[ { "id": "CF001", "displayNumber": "F1", "equation": "L_c = cT_c", "name": "Native causal length", "status": "DEFINITION", "sourcePage": "F1", "variableDefinitions": [ "L_c is the native causal length; c is the speed of light; T_c is intrinsic causal time." ], "explanation": "Defines the length obtained when the causal time scale is propagated at c.", "crossLinks": [ "CF002", "CF023", "CF039" ] }, { "id": "CF002", "displayNumber": "F2", "equation": "kₐ = L_c(1, nₐ)", "name": "Four future-directed null branches", "status": "MODEL GEOMETRY", "sourcePage": "F1", "variableDefinitions": [ "kₐ is branch a; L_c is native length; nₐ is one of four unit spatial branch directions." ], "explanation": "Encodes each future-directed null branch as a temporal component and a tetrahedral spatial direction.", "crossLinks": [ "CF001", "CF003", "CF004", "CF005" ] }, { "id": "CF003", "displayNumber": "F3", "equation": "nₐ·nᵦ = −1/3 (a ≠ b)", "name": "Regular tetrahedral directional balance", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "nₐ and nᵦ are distinct unit branch directions; a,b ∈ {1,2,3,4}." ], "explanation": "The four unit directions have the pairwise dot product required by a regular tetrahedron.", "crossLinks": [ "CF002", "CF004", "CF010" ] }, { "id": "CF004", "displayNumber": "F4", "equation": "Σₐ₌₁⁴ nₐ = 0", "name": "Tetrahedral centroid balance", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "nₐ are the four tetrahedral unit directions; Σ denotes the sum over all branches." ], "explanation": "The four spatial branch directions have a vanishing centroid.", "crossLinks": [ "CF002", "CF003", "CF010" ] }, { "id": "CF005", "displayNumber": "F5", "equation": "kₐ² = 0", "name": "Null character of each branch", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "kₐ is a branch four-vector; the square uses the model Lorentz metric." ], "explanation": "Each branch is null, so its temporal and spatial parts balance in the causal metric.", "crossLinks": [ "CF002", "CF010", "CF014" ] }, { "id": "CF006", "displayNumber": "F6", "equation": "C(4,2) = 6", "name": "Four branches generate six unordered pairs", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "4 is the number of branches; C(4,2) counts unordered two-branch pairs." ], "explanation": "Counts the six pair channels carried by four causal branches.", "crossLinks": [ "CF007", "CF018", "CF019", "CF020" ] }, { "id": "CF007", "displayNumber": "F7", "equation": "Σₐᵦ = kₐ ∧ kᵦ", "name": "Pairwise causal bivector", "status": "DEFINITION", "sourcePage": "F1", "variableDefinitions": [ "Σₐᵦ is the bivector for a branch pair; kₐ,kᵦ are branch four-vectors; ∧ is the exterior product." ], "explanation": "Defines each pair channel as the exterior product of its two null branches.", "crossLinks": [ "CF002", "CF006", "CF008", "CF009" ] }, { "id": "CF008", "displayNumber": "F8", "equation": "|kₐ ∧ kᵦ| = 3/2 L_c²", "name": "Neutral pair-area invariant", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "kₐ,kᵦ are distinct branches; L_c is native length; |·| is bivector magnitude." ], "explanation": "Every neutral branch pair has the same causal area magnitude.", "crossLinks": [ "CF001", "CF007", "CF009", "CF029" ] }, { "id": "CF009", "displayNumber": "F9", "equation": "AΣ,0 = 6(4L_c²/3) = 8L_c²", "name": "Aggregate neutral six-bivector magnitude", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "AΣ,0 is aggregate neutral pair area; the factor 6 counts pairs; L_c is native length." ], "explanation": "Sums the six pair-area magnitudes into the neutral aggregate area scale.", "crossLinks": [ "CF006", "CF008", "CF023" ] }, { "id": "CF010", "displayNumber": "F10", "equation": "Gₐₐ = 0, Gₐᵦ = −1/3 (a ≠ b)", "name": "Four-null Gram matrix", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "G is the four-branch Gram matrix; a,b index the four null branches." ], "explanation": "Records the diagonal and off-diagonal inner products of the four-null basis.", "crossLinks": [ "CF003", "CF005", "CF011", "CF012", "CF013" ] }, { "id": "CF011", "displayNumber": "F11", "equation": "spec(G₄) = {−4/3, 4/3, 4/3, 4/3}L_c²", "name": "Four-null Gram spectrum", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "G₄ is the four-null Gram matrix; spec denotes its eigenvalue multiset; L_c is native length." ], "explanation": "Gives the one longitudinal and three degenerate transverse Gram eigenvalues.", "crossLinks": [ "CF010", "CF012", "CF013" ] }, { "id": "CF012", "displayNumber": "F12", "equation": "det G₄ = −256/27 L_c⁸", "name": "Non-degenerate local causal basis", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "G₄ is the four-null Gram matrix; det is its determinant; L_c is native length." ], "explanation": "The nonzero determinant establishes that the four branch directions form a non-degenerate local basis.", "crossLinks": [ "CF010", "CF011", "CF014" ] }, { "id": "CF013", "displayNumber": "F13", "equation": "G₄⁻¹ = L_c⁻²(3/4 I₄ − 1/4 11ᵀ)", "name": "Inverse Gram", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "G₄⁻¹ is the inverse Gram matrix; I₄ is the 4×4 identity; 1 is the all-ones vector." ], "explanation": "Provides the explicit inverse needed to reconstruct the Minkowski metric from branch data.", "crossLinks": [ "CF010", "CF012", "CF014" ] }, { "id": "CF014", "displayNumber": "F14", "equation": "K G₄⁻¹ Kᵀ = η", "name": "Minkowski metric reconstruction", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "K contains the four branch vectors; G₄⁻¹ is the inverse Gram matrix; η is the Minkowski metric." ], "explanation": "Reconstructs the Minkowski metric by changing from branch coordinates to spacetime coordinates.", "crossLinks": [ "CF005", "CF010", "CF013", "CF015" ] }, { "id": "CF015", "displayNumber": "F15", "equation": "Dₐ = kₐ^μ∂_μ", "name": "Branch directional derivative", "status": "DEFINITION", "sourcePage": "F1", "variableDefinitions": [ "Dₐ is the derivative along branch a; kₐ^μ is its four-vector; ∂_μ is the spacetime derivative." ], "explanation": "Defines differentiation along each causal branch.", "crossLinks": [ "CF002", "CF014", "CF016" ] }, { "id": "CF016", "displayNumber": "F16", "equation": "(G₄⁻¹)ᵃᵇDₐDᵦ = □", "name": "Native d’Alembertian reconstruction", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "Dₐ are branch derivatives; G₄⁻¹ is the inverse branch Gram; □ is the Minkowski d’Alembertian." ], "explanation": "Combines branch derivatives with the inverse Gram to recover the wave operator.", "crossLinks": [ "CF013", "CF014", "CF015", "CF067" ] }, { "id": "CF017", "displayNumber": "F17", "equation": "BC = 0", "name": "Boundary-of-boundary incidence identity", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "B is the vertex-to-edge incidence matrix; C is the edge-to-face incidence matrix." ], "explanation": "The incidence composition vanishes because a boundary has no boundary.", "crossLinks": [ "CF018", "CF019", "CF020", "CF123" ] }, { "id": "CF018", "displayNumber": "F18", "equation": "BᵀB + CCᵀ = 4I₆", "name": "Exact six-edge Hodge decomposition", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "B,C are incidence matrices; I₆ is the identity on six edge channels." ], "explanation": "Relates vertex and face incidence operators on the six tetrahedral edge channels.", "crossLinks": [ "CF006", "CF017", "CF019", "CF020" ] }, { "id": "CF019", "displayNumber": "F19", "equation": "Pgrad = 1/4 BᵀB, Pcirc = 1/4 CCᵀ", "name": "Gradient/circulation projectors", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "Pgrad and Pcirc are six-channel projectors; B,C are incidence matrices." ], "explanation": "Normalizes the gradient and circulation components of edge-channel data.", "crossLinks": [ "CF017", "CF018", "CF020", "CF021", "CF022" ] }, { "id": "CF020", "displayNumber": "F20", "equation": "Pgrad + Pcirc = I₆", "name": "Complete six-channel decomposition", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "Pgrad and Pcirc are complementary projectors; I₆ is the six-channel identity." ], "explanation": "States that the gradient and circulation sectors exhaust the six pair channels.", "crossLinks": [ "CF018", "CF019", "CF022" ] }, { "id": "CF021", "displayNumber": "F21", "equation": "K₀ = BᵀB − 2I₆", "name": "Historical normalized six-channel operator convention", "status": "RETAINED", "sourcePage": "F1", "variableDefinitions": [ "K₀ is the retained normalized edge operator; B is incidence; I₆ is the six-channel identity." ], "explanation": "Records the normalized operator convention retained for the neutral channel spectrum.", "crossLinks": [ "CF018", "CF019", "CF022" ] }, { "id": "CF022", "displayNumber": "F22", "equation": "spec(K₀) = {4,0,0,0,−2,−2}", "name": "Six-channel neutral spectrum", "status": "EXACT", "sourcePage": "F1", "variableDefinitions": [ "K₀ is the normalized six-channel operator; spec denotes its eigenvalue multiset." ], "explanation": "Lists the three gradient-zero modes and the two negative circulation eigenvalues.", "crossLinks": [ "CF019", "CF020", "CF021" ] }, { "id": "CF023", "displayNumber": "F23", "equation": "V_c = (2/3)L_c³ = (√2/12)(cT_c)³", "name": "Native causal volume", "status": "DEFINITION", "sourcePage": "F1", "variableDefinitions": [ "V_c is native causal volume; L_c is native length; c and T_c are the causal speed and time." ], "explanation": "Defines the four-null causal cell volume in equivalent native-length and causal-time forms.", "crossLinks": [ "CF001", "CF008", "CF024", "CF025" ] }, { "id": "CF024", "displayNumber": "F24", "equation": "Γᵥ = dV/dt = 2cL_c²", "name": "Instantaneous causal-volume flux for dL/dt = c", "status": "CONDITIONAL", "sourcePage": "F1", "variableDefinitions": [ "Γᵥ is volume flux; V is causal volume; L_c is native length; c is propagation speed." ], "explanation": "Differentiates the native volume while assuming radial causal growth dL/dt = c.", "crossLinks": [ "CF023", "CF025", "CF040" ] }, { "id": "CF025", "displayNumber": "F25", "equation": "Γ_c = 2c³T_c²", "name": "Characteristic causal-volume flux", "status": "CONDITIONAL", "sourcePage": "F1", "variableDefinitions": [ "Γ_c is characteristic volume flux; c is light speed; T_c is intrinsic causal time." ], "explanation": "Expresses the characteristic flux after substituting L_c = cT_c.", "crossLinks": [ "CF001", "CF023", "CF024", "CF040" ] }, { "id": "CF026", "displayNumber": "F26", "equation": "E_f = ℏ/T_c", "name": "Intrinsic fabric energy scale", "status": "DEFINITION", "sourcePage": "F2", "variableDefinitions": [ "E_f is intrinsic fabric energy; ℏ is reduced Planck constant; T_c is intrinsic causal time." ], "explanation": "Defines the energy associated with one intrinsic causal update.", "crossLinks": [ "CF027", "CF028", "CF038" ] }, { "id": "CF027", "displayNumber": "F27", "equation": "E_fT_c = ℏ", "name": "One intrinsic fabric update carries one action quantum", "status": "MODEL NORMALIZATION", "sourcePage": "F2", "variableDefinitions": [ "E_f is fabric energy; T_c is update time; ℏ is reduced Planck constant." ], "explanation": "Sets the action of one intrinsic update to one quantum of action.", "crossLinks": [ "CF026", "CF044", "CF046" ] }, { "id": "CF028", "displayNumber": "F28", "equation": "m_c = ℏ/(c²T_c)", "name": "Native dimensional mass scale", "status": "DEFINITION", "sourcePage": "F2", "variableDefinitions": [ "m_c is native mass; ℏ is reduced Planck constant; c is light speed; T_c is causal time." ], "explanation": "Converts the intrinsic energy scale into its corresponding relativistic mass scale.", "crossLinks": [ "CF026", "CF037", "CF039" ] }, { "id": "CF029", "displayNumber": "F29", "equation": "A₀ = 4L_c²/3", "name": "Neutral pair-area scale", "status": "EXACT", "sourcePage": "F2", "variableDefinitions": [ "A₀ is the neutral pair-area scale; L_c is native causal length." ], "explanation": "Defines the single-area normalization associated with the neutral tetrahedral geometry.", "crossLinks": [ "CF008", "CF009", "CF030" ] }, { "id": "CF030", "displayNumber": "F30", "equation": "κ_C L_c² = ℏ", "name": "Candidate neutral area-stiffness closure", "status": "CONDITIONAL", "sourcePage": "F2", "variableDefinitions": [ "κ_C is candidate causal area stiffness; L_c is native length; ℏ is reduced Planck constant." ], "explanation": "Proposes an action-sized closure for the neutral area stiffness.", "crossLinks": [ "CF029", "CF031", "CF032" ] }, { "id": "CF031", "displayNumber": "F31", "equation": "σ₀ = κ_C/T_c", "name": "Candidate surface-tension scale", "status": "CONDITIONAL", "sourcePage": "F2", "variableDefinitions": [ "σ₀ is candidate surface tension; κ_C is area stiffness; T_c is causal time." ], "explanation": "Converts the candidate area stiffness into a surface-tension scale.", "crossLinks": [ "CF030", "CF032" ] }, { "id": "CF032", "displayNumber": "F32", "equation": "C_Σ = 12/(5π)", "name": "Retained causal-geometric coefficient", "status": "CONDITIONAL/REFERENCE", "sourcePage": "F2", "variableDefinitions": [ "C_Σ is the retained causal-geometric coefficient; π is the circular constant." ], "explanation": "Records the coefficient retained when the tetrahedral and spherical factors are combined.", "crossLinks": [ "CF033", "CF036", "CF067", "CF148" ] }, { "id": "CF033", "displayNumber": "F33", "equation": "C_Σ/(4π) = 3/(5π²)", "name": "Microscopic coefficient plus spherical spreading", "status": "EXACT GIVEN C_Σ", "sourcePage": "F2", "variableDefinitions": [ "C_Σ is the retained coefficient; π is the spherical spreading constant." ], "explanation": "Rewrites the retained coefficient after separating the 4π spherical factor.", "crossLinks": [ "CF032", "CF036", "CF151" ] }, { "id": "CF034", "displayNumber": "F34", "equation": "K_GD = 1/(5π²c)", "name": "G Model Kutsal Interface Constant; not Big G", "status": "HISTORICAL/INTERFACE", "sourcePage": "F2", "variableDefinitions": [ "K_GD is the historical interface constant; c is light speed; π is the circular constant." ], "explanation": "Records the historical interface normalization and explicitly distinguishes it from Newtonian G.", "crossLinks": [ "CF033", "CF035", "CF150" ] }, { "id": "CF035", "displayNumber": "F35", "equation": "4πG = 3cK_GD C_Σ", "name": "Exact algebraic interface identity", "status": "EXACT", "sourcePage": "F2", "variableDefinitions": [ "G is the interface coupling in this identity; K_GD is the interface constant; c is light speed; C_Σ is retained coefficient." ], "explanation": "Links the interface constant, retained geometric coefficient, and coupling by algebraic normalization.", "crossLinks": [ "CF032", "CF034", "CF036" ] }, { "id": "CF036", "displayNumber": "F36", "equation": "G_bridge = (C_Σ/4π)c⁵T_c²/ℏ = (3/5π²)c⁵T_c²/ℏ", "name": "Continuum matter-spacetime coupling bridge", "status": "CONDITIONAL", "sourcePage": "F2", "variableDefinitions": [ "G_bridge is continuum bridge coupling; C_Σ is retained coefficient; c is light speed; T_c is causal time; ℏ is action quantum." ], "explanation": "Proposes the continuum coupling obtained by matching the discrete source-transfer coefficient.", "crossLinks": [ "CF032", "CF033", "CF037", "CF151" ] }, { "id": "CF037", "displayNumber": "F37", "equation": "T_c = [ℏG/(C_Σc⁵)]¹ᐟ²", "name": "Big-G-calibrated fabric time", "status": "CALIBRATED", "sourcePage": "F2", "variableDefinitions": [ "T_c is fabric time; ℏ is reduced Planck constant; G is calibration coupling; C_Σ is retained coefficient; c is light speed." ], "explanation": "Inverts the continuum bridge using a supplied Big-G calibration.", "crossLinks": [ "CF032", "CF036", "CF038", "CF039" ] }, { "id": "CF038", "displayNumber": "F38", "equation": "T_c ≈ 2.18657 × 10⁻⁴³ s", "name": "Reference value for C_Σ = 12/(5π)", "status": "CALIBRATED", "sourcePage": "F2", "variableDefinitions": [ "T_c is calibrated fabric time in seconds; C_Σ is set to 12/(5π)." ], "explanation": "Gives the current numerical time reference after choosing the retained coefficient and Big-G calibration.", "crossLinks": [ "CF032", "CF037", "CF039", "CF197" ] }, { "id": "CF039", "displayNumber": "F39", "equation": "L_c ≈ 6.55517 × 10⁻³⁵ m", "name": "Reference causal length", "status": "CALIBRATED", "sourcePage": "F2", "variableDefinitions": [ "L_c is calibrated causal length in metres; c is light speed; T_c is calibrated time." ], "explanation": "Converts the calibrated intrinsic time to the corresponding causal length.", "crossLinks": [ "CF001", "CF037", "CF038" ] }, { "id": "CF040", "displayNumber": "F40", "equation": "6/T_c · c = √6 Γ_c", "name": "Neutral propagation normalization", "status": "CONDITIONAL", "sourcePage": "F2", "variableDefinitions": [ "T_c is causal time; c is light speed; Γ_c is characteristic causal flux." ], "explanation": "States the retained normalization connecting time, propagation, and neutral causal flux.", "crossLinks": [ "CF023", "CF025", "CF041" ] }, { "id": "CF041", "displayNumber": "F41", "equation": "E_cell = V_c T^{μν} t_μ t_ν", "name": "Covariant local causal-cell energy", "status": "DEFINITION", "sourcePage": "F2", "variableDefinitions": [ "E_cell is cell energy; V_c is cell volume; T^{μν} is stress-energy; t_μ is the cell timelike unit direction." ], "explanation": "Defines local cell energy as the stress-energy contraction integrated over the causal cell volume.", "crossLinks": [ "CF023", "CF042", "CF043" ] }, { "id": "CF042", "displayNumber": "F42", "equation": "E_cell = ρV_cc²", "name": "Stationary nonrelativistic limit", "status": "CONDITIONAL LIMIT", "sourcePage": "F2", "variableDefinitions": [ "E_cell is cell energy; ρ is mass density; V_c is cell volume; c is light speed." ], "explanation": "Reduces the covariant cell energy to rest-energy density for stationary matter.", "crossLinks": [ "CF041", "CF048", "CF051" ] }, { "id": "CF043", "displayNumber": "F43", "equation": "S_m(t) = ∫₀ᵗ E_cell(τ)dτ", "name": "Accumulated matter action", "status": "DEFINITION", "sourcePage": "F2", "variableDefinitions": [ "S_m is accumulated matter action; E_cell is cell energy; τ is integration time." ], "explanation": "Accumulates the local cell energy over the matter-response interval.", "crossLinks": [ "CF041", "CF044", "CF045", "CF050" ] }, { "id": "CF044", "displayNumber": "F44", "equation": "S_m(D₀) = ℏ", "name": "Aggregation definition of matter-response time", "status": "HYPOTHESIS", "sourcePage": "F2", "variableDefinitions": [ "S_m is matter action; D₀ is aggregation response time; ℏ is action quantum." ], "explanation": "Hypothesizes that one aggregation interval carries one action quantum.", "crossLinks": [ "CF027", "CF043", "CF045", "CF046" ] }, { "id": "CF045", "displayNumber": "F45", "equation": "D₀ = ℏ/E_cell", "name": "Constant-energy Aggregation law", "status": "EXACT WITHIN DEFINITION", "sourcePage": "F2", "variableDefinitions": [ "D₀ is aggregation response time; ℏ is action quantum; E_cell is constant cell energy." ], "explanation": "Solves the one-action condition for the duration of a constant-energy aggregation update.", "crossLinks": [ "CF041", "CF044", "CF047" ] }, { "id": "CF046", "displayNumber": "F46", "equation": "A_cell = E_cellT_c/ℏ", "name": "Dimensionless matter/fabric loading", "status": "DEFINITION", "sourcePage": "F2", "variableDefinitions": [ "A_cell is dimensionless loading; E_cell is cell energy; T_c is fabric time; ℏ is action quantum." ], "explanation": "Measures cell energy relative to the intrinsic one-update action.", "crossLinks": [ "CF027", "CF044", "CF047", "CF048" ] }, { "id": "CF047", "displayNumber": "F47", "equation": "D₀/T_c = 1/A_cell", "name": "Controlling Aggregation reciprocal relation", "status": "EXACT WITHIN DEFINITION", "sourcePage": "F2", "variableDefinitions": [ "D₀ is aggregation time; T_c is fabric time; A_cell is dimensionless loading." ], "explanation": "Shows that aggregation duration is the reciprocal of matter loading in intrinsic-time units.", "crossLinks": [ "CF045", "CF046", "CF049" ] }, { "id": "CF048", "displayNumber": "F48", "equation": "A_cell = (3/2)ρc⁵T_c⁴/ℏ", "name": "Stationary density loading", "status": "EXACT WITHIN DEFINITIONS", "sourcePage": "F2", "variableDefinitions": [ "A_cell is loading; ρ is stationary density; c is light speed; T_c is causal time; ℏ is action quantum." ], "explanation": "Substitutes the native cell volume and stationary energy into the loading definition.", "crossLinks": [ "CF023", "CF042", "CF046", "CF067" ] }, { "id": "CF049", "displayNumber": "F49", "equation": "A_cell = 1 ⇔ D₀ = T_c", "name": "Equal-action/equal-scale condition", "status": "CONDITIONAL PHYSICAL INTERPRETATION", "sourcePage": "F2", "variableDefinitions": [ "A_cell is dimensionless loading; D₀ is aggregation time; T_c is intrinsic time." ], "explanation": "Identifies equal loading with an aggregation interval equal to one intrinsic causal update.", "crossLinks": [ "CF046", "CF047", "CF048" ] }, { "id": "CF050", "displayNumber": "F50", "equation": "Γ_m = D₀⁻¹ = E_cell/ℏ", "name": "Additive matter-action accumulation rate", "status": "EXACT", "sourcePage": "F2", "variableDefinitions": [ "Γ_m is matter-action rate; D₀ is aggregation time; E_cell is cell energy; ℏ is action quantum." ], "explanation": "Expresses the additive update rate in equivalent response-time and energy units.", "crossLinks": [ "CF043", "CF045", "CF046", "CF051" ] }, { "id": "CF051", "displayNumber": "F51", "equation": "Γ_{m,tot} = Σ_i Γ_{m,i}", "name": "Independent energy contributions add in rate", "status": "EXACT", "sourcePage": "F3", "variableDefinitions": [ "Γ_{m,tot} is total matter-action rate; Γ_{m,i} is contribution i; Σ_i sums independent contributions." ], "explanation": "Adds independent cell or source contributions at the action-accumulation rate level.", "crossLinks": [ "CF042", "CF050", "CF052" ] }, { "id": "CF052", "displayNumber": "F52", "equation": "t^μ = K^μ/√(−K²), K^μ = Σ_a k_a^μ", "name": "Natural cell timelike direction", "status": "EXACT", "sourcePage": "F3", "variableDefinitions": [ "t^μ is normalized cell timelike direction; K^μ is the branch sum; k_a^μ are branch vectors." ], "explanation": "Normalizes the sum of the four null branches to obtain the cell time direction.", "crossLinks": [ "CF002", "CF005", "CF041" ] }, { "id": "CF053", "displayNumber": "F53", "equation": "χ_m ∈ {0,1}", "name": "Matter-occupancy bookkeeping variable", "status": "BOOKKEEPING", "sourcePage": "F3", "variableDefinitions": [ "χ_m is binary matter occupancy: 0 means absent and 1 means present." ], "explanation": "Introduces a binary indicator for whether matter occupies the local cell.", "crossLinks": [ "CF054", "CF055", "CF056", "CF057" ] }, { "id": "CF054", "displayNumber": "F54", "equation": "(χ_m,M) = (0,0)", "name": "Matter absent, neutral fabric", "status": "STATE", "sourcePage": "F3", "variableDefinitions": [ "χ_m is matter occupancy; M is deformation; both are zero in the neutral empty state." ], "explanation": "Labels the locally empty undeformed state.", "crossLinks": [ "CF053", "CF055", "CF056", "CF057" ] }, { "id": "CF055", "displayNumber": "F55", "equation": "(χ_m,M) = (1,M ≠ 0)", "name": "Matter present, deformed fabric", "status": "STATE", "sourcePage": "F3", "variableDefinitions": [ "χ_m=1 marks matter; M is a nonzero local deformation." ], "explanation": "Labels matter supported by a locally deformed fabric.", "crossLinks": [ "CF053", "CF054", "CF056", "CF061" ] }, { "id": "CF056", "displayNumber": "F56", "equation": "(χ_m,M) = (0,M ≠ 0)", "name": "Matter absent locally, deformation persists", "status": "STATE", "sourcePage": "F3", "variableDefinitions": [ "χ_m=0 marks local matter absence; M is a persistent nonzero deformation." ], "explanation": "Allows a deformation field to remain after local matter occupancy vanishes.", "crossLinks": [ "CF053", "CF054", "CF055", "CF061" ] }, { "id": "CF057", "displayNumber": "F57", "equation": "(χ_m,M) = (1,0)", "name": "Neutral M; does not by itself imply D₀ = T_c", "status": "SPECIAL STATE", "sourcePage": "F3", "variableDefinitions": [ "χ_m=1 marks matter; M=0 is neutral deformation; D₀ is aggregation time; T_c is intrinsic time." ], "explanation": "Separates matter occupancy from deformation and warns that neutral M alone does not impose equal times.", "crossLinks": [ "CF047", "CF049", "CF053", "CF055" ] }, { "id": "CF058", "displayNumber": "F58", "equation": "Σ_col = ρL_c", "name": "One-causal-depth column density", "status": "DEFINITION", "sourcePage": "F3", "variableDefinitions": [ "Σ_col is column density; ρ is mass density; L_c is one causal depth." ], "explanation": "Defines projected density through one native causal depth.", "crossLinks": [ "CF001", "CF059", "CF067" ] }, { "id": "CF059", "displayNumber": "F59", "equation": "∫Σ_col dA = M_source", "name": "Projected-area mass additivity", "status": "CONDITIONAL GEOMETRY", "sourcePage": "F3", "variableDefinitions": [ "Σ_col is column density; A is projected area; M_source is enclosed source mass." ], "explanation": "Relates projected column density to source mass under the chosen geometry.", "crossLinks": [ "CF058", "CF067", "CF068" ] }, { "id": "CF060", "displayNumber": "F60", "equation": "∂κ_C/∂m = ∂κ_C/∂ρ = ∂κ_C/∂Ω = 0", "name": "Baseline fabric universality requirement", "status": "POSTULATE", "sourcePage": "F3", "variableDefinitions": [ "κ_C is area stiffness; m is mass; ρ is density; Ω is orientation; partial derivatives test dependence." ], "explanation": "Postulates that baseline fabric stiffness is independent of mass, density, and orientation.", "crossLinks": [ "CF030", "CF061", "CF066" ] }, { "id": "CF061", "displayNumber": "F61", "equation": "M = ln(dτ/dt)", "name": "Operational deformation/clock relation", "status": "CONDITIONAL", "sourcePage": "F3", "variableDefinitions": [ "M is deformation; τ is local proper or fabric time; t is reference coordinate time." ], "explanation": "Defines deformation as the logarithm of the local clock-rate ratio.", "crossLinks": [ "CF055", "CF062", "CF063", "CF064" ] }, { "id": "CF062", "displayNumber": "F62", "equation": "dτ/dt = e^M", "name": "Local clock ratio", "status": "CONDITIONAL", "sourcePage": "F3", "variableDefinitions": [ "τ is local time; t is reference time; M is deformation." ], "explanation": "Rewrites the operational deformation relation as a local clock ratio.", "crossLinks": [ "CF061", "CF063", "CF076" ] }, { "id": "CF063", "displayNumber": "F63", "equation": "Φ = c²δM", "name": "Weak-field potential identification", "status": "CONDITIONAL", "sourcePage": "F3", "variableDefinitions": [ "Φ is weak-field potential; c is light speed; δM is a small deformation perturbation." ], "explanation": "Identifies the weak potential with the dimensionless deformation perturbation.", "crossLinks": [ "CF061", "CF064", "CF067", "CF068" ] }, { "id": "CF064", "displayNumber": "F64", "equation": "g = −c²∇M", "name": "Weak-field acceleration", "status": "CONDITIONAL", "sourcePage": "F3", "variableDefinitions": [ "g is weak gravitational acceleration; c is light speed; ∇M is the deformation gradient." ], "explanation": "Takes the negative spatial gradient of deformation as weak-field acceleration.", "crossLinks": [ "CF061", "CF063", "CF067", "CF069" ] }, { "id": "CF065", "displayNumber": "F65", "equation": "Ā[M] = M + (L_c²/6)∇²M + (L_c⁴/120)∇⁴M + ⋯", "name": "Orientation-balanced spherical mean", "status": "EXACT EXPANSION", "sourcePage": "F3", "variableDefinitions": [ "Ā[M] is the orientation-balanced mean; M is deformation; L_c is native length; ∇² is the Laplacian." ], "explanation": "Expands the isotropic average of M in even spatial derivatives through the causal scale.", "crossLinks": [ "CF066", "CF067", "CF070" ] }, { "id": "CF066", "displayNumber": "F66", "equation": "Ā[M] − M = (C_Σ/4)A_cell", "name": "Discrete mean-defect source rule", "status": "CONDITIONAL", "sourcePage": "F3", "variableDefinitions": [ "Ā[M] is spherical mean; M is local deformation; C_Σ is retained coefficient; A_cell is loading." ], "explanation": "Connects the orientation-averaging defect to the dimensionless matter loading.", "crossLinks": [ "CF032", "CF046", "CF060", "CF065" ] }, { "id": "CF067", "displayNumber": "F67", "equation": "∇²M = C_Σ(c³T_c²/ℏ)ρ", "name": "Weak continuum source law", "status": "CONDITIONAL", "sourcePage": "F3", "variableDefinitions": [ "M is deformation; C_Σ is retained coefficient; c is light speed; T_c is causal time; ρ is density; ℏ is action." ], "explanation": "Gives the retained weak continuum deformation response to mass density.", "crossLinks": [ "CF032", "CF048", "CF063", "CF068", "CF069", "CF098" ] }, { "id": "CF068", "displayNumber": "F68", "equation": "∇²M = (4πG/c²)ρ", "name": "Big-G-calibrated weak source law", "status": "CALIBRATED COMPATIBILITY", "sourcePage": "F3", "variableDefinitions": [ "M is deformation; G is calibrated gravitational coupling; c is light speed; ρ is density." ], "explanation": "Shows compatibility of the continuum source law with the Big-G Poisson normalization.", "crossLinks": [ "CF036", "CF067", "CF069", "CF111" ] }, { "id": "CF069", "displayNumber": "F69", "equation": "∇²Φ = 4πGρ", "name": "Poisson normalization", "status": "CALIBRATED COMPATIBILITY", "sourcePage": "F3", "variableDefinitions": [ "Φ is weak potential; G is calibrated gravitational coupling; ρ is density." ], "explanation": "Recovers the conventional weak-field Poisson equation after the potential identification.", "crossLinks": [ "CF063", "CF068", "CF098" ] }, { "id": "CF070", "displayNumber": "F70", "equation": "Q₆(M) = 16e²M + 8e⁻²M", "name": "Reduced reciprocal quadratic spectrum", "status": "EXACT WITHIN REDUCED MODEL", "sourcePage": "F3", "variableDefinitions": [ "Q₆ is reduced reciprocal spectrum; M is deformation; e is the exponential base." ], "explanation": "Defines the reduced six-channel reciprocal quadratic used for the spectral minimum.", "crossLinks": [ "CF071", "CF072", "CF073", "CF074", "CF075" ] }, { "id": "CF071", "displayNumber": "F71", "equation": "M_* = −(1/4)ln 2", "name": "Mathematical minimum of Q₆", "status": "EXACT", "sourcePage": "F3", "variableDefinitions": [ "M_* is minimizing deformation; Q₆ is the reduced spectrum; ln is natural logarithm." ], "explanation": "Locates the stationary minimum of the reduced reciprocal spectrum.", "crossLinks": [ "CF070", "CF072", "CF073" ] }, { "id": "CF072", "displayNumber": "F72", "equation": "Q₆,min = 16√2", "name": "Minimum reciprocal spectral value", "status": "EXACT", "sourcePage": "F3", "variableDefinitions": [ "Q₆,min is the minimum spectrum value; Q₆ is the reduced reciprocal spectrum." ], "explanation": "Evaluates Q₆ at its mathematical minimum.", "crossLinks": [ "CF070", "CF071", "CF073" ] }, { "id": "CF073", "displayNumber": "F73", "equation": "Q₊Q₋ = 128", "name": "Reciprocal spectral-product invariant", "status": "EXACT", "sourcePage": "F3", "variableDefinitions": [ "Q₊ and Q₋ are the reciprocal spectral sectors." ], "explanation": "States the deformation-independent product of the paired reciprocal sectors.", "crossLinks": [ "CF070", "CF072", "CF074" ] }, { "id": "CF074", "displayNumber": "F74", "equation": "S₆² − D₆² = 512", "name": "Hyperbolic spectral invariant", "status": "EXACT", "sourcePage": "F3", "variableDefinitions": [ "S₆ and D₆ are the sum and difference spectral combinations." ], "explanation": "Records the invariant difference of squared sum and difference sectors.", "crossLinks": [ "CF070", "CF073", "CF075" ] }, { "id": "CF075", "displayNumber": "F75", "equation": "Tr|K(M)| = 8 cosh M", "name": "Linear reciprocal spectral magnitude", "status": "EXACT IN STATED SPECTRUM", "sourcePage": "F3", "variableDefinitions": [ "K(M) is the deformation-dependent channel operator; Tr is trace; M is deformation." ], "explanation": "Gives the trace magnitude of the stated reciprocal spectrum in hyperbolic form.", "crossLinks": [ "CF070", "CF073", "CF074" ] }, { "id": "CF076", "displayNumber": "F76", "equation": "T₊T₋ = T_c²", "name": "Dolu reciprocal-time invariant", "status": "EXACT WITHIN CONSTRUCTION", "sourcePage": "F4", "variableDefinitions": [ "T₊ and T₋ are reciprocal local times; T_c is intrinsic causal time." ], "explanation": "Constrains the product of the reciprocal time sectors to the intrinsic-time square.", "crossLinks": [ "CF037", "CF077", "CF078" ] }, { "id": "CF077", "displayNumber": "F77", "equation": "δln T₊ + δln T₋ = 0", "name": "Differential reciprocal-time form", "status": "EXACT WITHIN CONSTRUCTION", "sourcePage": "F4", "variableDefinitions": [ "T₊,T₋ are reciprocal times; δ denotes a variation." ], "explanation": "The logarithmic variations of the two reciprocal sectors cancel.", "crossLinks": [ "CF076", "CF078" ] }, { "id": "CF078", "displayNumber": "F78", "equation": "T₊ = T_ce⁺²ᵘ⁄³", "name": "Three-dimensional reciprocal time sectors", "status": "CONDITIONAL", "sourcePage": "F4", "variableDefinitions": [ "T₊ is the plus sector; T_c is intrinsic time; u is the reciprocal-sector coordinate." ], "explanation": "Parametrizes one reciprocal time sector by a three-dimensional exponential coordinate.", "crossLinks": [ "CF076", "CF079", "CF080" ] }, { "id": "CF079", "displayNumber": "F79", "equation": "V₊ = V_ce⁺²ᵘ", "name": "Reciprocal volume sectors", "status": "CONDITIONAL", "sourcePage": "F4", "variableDefinitions": [ "V₊ is plus-sector volume; V_c is native volume; u is shared reciprocal coordinate." ], "explanation": "Uses the shared coordinate to scale the causal volume sectors.", "crossLinks": [ "CF023", "CF078", "CF080" ] }, { "id": "CF080", "displayNumber": "F80", "equation": "2V_cQ₊V₊ = Q₋V₋ = 8√2V_c", "name": "Spectral-volume equalization if shared-u bridge is adopted", "status": "CONDITIONAL BRIDGE", "sourcePage": "F4", "variableDefinitions": [ "V_c,V₊,V₋ are causal volumes; Q₊,Q₋ are reciprocal spectra; u is shared sector coordinate." ], "explanation": "States the proposed equalization between reciprocal spectra and volumes when the shared-u bridge is assumed.", "crossLinks": [ "CF073", "CF078", "CF079" ] }, { "id": "CF081", "displayNumber": "F81", "equation": "Kₓ(θ) = Λₓ(θ)Kₓ(0)", "name": "Jadvyga local reorientation", "status": "DEFINITION", "sourcePage": "F4", "variableDefinitions": [ "Kₓ is a local branch frame; Λₓ(θ) is the local Lorentz reorientation at x and angle θ." ], "explanation": "Defines local frame reorientation by acting with a Lorentz transformation.", "crossLinks": [ "CF082", "CF083", "CF084" ] }, { "id": "CF082", "displayNumber": "F82", "equation": "ΛₓᵀηΛₓ = η", "name": "Lorentz-frame preservation", "status": "EXACT", "sourcePage": "F4", "variableDefinitions": [ "Λₓ is the local frame transformation; η is the Minkowski metric." ], "explanation": "Requires the local reorientation to preserve the Lorentz metric.", "crossLinks": [ "CF014", "CF081", "CF083" ] }, { "id": "CF083", "displayNumber": "F83", "equation": "W_f → ΛₓW_fΛₓ⁻¹", "name": "Holonomy conjugation under local reorientation", "status": "EXACT", "sourcePage": "F4", "variableDefinitions": [ "W_f is face holonomy; Λₓ is local Lorentz reorientation." ], "explanation": "Transforms face holonomy by conjugation under a local frame change.", "crossLinks": [ "CF081", "CF082", "CF123" ] }, { "id": "CF084", "displayNumber": "F84", "equation": "C_xy(Π) = 1/2‖ΠᵀΠ_xy − Π‖²_F", "name": "Celal invariant mismatch cost", "status": "DEFINITION", "sourcePage": "F4", "variableDefinitions": [ "C_xy is mismatch cost; Π is a branch correspondence; ‖·‖_F is Frobenius norm." ], "explanation": "Measures the squared mismatch between transported and target branch correspondences.", "crossLinks": [ "CF085", "CF086", "CF092" ] }, { "id": "CF085", "displayNumber": "F85", "equation": "Π_Celal ∈ arg min_Π∈S₄ C_xy(Π)", "name": "Celal branch correspondence", "status": "EXACT GEOMETRIC SELECTION", "sourcePage": "F4", "variableDefinitions": [ "Π_Celal is selected correspondence; S₄ is the permutation group on four branches; C_xy is mismatch cost." ], "explanation": "Selects the permutation that minimizes the geometric mismatch between neighboring cells.", "crossLinks": [ "CF006", "CF084", "CF086", "CF092" ] }, { "id": "CF086", "displayNumber": "F86", "equation": "q = (a,b,c,c,b,a), a+b+c = 0", "name": "Centered physical shape sector", "status": "DEFINITION", "sourcePage": "F4", "variableDefinitions": [ "q is a centered six-channel shape vector; a,b,c are pair amplitudes satisfying zero sum." ], "explanation": "Restricts the six-channel shape to the centered reciprocal sector.", "crossLinks": [ "CF070", "CF087", "CF088", "CF158" ] }, { "id": "CF087", "displayNumber": "F87", "equation": "T_c dq/dτ = −(I−P)q", "name": "Candidate reconnect relaxation law", "status": "CONDITIONAL", "sourcePage": "F4", "variableDefinitions": [ "q is shape; τ is update time; P is the retained target projector; I is identity." ], "explanation": "Proposes relaxation of shape toward the selected target subspace during reconnection.", "crossLinks": [ "CF086", "CF088", "CF089", "CF157" ] }, { "id": "CF088", "displayNumber": "F88", "equation": "q(τ) = q₀e⁻³ᵗ⁄²ᵀ_c", "name": "Centered relaxation solution", "status": "CONDITIONAL", "sourcePage": "F4", "variableDefinitions": [ "q is centered shape; q₀ is initial shape; τ is reconnect time; T_c is intrinsic time." ], "explanation": "Gives the exponential solution of the centered relaxation candidate.", "crossLinks": [ "CF087", "CF089", "CF157" ] }, { "id": "CF089", "displayNumber": "F89", "equation": "τ_rec = 2T_c/3", "name": "Candidate reconnect timescale", "status": "CONDITIONAL", "sourcePage": "F4", "variableDefinitions": [ "τ_rec is reconnect timescale; T_c is intrinsic causal time." ], "explanation": "Defines the characteristic e-folding timescale associated with the candidate relaxation.", "crossLinks": [ "CF087", "CF088", "CF157" ] }, { "id": "CF090", "displayNumber": "F90", "equation": "J_*² = −I₆", "name": "Lorentz bivector Hodge structure", "status": "EXACT", "sourcePage": "F4", "variableDefinitions": [ "J_* is the Hodge-star operator on six bivectors; I₆ is the six-channel identity." ], "explanation": "Records the complex-structure square of the Lorentz bivector Hodge operator.", "crossLinks": [ "CF019", "CF091", "CF132" ] }, { "id": "CF091", "displayNumber": "F91", "equation": "J_*ᵀΓJ_* = −Γ", "name": "Hodge compatibility with bivector metric", "status": "EXACT", "sourcePage": "F4", "variableDefinitions": [ "J_* is the bivector Hodge operator; Γ is the six-space bivector metric." ], "explanation": "States how the Hodge operator acts on the bivector metric.", "crossLinks": [ "CF090", "CF144", "CF145" ] }, { "id": "CF092", "displayNumber": "F92", "equation": "Ω_yx = K_yΠ_yxKₓ⁻¹", "name": "Inter-cell branch-frame transporter", "status": "CONDITIONAL/GEOMETRIC", "sourcePage": "F4", "variableDefinitions": [ "Ω_yx transports frame x to y; Kₓ,K_y are branch frames; Π_yx is selected correspondence." ], "explanation": "Builds the transporter from neighboring branch frames and their selected matching.", "crossLinks": [ "CF081", "CF085", "CF093" ] }, { "id": "CF093", "displayNumber": "F93", "equation": "Ω_yxᵀηΩ_yx = η", "name": "Lorentz transport on matched frames", "status": "EXACT WHEN GRAMS MATCH", "sourcePage": "F4", "variableDefinitions": [ "Ω_yx is inter-cell transporter; η is Minkowski metric; matched frames have equal Gram data." ], "explanation": "The inter-cell map is Lorentz when the matched branch Gram structures agree.", "crossLinks": [ "CF082", "CF092" ] }, { "id": "CF094", "displayNumber": "F94", "equation": "C_b = (G₄⁻¹)ᵃ_cDₐH_cb", "name": "Four-component source-divergence candidate", "status": "EXACT REPRESENTATION", "sourcePage": "F4", "variableDefinitions": [ "C_b is branch source divergence; G₄⁻¹ is inverse Gram; Dₐ is branch derivative; H_cb is source data." ], "explanation": "Represents four source-divergence components in branch variables.", "crossLinks": [ "CF013", "CF015", "CF095", "CF102" ] }, { "id": "CF095", "displayNumber": "F95", "equation": "rank(H ↦ C_b) = 4", "name": "Capacity for four independent continuity equations", "status": "EXACT REPRESENTATION", "sourcePage": "F4", "variableDefinitions": [ "H is branch source data; C_b is its four-component divergence; rank is linear-map rank." ], "explanation": "States that the represented source map can carry four independent continuity constraints.", "crossLinks": [ "CF094", "CF102", "CF107" ] }, { "id": "CF096", "displayNumber": "F96", "equation": "Tᵢⱼ^(M) = K_M(∂ᵢM∂ⱼM − 1/2δᵢⱼ|∇M|²)", "name": "Weak scalar gradient-stress tensor", "status": "DEFINITION", "sourcePage": "F4", "variableDefinitions": [ "Tᵢⱼ^(M) is deformation stress; K_M is source modulus; M is deformation; δᵢⱼ is spatial identity." ], "explanation": "Defines the weak deformation field stress from its spatial gradient.", "crossLinks": [ "CF064", "CF097", "CF098" ] }, { "id": "CF097", "displayNumber": "F97", "equation": "∂ⱼTᵢⱼ^(M) = K_M(∂ᵢM)∇²M", "name": "Field-stress divergence; matter force is negative divergence", "status": "EXACT", "sourcePage": "F4", "variableDefinitions": [ "Tᵢⱼ^(M) is field stress; K_M is source modulus; M is deformation; ∂ᵢ is spatial derivative." ], "explanation": "Relates the divergence of the scalar gradient stress to the source Laplacian.", "crossLinks": [ "CF096", "CF098", "CF111" ] }, { "id": "CF098", "displayNumber": "F98", "equation": "K_M∇²M = ρc²", "name": "Weak source-modulus form", "status": "CONDITIONAL", "sourcePage": "F4", "variableDefinitions": [ "K_M is source modulus; M is deformation; ρ is mass density; c is light speed." ], "explanation": "Expresses the weak deformation source equation in terms of the source modulus.", "crossLinks": [ "CF067", "CF096", "CF097", "CF099", "CF201" ] }, { "id": "CF099", "displayNumber": "F99", "equation": "3 reciprocal strains → 1 trace + 2 anisotropic", "name": "Six-channel strain hierarchy", "status": "EXACT ALGEBRAIC", "sourcePage": "F4", "variableDefinitions": [ "The three reciprocal strains are the independent paired deformation channels; trace and anisotropic sectors are their decomposition." ], "explanation": "Decomposes three reciprocal strains into one trace mode and two anisotropic modes.", "crossLinks": [ "CF070", "CF073", "CF096" ] }, { "id": "CF100", "displayNumber": "F100", "equation": "ρ_f^rest = 0, u_f0 ≥ 0", "name": "Massless-fabric/background-energy candidate", "status": "OPEN HYPOTHESIS", "sourcePage": "F4", "variableDefinitions": [ "ρ_f^rest is rest-mass density of fabric; u_f0 is background fabric energy density." ], "explanation": "Keeps the fabric rest-massless while allowing a nonnegative background-energy candidate.", "crossLinks": [ "CF098", "CF126", "CF129" ] }, { "id": "CF101", "displayNumber": "F101", "equation": "Q_ab = k_a^μ k_b^ν h_μν", "name": "Branch encoding of metric perturbation", "status": "EXACT REPRESENTATION", "sourcePage": "F5", "variableDefinitions": [ "Q_ab is branch metric data; k_a^μ,k_b^ν are branch vectors; h_μν is metric perturbation." ], "explanation": "Projects a spacetime metric perturbation into the four-branch basis.", "crossLinks": [ "CF014", "CF103", "CF105", "CF107" ] }, { "id": "CF102", "displayNumber": "F102", "equation": "H_ab = k_a^μ k_b^ν T_μν", "name": "Branch encoding of stress-energy", "status": "EXACT REPRESENTATION", "sourcePage": "F5", "variableDefinitions": [ "H_ab is branch stress data; k_a^μ,k_b^ν are branch vectors; T_μν is stress-energy." ], "explanation": "Projects stress-energy into the same branch basis used for metric reconstruction.", "crossLinks": [ "CF094", "CF101", "CF104", "CF111" ] }, { "id": "CF103", "displayNumber": "F103", "equation": "h = (K⁻¹)ᵀQK⁻¹", "name": "Reconstruction of h_μν", "status": "EXACT", "sourcePage": "F5", "variableDefinitions": [ "h is spacetime metric perturbation; K is branch-frame matrix; Q is branch metric data." ], "explanation": "Reconstructs the spacetime perturbation from branch metric components.", "crossLinks": [ "CF101", "CF104", "CF105" ] }, { "id": "CF104", "displayNumber": "F104", "equation": "T = (K⁻¹)ᵀHK⁻¹", "name": "Reconstruction of T_μν", "status": "EXACT", "sourcePage": "F5", "variableDefinitions": [ "T is spacetime stress-energy; K is branch-frame matrix; H is branch stress data." ], "explanation": "Reconstructs stress-energy from its branch-space encoding.", "crossLinks": [ "CF102", "CF103", "CF111" ] }, { "id": "CF105", "displayNumber": "F105", "equation": "Q = (G₄⁻¹)ᵃᵇQ_ab", "name": "Branch-space trace", "status": "EXACT", "sourcePage": "F5", "variableDefinitions": [ "Q is branch trace; G₄⁻¹ is inverse branch Gram; Q_ab is branch metric data." ], "explanation": "Contracts branch metric data with the inverse Gram.", "crossLinks": [ "CF010", "CF013", "CF101", "CF106" ] }, { "id": "CF106", "displayNumber": "F106", "equation": "Q̄_ab = Q_ab − 1/2(G₄)_abQ", "name": "Branch trace reversal", "status": "EXACT", "sourcePage": "F5", "variableDefinitions": [ "Q̄_ab is trace-reversed branch metric data; Q_ab is branch data; Q is its trace; G₄ is Gram." ], "explanation": "Removes the branch trace in the standard trace-reversal combination.", "crossLinks": [ "CF101", "CF105", "CF107" ] }, { "id": "CF107", "displayNumber": "F107", "equation": "(G₄⁻¹)ᵃ_cDₐQ̄_cb = 0", "name": "Harmonic condition in branch variables", "status": "CONDITIONAL GAUGE", "sourcePage": "F5", "variableDefinitions": [ "Q̄_cb is trace-reversed branch perturbation; Dₐ is branch derivative; G₄⁻¹ is inverse Gram." ], "explanation": "Imposes the harmonic gauge condition after translating it to branch variables.", "crossLinks": [ "CF015", "CF016", "CF103", "CF106" ] }, { "id": "CF108", "displayNumber": "F108", "equation": "E_ab^(1)[Q] = 1/2[−□Q_ab + DₐD_cQ_cb + D_bD_cQ_ca − DₐD_bQ − (G₄)_abD_cDᵈQ_cd]", "name": "Linearized Einstein operator", "status": "EXACT UNDER CONTINUUM ASSUMPTIONS", "sourcePage": "F5", "variableDefinitions": [ "E_ab^(1) is linearized Einstein operator; Q_ab is branch perturbation; Dₐ are branch derivatives; □ is d’Alembertian." ], "explanation": "Writes the linearized Einstein operator in branch variables before gauge simplification.", "crossLinks": [ "CF016", "CF101", "CF105", "CF107" ] }, { "id": "CF109", "displayNumber": "F109", "equation": "DᵃE_ab^(1) = 0", "name": "Identical conservation of linearized operator", "status": "EXACT", "sourcePage": "F5", "variableDefinitions": [ "E_ab^(1) is the linearized Einstein operator; Dᵃ is the branch derivative with raised index." ], "explanation": "States the linearized Bianchi identity for the branch-space operator.", "crossLinks": [ "CF107", "CF108", "CF110" ] }, { "id": "CF110", "displayNumber": "F110", "equation": "δQ_ab = Dₐξ_b + D_bξₐ", "name": "Linearized gauge transformation", "status": "CONDITIONAL CONTINUUM", "sourcePage": "F5", "variableDefinitions": [ "δQ_ab is metric-data variation; ξ_a is infinitesimal branch gauge vector; Dₐ is branch derivative." ], "explanation": "Defines the infinitesimal gauge transformation of branch metric data.", "crossLinks": [ "CF101", "CF107", "CF108", "CF109" ] }, { "id": "CF111", "displayNumber": "F111", "equation": "E_ab = (8πG/c⁴)H_ab", "name": "Big-G-calibrated linearized source equation", "status": "CALIBRATED COMPATIBILITY", "sourcePage": "F5", "variableDefinitions": [ "E_ab is linearized field tensor; G is calibrated coupling; c is light speed; H_ab is branch stress data." ], "explanation": "Matches the linearized branch source equation to the Big-G normalization.", "crossLinks": [ "CF068", "CF102", "CF108", "CF109" ] }, { "id": "CF112", "displayNumber": "F112", "equation": "ψ = 2 tanh(M/2)", "name": "Two-function strong-field variable", "status": "CONDITIONAL COMPLETION", "sourcePage": "F5", "variableDefinitions": [ "ψ is bounded strong-field variable; M is deformation." ], "explanation": "Maps unbounded deformation M to a bounded variable for the strong-field completion.", "crossLinks": [ "CF061", "CF113", "CF114", "CF115" ] }, { "id": "CF113", "displayNumber": "F113", "equation": "T = e^M = (1+ψ/2)/(1−ψ/2)", "name": "Temporal metric factor", "status": "CONDITIONAL COMPLETION", "sourcePage": "F5", "variableDefinitions": [ "T is temporal metric factor; M is deformation; ψ is bounded strong-field variable." ], "explanation": "Defines the temporal metric factor in both deformation and bounded-variable forms.", "crossLinks": [ "CF061", "CF112", "CF114" ] }, { "id": "CF114", "displayNumber": "F114", "equation": "X = (1−ψ/2)²", "name": "Spatial metric factor", "status": "CONDITIONAL COMPLETION", "sourcePage": "F5", "variableDefinitions": [ "X is spatial metric factor; ψ is bounded strong-field variable." ], "explanation": "Defines the companion spatial factor used by the isotropic strong-field metric.", "crossLinks": [ "CF112", "CF113", "CF115" ] }, { "id": "CF115", "displayNumber": "F115", "equation": "ψ = −μ/r", "name": "Static spherical vacuum solution", "status": "EXACT WITHIN COMPLETION", "sourcePage": "F5", "variableDefinitions": [ "ψ is bounded strong-field variable; μ is spherical mass-length parameter; r is isotropic radius." ], "explanation": "Uses the inverse-radius vacuum solution for the bounded strong-field variable.", "crossLinks": [ "CF112", "CF116", "CF117", "CF118" ] }, { "id": "CF116", "displayNumber": "F116", "equation": "ds² = (1−μ/(2r))²/(1+μ/(2r))² c²dt² + (1+μ/(2r))⁴(dr²+r²dΩ²)", "name": "Isotropic Schwarzschild exterior", "status": "EXACT WITHIN COMPLETION", "sourcePage": "F5", "variableDefinitions": [ "ds² is spherical line element; μ is mass-length parameter; r is radius; dΩ² is unit-sphere metric." ], "explanation": "Gives the isotropic-coordinate spherical exterior produced by the two-function completion.", "crossLinks": [ "CF113", "CF114", "CF115", "CF117", "CF119" ] }, { "id": "CF117", "displayNumber": "F117", "equation": "R = r(1+μ/2r)²", "name": "Areal-radius transformation", "status": "EXACT", "sourcePage": "F5", "variableDefinitions": [ "R is areal radius; r is isotropic radius; μ is mass-length parameter." ], "explanation": "Converts the isotropic radius to the Schwarzschild areal radius.", "crossLinks": [ "CF115", "CF116", "CF118" ] }, { "id": "CF118", "displayNumber": "F118", "equation": "C_F = 1 − 2μ/R", "name": "Invariant areal-radius causal-support scalar", "status": "EXACT WITHIN SCHWARZSCHILD BRANCH", "sourcePage": "F5", "variableDefinitions": [ "C_F is causal-support scalar; μ is mass-length parameter; R is areal radius." ], "explanation": "Defines the invariant radial factor that vanishes at the spherical causal boundary.", "crossLinks": [ "CF117", "CF119", "CF120" ] }, { "id": "CF119", "displayNumber": "F119", "equation": "R_h = 2μ = 2GM/c²", "name": "Horizon radius with empirical Big G", "status": "CALIBRATED/GR COMPATIBILITY", "sourcePage": "F5", "variableDefinitions": [ "R_h is horizon radius; μ is mass-length parameter; G is empirical coupling; M is gravitating mass; c is light speed." ], "explanation": "Identifies the spherical horizon radius with the empirical Schwarzschild value.", "crossLinks": [ "CF116", "CF118", "CF120" ] }, { "id": "CF120", "displayNumber": "F120", "equation": "K = 48μ²/R⁶", "name": "Schwarzschild Kretschmann scalar", "status": "EXACT", "sourcePage": "F5", "variableDefinitions": [ "K is Kretschmann scalar; μ is mass-length parameter; R is areal radius." ], "explanation": "Gives the curvature invariant for the spherical vacuum branch.", "crossLinks": [ "CF117", "CF118", "CF184" ] }, { "id": "CF121", "displayNumber": "F121", "equation": "A → A + Bᵀχ", "name": "Finite-cell abelian-type gauge transformation", "status": "EXACT KINEMATICS", "sourcePage": "F5", "variableDefinitions": [ "A is edge potential; B is incidence matrix; χ is vertex gauge parameter." ], "explanation": "Defines the discrete gauge shift of an edge potential by a vertex coboundary.", "crossLinks": [ "CF017", "CF122", "CF123" ] }, { "id": "CF122", "displayNumber": "F122", "equation": "F = CᵀA", "name": "Discrete face/loop field strength", "status": "DEFINITION", "sourcePage": "F5", "variableDefinitions": [ "F is face/loop field strength; C is edge-to-face incidence; A is edge potential." ], "explanation": "Defines discrete curvature or field strength as the face incidence of the edge potential.", "crossLinks": [ "CF017", "CF121", "CF123" ] }, { "id": "CF123", "displayNumber": "F123", "equation": "F′ = F", "name": "Gauge invariance from BC = 0", "status": "EXACT KINEMATICS", "sourcePage": "F5", "variableDefinitions": [ "F is discrete field strength; F′ is its gauge-transformed value; B,C are incidence matrices." ], "explanation": "The field strength is unchanged because the boundary-of-boundary composition vanishes.", "crossLinks": [ "CF017", "CF121", "CF122" ] }, { "id": "CF124", "displayNumber": "F124", "equation": "F = dA, d²A = 0", "name": "Continuum closure", "status": "EXACT KINEMATICS", "sourcePage": "F5", "variableDefinitions": [ "A is a continuum gauge potential; F is its field strength; d is exterior derivative." ], "explanation": "States the continuum differential closure corresponding to the discrete incidence identity.", "crossLinks": [ "CF121", "CF122", "CF123" ] }, { "id": "CF125", "displayNumber": "F125", "equation": "S_γ = −(κ_γ/4)∫F_μνF^μν d⁴x", "name": "Minimal parity-even light action", "status": "CONDITIONAL COMPLETION", "sourcePage": "F5", "variableDefinitions": [ "S_γ is light action; κ_γ is coupling; F_μν is field strength; d⁴x is spacetime volume." ], "explanation": "Proposes the parity-even quadratic action for the continuum light-sector field.", "crossLinks": [ "CF122", "CF124", "CF126", "CF135" ] }, { "id": "CF126", "displayNumber": "F126", "equation": "∂_μF^{μν} = 0", "name": "Vacuum field equation", "status": "CONDITIONAL CONSEQUENCE", "sourcePage": "F6", "variableDefinitions": [ "F^{μν} is the raised light-sector field strength; ∂_μ is the spacetime derivative; ν is a spacetime index." ], "explanation": "The source-free Euler-Lagrange equation follows from the proposed light action in vacuum.", "crossLinks": [ "CF124", "CF125", "CF127", "CF129" ] }, { "id": "CF127", "displayNumber": "F127", "equation": "F = q ∧ a", "name": "Plane-wave closure form", "status": "EXACT CONSEQUENCE", "sourcePage": "F6", "variableDefinitions": [ "F is plane-wave field strength; q is wave covector; a is polarization one-form; ∧ is exterior product." ], "explanation": "Represents a closed plane-wave field as the wedge of its wave covector and polarization.", "crossLinks": [ "CF126", "CF128", "CF129", "CF132" ] }, { "id": "CF128", "displayNumber": "F128", "equation": "q_μF^{μν} = q²a^ν − (q·a)q^ν", "name": "Plane-wave co-closure", "status": "EXACT", "sourcePage": "F6", "variableDefinitions": [ "q is wave covector; F is plane-wave field; a is polarization; q² and q·a use the Lorentz metric." ], "explanation": "Contracts the plane-wave field with its wave covector and exposes the transverse condition.", "crossLinks": [ "CF127", "CF129", "CF130" ] }, { "id": "CF129", "displayNumber": "F129", "equation": "q² = 0", "name": "Nontrivial vacuum light is null", "status": "EXACT WITHIN CANDIDATE DYNAMICS", "sourcePage": "F6", "variableDefinitions": [ "q is the wave covector; q² is its Lorentz norm." ], "explanation": "Requires a nontrivial transverse vacuum wave to have a null wave covector.", "crossLinks": [ "CF126", "CF128", "CF130", "CF131" ] }, { "id": "CF130", "displayNumber": "F130", "equation": "ω² = c²k²", "name": "Vacuum light dispersion", "status": "EXACT WITHIN CANDIDATE DYNAMICS", "sourcePage": "F6", "variableDefinitions": [ "ω is angular frequency; k is spatial wave number; c is light speed." ], "explanation": "Writes the null-wave condition as the standard vacuum dispersion relation.", "crossLinks": [ "CF129", "CF131", "CF137" ] }, { "id": "CF131", "displayNumber": "F131", "equation": "dim(q⊥/span{q}) = 2", "name": "Two physical transverse modes", "status": "EXACT WITHIN CANDIDATE DYNAMICS", "sourcePage": "F6", "variableDefinitions": [ "q⊥ is the Lorentz-orthogonal polarization space; span{q} is the gauge direction." ], "explanation": "After quotienting the gauge direction, two independent transverse polarization modes remain.", "crossLinks": [ "CF127", "CF129", "CF132", "CF133" ] }, { "id": "CF132", "displayNumber": "F132", "equation": "F_± = F ∓ iJ_*F", "name": "Hodge helicity sectors", "status": "DEFINITION", "sourcePage": "F6", "variableDefinitions": [ "F_± are helicity sectors; F is field strength; J_* is the bivector Hodge operator; i²=−1." ], "explanation": "Splits the field into the two complex Hodge-helicity sectors.", "crossLinks": [ "CF090", "CF127", "CF133" ] }, { "id": "CF133", "displayNumber": "F133", "equation": "J_*F_± = ±iF_±", "name": "Helicity eigenvalue relation", "status": "EXACT", "sourcePage": "F6", "variableDefinitions": [ "J_* is Hodge star; F_± are helicity fields; i is the imaginary unit." ], "explanation": "Shows that each helicity sector is an eigenvector of the Hodge operator.", "crossLinks": [ "CF090", "CF132" ] }, { "id": "CF134", "displayNumber": "F134", "equation": "BJ = 0", "name": "Vertex conservation of edge-transfer current", "status": "EXACT", "sourcePage": "F6", "variableDefinitions": [ "B is vertex-edge incidence; J is edge-transfer current." ], "explanation": "Imposes vanishing net current at every vertex in the discrete cell.", "crossLinks": [ "CF017", "CF121", "CF135" ] }, { "id": "CF135", "displayNumber": "F135", "equation": "S_int = −g_AJᵀA", "name": "Minimal source coupling", "status": "CONDITIONAL IDENTIFICATION", "sourcePage": "F6", "variableDefinitions": [ "S_int is interaction action; g_A is coupling; J is edge current; A is edge potential." ], "explanation": "Couples the discrete current linearly to the edge potential.", "crossLinks": [ "CF121", "CF125", "CF134", "CF136" ] }, { "id": "CF136", "displayNumber": "F136", "equation": "κ_γCCᵀJ = g_AJ", "name": "Discrete sourced light equation", "status": "CONDITIONAL", "sourcePage": "F6", "variableDefinitions": [ "κ_γ is light stiffness; C is edge-face incidence; J is edge current; g_A is source coupling." ], "explanation": "Gives the sourced discrete light equation from the quadratic action and current coupling.", "crossLinks": [ "CF122", "CF125", "CF135", "CF137" ] }, { "id": "CF137", "displayNumber": "F137", "equation": "E(r) = g_AQ/(4πκ_γr²)", "name": "Coulomb-form static field", "status": "CONDITIONAL", "sourcePage": "F6", "variableDefinitions": [ "E(r) is radial static field; g_A is source coupling; Q is source charge; κ_γ is stiffness; r is radius." ], "explanation": "Provides the inverse-square static solution for a point source in the candidate light sector.", "crossLinks": [ "CF136", "CF138" ] }, { "id": "CF138", "displayNumber": "F138", "equation": "U_ij = e^{i a_ij}", "name": "Compact link variable", "status": "CONDITIONAL U(1) COMPLETION", "sourcePage": "F6", "variableDefinitions": [ "U_ij is a compact link; a_ij is a real link phase; i is the imaginary unit." ], "explanation": "Exponentiates the link phase to make the discrete variable compact and U(1)-valued.", "crossLinks": [ "CF139", "CF140" ] }, { "id": "CF139", "displayNumber": "F139", "equation": "ψ_i → g_i^nψ_i, n ∈ ℤ", "name": "Compact U(1) integer representation weights", "status": "EXACT GROUP THEORY", "sourcePage": "F6", "variableDefinitions": [ "ψ_i is a site field; g_i is a U(1) group element; n is an integer representation weight." ], "explanation": "Records the integer-weight action of compact U(1) on site fields.", "crossLinks": [ "CF138", "CF140" ] }, { "id": "CF140", "displayNumber": "F140", "equation": "A₄/V₄ ≅ ℤ₃", "name": "Native tetrahedral residual triality", "status": "EXACT GROUP THEORY", "sourcePage": "F6", "variableDefinitions": [ "A₄ is the tetrahedral rotation group; V₄ is its Klein-four normal subgroup; ℤ₃ is the quotient." ], "explanation": "Identifies the residual three-class quotient of the tetrahedral rotation group.", "crossLinks": [ "CF006", "CF138", "CF139" ] }, { "id": "CF141", "displayNumber": "F141", "equation": "W₀₁₂W₀₂₃W₀₃₁(Ω₀₁W₁₂₃Ω₁₀⁻¹) = I", "name": "Exact tetrahedral group-valued Bianchi closure", "status": "EXACT ALGEBRAIC", "sourcePage": "F7", "variableDefinitions": [ "W_ijk are face holonomies; Ω₀₁ transports between cells; I is the group identity." ], "explanation": "States exact ordered holonomy closure around the tetrahedral boundary.", "crossLinks": [ "CF083", "CF142", "CF146" ] }, { "id": "CF142", "displayNumber": "F142", "equation": "log(eᴷ¹eᴷ²⋯) = ΣK_i + 1/2Σ[K_i,K_j] + O(K³)", "name": "Formal BCH relation; additive curvature is the near-identity/small-curvature limit", "status": "LOCAL / SMALL-CURVATURE BCH", "sourcePage": "F7", "variableDefinitions": [ "K_i are near-identity Lie-algebra generators; [·,·] is a commutator; O(K³) is higher order." ], "explanation": "Relates group-valued holonomy products to additive curvature only in the local small-curvature limit.", "crossLinks": [ "CF141", "CF146", "CF147" ] }, { "id": "CF143", "displayNumber": "F143", "equation": "V₄,x = (2/3)L_c⁴[|detγ_x|/|detγ₀|]¹ᐟ²", "name": "Invariant G Model causal four-volume extension; not uniquely canonical", "status": "CANDIDATE INVARIANT EXTENSION", "sourcePage": "F7", "variableDefinitions": [ "V₄,x is causal four-volume at x; L_c is native length; γ_x and γ₀ are local and neutral metrics." ], "explanation": "Proposes a determinant-weighted four-volume while explicitly retaining non-uniqueness.", "crossLinks": [ "CF023", "CF144", "CF145" ] }, { "id": "CF144", "displayNumber": "F144", "equation": "Σ̂_A = Σ_A/L_c²; Γ̂_AB = ⟨Σ̂_A,Σ̂_B⟩", "name": "Dimensionless bivectors and six-space metric remove unit ambiguity", "status": "DEFINITION / DIMENSIONAL CLEANUP", "sourcePage": "F7", "variableDefinitions": [ "Σ_A is a bivector; Σ̂_A is dimensionless; Γ̂_AB is the six-space metric; L_c is native length." ], "explanation": "Normalizes bivectors by the native area so six-space contractions are dimensionless.", "crossLinks": [ "CF007", "CF091", "CF143", "CF145" ] }, { "id": "CF145", "displayNumber": "F145", "equation": "B^dual_B = −2(V₄,x/L_c²)Γ̂_ABJ_*Σ̂_A", "name": "Dimensionally explicit candidate dual-area bivector; [B^dual]=L²", "status": "CANDIDATE DEFINITION; UNIQUENESS OPEN", "sourcePage": "F7", "variableDefinitions": [ "B^dual is candidate dual bivector; V₄,x is four-volume; Γ̂ is six-metric; J_* is Hodge star; Σ̂ is normalized bivector." ], "explanation": "Defines a dimensionally explicit dual-area candidate but leaves uniqueness open.", "crossLinks": [ "CF090", "CF143", "CF144", "CF146" ] }, { "id": "CF146", "displayNumber": "F146", "equation": "R^disc_x = (2/L_c²)Γ̂^AB⟨Σ̂_A,K_B⟩; Σ_B⟨B^dual,J_*K⟩ = V₄,xR^disc_x", "name": "Discrete curvature has L⁻² units; contraction identity follows from definitions", "status": "IDENTITY EXACT GIVEN DEFINITIONS; R_disc→R OPEN", "sourcePage": "F7", "variableDefinitions": [ "R^disc_x is discrete scalar curvature; K_B is curvature bivector; Γ̂ is six-metric; V₄,x is four-volume." ], "explanation": "Defines discrete curvature and its dual-area contraction, while leaving the continuum limit open.", "crossLinks": [ "CF141", "CF143", "CF145", "CF147" ] }, { "id": "CF147", "displayNumber": "F147", "equation": "S_micro^G/ℏ = ζ_microΣ⟨B^dual/L_c²,(J_*+βI)K(W)⟩_Γ", "name": "Microscopic holonomy-curvature action with explicit unfixed normalization", "status": "CANDIDATE ACTION", "sourcePage": "F7", "variableDefinitions": [ "S_micro^G is microscopic gravitational action; ζ_micro is normalization; β is parity parameter; K(W) is holonomy curvature." ], "explanation": "Proposes a microscopic holonomy-curvature action with normalization still to be fixed.", "crossLinks": [ "CF142", "CF145", "CF146", "CF148" ] }, { "id": "CF148", "displayNumber": "F148", "equation": "ζ_eff ≡ ζ_microN_disc = 1/(4C_Σ) = 5π/48 ≈ 0.327249", "name": "Effective continuum normalization required by internal source matching", "status": "DERIVED MATCHING; MICRO FACTORS/UNIQUENESS OPEN", "sourcePage": "F7", "variableDefinitions": [ "ζ_eff is effective normalization; ζ_micro is microscopic normalization; N_disc is discrete count; C_Σ is retained coefficient." ], "explanation": "Matches the microscopic normalization to the continuum source coefficient, without claiming microscopic uniqueness.", "crossLinks": [ "CF032", "CF067", "CF147", "CF150" ] }, { "id": "CF149", "displayNumber": "F149", "equation": "|ΔS_m| = E_cellT_c = ℏA_cell", "name": "One-update matter action in constant-energy Aggregation", "status": "EXACT WITHIN AGGREGATION DEFINITIONS", "sourcePage": "F7", "variableDefinitions": [ "ΔS_m is one-update matter action; E_cell is cell energy; T_c is causal time; A_cell is loading." ], "explanation": "Relates one intrinsic-time matter update to both cell energy and dimensionless loading.", "crossLinks": [ "CF027", "CF041", "CF046", "CF050" ] }, { "id": "CF150", "displayNumber": "F150", "equation": "E^G_ab = R_ab − (1/2)γ_abR = 2C_Σ(cT_c²/ℏ)H_ab", "name": "Continuum source equation matched by action normalization", "status": "CONDITIONAL STRUCTURAL", "sourcePage": "F7", "variableDefinitions": [ "E^G_ab is Einstein tensor; R_ab and R are curvature; γ_ab is metric; H_ab is source; C_Σ is coefficient." ], "explanation": "States the continuum tensor source equation obtained from the retained action matching.", "crossLinks": [ "CF032", "CF102", "CF111", "CF148" ] }, { "id": "CF151", "displayNumber": "F151", "equation": "G_G = [3/(5π²)]c⁵T_c²/ℏ", "name": "G Model gravitational coupling used in Mercury benchmark", "status": "CONDITIONAL; CALIBRATED WHEN Tᶜ FROM BIG G", "sourcePage": "F7", "variableDefinitions": [ "G_G is G Model coupling; c is light speed; T_c is causal time; ℏ is action quantum." ], "explanation": "Defines the model coupling used by the orbital benchmark, with its numerical time calibration conditional.", "crossLinks": [ "CF036", "CF037", "CF153", "CF154" ] }, { "id": "CF152", "displayNumber": "F152", "equation": "β_G = 1, γ_G = 1", "name": "First post-Newtonian coefficients in the smooth metric-compatible vacuum closure", "status": "CONDITIONAL CONTINUUM CONSEQUENCE", "sourcePage": "F7", "variableDefinitions": [ "β_G and γ_G are post-Newtonian coefficients in the smooth metric closure." ], "explanation": "Records the retained first post-Newtonian coefficients of the smooth vacuum branch.", "crossLinks": [ "CF116", "CF151", "CF153" ] }, { "id": "CF153", "displayNumber": "F153", "equation": "Δϖ_G = [18/(5π)]c³T_c²M_Sun/[ℏa(1−e²)]", "name": "Mercury perihelion advance; M_Sun avoids conflict with deformation variable M", "status": "CONDITIONAL FORM; Tᶜ CALIBRATION MAY CONTAIN BIG G", "sourcePage": "F7", "variableDefinitions": [ "Δϖ_G is perihelion advance; M_Sun is solar mass; a is semimajor axis; e is eccentricity; T_c is causal time." ], "explanation": "Gives the retained Mercury advance in model variables and flags its calibration dependence.", "crossLinks": [ "CF119", "CF151", "CF152", "CF154" ] }, { "id": "CF154", "displayNumber": "F154", "equation": "T_c = 2.18657×10⁻⁴³ s ⇒ Δϖ_G = 42.9826 arcsec/Julian century", "name": "Mercury numerical benchmark", "status": "CALIBRATED COMPATIBILITY", "sourcePage": "F7", "variableDefinitions": [ "T_c is calibrated causal time; Δϖ_G is predicted perihelion advance; arcsec is angular arcsecond." ], "explanation": "Reports the numerical Mercury comparison after inserting the calibrated intrinsic time.", "crossLinks": [ "CF038", "CF151", "CF153" ] }, { "id": "CF155", "displayNumber": "F155", "equation": "w(x)=x ⇒ ∫₀¹4x³w(x)dx=4/5; ⟨|cosφ|+|sinφ|⟩=4/π", "name": "Post-7 Sep rationale for retained depth and Hodge-throughput factors", "status": "CONDITIONAL PHYSICAL-SELECTION RATIONALE", "sourcePage": "F7", "variableDefinitions": [ "w(x) is a radial weight; x is normalized radius; φ is angular coordinate." ], "explanation": "Records the weighting and angular averages used as a physical-selection rationale for retained geometric factors.", "crossLinks": [ "CF032", "CF148" ] }, { "id": "CF156", "displayNumber": "F156", "equation": "e = q − q_eq", "name": "Centered mismatch definition", "status": "DEFINITION", "sourcePage": "F8", "variableDefinitions": [ "e is mismatch; q is current centered shape; q_eq is target or equilibrium shape." ], "explanation": "Defines the deviation of the current shape from its selected target.", "crossLinks": [ "CF086", "CF157", "CF158" ] }, { "id": "CF157", "displayNumber": "F157", "equation": "T_c dq/dτ = −(I₆−P)e", "name": "Causal re-equalization flow", "status": "STRUCTURAL EFFECTIVE STRUCTURAL CANDIDATE", "sourcePage": "F8", "variableDefinitions": [ "q is shape; τ is update time; P is target projector; e is mismatch; I₆ is channel identity." ], "explanation": "Provides the effective re-equalization flow retained for the special tracking construction.", "crossLinks": [ "CF087", "CF156", "CF158", "CF159", "CF161" ] }, { "id": "CF158", "displayNumber": "F158", "equation": "q=(α,β,γ,γ,β,α), α+β+γ=0; Pq=−q/2", "name": "Centered physical shape sector", "status": "EXACT IN CENTERED SECTOR", "sourcePage": "F8", "variableDefinitions": [ "q is centered six-channel shape; α,β,γ are paired shape coordinates; P is target projector." ], "explanation": "Specifies the paired centered sector and its target-projector eigenvalue.", "crossLinks": [ "CF086", "CF156", "CF157", "CF162", "CF167" ] }, { "id": "CF159", "displayNumber": "F159", "equation": "e(τ)=e(0)e^{−3τ/(2T_c)}; τ_rec=2T_c/3", "name": "Fixed-target centered tracking / recovery under CF157", "status": "EXACT GIVEN CF157", "sourcePage": "F8", "variableDefinitions": [ "e is mismatch; τ is tracking time; T_c is causal time; τ_rec is recovery timescale." ], "explanation": "Solves the fixed-target mismatch flow and identifies its decay timescale.", "crossLinks": [ "CF157", "CF156", "CF160", "CF161" ] }, { "id": "CF160", "displayNumber": "F160", "equation": "R(t)=exp[−(I₆−P)t]", "name": "Effective tracking operator", "status": "DEFINITION", "sourcePage": "F8", "variableDefinitions": [ "R(t) is tracking operator; P is target projector; I₆ is six-channel identity." ], "explanation": "Defines the operator exponential implementing the effective centered tracking flow.", "crossLinks": [ "CF157", "CF159", "CF161" ] }, { "id": "CF161", "displayNumber": "F161", "equation": "R(δ₁)R(δ₂)=R(δ₁+δ₂)", "name": "Tracking semigroup", "status": "EXACT GIVEN CF157", "sourcePage": "F8", "variableDefinitions": [ "R is effective tracking operator; δ₁ and δ₂ are successive tracking intervals." ], "explanation": "The fixed-target recovery operators compose as a one-parameter semigroup.", "crossLinks": [ "CF159", "CF160" ] }, { "id": "CF162", "displayNumber": "F162", "equation": "E₁=(2/3)P+ (1/3)P̄; E₀=I+P−2P²; E_minus=(4/3)(P²−P)", "name": "Exact spectral projectors", "status": "EXACT", "sourcePage": "F8", "variableDefinitions": [ "P is target projector; P̄ is its complementary sector; E₁,E₀,E_minus are spectral operators." ], "explanation": "Lists the retained projector decomposition of the centered tracking operator.", "crossLinks": [ "CF158", "CF160", "CF163" ] }, { "id": "CF163", "displayNumber": "F163", "equation": "R=E₀+e^{−δ}E₁+e^{−3δ/2}E_minus", "name": "Spectral tracking decomposition", "status": "EXACT GIVEN CF157", "sourcePage": "F8", "variableDefinitions": [ "R is tracking operator; δ is normalized time; E₀,E₁,E_minus are spectral projectors." ], "explanation": "Expresses recovery as decay in the nonzero spectral sectors while retaining the neutral sector.", "crossLinks": [ "CF159", "CF160", "CF162", "CF164" ] }, { "id": "CF164", "displayNumber": "F164", "equation": "dV_L/dτ = −(1/T_c)eᵀ(I−P)e ≤ 0", "name": "Lyapunov stability of the effective tracking candidate", "status": "EXACT GIVEN CF157", "sourcePage": "F8", "variableDefinitions": [ "V_L is Lyapunov function; e is mismatch; P is target projector; T_c is causal time." ], "explanation": "Shows non-increasing mismatch energy for the effective candidate flow.", "crossLinks": [ "CF156", "CF157", "CF163" ] }, { "id": "CF165", "displayNumber": "F165", "equation": "L_c²∇²M = (3C_Σ/2)A_cell = [18/(5π)]T_c/D₀", "name": "Aggregation-curvature bridge", "status": "EXACT WITHIN COMBINED BRANCH", "sourcePage": "F8", "variableDefinitions": [ "L_c is causal length; M is deformation; C_Σ is coefficient; A_cell is loading; D₀ is aggregation time." ], "explanation": "Connects deformation curvature to cell loading and the aggregation-time ratio.", "crossLinks": [ "CF032", "CF046", "CF047", "CF067" ] }, { "id": "CF166", "displayNumber": "F166", "equation": "E_ij = ∂_i∂_jM − (1/6)δ_ij∇²M", "name": "Weak-static traceless tidal tensor", "status": "DEFINITION", "sourcePage": "F8", "variableDefinitions": [ "E_ij is traceless tidal tensor; M is deformation; δ_ij is spatial identity; ∇² is Laplacian." ], "explanation": "Removes the scalar trace from the Hessian of the weak deformation field.", "crossLinks": [ "CF064", "CF096", "CF167" ] }, { "id": "CF167", "displayNumber": "F167", "equation": "q_eq = L_c²(ε₁, ε₂, ε₃, ε₃, ε₂, ε₁)", "name": "Tidal source-to-target shape direction", "status": "CONDITIONAL STRONG PASS; ρ OPEN", "sourcePage": "F8", "variableDefinitions": [ "q_eq is target shape; ε_i are eigenvalues of the tidal tensor; L_c is causal length; ρ is density contrast." ], "explanation": "Maps the ordered tidal eigenvalues into the paired target channel direction; its source weighting remains open.", "crossLinks": [ "CF158", "CF166", "CF168", "CF192" ] }, { "id": "CF168", "displayNumber": "F168", "equation": "β_A=2/3; β_B=1/6; β_A/β_B=4", "name": "Two native target-normalization routes", "status": "OPEN NORMALIZATION AUDIT", "sourcePage": "F9", "variableDefinitions": [ "β_A and β_B are candidate normalization coefficients for two target routes." ], "explanation": "Records two retained normalization choices and their factor-four ratio without selecting one.", "crossLinks": [ "CF167", "CF173", "CF187" ] }, { "id": "CF169", "displayNumber": "F169", "equation": "E_minus(A_cell 1_6)=0", "name": "Scalar loading cannot directly create centered shape", "status": "EXACT NO-GO BY SYMMETRY", "sourcePage": "F9", "variableDefinitions": [ "E_minus is anisotropic spectral projector; A_cell1₆ is uniform scalar loading." ], "explanation": "Uniform scalar loading lies outside the centered anisotropic target sector.", "crossLinks": [ "CF046", "CF162", "CF167" ] }, { "id": "CF170", "displayNumber": "F170", "equation": "q_{n+1}=q_eq,n+1+R(δ_n)(q_n−q_eq,n+1)", "name": "Layered Celal + effective tracking update", "status": "CANDIDATE CF157 UPDATE; DEPENDS ON CF157", "sourcePage": "F9", "variableDefinitions": [ "q_n is current shape; q_eq,n+1 is next target; R is recovery operator; δ_n is update interval." ], "explanation": "Updates toward a changing target by applying the fixed-target recovery operator at each layer.", "crossLinks": [ "CF157", "CF159", "CF160", "CF167" ] }, { "id": "CF171", "displayNumber": "F171", "equation": "de/dτ + [3/(2T_c)]e = −dq_eq/dτ", "name": "Moving-target tracking equation", "status": "EXACT GIVEN CF157", "sourcePage": "F9", "variableDefinitions": [ "e is target mismatch; q_eq is moving target; τ is time; T_c is causal time." ], "explanation": "Adds the target-motion forcing term to the fixed-target mismatch decay equation.", "crossLinks": [ "CF157", "CF159", "CF170" ] }, { "id": "CF172", "displayNumber": "F172", "equation": "R_rate ≡ τ_rec/D₀ = (2/3)A_cell", "name": "Effective tracking-timescale to Aggregation ratio", "status": "EXACT GIVEN COMBINED CANDIDATES", "sourcePage": "F9", "variableDefinitions": [ "R_rate is timescale ratio; τ_rec is recovery time; D₀ is aggregation time; A_cell is loading." ], "explanation": "Relates recovery speed to aggregation through the intrinsic loading ratio.", "crossLinks": [ "CF047", "CF049", "CF089", "CF159" ] }, { "id": "CF173", "displayNumber": "F173", "equation": "R_rate=1 ⇔ A_cell=3/2 ⇔ D₀=2T_c/3", "name": "Candidate rate-matching point; not a fundamental-physics threshold", "status": "CONDITIONAL COMBINED-CANDIDATE RESULT", "sourcePage": "F9", "variableDefinitions": [ "R_rate is rate ratio; A_cell is loading; D₀ is aggregation time; T_c is intrinsic time." ], "explanation": "Identifies the candidate point where recovery and aggregation rates coincide, without promoting it to a universal threshold.", "crossLinks": [ "CF047", "CF089", "CF172" ] }, { "id": "CF174", "displayNumber": "F174", "equation": "A₁₂=A₃₄=xL_c²; A₁₃=A₂₄=yL_c²; A₁₄=A₂₃=zL_c²; x+y+z=4", "name": "Centered opposite-pair coordinates", "status": "EXACT CENTERED GEOMETRY", "sourcePage": "F9", "variableDefinitions": [ "A_ij are opposite pair areas; x,y,z are normalized pair coordinates; L_c is native length." ], "explanation": "Parametrizes the three opposite pair-area classes in the centered tetrahedral geometry.", "crossLinks": [ "CF175", "CF176", "CF177", "CF179" ] }, { "id": "CF175", "displayNumber": "F175", "equation": "γ/L_c² = [[0,x,y],[x,0,z],[y,z,0]]", "name": "Centered four-null Gram matrix", "status": "EXACT GIVEN CF174", "sourcePage": "F9", "variableDefinitions": [ "γ is centered three-coordinate Gram matrix; x,y,z are pair coordinates; L_c is native length." ], "explanation": "Gives the centered Gram matrix in the opposite-pair coordinates.", "crossLinks": [ "CF174", "CF176", "CF180" ] }, { "id": "CF176", "displayNumber": "F176", "equation": "spec(γ/L_c²) = {−(x+y+z), x+y−z, x−y+z, −x+y+z}", "name": "Centered Gram spectrum", "status": "EXACT", "sourcePage": "F9", "variableDefinitions": [ "γ is centered Gram matrix; x,y,z are opposite-pair coordinates; spec denotes eigenvalues." ], "explanation": "Lists the eigenvalues governing centered Xyz causal integrity.", "crossLinks": [ "CF175", "CF177", "CF180", "CF181" ] }, { "id": "CF177", "displayNumber": "F177", "equation": "−det(γ/L_c²) = (x+y−z)(x−y+z)(−x+y+z)", "name": "Xyz determinant factorization", "status": "EXACT", "sourcePage": "F9", "variableDefinitions": [ "γ is centered Gram matrix; x,y,z are pair coordinates; det is determinant." ], "explanation": "Factors the centered Gram determinant into the three causal-integrity margins.", "crossLinks": [ "CF175", "CF176", "CF178", "CF181" ] }, { "id": "CF178", "displayNumber": "F178", "equation": "−det(γ/L_c²) = 16A_tri²", "name": "Xyz Law / Heron form", "status": "EXACT CENTERED GEOMETRY", "sourcePage": "F9", "variableDefinitions": [ "A_tri is triangle area; x,y,z are centered pair coordinates; γ is centered Gram matrix." ], "explanation": "Identifies the determinant with the Heron-form squared area of the associated triangle.", "crossLinks": [ "CF177", "CF181", "CF182" ] }, { "id": "CF179", "displayNumber": "F179", "equation": "x=4/3+α; y=4/3+β; z=4/3+γ; α+β+γ=0", "name": "Centered deviation coordinates", "status": "EXACT PARAMETRIZATION", "sourcePage": "F9", "variableDefinitions": [ "x,y,z are pair coordinates; α,β,γ are zero-sum deviations." ], "explanation": "Centers the pair coordinates around the neutral value 4/3.", "crossLinks": [ "CF174", "CF177", "CF178", "CF180" ] }, { "id": "CF180", "displayNumber": "F180", "equation": "spec(γ)=L_c²{−4, 4/3−2α, 4/3−2β, 4/3−2γ}", "name": "Centered Xyz Gram spectrum", "status": "EXACT", "sourcePage": "F10", "variableDefinitions": [ "γ is centered Gram matrix; α,β,γ are centered deviations; L_c is native length." ], "explanation": "Rewrites the centered Gram spectrum in deviation coordinates.", "crossLinks": [ "CF176", "CF179", "CF181", "CF182" ] }, { "id": "CF181", "displayNumber": "F181", "equation": "detγ=0 ⇔ α=2/3 or β=2/3 or γ=2/3", "name": "Centered Xyz degeneracy boundary", "status": "EXACT", "sourcePage": "F10", "variableDefinitions": [ "γ is centered Gram matrix; α,β,γ are deviation coordinates." ], "explanation": "Locates the three rank-loss boundaries of centered Xyz geometry.", "crossLinks": [ "CF177", "CF180", "CF182", "CF204" ] }, { "id": "CF182", "displayNumber": "F182", "equation": "detγ = −3s(s−2α)(s−2β)(s−2γ)L_c⁸", "name": "General common-mode Xyz determinant", "status": "EXACT GIVEN x=s+α", "sourcePage": "F10", "variableDefinitions": [ "s is common mode; α,β,γ are deviations; γ is centered Gram matrix; L_c is native length." ], "explanation": "Factors the determinant for a general common mode and isolates the three causal margins.", "crossLinks": [ "CF179", "CF181", "CF183", "CF204" ] }, { "id": "CF183", "displayNumber": "F183", "equation": "det(ΓᵀγΓ)=detγ", "name": "Celal/permutation invariance of centered degeneracy", "status": "EXACT", "sourcePage": "F10", "variableDefinitions": [ "Γ is a permutation or Celal frame map; γ is centered Gram matrix." ], "explanation": "Shows that centered degeneracy is unchanged by the permitted branch permutation.", "crossLinks": [ "CF085", "CF181", "CF182" ] }, { "id": "CF184", "displayNumber": "F184", "equation": "{ε_r,ε_t,ε_t}={r_s/r, r_s/(2r), −r_s/(2r)}", "name": "Schwarzschild principal tidal pattern used in Xyz test", "status": "CONDITIONAL STRONG-FIELD INPUT", "sourcePage": "F10", "variableDefinitions": [ "ε_r and ε_t are radial and tangential tidal eigenvalues; r_s is Schwarzschild radius; r is radius." ], "explanation": "Supplies the radial/tangential tidal eigenvalue pattern used to test Xyz geometry.", "crossLinks": [ "CF166", "CF185", "CF188" ] }, { "id": "CF185", "displayNumber": "F185", "equation": "r_Xyz = (3β/2)r_sL_c²", "name": "Schwarzschild-Xyz radius", "status": "CONDITIONAL COROLLARY", "sourcePage": "F10", "variableDefinitions": [ "r_Xyz is Xyz radius; β is normalization; r_s is Schwarzschild radius; L_c is causal length." ], "explanation": "Defines the radius at which the Schwarzschild tidal pattern is compared with the Xyz scale.", "crossLinks": [ "CF168", "CF184", "CF186" ] }, { "id": "CF186", "displayNumber": "F186", "equation": "r_Xyz,A=(r_s/β)^(1/3); r_Xyz,B=r_s(L_c/4)^(1/3)", "name": "Route-A / Route-B-Xyz radii", "status": "CONDITIONAL; β AMBIGUITY", "sourcePage": "F10", "variableDefinitions": [ "r_Xyz,A and r_Xyz,B are candidate radii; r_s is Schwarzschild radius; β is route coefficient; L_c is causal length." ], "explanation": "Lists the two radius scalings generated by the competing normalization routes.", "crossLinks": [ "CF168", "CF185", "CF187" ] }, { "id": "CF187", "displayNumber": "F187", "equation": "K_Xyz=[16/(3β²)]L_c⁻⁴ ⇒ 12/L_c⁴ (A), 192/L_c⁴ (B)", "name": "Mass-independent Xyz curvature threshold", "status": "CONDITIONAL COROLLARY", "sourcePage": "F10", "variableDefinitions": [ "K_Xyz is threshold curvature; β is route coefficient; L_c is causal length; A and B label routes." ], "explanation": "Gives the route-dependent curvature threshold after eliminating source mass.", "crossLinks": [ "CF168", "CF185", "CF186", "CF190" ] }, { "id": "CF188", "displayNumber": "F188", "equation": "ψ₂=−m_0/(r−ia cosθ)³; 2|ψ₂|=r_s/(r²+ a²cos²θ)³ᐟ²", "name": "Kerr invariant principal tidal amplitude", "status": "EXACT KERR GEOMETRY / INPUT", "sourcePage": "F10", "variableDefinitions": [ "ψ₂ is Kerr Weyl scalar; m₀ is mass parameter; r is radius; a is spin; θ is polar angle; r_s is Schwarzschild radius." ], "explanation": "Supplies the Kerr principal curvature amplitude used in the rotating extension.", "crossLinks": [ "CF184", "CF189", "CF190" ] }, { "id": "CF189", "displayNumber": "F189", "equation": "r²+a²cos²θ=R_Xyz²; R_Xyz=[(3β/2)r_sL_c²]¹ᐟ³", "name": "Kerr Xyz candidate surface", "status": "CONDITIONAL INVARIANT EXTENSION", "sourcePage": "F10", "variableDefinitions": [ "R_Xyz is candidate surface radius; r,a,θ are Kerr coordinates; β is normalization; r_s is Schwarzschild radius." ], "explanation": "Extends the spherical Xyz surface condition to the Kerr invariant radius.", "crossLinks": [ "CF185", "CF186", "CF188", "CF190" ] }, { "id": "CF190", "displayNumber": "F190", "equation": "W=48|ψ₂|²=12r_s²/Σ³; W_Xyz=[16/(3β²)]L_c⁻⁴", "name": "Kerr invariant curvature at Xyz boundary", "status": "CONDITIONAL COROLLARY", "sourcePage": "F10", "variableDefinitions": [ "W is Weyl curvature invariant; ψ₂ is Kerr scalar; Σ=r²+a²cos²θ; β is normalization." ], "explanation": "Evaluates the Kerr invariant at the candidate Xyz boundary.", "crossLinks": [ "CF187", "CF188", "CF189" ] }, { "id": "CF191", "displayNumber": "F191", "equation": "ε_r=(2A/3)(ρ−ρ̄); ε_t=−(A/3)(ρ−ρ̄), A=C_Σc³T_c²/ℏ", "name": "Spherical traceless tidal density-contrast law", "status": "EXACT WITHIN WEAK SPHERICAL BRANCH", "sourcePage": "F10", "variableDefinitions": [ "ε_r,ε_t are radial/tangential tidal strains; ρ,ρ̄ are local and mean densities; A is source coefficient." ], "explanation": "Relates the traceless spherical tidal pattern to density contrast in the weak branch.", "crossLinks": [ "CF067", "CF184", "CF192", "CF193" ] }, { "id": "CF192", "displayNumber": "F192", "equation": "|ρ̄−ρ|_Xyz = ℏ/(βC_Σc⁵T_c⁴)", "name": "Spherical Xyz density-contrast threshold", "status": "CONDITIONAL SOURCE-TO-SHAPE COROLLARY", "sourcePage": "F11", "variableDefinitions": [ "ρ̄−ρ is density contrast; β is normalization; C_Σ is coefficient; c is light speed; T_c is causal time." ], "explanation": "Converts the Xyz curvature condition into a spherical density-contrast threshold.", "crossLinks": [ "CF032", "CF167", "CF191", "CF193" ] }, { "id": "CF193", "displayNumber": "F193", "equation": "A_Δ≡(2/3)|ρ̄−ρ|c⁵T_c⁴/ℏ; A_Δ,Xyz=2/(3βC_Σ)", "name": "Dimensionless contrast loading and threshold", "status": "EXACT GIVEN DEFINITIONS / β OPEN", "sourcePage": "F11", "variableDefinitions": [ "A_Δ is contrast loading; ρ̄−ρ is density contrast; β and C_Σ are normalization coefficients." ], "explanation": "Defines dimensionless density-contrast loading and its Xyz threshold.", "crossLinks": [ "CF032", "CF168", "CF192" ] }, { "id": "CF194", "displayNumber": "F194", "equation": "r_s^ν=(3β/2)L_c; r_Xyz=r_s", "name": "Horizon-Xyz crossover", "status": "CONDITIONAL", "sourcePage": "F11", "variableDefinitions": [ "r_s is Schwarzschild radius; r_s^ν is model crossover scale; β is normalization; L_c is causal length." ], "explanation": "States the conditional radius at which the Xyz scale coincides with a Schwarzschild scale.", "crossLinks": [ "CF185", "CF195", "CF196" ] }, { "id": "CF195", "displayNumber": "F195", "equation": "M_X=(L_c/2G_G)[(3β/2)^(1/3)]", "name": "Conditional black-hole crossover mass", "status": "CONDITIONAL; CALIBRATED BRANCH", "sourcePage": "F11", "variableDefinitions": [ "M_X is crossover mass; L_c is causal length; G_G is model coupling; β is normalization." ], "explanation": "Converts the crossover radius into a mass using the calibrated model coupling.", "crossLinks": [ "CF151", "CF194", "CF196" ] }, { "id": "CF196", "displayNumber": "F196", "equation": "ρ̄_X = 3M_X/(4πr_s³) = ℏ/(βC_Σc⁵T_c⁴) = Δρ_Xyz", "name": "Density/horizon form of same Xyz crossover", "status": "EXACT ALGEBRA GIVEN CF194-CF195", "sourcePage": "F11", "variableDefinitions": [ "ρ̄_X is crossover mean density; M_X is crossover mass; r_s is Schwarzschild radius; Δρ_Xyz is threshold contrast." ], "explanation": "Shows the equivalent density, horizon, and Xyz threshold forms of the crossover.", "crossLinks": [ "CF192", "CF194", "CF195" ] }, { "id": "CF197", "displayNumber": "F197", "equation": "T_c = [625π²ℏG_G³a(1−e²)Δϖ/(18M_Sun)]¹ᐟ²", "name": "Mercury back-inversion for T_c", "status": "OBSERVATIONAL INVERSION / CONSISTENCY", "sourcePage": "F11", "variableDefinitions": [ "T_c is inferred causal time; G_G is coupling; a is semimajor axis; e is eccentricity; Δϖ is perihelion advance." ], "explanation": "Inverts the Mercury expression to infer the intrinsic time from orbital observables.", "crossLinks": [ "CF038", "CF151", "CF153", "CF198" ] }, { "id": "CF198", "displayNumber": "F198", "equation": "K_M=(25π³/12)ℏK_GD/T_c²", "name": "K_M reconstruction from K_GD and T_c", "status": "EXACT ALGEBRA GIVEN RETAINED K_M LAW", "sourcePage": "F11", "variableDefinitions": [ "K_M is source modulus; K_GD is interface constant; ℏ is action; T_c is causal time." ], "explanation": "Reconstructs the source modulus from the retained interface constant and intrinsic time.", "crossLinks": [ "CF034", "CF151", "CF197", "CF201" ] }, { "id": "CF199", "displayNumber": "F199", "equation": "G_eff = Δa_mod/[2M_mod(r_s⁻²−r_f⁻²)]", "name": "Two-cloud measured effective gravitational coupling", "status": "EXPERIMENTAL RECONSTRUCTION", "sourcePage": "F11", "variableDefinitions": [ "G_eff is measured coupling; Δa_mod is acceleration difference; M_mod is source mass; r_s,r_f are source and far radii." ], "explanation": "Defines an experimental two-radius reconstruction of effective coupling from differential acceleration.", "crossLinks": [ "CF151", "CF200", "CF201" ] }, { "id": "CF200", "displayNumber": "F200", "equation": "T_c = [15625π²ℏK_GD⁵Δa_mod/(96M_mod(r_s⁻²−r_f⁻²))]¹ᐟ²", "name": "Laboratory T_c reconstruction", "status": "EXPERIMENTAL RECONSTRUCTION FORM", "sourcePage": "F11", "variableDefinitions": [ "T_c is reconstructed causal time; K_GD is interface constant; Δa_mod and M_mod are measured source data; r_s,r_f are radii." ], "explanation": "Combines the two-cloud reconstruction with the retained modulus law to infer intrinsic time.", "crossLinks": [ "CF198", "CF199", "CF201" ] }, { "id": "CF201", "displayNumber": "F201", "equation": "K_M = M_s(r_s⁻²−r_f⁻²)/[1250π⁹K_GD⁴Δa_mod]", "name": "Laboratory K_M reconstruction", "status": "EXPERIMENTAL RECONSTRUCTION FORM", "sourcePage": "F11", "variableDefinitions": [ "K_M is source modulus; M_s is source mass; r_s,r_f are radii; K_GD is interface constant; Δa_mod is measured acceleration." ], "explanation": "Provides the corresponding direct laboratory reconstruction of source modulus.", "crossLinks": [ "CF098", "CF199", "CF200" ] }, { "id": "CF202", "displayNumber": "F202", "equation": "T_H,X = ℏ/[4πk_BT_C√(3β/2)]", "name": "Hawking temperature at conditional Xyz crossover", "status": "SEMICLASSICAL COMPARISON ONLY", "sourcePage": "F11", "variableDefinitions": [ "T_H,X is Hawking temperature; ℏ is action; k_B is Boltzmann constant; T_C is crossover time; β is normalization." ], "explanation": "Compares the conditional Xyz crossover with a semiclassical Hawking temperature scale.", "crossLinks": [ "CF194", "CF195", "CF203" ] }, { "id": "CF203", "displayNumber": "F203", "equation": "T_H,X = (3200π³/3)(3β/2)³ᐟ²T_c", "name": "Standard Schwarzschild lifetime evaluated at crossover", "status": "SEMICLASSICAL COMPARISON ONLY", "sourcePage": "F11", "variableDefinitions": [ "T_H,X is comparison temperature; β is normalization; T_c is intrinsic causal time." ], "explanation": "Re-expresses the crossover Hawking comparison solely in the intrinsic-time parameterization.", "crossLinks": [ "CF202", "CF194" ] }, { "id": "CF204", "displayNumber": "F204", "equation": "χ_Xyz ≡ min{s−2α, s−2β, s−2γ_d}", "name": "Xyz causal-integrity margin", "status": "DEFINITION GIVEN CF182", "sourcePage": "F12", "variableDefinitions": [ "χ_Xyz is causal-integrity margin; s is common mode; α,β,γ_d are deviation coordinates." ], "explanation": "Defines the smallest distance to any of the three determinant rank-loss boundaries.", "crossLinks": [ "CF181", "CF182", "CF205", "CF207" ] }, { "id": "CF205", "displayNumber": "F205", "equation": "u_x=(1,1,−1,−1), γu_x=0; k₁+k₂−k₃−k₄=0 at α=s/2", "name": "Failed simple α-edge Xyz mode and branch dependence", "status": "EXACT AT SIMPLE α EDGE", "sourcePage": "F12", "variableDefinitions": [ "u_x is candidate null mode; γ is centered Gram matrix; k_i are branch vectors; α,s are shape coordinates." ], "explanation": "Identifies a null mode at a simple edge and shows the corresponding branch-sum relation.", "crossLinks": [ "CF181", "CF204", "CF206" ] }, { "id": "CF206", "displayNumber": "F206", "equation": "λ_α=s−2α; (dλ_α/dτ)_CF157=(3/T_c)(s/2−α_eq) at α=s/2; dλ/dτ=0", "name": "Boundary-flow sign under the effective CF157 tracking candidate", "status": "EXACT GIVEN CF157; NOT GENERAL TRIGGER", "sourcePage": "F12", "variableDefinitions": [ "λ_α is α-edge margin; α_eq is target coordinate; τ is time; T_c is causal time." ], "explanation": "Evaluates the effective candidate flow at the α-edge and explicitly does not promote it to a general trigger.", "crossLinks": [ "CF157", "CF204", "CF205", "CF207" ] }, { "id": "CF207", "displayNumber": "F207", "equation": "χ_Xyz=0 and D_τχ_Xyz<0 ⇒ smooth ordinary 1+3 continuation cannot remain in retained phase unchanged", "name": "General Xyz outward-crossing causal-integrity condition", "status": "GENERAL GEOMETRIC / KINEMATIC NECESSARY CONDITION; DYNAMICS OPEN", "sourcePage": "F12", "variableDefinitions": [ "χ_Xyz is integrity margin; D_τ is material derivative; τ is evolution time." ], "explanation": "States the necessary geometric condition for an outward crossing while leaving the dynamical mechanism open.", "crossLinks": [ "CF204", "CF206", "CF208", "CF217" ] }, { "id": "CF208", "displayNumber": "F208", "equation": "Δq=(Δα,Δβ,Δγ,Δβ,Δα); V_geom=(1/2)||Δq||²", "name": "Centered geometric mismatch norm", "status": "DEFINITION", "sourcePage": "F12", "variableDefinitions": [ "Δq is centered shape mismatch; Δα,Δβ,Δγ are coordinate differences; V_geom is mismatch energy." ], "explanation": "Defines a permutation-symmetric mismatch measure in the centered shape sector.", "crossLinks": [ "CF156", "CF157", "CF209", "CF211" ] }, { "id": "CF209", "displayNumber": "F209", "equation": "(1/2)||Δγ||_F² = (1/2)L_c⁴V_geom", "name": "Celal Gram mismatch equals centered geometric mismatch up to fixed factor", "status": "EXACT IN CENTERED FIXED-COMMON-MODE SECTOR", "sourcePage": "F12", "variableDefinitions": [ "Δγ is Gram-matrix mismatch; L_c is causal length; V_geom is geometric mismatch." ], "explanation": "Relates the matrix Frobenius mismatch to the coordinate mismatch in the fixed common-mode sector.", "crossLinks": [ "CF175", "CF182", "CF208", "CF210" ] }, { "id": "CF210", "displayNumber": "F210", "equation": "e_r=U_(n,r)q_eq,r; C_r=(1/2)‖r−U_(n,r)U_(n,r)ᵀ‖_F²", "name": "Candidate post-Xyz transported mismatch / discontinuity bookkeeping", "status": "DEFINITION; PHYSICAL MOVE SET OPEN", "sourcePage": "F12", "variableDefinitions": [ "e_r is transported mismatch; U_(n,r) is transport basis; q_eq,r is target; C_r is discontinuity cost." ], "explanation": "Defines candidate transported mismatch and a projector-based discontinuity cost after an Xyz transition.", "crossLinks": [ "CF085", "CF167", "CF208", "CF211" ] }, { "id": "CF211", "displayNumber": "F211", "equation": "r_* ∈ arg min_r(C_r+R_r), R_r admissible", "name": "Minimum-discontinuity continuation selector", "status": "STRONG RATIONALE; admissible R_r remains OPEN", "sourcePage": "F12", "variableDefinitions": [ "r_* is selected continuation; C_r is mismatch cost; R_r is admissibility penalty." ], "explanation": "Selects the continuation minimizing discontinuity plus an admissibility penalty, without specifying that penalty fully.", "crossLinks": [ "CF210", "CF217" ] }, { "id": "CF212", "displayNumber": "F212", "equation": "V_L(τ)=V_L(0)e^(−3τ/T_c)", "name": "Observed fixed-target recovery-cost decay under CF157", "status": "EXACT GIVEN CF157; EFFECTIVE TRACKING CANDIDATE ONLY", "sourcePage": "F12", "variableDefinitions": [ "V_L is recovery cost; τ is time; T_c is causal time." ], "explanation": "States the exponential decay of the fixed-target recovery cost in the retained effective candidate.", "crossLinks": [ "CF157", "CF159", "CF164" ] }, { "id": "CF213", "displayNumber": "F213", "equation": "c_X=BᵀP_Xu_x, c_X∈(0,±2,±2,±2,±2,0)", "name": "Xyz 2+2 incidence cut selected by failed mode", "status": "EXACT UP TO EDGE ORIENTATION / ORDERING", "sourcePage": "F12", "variableDefinitions": [ "c_X is edge cut vector; B is incidence; P_X is Xyz projector; u_x is failed mode." ], "explanation": "Constructs the discrete 2+2 edge cut associated with the failed simple-edge mode.", "crossLinks": [ "CF017", "CF205", "CF214", "CF215" ] }, { "id": "CF214", "displayNumber": "F214", "equation": "P_grad c_X ≠ c_X; P_circ c_X = c_X", "name": "Failed Xyz cut is pure gradient in six-channel decomposition", "status": "EXACT", "sourcePage": "F12", "variableDefinitions": [ "P_grad and P_circ are gradient/circulation projectors; c_X is Xyz edge cut." ], "explanation": "Classifies the selected cut entirely in the circulation sector rather than the gradient sector.", "crossLinks": [ "CF019", "CF020", "CF213" ] }, { "id": "CF215", "displayNumber": "F215", "equation": "|E(K₄)| = C(4,2) = 6", "name": "No unused eternal edge exists inside the one undirected four-vertex cell", "status": "EXACT COMBINATORIAL NO-GO", "sourcePage": "F12", "variableDefinitions": [ "E(K₄) is the edge set of the complete four-vertex graph; C(4,2) counts its six edges." ], "explanation": "The complete four-vertex cell already uses all six unordered branch pairs.", "crossLinks": [ "CF006", "CF018", "CF213" ] }, { "id": "CF216", "displayNumber": "F216", "equation": "detγ_X=0 ⇒ V₄,x→0 ⇒ B^dual_X→0", "name": "Local candidate dual-area/action weight collapses at Xyz rank loss while 4-volume remains finite", "status": "EXACT CONSEQUENCE WITHIN CF143-CF147 CANDIDATE ACTION", "sourcePage": "F12", "variableDefinitions": [ "γ_X is degenerate Xyz Gram; V₄,x is four-volume; B^dual_X is candidate dual area." ], "explanation": "Propagates Gram rank loss through the candidate determinant-weighted dual area, without claiming physical collapse of volume.", "crossLinks": [ "CF143", "CF145", "CF146", "CF147", "CF181" ] }, { "id": "CF217", "displayNumber": "F217", "equation": "χ_Xyz>0: ordinary evolution may remain on existing causal relations; χ_Xyz=0,D_τχ_Xyz≥0: retained phase may remain admissible; χ_Xyz=0,D_τχ_Xyz<0: ordinary smooth continuation is inadmissible and an admissible intervention is required", "name": "Consolidated causal-integrity law; Celal may select among a separately derived admissible continuation set", "status": "EXACT BOUNDARY + CONDITIONAL SELECTOR; ORDINARY TRANSPORT AND POST-Xyz MOVE SET OPEN", "sourcePage": "F12", "variableDefinitions": [ "χ_Xyz is integrity margin; D_τ is its material derivative; Celal is the branch-selection procedure." ], "explanation": "Summarizes interior, tangent-boundary, and outward-crossing cases while preserving the open continuation dynamics.", "crossLinks": [ "CF085", "CF204", "CF207", "CF210", "CF211" ] }, { "id": "CF218", "displayNumber": "F218", "equation": "∂_tχ_m + v_m·∇χ_m = 0", "name": "Advection of a sharp material-support indicator with the matter flow", "status": "EXACT KINEMATICS FOR MATERIAL SUPPORT", "sourcePage": "F13", "variableDefinitions": [ "χ_m is material-support indicator; v_m is material velocity; t is time; ∇ is spatial gradient." ], "explanation": "Advects a sharp matter-support field with the material velocity without making it a graph-topology boundary.", "crossLinks": [ "CF053", "CF219", "CF222" ] }, { "id": "CF219", "displayNumber": "F219", "equation": "v_m(x,t)=V(t)+ω(t)×[x−X(t)]", "name": "Rigid-body matter velocity: translation plus axial rotation", "status": "EXACT NONRELATIVISTIC RIGID-BODY KINEMATICS", "sourcePage": "F13", "variableDefinitions": [ "v_m is matter velocity; V is translation velocity; ω is angular velocity; X is center; x is position." ], "explanation": "Decomposes rigid-body material motion into translation and rotation.", "crossLinks": [ "CF218", "CF220", "CF221" ] }, { "id": "CF220", "displayNumber": "F220", "equation": "(ω×r)·n = 0 for n ∥ r", "name": "Rotation is tangent to an ideal spherical matter boundary, including a tilted axis", "status": "EXACT GEOMETRY", "sourcePage": "F13", "variableDefinitions": [ "ω is angular velocity; r is radial position; n is spherical outward normal." ], "explanation": "Shows that rigid rotation has no normal component at an ideal spherical boundary.", "crossLinks": [ "CF219", "CF221" ] }, { "id": "CF221", "displayNumber": "F221", "equation": "T^{00}≈ρc²; T^{0i}≈ρcv^i; T^{ij}≈ρv^iv^j", "name": "Low-velocity dust source decomposition; rotation/tilt enters through momentum and stress", "status": "STANDARD NONRELATIVISTIC SOURCE LIMIT / INPUT", "sourcePage": "F13", "variableDefinitions": [ "T^{μν} is stress-energy; ρ is density; v^i is material velocity; c is light speed." ], "explanation": "Provides the nonrelativistic stress-energy components used as the matter-layer source input.", "crossLinks": [ "CF041", "CF042", "CF102", "CF218", "CF219" ] }, { "id": "CF222", "displayNumber": "F222", "equation": "[M]_S=0; [∂_nM]_S=0; [∇²M]_S=C_Σ(c³T_c²/ℏ)ρ|_S", "name": "Ordinary weak matter-boundary matching without a delta-function surface layer", "status": "EXACT WITHIN RETAINED WEAK SOURCE EQUATION AND NO-SURFACE-LAYER ASSUMPTION", "sourcePage": "F13", "variableDefinitions": [ "[·]_S denotes the matter-boundary value; n is boundary normal; M is deformation; C_Σ is retained coefficient; ρ is boundary density." ], "explanation": "Matches deformation, normal derivative, and weak-source curvature at the material boundary while excluding a delta-function surface source.", "crossLinks": [ "CF032", "CF053", "CF067", "CF218", "CF220" ] } ]