+log(1-z)/2+O(epsilon).", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [ "epsilon>0; |z|<=rho<1; logarithm continued from0." ], "dependencies": [], "derivation": "Absolute convergence permits expanding log; the scalar remainder (1-e^-t)^-1-t^-1-1/2 is bounded by C*t for all t>0, giving an error bounded by C*epsilon*sum rho^m.", "scope_limit": "No unit-circle product convergence or root-of-unity substitution is claimed.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.quantum_pentagon", "labels": [], "statement": "The cited quantum pentagon has the classical Rogers five-term limit.", "status": "external_statement_checked", "source_keys": [ "FaddeevKashaev" ], "hypotheses": [], "dependencies": [], "derivation": "", "scope_limit": "An explanatory external identity, not a theorem identifying all quantum dilogarithm conventions.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.bloch_wigner", "labels": [ "eq:introBlochWigner" ], "statement": "D(z) gives the oriented upper-half-plane ideal-tetrahedron volume and dLi2=-log(1-z)dlog z.", "status": "derived_from_reviewed_dependencies", "source_keys": [], "hypotheses": [ "Im z>0; compatible branches." ], "dependencies": [ "book/chapters/03-ideal-tetrahedron.tex" ], "derivation": "", "scope_limit": "Boundary logarithms are required; an arbitrary raw dilogarithm imaginary part is not the geometric volume.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.quantum_shape", "labels": [ "eq:introquantumshape" ], "statement": "The tetrahedral shift equation and its leading phase produce the companion shape1-z^-1.", "status": "external_statement_checked", "source_keys": [ "DimofteGaroufalidis" ], "hypotheses": [ "Fixed branch away from limiting pole cuts; q=exp(hbar)." ], "dependencies": [], "derivation": "In the source product, the shift Z->Z+hbar extracts1-e^-Z; differentiate the leading dilogarithm.", "scope_limit": "Only local formal/special-function correspondence; no global quantum invariant or unique WKB solution is inferred.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.state_integral_gluing", "labels": [], "statement": "A specified tetrahedral state-integral phase differentiates to logarithmic gluing equations.", "status": "formal_framework", "source_keys": [ "DimofteGaroufalidis" ], "hypotheses": [ "Quadratic gluing term, logarithm lifts and contour are supplied." ], "dependencies": [], "derivation": "", "scope_limit": "No existence or convergence theorem for all such integrals is asserted.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.stationary_phase", "labels": [ "sec:introsemiclassical", "eq:introsaddleintegral", "eq:introgaussian" ], "statement": "A localized nondegenerate steepest-descent integral has the displayed Gaussian expansion.", "status": "conditional_analytic_statement", "source_keys": [], "hypotheses": [ "Holomorphic phase near critical point; classical-symbol amplitude with derivative bounds and nonzero a0; fixed hbar ray; oriented real m-dimensional descent contour; local strict quadratic maximum." ], "dependencies": [], "derivation": "Localize, rescale by sqrt(hbar), expand, and integrate Gaussian moments. Contour/ray fixes square roots.", "scope_limit": "Extension to the full contour needs complement bounds; this statement does not include zero modes.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.loop_dictionary", "labels": [], "statement": "Tree, one-loop and higher-loop terms are phase, Gaussian/amplitude and higher contractions; logarithmic powers depend on normalization.", "status": "formal_framework", "source_keys": [], "hypotheses": [], "dependencies": [ "s10.stationary_phase" ], "derivation": "", "scope_limit": "Zero modes require a separate reduction. Classical area or inverse-slope expansions are not quantum loop expansions.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.torsion_scope", "labels": [], "statement": "Adjoint cellular cohomology and chosen peripheral/cohomology bases govern torsion; a bare Hessian is insufficient.", "status": "external_statement_checked", "source_keys": [ "DimofteGaroufalidis", "WongYangDehn" ], "hypotheses": [ "Specified cellular and cohomology bases; regular/acyclic domain when used in surgery." ], "dependencies": [ "book/claims/section16.json" ], "derivation": "", "scope_limit": "Conjecture1.8 in Dimofte-Garoufalidis is not cited as a general torsion theorem; the separate figure-eight asymptotic is an external theorem.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.thimbles", "labels": [], "statement": "Relative-cycle coefficients select saddle contributions and change at Stokes loci.", "status": "external_statement_checked", "source_keys": [ "WittenContinuation" ], "hypotheses": [ "Suitable analytic Morse/relative-homology setup." ], "dependencies": [], "derivation": "", "scope_limit": "No universal decomposition or noncancellation theorem is assumed for every quantum state integral.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.fold_scale", "labels": [ "fig:mono-airy" ], "statement": "Cubic balance gives x~hbar^(1/3), t~hbar^(2/3); the regular-family theorem has h=2pi*b^2 and volume-action parameter h/2.", "status": "derived_from_reviewed_dependencies", "source_keys": [], "hypotheses": [], "dependencies": [ "book/claims/section15.json", "book/claims/sections12-14.json" ], "derivation": "Substitute x=hbar^(1/3)y,t=hbar^(2/3)s; W~A^(3/2) gives the same scale.", "scope_limit": "Figure is a schematic real cubic, not a quantum contour; Section15 is restricted to its cleared integral and bounded real transition parameters.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.cs_functional", "labels": [ "sec:introCS", "eq:introCSfunctional" ], "statement": "The displayed Chern-Simons action has variation(2pi)^-1 integral tr(delta A wedge F) on a closed manifold.", "status": "derived_in_chapter", "source_keys": [ "GukovApoly", "DimofteQuantumRS" ], "hypotheses": [ "Chosen trace normalization and trivialization; closed M for the displayed variation." ], "dependencies": [], "derivation": "Vary A wedge dA, integrate by parts with no boundary, and use cyclicity of trace for the cubic term.", "scope_limit": "Gauge periods and functional-integral existence are not fixed by this elementary variation.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.complex_cs", "labels": [], "statement": "Hyperbolic holonomy determines a complex flat connection; analytic continuation requires a cycle and an action convention.", "status": "formal_framework", "source_keys": [ "GukovApoly", "WittenContinuation" ], "hypotheses": [], "dependencies": [], "derivation": "", "scope_limit": "The formal infinite-dimensional integral is motivation, not an ordinary convergent integral or an automatic proof of RT equality.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.boundary_pairing", "labels": [], "statement": "TQFT gluing pairs boundary states; orientation reversal supplies the dual, while analytic models need convergence.", "status": "external_statement_checked", "source_keys": [ "TuraevVirelizier", "DimofteQuantumRS" ], "hypotheses": [], "dependencies": [], "derivation": "", "scope_limit": "Finite categorical pairing and continuous analytic pairing are different constructions.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.nz_polarization", "labels": [ "eq:introNZpolarization", "eq:introWKBstate" ], "statement": "Multiplier logs give Omega=dv wedge du/2 and Phi prime=v/2 in the specified WKB convention.", "status": "derived_from_reviewed_dependencies", "source_keys": [], "hypotheses": [ "Local one-cusp geometric branch; smooth boundary chart." ], "dependencies": [ "book/claims/section16.json" ], "derivation": "", "scope_limit": "WKB existence is conditional; the symplectic and primitive identities are classical.", "computational_coverage": { "status": "finite_fixture_support", "implemented_result_groups": [ "Q3_peripheral_filling" ], "result_pointers": [ "/groups/Q3_peripheral_filling" ], "evidence_reference": "computational_companion.reviewed_author_result", "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.volume_boundary", "labels": [ "eq:introvolumeboundary" ], "statement": "Incomplete volume has the boundary correction -Re(u)Im(v)/2 in addition to Im(F)/2.", "status": "derived_from_reviewed_dependencies", "source_keys": [], "hypotheses": [ "F prime=v,F(0)=0; marked local geometric branch." ], "dependencies": [ "book/claims/section16.json" ], "derivation": "", "scope_limit": "Geometric completion is separate from the analytic volume germ.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.weyl_operators", "labels": [ "sec:introquantization", "eq:introcommutator" ], "statement": "P=hbar*d/du, m=e^(u/2), l=shift_hbar give [P,u]=hbar and lm=e^(hbar/2)ml.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [ "Formal local operator convention with physical i absorbed in hbar." ], "dependencies": [], "derivation": "Apply each operator to an arbitrary test function; fixed lift signs commute.", "scope_limit": "The Weyl factor need not equal every variable called q in the literature.", "computational_coverage": { "status": "finite_fixture_support", "implemented_result_groups": [ "Q3_peripheral_filling" ], "result_pointers": [ "/groups/Q3_peripheral_filling" ], "evidence_reference": "computational_companion.reviewed_author_result", "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.quantum_curve", "labels": [], "statement": "A difference operator with principal symbol A(l,m) recovers A(exp(Phi prime),exp(u/2))=0 from WKB.", "status": "conditional_analytic_statement", "source_keys": [ "GukovApoly", "DimofteQuantumRS" ], "hypotheses": [ "Appropriate boundary character component, chosen lift and symbol limit." ], "dependencies": [], "derivation": "", "scope_limit": "Ordering and subleading terms are undetermined; no universal colored-Jones recursion theorem is asserted.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.filling_generator", "labels": [ "eq:introfillingkernel" ], "statement": "The determinant-one slope transform preserves dv wedge du and dG=b_core da/2-v du/2; stationarity imposes a=pu+qv.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [ "Integer p,q,rho,sigma with p*sigma-q*rho=1; q!=0 for this chart." ], "dependencies": [ "book/claims/section16.json" ], "derivation": "Solve v=(a-pu)/q and b_core=(-u+sigma*a)/q using p*sigma-q*rho=1; compare coefficients of du and da.", "scope_limit": "Classical generating function does not fix the full quantum surgery kernel.", "computational_coverage": { "status": "finite_fixture_support", "implemented_result_groups": [ "Q3_peripheral_filling" ], "result_pointers": [ "/groups/Q3_peripheral_filling" ], "evidence_reference": "computational_companion.reviewed_author_result", "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.rt_exponent", "labels": [ "eq:introRTgrowth" ], "statement": "With hbar_r=4pi*i/r and Im(Phi_filled)=V, exponential modulus is exp(rV/(4pi)).", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [], "dependencies": [ "s10.filling_generator", "book/claims/section16.json" ], "derivation": "Compute real part of Phi_filled/(4pi*i/r).", "scope_limit": "The Gaussian D^-1/2 presupposes Fourier normalization and D!=0; remaining amplitudes and global dominance are separate.", "computational_coverage": { "status": "finite_fixture_support", "implemented_result_groups": [ "Q3_peripheral_filling" ], "result_pointers": [ "/groups/Q3_peripheral_filling" ], "evidence_reference": "computational_companion.reviewed_author_result", "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.liouville_construction", "labels": [ "sec:introLiouville" ], "statement": "Subcritical GMC renormalizes the area exponential and point insertions; correlations are marked-surface functions.", "status": "external_statement_checked", "source_keys": [ "KRV" ], "hypotheses": [ "0Q/2, alpha2+alpha4>Q/2 for marks0,1,z,infinity.", "status": "external_statement_checked", "source_keys": [ "GKRVBootstrap", "KRV" ], "hypotheses": [ "Gamma=2b in probabilistic notation; external source charges are2alpha_i; (0,z) is the s-channel pair." ], "dependencies": [], "derivation": "", "scope_limit": "Absolute squares describe the real domain. Complex charge continuation analytically continues both factors; crossings may add discrete terms.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.fusion_kernel", "labels": [ "eq:introfusion" ], "statement": "Changing pants decomposition gives continuous fusion with explicit trivalent and spectral normalizations.", "status": "external_statement_checked", "source_keys": [ "TeschnerVartanov", "TeschnerRevisited" ], "hypotheses": [ "Principal labels alpha=Q/2+iP and modular-double parameters exp(pi*i*b^2),exp(pi*i/b^2)." ], "dependencies": [], "derivation": "", "scope_limit": "The scalar fusion kernel is not an unnormalized b-6j; continued contours have separate conditions.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.double_sine", "labels": [ "eq:introdoublesine" ], "statement": "Double-Gamma quotient, two sine shifts and reflection are the stated double-sine conventions.", "status": "external_statement_checked", "source_keys": [ "TeschnerVartanov" ], "hypotheses": [], "dependencies": [], "derivation": "", "scope_limit": "Special-function identities alone do not justify a uniform semiclassical limit.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.double_sine_limit", "labels": [], "statement": "The shift quotient formally forces L prime=log(2sin pi*x), and reflection fixes L=-Lambda(pi*x)/pi.", "status": "formal_framework", "source_keys": [], "hypotheses": [ "01+b^2-x; Lambda(pi/2)=0.", "scope_limit": "Uniform estimate is supplied by the reviewed centered-strip lemma in Section15, not proved from a finite difference alone.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.zero_mode", "labels": [ "sec:introscreening", "eq:introzeromode" ], "statement": "The constant-mode integral is b^-1 Lambda_L^-s Gamma(s) Z^-s.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [ "Lambda_L,Z>0; Re s>0." ], "dependencies": [], "derivation": "Use t=Lambda_L*Z*exp(2b*c0), retaining the leading factor2.", "scope_limit": "", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s10.screening_first_moment", "labels": [ "eq:introfirstmoment" ], "statement": "Flat charges give s=-1; the continued unit-volume expression uses the first moment, equal to the finite flat-cone area.", "status": "derived_from_reviewed_dependencies", "source_keys": [ "KRV" ], "hypotheses": [ "00 and r*sup|y|<1." ], "dependencies": [], "derivation": "Klein tangential/radial eigenvalues give the density; expand(1-r^2|y|^2)^-2 uniformly and integrate.", "scope_limit": "Specified homothetic path only; no global degeneration or quantum conclusion.", "computational_coverage": { "status": "finite_fixture_support", "implemented_result_groups": [ "Q2_klein_geometry" ], "result_pointers": [ "/groups/Q2_klein_geometry" ], "evidence_reference": "computational_companion.reviewed_author_result", "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s11.klein_remainder", "labels": [], "statement": "The geometric-series tail is nonnegative and bounded by r^3 Vol(Q)t^(m+1)(m+2-(m+1)t)/(1-t)^2.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [ "t=r^2 R^2<1;m nonnegative integer." ], "dependencies": [], "derivation": "Sum the tail of sum(k+1)t^k and majorize |y| by R.", "scope_limit": "An analytic series-truncation bound, not a certification of any numerical quadrature.", "computational_coverage": { "status": "finite_fixture_support", "implemented_result_groups": [ "Q2_klein_geometry" ], "result_pointers": [ "/groups/Q2_klein_geometry" ], "evidence_reference": "computational_companion.reviewed_author_result", "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s11.schlafli", "labels": [ "eq:Schlafli" ], "statement": "For compact curvature-minus-one polyhedra, dW=-1/2 sum ell_e d alpha_e for interior dihedral angles.", "status": "external_statement_checked", "source_keys": [ "RivinDegeneration" ], "hypotheses": [ "Differentiable fixed-combinatorics compact polyhedral family." ], "dependencies": [], "derivation": "", "scope_limit": "Exterior-angle convention reverses the sign; no ideal signed-length formula is substituted.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s11.face_coefficients", "labels": [], "statement": "Exact Lorentzian face normals give the displayed cosine and b_e formulas and even-angle/odd-length expansions.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [ "Actual adjacent faces n_f dot y=h_f, unit outward normals;0(pi-theta_e)/(4pi); four admissible triples; selected vertex label sum>=(r-3)/2." ], "dependencies": [], "derivation": "", "scope_limit": "This is the broader leading theorem, not the refined prefactor or a uniform shrinking-family result.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s11.cm_refined", "labels": [ "eq:quantum6jstructure" ], "statement": "The refined equivalent has sqrt(2)*pi*r^-3/2*(-detG)^-1/4 times exp(rW/(2pi)) only on the exact arithmetic sequence.", "status": "external_statement_checked", "source_keys": [ "ChenMurakami" ], "hypotheses": [ "Same fixed ideal/ultraideal geometry; eta=(pi-theta)/(2pi) rational with odd numerator and denominator; r odd common denominator multiple;j=r*eta/2-1/2; RW normalization and specified q-half-power." ], "dependencies": [], "derivation": "", "scope_limit": "Arbitrary convergent label sequences are not silently assigned this equivalent. The refinement theorem does not cover compact shrinking tetrahedra.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s11.roberts", "labels": [], "statement": "Classical large-label Euclidean 6j asymptotics have oscillatory Regge action and Euclidean-volume amplitude.", "status": "external_statement_checked", "source_keys": [ "Roberts" ], "hypotheses": [ "Admissibly labeled nondegenerate Euclidean tetrahedron; exterior dihedrals in the source." ], "dependencies": [], "derivation": "", "scope_limit": "No automatic identification with root-of-unity hyperbolic or b->0 limits.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s11.gram_degeneracy", "labels": [], "statement": "At a nondegenerate Euclidean tetrahedron, the face-normal Gram matrix has rank3 and determinant0.", "status": "derived_in_chapter", "source_keys": [], "hypotheses": [], "dependencies": [], "derivation": "The four Euclidean unit outward normals span R3 and have the area-weighted linear dependence; their Gram has rank3.", "scope_limit": "A vanishing determinant warns that the fixed-shape prefactor is nonuniform; it does not by itself prove an Airy theorem.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s11.joint_scale", "labels": [ "eq:jointscale" ], "statement": "A contribution exp(W/hbar) with W~A^(3/2) selects A~hbar^(2/3).", "status": "formal_framework", "source_keys": [], "hypotheses": [ "A specified geometric branch and chosen action variable; positive volume-action scale for the displayed balance." ], "dependencies": [ "s11.individual_volume", "book/claims/section15.json" ], "derivation": "", "scope_limit": "Scaling prediction only in general; reviewed Section15 covers the cleared regular family at bounded real transition parameter, with hbar_vol=h/2.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } }, { "id": "s11.compact_model", "labels": [ "eq:modelpartition", "eq:modelpartitionlimit" ], "statement": "Uniform geometry and the local WP coefficient limit prove the compact constructed integral limit.", "status": "conditional_analytic_statement", "source_keys": [], "hypotheses": [ "Fixed weights and intrinsic path; compact measurable K; continuous nondegenerate limiting realizations; W/A^(3/2) converges uniformly; positive hbar,A; A^(3/2)/hbar->lambda0 in(0,infinity)." ], "dependencies": [ "s11.individual_volume", "book/claims/sections7-8.json" ], "derivation": "Uniform exponential convergence and locally uniform coefficient convergence of(2omega_WP/A)^d combine with a finite coordinate cover and an integrable bounded density.", "scope_limit": "No whole noncompact-moduli limit, collision/flattening control, or quantum-invariant identification follows.", "computational_coverage": { "status": "no_direct_fixture_in_this_companion", "implemented_result_groups": [], "result_pointers": [], "evidence_reference": null, "limit": "Only the named finite formulas or normalization factors are tested. The mathematical statement and its hypotheses retain their independently reviewed status; convergence, existence and asymptotics are not proved by these fixtures." } } ], "integration": { "stage": "author pointer/metadata integration for independent review", "change": "Computational paragraphs and current source hashes added; mathematical claims unchanged.", "checkpoint": "docs/execution/final-chapters-review-2026-11-01/README.md", "accepted_preintegration_ledger_sha256": "34dbb6255dc468cb796bb3a129eeaf37ac0c5f4dc358abfd73030e64331b3539" } }