# G Model Complete Formula Register **Current register:** CF001–CF222 **Author:** Gurbuz Dolu **Encoding:** UTF-8 ## F1 / CF001 — Native causal length **Expression** ```text L_c = cT_c ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF002, CF023, CF039 **Source page:** F1 ## F2 / CF002 — Four future-directed null branches **Expression** ```text kₐ = L_c(1, nₐ) ``` **Status:** MODEL GEOMETRY **Definitions** - 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. **Cross-references:** CF001, CF003, CF004, CF005 **Source page:** F1 ## F3 / CF003 — Regular tetrahedral directional balance **Expression** ```text nₐ·nᵦ = −1/3 (a ≠ b) ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF002, CF004, CF010 **Source page:** F1 ## F4 / CF004 — Tetrahedral centroid balance **Expression** ```text Σₐ₌₁⁴ nₐ = 0 ``` **Status:** EXACT **Definitions** - nₐ are the four tetrahedral unit directions; Σ denotes the sum over all branches. **Explanation:** The four spatial branch directions have a vanishing centroid. **Cross-references:** CF002, CF003, CF010 **Source page:** F1 ## F5 / CF005 — Null character of each branch **Expression** ```text kₐ² = 0 ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF002, CF010, CF014 **Source page:** F1 ## F6 / CF006 — Four branches generate six unordered pairs **Expression** ```text C(4,2) = 6 ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF007, CF018, CF019, CF020 **Source page:** F1 ## F7 / CF007 — Pairwise causal bivector **Expression** ```text Σₐᵦ = kₐ ∧ kᵦ ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF002, CF006, CF008, CF009 **Source page:** F1 ## F8 / CF008 — Neutral pair-area invariant **Expression** ```text |kₐ ∧ kᵦ| = 3/2 L_c² ``` **Status:** EXACT **Definitions** - kₐ,kᵦ are distinct branches; L_c is native length; |·| is bivector magnitude. **Explanation:** Every neutral branch pair has the same causal area magnitude. **Cross-references:** CF001, CF007, CF009, CF029 **Source page:** F1 ## F9 / CF009 — Aggregate neutral six-bivector magnitude **Expression** ```text AΣ,0 = 6(4L_c²/3) = 8L_c² ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF006, CF008, CF023 **Source page:** F1 ## F10 / CF010 — Four-null Gram matrix **Expression** ```text Gₐₐ = 0, Gₐᵦ = −1/3 (a ≠ b) ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF003, CF005, CF011, CF012, CF013 **Source page:** F1 ## F11 / CF011 — Four-null Gram spectrum **Expression** ```text spec(G₄) = {−4/3, 4/3, 4/3, 4/3}L_c² ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF010, CF012, CF013 **Source page:** F1 ## F12 / CF012 — Non-degenerate local causal basis **Expression** ```text det G₄ = −256/27 L_c⁸ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF010, CF011, CF014 **Source page:** F1 ## F13 / CF013 — Inverse Gram **Expression** ```text G₄⁻¹ = L_c⁻²(3/4 I₄ − 1/4 11ᵀ) ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF010, CF012, CF014 **Source page:** F1 ## F14 / CF014 — Minkowski metric reconstruction **Expression** ```text K G₄⁻¹ Kᵀ = η ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF005, CF010, CF013, CF015 **Source page:** F1 ## F15 / CF015 — Branch directional derivative **Expression** ```text Dₐ = kₐ^μ∂_μ ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF002, CF014, CF016 **Source page:** F1 ## F16 / CF016 — Native d’Alembertian reconstruction **Expression** ```text (G₄⁻¹)ᵃᵇDₐDᵦ = □ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF013, CF014, CF015, CF067 **Source page:** F1 ## F17 / CF017 — Boundary-of-boundary incidence identity **Expression** ```text BC = 0 ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF018, CF019, CF020, CF123 **Source page:** F1 ## F18 / CF018 — Exact six-edge Hodge decomposition **Expression** ```text BᵀB + CCᵀ = 4I₆ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF006, CF017, CF019, CF020 **Source page:** F1 ## F19 / CF019 — Gradient/circulation projectors **Expression** ```text Pgrad = 1/4 BᵀB, Pcirc = 1/4 CCᵀ ``` **Status:** EXACT **Definitions** - Pgrad and Pcirc are six-channel projectors; B,C are incidence matrices. **Explanation:** Normalizes the gradient and circulation components of edge-channel data. **Cross-references:** CF017, CF018, CF020, CF021, CF022 **Source page:** F1 ## F20 / CF020 — Complete six-channel decomposition **Expression** ```text Pgrad + Pcirc = I₆ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF018, CF019, CF022 **Source page:** F1 ## F21 / CF021 — Historical normalized six-channel operator convention **Expression** ```text K₀ = BᵀB − 2I₆ ``` **Status:** RETAINED **Definitions** - 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. **Cross-references:** CF018, CF019, CF022 **Source page:** F1 ## F22 / CF022 — Six-channel neutral spectrum **Expression** ```text spec(K₀) = {4,0,0,0,−2,−2} ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF019, CF020, CF021 **Source page:** F1 ## F23 / CF023 — Native causal volume **Expression** ```text V_c = (2/3)L_c³ = (√2/12)(cT_c)³ ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF001, CF008, CF024, CF025 **Source page:** F1 ## F24 / CF024 — Instantaneous causal-volume flux for dL/dt = c **Expression** ```text Γᵥ = dV/dt = 2cL_c² ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF023, CF025, CF040 **Source page:** F1 ## F25 / CF025 — Characteristic causal-volume flux **Expression** ```text Γ_c = 2c³T_c² ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF001, CF023, CF024, CF040 **Source page:** F1 ## F26 / CF026 — Intrinsic fabric energy scale **Expression** ```text E_f = ℏ/T_c ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF027, CF028, CF038 **Source page:** F2 ## F27 / CF027 — One intrinsic fabric update carries one action quantum **Expression** ```text E_fT_c = ℏ ``` **Status:** MODEL NORMALIZATION **Definitions** - 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. **Cross-references:** CF026, CF044, CF046 **Source page:** F2 ## F28 / CF028 — Native dimensional mass scale **Expression** ```text m_c = ℏ/(c²T_c) ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF026, CF037, CF039 **Source page:** F2 ## F29 / CF029 — Neutral pair-area scale **Expression** ```text A₀ = 4L_c²/3 ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF008, CF009, CF030 **Source page:** F2 ## F30 / CF030 — Candidate neutral area-stiffness closure **Expression** ```text κ_C L_c² = ℏ ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF029, CF031, CF032 **Source page:** F2 ## F31 / CF031 — Candidate surface-tension scale **Expression** ```text σ₀ = κ_C/T_c ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF030, CF032 **Source page:** F2 ## F32 / CF032 — Retained causal-geometric coefficient **Expression** ```text C_Σ = 12/(5π) ``` **Status:** CONDITIONAL/REFERENCE **Definitions** - 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. **Cross-references:** CF033, CF036, CF067, CF148 **Source page:** F2 ## F33 / CF033 — Microscopic coefficient plus spherical spreading **Expression** ```text C_Σ/(4π) = 3/(5π²) ``` **Status:** EXACT GIVEN C_Σ **Definitions** - C_Σ is the retained coefficient; π is the spherical spreading constant. **Explanation:** Rewrites the retained coefficient after separating the 4π spherical factor. **Cross-references:** CF032, CF036, CF151 **Source page:** F2 ## F34 / CF034 — G Model Kutsal Interface Constant; not Big G **Expression** ```text K_GD = 1/(5π²c) ``` **Status:** HISTORICAL/INTERFACE **Definitions** - 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. **Cross-references:** CF033, CF035, CF150 **Source page:** F2 ## F35 / CF035 — Exact algebraic interface identity **Expression** ```text 4πG = 3cK_GD C_Σ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF032, CF034, CF036 **Source page:** F2 ## F36 / CF036 — Continuum matter-spacetime coupling bridge **Expression** ```text G_bridge = (C_Σ/4π)c⁵T_c²/ℏ = (3/5π²)c⁵T_c²/ℏ ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF032, CF033, CF037, CF151 **Source page:** F2 ## F37 / CF037 — Big-G-calibrated fabric time **Expression** ```text T_c = [ℏG/(C_Σc⁵)]¹ᐟ² ``` **Status:** CALIBRATED **Definitions** - 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. **Cross-references:** CF032, CF036, CF038, CF039 **Source page:** F2 ## F38 / CF038 — Reference value for C_Σ = 12/(5π) **Expression** ```text T_c ≈ 2.18657 × 10⁻⁴³ s ``` **Status:** CALIBRATED **Definitions** - 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. **Cross-references:** CF032, CF037, CF039, CF197 **Source page:** F2 ## F39 / CF039 — Reference causal length **Expression** ```text L_c ≈ 6.55517 × 10⁻³⁵ m ``` **Status:** CALIBRATED **Definitions** - 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. **Cross-references:** CF001, CF037, CF038 **Source page:** F2 ## F40 / CF040 — Neutral propagation normalization **Expression** ```text 6/T_c · c = √6 Γ_c ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF023, CF025, CF041 **Source page:** F2 ## F41 / CF041 — Covariant local causal-cell energy **Expression** ```text E_cell = V_c T^{μν} t_μ t_ν ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF023, CF042, CF043 **Source page:** F2 ## F42 / CF042 — Stationary nonrelativistic limit **Expression** ```text E_cell = ρV_cc² ``` **Status:** CONDITIONAL LIMIT **Definitions** - 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. **Cross-references:** CF041, CF048, CF051 **Source page:** F2 ## F43 / CF043 — Accumulated matter action **Expression** ```text S_m(t) = ∫₀ᵗ E_cell(τ)dτ ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF041, CF044, CF045, CF050 **Source page:** F2 ## F44 / CF044 — Aggregation definition of matter-response time **Expression** ```text S_m(D₀) = ℏ ``` **Status:** HYPOTHESIS **Definitions** - S_m is matter action; D₀ is aggregation response time; ℏ is action quantum. **Explanation:** Hypothesizes that one aggregation interval carries one action quantum. **Cross-references:** CF027, CF043, CF045, CF046 **Source page:** F2 ## F45 / CF045 — Constant-energy Aggregation law **Expression** ```text D₀ = ℏ/E_cell ``` **Status:** EXACT WITHIN DEFINITION **Definitions** - 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. **Cross-references:** CF041, CF044, CF047 **Source page:** F2 ## F46 / CF046 — Dimensionless matter/fabric loading **Expression** ```text A_cell = E_cellT_c/ℏ ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF027, CF044, CF047, CF048 **Source page:** F2 ## F47 / CF047 — Controlling Aggregation reciprocal relation **Expression** ```text D₀/T_c = 1/A_cell ``` **Status:** EXACT WITHIN DEFINITION **Definitions** - 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. **Cross-references:** CF045, CF046, CF049 **Source page:** F2 ## F48 / CF048 — Stationary density loading **Expression** ```text A_cell = (3/2)ρc⁵T_c⁴/ℏ ``` **Status:** EXACT WITHIN DEFINITIONS **Definitions** - 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. **Cross-references:** CF023, CF042, CF046, CF067 **Source page:** F2 ## F49 / CF049 — Equal-action/equal-scale condition **Expression** ```text A_cell = 1 ⇔ D₀ = T_c ``` **Status:** CONDITIONAL PHYSICAL INTERPRETATION **Definitions** - 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. **Cross-references:** CF046, CF047, CF048 **Source page:** F2 ## F50 / CF050 — Additive matter-action accumulation rate **Expression** ```text Γ_m = D₀⁻¹ = E_cell/ℏ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF043, CF045, CF046, CF051 **Source page:** F2 ## F51 / CF051 — Independent energy contributions add in rate **Expression** ```text Γ_{m,tot} = Σ_i Γ_{m,i} ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF042, CF050, CF052 **Source page:** F3 ## F52 / CF052 — Natural cell timelike direction **Expression** ```text t^μ = K^μ/√(−K²), K^μ = Σ_a k_a^μ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF002, CF005, CF041 **Source page:** F3 ## F53 / CF053 — Matter-occupancy bookkeeping variable **Expression** ```text χ_m ∈ {0,1} ``` **Status:** BOOKKEEPING **Definitions** - 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. **Cross-references:** CF054, CF055, CF056, CF057 **Source page:** F3 ## F54 / CF054 — Matter absent, neutral fabric **Expression** ```text (χ_m,M) = (0,0) ``` **Status:** STATE **Definitions** - χ_m is matter occupancy; M is deformation; both are zero in the neutral empty state. **Explanation:** Labels the locally empty undeformed state. **Cross-references:** CF053, CF055, CF056, CF057 **Source page:** F3 ## F55 / CF055 — Matter present, deformed fabric **Expression** ```text (χ_m,M) = (1,M ≠ 0) ``` **Status:** STATE **Definitions** - χ_m=1 marks matter; M is a nonzero local deformation. **Explanation:** Labels matter supported by a locally deformed fabric. **Cross-references:** CF053, CF054, CF056, CF061 **Source page:** F3 ## F56 / CF056 — Matter absent locally, deformation persists **Expression** ```text (χ_m,M) = (0,M ≠ 0) ``` **Status:** STATE **Definitions** - 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. **Cross-references:** CF053, CF054, CF055, CF061 **Source page:** F3 ## F57 / CF057 — Neutral M; does not by itself imply D₀ = T_c **Expression** ```text (χ_m,M) = (1,0) ``` **Status:** SPECIAL STATE **Definitions** - 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. **Cross-references:** CF047, CF049, CF053, CF055 **Source page:** F3 ## F58 / CF058 — One-causal-depth column density **Expression** ```text Σ_col = ρL_c ``` **Status:** DEFINITION **Definitions** - Σ_col is column density; ρ is mass density; L_c is one causal depth. **Explanation:** Defines projected density through one native causal depth. **Cross-references:** CF001, CF059, CF067 **Source page:** F3 ## F59 / CF059 — Projected-area mass additivity **Expression** ```text ∫Σ_col dA = M_source ``` **Status:** CONDITIONAL GEOMETRY **Definitions** - 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. **Cross-references:** CF058, CF067, CF068 **Source page:** F3 ## F60 / CF060 — Baseline fabric universality requirement **Expression** ```text ∂κ_C/∂m = ∂κ_C/∂ρ = ∂κ_C/∂Ω = 0 ``` **Status:** POSTULATE **Definitions** - 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. **Cross-references:** CF030, CF061, CF066 **Source page:** F3 ## F61 / CF061 — Operational deformation/clock relation **Expression** ```text M = ln(dτ/dt) ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF055, CF062, CF063, CF064 **Source page:** F3 ## F62 / CF062 — Local clock ratio **Expression** ```text dτ/dt = e^M ``` **Status:** CONDITIONAL **Definitions** - τ is local time; t is reference time; M is deformation. **Explanation:** Rewrites the operational deformation relation as a local clock ratio. **Cross-references:** CF061, CF063, CF076 **Source page:** F3 ## F63 / CF063 — Weak-field potential identification **Expression** ```text Φ = c²δM ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF061, CF064, CF067, CF068 **Source page:** F3 ## F64 / CF064 — Weak-field acceleration **Expression** ```text g = −c²∇M ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF061, CF063, CF067, CF069 **Source page:** F3 ## F65 / CF065 — Orientation-balanced spherical mean **Expression** ```text Ā[M] = M + (L_c²/6)∇²M + (L_c⁴/120)∇⁴M + ⋯ ``` **Status:** EXACT EXPANSION **Definitions** - 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. **Cross-references:** CF066, CF067, CF070 **Source page:** F3 ## F66 / CF066 — Discrete mean-defect source rule **Expression** ```text Ā[M] − M = (C_Σ/4)A_cell ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF032, CF046, CF060, CF065 **Source page:** F3 ## F67 / CF067 — Weak continuum source law **Expression** ```text ∇²M = C_Σ(c³T_c²/ℏ)ρ ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF032, CF048, CF063, CF068, CF069, CF098 **Source page:** F3 ## F68 / CF068 — Big-G-calibrated weak source law **Expression** ```text ∇²M = (4πG/c²)ρ ``` **Status:** CALIBRATED COMPATIBILITY **Definitions** - 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. **Cross-references:** CF036, CF067, CF069, CF111 **Source page:** F3 ## F69 / CF069 — Poisson normalization **Expression** ```text ∇²Φ = 4πGρ ``` **Status:** CALIBRATED COMPATIBILITY **Definitions** - Φ is weak potential; G is calibrated gravitational coupling; ρ is density. **Explanation:** Recovers the conventional weak-field Poisson equation after the potential identification. **Cross-references:** CF063, CF068, CF098 **Source page:** F3 ## F70 / CF070 — Reduced reciprocal quadratic spectrum **Expression** ```text Q₆(M) = 16e²M + 8e⁻²M ``` **Status:** EXACT WITHIN REDUCED MODEL **Definitions** - 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. **Cross-references:** CF071, CF072, CF073, CF074, CF075 **Source page:** F3 ## F71 / CF071 — Mathematical minimum of Q₆ **Expression** ```text M_* = −(1/4)ln 2 ``` **Status:** EXACT **Definitions** - M_* is minimizing deformation; Q₆ is the reduced spectrum; ln is natural logarithm. **Explanation:** Locates the stationary minimum of the reduced reciprocal spectrum. **Cross-references:** CF070, CF072, CF073 **Source page:** F3 ## F72 / CF072 — Minimum reciprocal spectral value **Expression** ```text Q₆,min = 16√2 ``` **Status:** EXACT **Definitions** - Q₆,min is the minimum spectrum value; Q₆ is the reduced reciprocal spectrum. **Explanation:** Evaluates Q₆ at its mathematical minimum. **Cross-references:** CF070, CF071, CF073 **Source page:** F3 ## F73 / CF073 — Reciprocal spectral-product invariant **Expression** ```text Q₊Q₋ = 128 ``` **Status:** EXACT **Definitions** - Q₊ and Q₋ are the reciprocal spectral sectors. **Explanation:** States the deformation-independent product of the paired reciprocal sectors. **Cross-references:** CF070, CF072, CF074 **Source page:** F3 ## F74 / CF074 — Hyperbolic spectral invariant **Expression** ```text S₆² − D₆² = 512 ``` **Status:** EXACT **Definitions** - S₆ and D₆ are the sum and difference spectral combinations. **Explanation:** Records the invariant difference of squared sum and difference sectors. **Cross-references:** CF070, CF073, CF075 **Source page:** F3 ## F75 / CF075 — Linear reciprocal spectral magnitude **Expression** ```text Tr|K(M)| = 8 cosh M ``` **Status:** EXACT IN STATED SPECTRUM **Definitions** - 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. **Cross-references:** CF070, CF073, CF074 **Source page:** F3 ## F76 / CF076 — Dolu reciprocal-time invariant **Expression** ```text T₊T₋ = T_c² ``` **Status:** EXACT WITHIN CONSTRUCTION **Definitions** - 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. **Cross-references:** CF037, CF077, CF078 **Source page:** F4 ## F77 / CF077 — Differential reciprocal-time form **Expression** ```text δln T₊ + δln T₋ = 0 ``` **Status:** EXACT WITHIN CONSTRUCTION **Definitions** - T₊,T₋ are reciprocal times; δ denotes a variation. **Explanation:** The logarithmic variations of the two reciprocal sectors cancel. **Cross-references:** CF076, CF078 **Source page:** F4 ## F78 / CF078 — Three-dimensional reciprocal time sectors **Expression** ```text T₊ = T_ce⁺²ᵘ⁄³ ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF076, CF079, CF080 **Source page:** F4 ## F79 / CF079 — Reciprocal volume sectors **Expression** ```text V₊ = V_ce⁺²ᵘ ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF023, CF078, CF080 **Source page:** F4 ## F80 / CF080 — Spectral-volume equalization if shared-u bridge is adopted **Expression** ```text 2V_cQ₊V₊ = Q₋V₋ = 8√2V_c ``` **Status:** CONDITIONAL BRIDGE **Definitions** - 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. **Cross-references:** CF073, CF078, CF079 **Source page:** F4 ## F81 / CF081 — Jadvyga local reorientation **Expression** ```text Kₓ(θ) = Λₓ(θ)Kₓ(0) ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF082, CF083, CF084 **Source page:** F4 ## F82 / CF082 — Lorentz-frame preservation **Expression** ```text ΛₓᵀηΛₓ = η ``` **Status:** EXACT **Definitions** - Λₓ is the local frame transformation; η is the Minkowski metric. **Explanation:** Requires the local reorientation to preserve the Lorentz metric. **Cross-references:** CF014, CF081, CF083 **Source page:** F4 ## F83 / CF083 — Holonomy conjugation under local reorientation **Expression** ```text W_f → ΛₓW_fΛₓ⁻¹ ``` **Status:** EXACT **Definitions** - W_f is face holonomy; Λₓ is local Lorentz reorientation. **Explanation:** Transforms face holonomy by conjugation under a local frame change. **Cross-references:** CF081, CF082, CF123 **Source page:** F4 ## F84 / CF084 — Celal invariant mismatch cost **Expression** ```text C_xy(Π) = 1/2‖ΠᵀΠ_xy − Π‖²_F ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF085, CF086, CF092 **Source page:** F4 ## F85 / CF085 — Celal branch correspondence **Expression** ```text Π_Celal ∈ arg min_Π∈S₄ C_xy(Π) ``` **Status:** EXACT GEOMETRIC SELECTION **Definitions** - 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. **Cross-references:** CF006, CF084, CF086, CF092 **Source page:** F4 ## F86 / CF086 — Centered physical shape sector **Expression** ```text q = (a,b,c,c,b,a), a+b+c = 0 ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF070, CF087, CF088, CF158 **Source page:** F4 ## F87 / CF087 — Candidate reconnect relaxation law **Expression** ```text T_c dq/dτ = −(I−P)q ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF086, CF088, CF089, CF157 **Source page:** F4 ## F88 / CF088 — Centered relaxation solution **Expression** ```text q(τ) = q₀e⁻³ᵗ⁄²ᵀ_c ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF087, CF089, CF157 **Source page:** F4 ## F89 / CF089 — Candidate reconnect timescale **Expression** ```text τ_rec = 2T_c/3 ``` **Status:** CONDITIONAL **Definitions** - τ_rec is reconnect timescale; T_c is intrinsic causal time. **Explanation:** Defines the characteristic e-folding timescale associated with the candidate relaxation. **Cross-references:** CF087, CF088, CF157 **Source page:** F4 ## F90 / CF090 — Lorentz bivector Hodge structure **Expression** ```text J_*² = −I₆ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF019, CF091, CF132 **Source page:** F4 ## F91 / CF091 — Hodge compatibility with bivector metric **Expression** ```text J_*ᵀΓJ_* = −Γ ``` **Status:** EXACT **Definitions** - J_* is the bivector Hodge operator; Γ is the six-space bivector metric. **Explanation:** States how the Hodge operator acts on the bivector metric. **Cross-references:** CF090, CF144, CF145 **Source page:** F4 ## F92 / CF092 — Inter-cell branch-frame transporter **Expression** ```text Ω_yx = K_yΠ_yxKₓ⁻¹ ``` **Status:** CONDITIONAL/GEOMETRIC **Definitions** - 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. **Cross-references:** CF081, CF085, CF093 **Source page:** F4 ## F93 / CF093 — Lorentz transport on matched frames **Expression** ```text Ω_yxᵀηΩ_yx = η ``` **Status:** EXACT WHEN GRAMS MATCH **Definitions** - 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. **Cross-references:** CF082, CF092 **Source page:** F4 ## F94 / CF094 — Four-component source-divergence candidate **Expression** ```text C_b = (G₄⁻¹)ᵃ_cDₐH_cb ``` **Status:** EXACT REPRESENTATION **Definitions** - 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. **Cross-references:** CF013, CF015, CF095, CF102 **Source page:** F4 ## F95 / CF095 — Capacity for four independent continuity equations **Expression** ```text rank(H ↦ C_b) = 4 ``` **Status:** EXACT REPRESENTATION **Definitions** - 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. **Cross-references:** CF094, CF102, CF107 **Source page:** F4 ## F96 / CF096 — Weak scalar gradient-stress tensor **Expression** ```text Tᵢⱼ^(M) = K_M(∂ᵢM∂ⱼM − 1/2δᵢⱼ|∇M|²) ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF064, CF097, CF098 **Source page:** F4 ## F97 / CF097 — Field-stress divergence; matter force is negative divergence **Expression** ```text ∂ⱼTᵢⱼ^(M) = K_M(∂ᵢM)∇²M ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF096, CF098, CF111 **Source page:** F4 ## F98 / CF098 — Weak source-modulus form **Expression** ```text K_M∇²M = ρc² ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF067, CF096, CF097, CF099, CF201 **Source page:** F4 ## F99 / CF099 — Six-channel strain hierarchy **Expression** ```text 3 reciprocal strains → 1 trace + 2 anisotropic ``` **Status:** EXACT ALGEBRAIC **Definitions** - 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. **Cross-references:** CF070, CF073, CF096 **Source page:** F4 ## F100 / CF100 — Massless-fabric/background-energy candidate **Expression** ```text ρ_f^rest = 0, u_f0 ≥ 0 ``` **Status:** OPEN HYPOTHESIS **Definitions** - 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. **Cross-references:** CF098, CF126, CF129 **Source page:** F4 ## F101 / CF101 — Branch encoding of metric perturbation **Expression** ```text Q_ab = k_a^μ k_b^ν h_μν ``` **Status:** EXACT REPRESENTATION **Definitions** - 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. **Cross-references:** CF014, CF103, CF105, CF107 **Source page:** F5 ## F102 / CF102 — Branch encoding of stress-energy **Expression** ```text H_ab = k_a^μ k_b^ν T_μν ``` **Status:** EXACT REPRESENTATION **Definitions** - 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. **Cross-references:** CF094, CF101, CF104, CF111 **Source page:** F5 ## F103 / CF103 — Reconstruction of h_μν **Expression** ```text h = (K⁻¹)ᵀQK⁻¹ ``` **Status:** EXACT **Definitions** - h is spacetime metric perturbation; K is branch-frame matrix; Q is branch metric data. **Explanation:** Reconstructs the spacetime perturbation from branch metric components. **Cross-references:** CF101, CF104, CF105 **Source page:** F5 ## F104 / CF104 — Reconstruction of T_μν **Expression** ```text T = (K⁻¹)ᵀHK⁻¹ ``` **Status:** EXACT **Definitions** - T is spacetime stress-energy; K is branch-frame matrix; H is branch stress data. **Explanation:** Reconstructs stress-energy from its branch-space encoding. **Cross-references:** CF102, CF103, CF111 **Source page:** F5 ## F105 / CF105 — Branch-space trace **Expression** ```text Q = (G₄⁻¹)ᵃᵇQ_ab ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF010, CF013, CF101, CF106 **Source page:** F5 ## F106 / CF106 — Branch trace reversal **Expression** ```text Q̄_ab = Q_ab − 1/2(G₄)_abQ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF101, CF105, CF107 **Source page:** F5 ## F107 / CF107 — Harmonic condition in branch variables **Expression** ```text (G₄⁻¹)ᵃ_cDₐQ̄_cb = 0 ``` **Status:** CONDITIONAL GAUGE **Definitions** - 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. **Cross-references:** CF015, CF016, CF103, CF106 **Source page:** F5 ## F108 / CF108 — Linearized Einstein operator **Expression** ```text E_ab^(1)[Q] = 1/2[−□Q_ab + DₐD_cQ_cb + D_bD_cQ_ca − DₐD_bQ − (G₄)_abD_cDᵈQ_cd] ``` **Status:** EXACT UNDER CONTINUUM ASSUMPTIONS **Definitions** - 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. **Cross-references:** CF016, CF101, CF105, CF107 **Source page:** F5 ## F109 / CF109 — Identical conservation of linearized operator **Expression** ```text DᵃE_ab^(1) = 0 ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF107, CF108, CF110 **Source page:** F5 ## F110 / CF110 — Linearized gauge transformation **Expression** ```text δQ_ab = Dₐξ_b + D_bξₐ ``` **Status:** CONDITIONAL CONTINUUM **Definitions** - 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. **Cross-references:** CF101, CF107, CF108, CF109 **Source page:** F5 ## F111 / CF111 — Big-G-calibrated linearized source equation **Expression** ```text E_ab = (8πG/c⁴)H_ab ``` **Status:** CALIBRATED COMPATIBILITY **Definitions** - 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. **Cross-references:** CF068, CF102, CF108, CF109 **Source page:** F5 ## F112 / CF112 — Two-function strong-field variable **Expression** ```text ψ = 2 tanh(M/2) ``` **Status:** CONDITIONAL COMPLETION **Definitions** - ψ is bounded strong-field variable; M is deformation. **Explanation:** Maps unbounded deformation M to a bounded variable for the strong-field completion. **Cross-references:** CF061, CF113, CF114, CF115 **Source page:** F5 ## F113 / CF113 — Temporal metric factor **Expression** ```text T = e^M = (1+ψ/2)/(1−ψ/2) ``` **Status:** CONDITIONAL COMPLETION **Definitions** - 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. **Cross-references:** CF061, CF112, CF114 **Source page:** F5 ## F114 / CF114 — Spatial metric factor **Expression** ```text X = (1−ψ/2)² ``` **Status:** CONDITIONAL COMPLETION **Definitions** - X is spatial metric factor; ψ is bounded strong-field variable. **Explanation:** Defines the companion spatial factor used by the isotropic strong-field metric. **Cross-references:** CF112, CF113, CF115 **Source page:** F5 ## F115 / CF115 — Static spherical vacuum solution **Expression** ```text ψ = −μ/r ``` **Status:** EXACT WITHIN COMPLETION **Definitions** - 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. **Cross-references:** CF112, CF116, CF117, CF118 **Source page:** F5 ## F116 / CF116 — Isotropic Schwarzschild exterior **Expression** ```text ds² = (1−μ/(2r))²/(1+μ/(2r))² c²dt² + (1+μ/(2r))⁴(dr²+r²dΩ²) ``` **Status:** EXACT WITHIN COMPLETION **Definitions** - 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. **Cross-references:** CF113, CF114, CF115, CF117, CF119 **Source page:** F5 ## F117 / CF117 — Areal-radius transformation **Expression** ```text R = r(1+μ/2r)² ``` **Status:** EXACT **Definitions** - R is areal radius; r is isotropic radius; μ is mass-length parameter. **Explanation:** Converts the isotropic radius to the Schwarzschild areal radius. **Cross-references:** CF115, CF116, CF118 **Source page:** F5 ## F118 / CF118 — Invariant areal-radius causal-support scalar **Expression** ```text C_F = 1 − 2μ/R ``` **Status:** EXACT WITHIN SCHWARZSCHILD BRANCH **Definitions** - 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. **Cross-references:** CF117, CF119, CF120 **Source page:** F5 ## F119 / CF119 — Horizon radius with empirical Big G **Expression** ```text R_h = 2μ = 2GM/c² ``` **Status:** CALIBRATED/GR COMPATIBILITY **Definitions** - 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. **Cross-references:** CF116, CF118, CF120 **Source page:** F5 ## F120 / CF120 — Schwarzschild Kretschmann scalar **Expression** ```text K = 48μ²/R⁶ ``` **Status:** EXACT **Definitions** - K is Kretschmann scalar; μ is mass-length parameter; R is areal radius. **Explanation:** Gives the curvature invariant for the spherical vacuum branch. **Cross-references:** CF117, CF118, CF184 **Source page:** F5 ## F121 / CF121 — Finite-cell abelian-type gauge transformation **Expression** ```text A → A + Bᵀχ ``` **Status:** EXACT KINEMATICS **Definitions** - 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. **Cross-references:** CF017, CF122, CF123 **Source page:** F5 ## F122 / CF122 — Discrete face/loop field strength **Expression** ```text F = CᵀA ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF017, CF121, CF123 **Source page:** F5 ## F123 / CF123 — Gauge invariance from BC = 0 **Expression** ```text F′ = F ``` **Status:** EXACT KINEMATICS **Definitions** - 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. **Cross-references:** CF017, CF121, CF122 **Source page:** F5 ## F124 / CF124 — Continuum closure **Expression** ```text F = dA, d²A = 0 ``` **Status:** EXACT KINEMATICS **Definitions** - 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. **Cross-references:** CF121, CF122, CF123 **Source page:** F5 ## F125 / CF125 — Minimal parity-even light action **Expression** ```text S_γ = −(κ_γ/4)∫F_μνF^μν d⁴x ``` **Status:** CONDITIONAL COMPLETION **Definitions** - 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. **Cross-references:** CF122, CF124, CF126, CF135 **Source page:** F5 ## F126 / CF126 — Vacuum field equation **Expression** ```text ∂_μF^{μν} = 0 ``` **Status:** CONDITIONAL CONSEQUENCE **Definitions** - 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. **Cross-references:** CF124, CF125, CF127, CF129 **Source page:** F6 ## F127 / CF127 — Plane-wave closure form **Expression** ```text F = q ∧ a ``` **Status:** EXACT CONSEQUENCE **Definitions** - 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. **Cross-references:** CF126, CF128, CF129, CF132 **Source page:** F6 ## F128 / CF128 — Plane-wave co-closure **Expression** ```text q_μF^{μν} = q²a^ν − (q·a)q^ν ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF127, CF129, CF130 **Source page:** F6 ## F129 / CF129 — Nontrivial vacuum light is null **Expression** ```text q² = 0 ``` **Status:** EXACT WITHIN CANDIDATE DYNAMICS **Definitions** - q is the wave covector; q² is its Lorentz norm. **Explanation:** Requires a nontrivial transverse vacuum wave to have a null wave covector. **Cross-references:** CF126, CF128, CF130, CF131 **Source page:** F6 ## F130 / CF130 — Vacuum light dispersion **Expression** ```text ω² = c²k² ``` **Status:** EXACT WITHIN CANDIDATE DYNAMICS **Definitions** - 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. **Cross-references:** CF129, CF131, CF137 **Source page:** F6 ## F131 / CF131 — Two physical transverse modes **Expression** ```text dim(q⊥/span{q}) = 2 ``` **Status:** EXACT WITHIN CANDIDATE DYNAMICS **Definitions** - 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. **Cross-references:** CF127, CF129, CF132, CF133 **Source page:** F6 ## F132 / CF132 — Hodge helicity sectors **Expression** ```text F_± = F ∓ iJ_*F ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF090, CF127, CF133 **Source page:** F6 ## F133 / CF133 — Helicity eigenvalue relation **Expression** ```text J_*F_± = ±iF_± ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF090, CF132 **Source page:** F6 ## F134 / CF134 — Vertex conservation of edge-transfer current **Expression** ```text BJ = 0 ``` **Status:** EXACT **Definitions** - B is vertex-edge incidence; J is edge-transfer current. **Explanation:** Imposes vanishing net current at every vertex in the discrete cell. **Cross-references:** CF017, CF121, CF135 **Source page:** F6 ## F135 / CF135 — Minimal source coupling **Expression** ```text S_int = −g_AJᵀA ``` **Status:** CONDITIONAL IDENTIFICATION **Definitions** - 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. **Cross-references:** CF121, CF125, CF134, CF136 **Source page:** F6 ## F136 / CF136 — Discrete sourced light equation **Expression** ```text κ_γCCᵀJ = g_AJ ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF122, CF125, CF135, CF137 **Source page:** F6 ## F137 / CF137 — Coulomb-form static field **Expression** ```text E(r) = g_AQ/(4πκ_γr²) ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF136, CF138 **Source page:** F6 ## F138 / CF138 — Compact link variable **Expression** ```text U_ij = e^{i a_ij} ``` **Status:** CONDITIONAL U(1) COMPLETION **Definitions** - 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. **Cross-references:** CF139, CF140 **Source page:** F6 ## F139 / CF139 — Compact U(1) integer representation weights **Expression** ```text ψ_i → g_i^nψ_i, n ∈ ℤ ``` **Status:** EXACT GROUP THEORY **Definitions** - 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. **Cross-references:** CF138, CF140 **Source page:** F6 ## F140 / CF140 — Native tetrahedral residual triality **Expression** ```text A₄/V₄ ≅ ℤ₃ ``` **Status:** EXACT GROUP THEORY **Definitions** - 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. **Cross-references:** CF006, CF138, CF139 **Source page:** F6 ## F141 / CF141 — Exact tetrahedral group-valued Bianchi closure **Expression** ```text W₀₁₂W₀₂₃W₀₃₁(Ω₀₁W₁₂₃Ω₁₀⁻¹) = I ``` **Status:** EXACT ALGEBRAIC **Definitions** - W_ijk are face holonomies; Ω₀₁ transports between cells; I is the group identity. **Explanation:** States exact ordered holonomy closure around the tetrahedral boundary. **Cross-references:** CF083, CF142, CF146 **Source page:** F7 ## F142 / CF142 — Formal BCH relation; additive curvature is the near-identity/small-curvature limit **Expression** ```text log(eᴷ¹eᴷ²⋯) = ΣK_i + 1/2Σ[K_i,K_j] + O(K³) ``` **Status:** LOCAL / SMALL-CURVATURE BCH **Definitions** - 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. **Cross-references:** CF141, CF146, CF147 **Source page:** F7 ## F143 / CF143 — Invariant G Model causal four-volume extension; not uniquely canonical **Expression** ```text V₄,x = (2/3)L_c⁴[|detγ_x|/|detγ₀|]¹ᐟ² ``` **Status:** CANDIDATE INVARIANT EXTENSION **Definitions** - 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. **Cross-references:** CF023, CF144, CF145 **Source page:** F7 ## F144 / CF144 — Dimensionless bivectors and six-space metric remove unit ambiguity **Expression** ```text Σ̂_A = Σ_A/L_c²; Γ̂_AB = ⟨Σ̂_A,Σ̂_B⟩ ``` **Status:** DEFINITION / DIMENSIONAL CLEANUP **Definitions** - 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. **Cross-references:** CF007, CF091, CF143, CF145 **Source page:** F7 ## F145 / CF145 — Dimensionally explicit candidate dual-area bivector; [B^dual]=L² **Expression** ```text B^dual_B = −2(V₄,x/L_c²)Γ̂_ABJ_*Σ̂_A ``` **Status:** CANDIDATE DEFINITION; UNIQUENESS OPEN **Definitions** - 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. **Cross-references:** CF090, CF143, CF144, CF146 **Source page:** F7 ## F146 / CF146 — Discrete curvature has L⁻² units; contraction identity follows from definitions **Expression** ```text R^disc_x = (2/L_c²)Γ̂^AB⟨Σ̂_A,K_B⟩; Σ_B⟨B^dual,J_*K⟩ = V₄,xR^disc_x ``` **Status:** IDENTITY EXACT GIVEN DEFINITIONS; R_disc→R OPEN **Definitions** - 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. **Cross-references:** CF141, CF143, CF145, CF147 **Source page:** F7 ## F147 / CF147 — Microscopic holonomy-curvature action with explicit unfixed normalization **Expression** ```text S_micro^G/ℏ = ζ_microΣ⟨B^dual/L_c²,(J_*+βI)K(W)⟩_Γ ``` **Status:** CANDIDATE ACTION **Definitions** - 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. **Cross-references:** CF142, CF145, CF146, CF148 **Source page:** F7 ## F148 / CF148 — Effective continuum normalization required by internal source matching **Expression** ```text ζ_eff ≡ ζ_microN_disc = 1/(4C_Σ) = 5π/48 ≈ 0.327249 ``` **Status:** DERIVED MATCHING; MICRO FACTORS/UNIQUENESS OPEN **Definitions** - 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. **Cross-references:** CF032, CF067, CF147, CF150 **Source page:** F7 ## F149 / CF149 — One-update matter action in constant-energy Aggregation **Expression** ```text |ΔS_m| = E_cellT_c = ℏA_cell ``` **Status:** EXACT WITHIN AGGREGATION DEFINITIONS **Definitions** - 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. **Cross-references:** CF027, CF041, CF046, CF050 **Source page:** F7 ## F150 / CF150 — Continuum source equation matched by action normalization **Expression** ```text E^G_ab = R_ab − (1/2)γ_abR = 2C_Σ(cT_c²/ℏ)H_ab ``` **Status:** CONDITIONAL STRUCTURAL **Definitions** - 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. **Cross-references:** CF032, CF102, CF111, CF148 **Source page:** F7 ## F151 / CF151 — G Model gravitational coupling used in Mercury benchmark **Expression** ```text G_G = [3/(5π²)]c⁵T_c²/ℏ ``` **Status:** CONDITIONAL; CALIBRATED WHEN Tᶜ FROM BIG G **Definitions** - 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. **Cross-references:** CF036, CF037, CF153, CF154 **Source page:** F7 ## F152 / CF152 — First post-Newtonian coefficients in the smooth metric-compatible vacuum closure **Expression** ```text β_G = 1, γ_G = 1 ``` **Status:** CONDITIONAL CONTINUUM CONSEQUENCE **Definitions** - 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. **Cross-references:** CF116, CF151, CF153 **Source page:** F7 ## F153 / CF153 — Mercury perihelion advance; M_Sun avoids conflict with deformation variable M **Expression** ```text Δϖ_G = [18/(5π)]c³T_c²M_Sun/[ℏa(1−e²)] ``` **Status:** CONDITIONAL FORM; Tᶜ CALIBRATION MAY CONTAIN BIG G **Definitions** - 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. **Cross-references:** CF119, CF151, CF152, CF154 **Source page:** F7 ## F154 / CF154 — Mercury numerical benchmark **Expression** ```text T_c = 2.18657×10⁻⁴³ s ⇒ Δϖ_G = 42.9826 arcsec/Julian century ``` **Status:** CALIBRATED COMPATIBILITY **Definitions** - 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. **Cross-references:** CF038, CF151, CF153 **Source page:** F7 ## F155 / CF155 — Post-7 Sep rationale for retained depth and Hodge-throughput factors **Expression** ```text w(x)=x ⇒ ∫₀¹4x³w(x)dx=4/5; ⟨|cosφ|+|sinφ|⟩=4/π ``` **Status:** CONDITIONAL PHYSICAL-SELECTION RATIONALE **Definitions** - 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. **Cross-references:** CF032, CF148 **Source page:** F7 ## F156 / CF156 — Centered mismatch definition **Expression** ```text e = q − q_eq ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF086, CF157, CF158 **Source page:** F8 ## F157 / CF157 — Causal re-equalization flow **Expression** ```text T_c dq/dτ = −(I₆−P)e ``` **Status:** STRUCTURAL EFFECTIVE STRUCTURAL CANDIDATE **Definitions** - 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. **Cross-references:** CF087, CF156, CF158, CF159, CF161 **Source page:** F8 ## F158 / CF158 — Centered physical shape sector **Expression** ```text q=(α,β,γ,γ,β,α), α+β+γ=0; Pq=−q/2 ``` **Status:** EXACT IN CENTERED SECTOR **Definitions** - 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. **Cross-references:** CF086, CF156, CF157, CF162, CF167 **Source page:** F8 ## F159 / CF159 — Fixed-target centered tracking / recovery under CF157 **Expression** ```text e(τ)=e(0)e^{−3τ/(2T_c)}; τ_rec=2T_c/3 ``` **Status:** EXACT GIVEN CF157 **Definitions** - 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. **Cross-references:** CF157, CF156, CF160, CF161 **Source page:** F8 ## F160 / CF160 — Effective tracking operator **Expression** ```text R(t)=exp[−(I₆−P)t] ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF157, CF159, CF161 **Source page:** F8 ## F161 / CF161 — Tracking semigroup **Expression** ```text R(δ₁)R(δ₂)=R(δ₁+δ₂) ``` **Status:** EXACT GIVEN CF157 **Definitions** - R is effective tracking operator; δ₁ and δ₂ are successive tracking intervals. **Explanation:** The fixed-target recovery operators compose as a one-parameter semigroup. **Cross-references:** CF159, CF160 **Source page:** F8 ## F162 / CF162 — Exact spectral projectors **Expression** ```text E₁=(2/3)P+ (1/3)P̄; E₀=I+P−2P²; E_minus=(4/3)(P²−P) ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF158, CF160, CF163 **Source page:** F8 ## F163 / CF163 — Spectral tracking decomposition **Expression** ```text R=E₀+e^{−δ}E₁+e^{−3δ/2}E_minus ``` **Status:** EXACT GIVEN CF157 **Definitions** - 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. **Cross-references:** CF159, CF160, CF162, CF164 **Source page:** F8 ## F164 / CF164 — Lyapunov stability of the effective tracking candidate **Expression** ```text dV_L/dτ = −(1/T_c)eᵀ(I−P)e ≤ 0 ``` **Status:** EXACT GIVEN CF157 **Definitions** - 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. **Cross-references:** CF156, CF157, CF163 **Source page:** F8 ## F165 / CF165 — Aggregation-curvature bridge **Expression** ```text L_c²∇²M = (3C_Σ/2)A_cell = [18/(5π)]T_c/D₀ ``` **Status:** EXACT WITHIN COMBINED BRANCH **Definitions** - 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. **Cross-references:** CF032, CF046, CF047, CF067 **Source page:** F8 ## F166 / CF166 — Weak-static traceless tidal tensor **Expression** ```text E_ij = ∂_i∂_jM − (1/6)δ_ij∇²M ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF064, CF096, CF167 **Source page:** F8 ## F167 / CF167 — Tidal source-to-target shape direction **Expression** ```text q_eq = L_c²(ε₁, ε₂, ε₃, ε₃, ε₂, ε₁) ``` **Status:** CONDITIONAL STRONG PASS; ρ OPEN **Definitions** - 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. **Cross-references:** CF158, CF166, CF168, CF192 **Source page:** F8 ## F168 / CF168 — Two native target-normalization routes **Expression** ```text β_A=2/3; β_B=1/6; β_A/β_B=4 ``` **Status:** OPEN NORMALIZATION AUDIT **Definitions** - 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. **Cross-references:** CF167, CF173, CF187 **Source page:** F9 ## F169 / CF169 — Scalar loading cannot directly create centered shape **Expression** ```text E_minus(A_cell 1_6)=0 ``` **Status:** EXACT NO-GO BY SYMMETRY **Definitions** - E_minus is anisotropic spectral projector; A_cell1₆ is uniform scalar loading. **Explanation:** Uniform scalar loading lies outside the centered anisotropic target sector. **Cross-references:** CF046, CF162, CF167 **Source page:** F9 ## F170 / CF170 — Layered Celal + effective tracking update **Expression** ```text q_{n+1}=q_eq,n+1+R(δ_n)(q_n−q_eq,n+1) ``` **Status:** CANDIDATE CF157 UPDATE; DEPENDS ON CF157 **Definitions** - 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. **Cross-references:** CF157, CF159, CF160, CF167 **Source page:** F9 ## F171 / CF171 — Moving-target tracking equation **Expression** ```text de/dτ + [3/(2T_c)]e = −dq_eq/dτ ``` **Status:** EXACT GIVEN CF157 **Definitions** - 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. **Cross-references:** CF157, CF159, CF170 **Source page:** F9 ## F172 / CF172 — Effective tracking-timescale to Aggregation ratio **Expression** ```text R_rate ≡ τ_rec/D₀ = (2/3)A_cell ``` **Status:** EXACT GIVEN COMBINED CANDIDATES **Definitions** - 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. **Cross-references:** CF047, CF049, CF089, CF159 **Source page:** F9 ## F173 / CF173 — Candidate rate-matching point; not a fundamental-physics threshold **Expression** ```text R_rate=1 ⇔ A_cell=3/2 ⇔ D₀=2T_c/3 ``` **Status:** CONDITIONAL COMBINED-CANDIDATE RESULT **Definitions** - 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. **Cross-references:** CF047, CF089, CF172 **Source page:** F9 ## F174 / CF174 — Centered opposite-pair coordinates **Expression** ```text A₁₂=A₃₄=xL_c²; A₁₃=A₂₄=yL_c²; A₁₄=A₂₃=zL_c²; x+y+z=4 ``` **Status:** EXACT CENTERED GEOMETRY **Definitions** - 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. **Cross-references:** CF175, CF176, CF177, CF179 **Source page:** F9 ## F175 / CF175 — Centered four-null Gram matrix **Expression** ```text γ/L_c² = [[0,x,y],[x,0,z],[y,z,0]] ``` **Status:** EXACT GIVEN CF174 **Definitions** - 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. **Cross-references:** CF174, CF176, CF180 **Source page:** F9 ## F176 / CF176 — Centered Gram spectrum **Expression** ```text spec(γ/L_c²) = {−(x+y+z), x+y−z, x−y+z, −x+y+z} ``` **Status:** EXACT **Definitions** - γ is centered Gram matrix; x,y,z are opposite-pair coordinates; spec denotes eigenvalues. **Explanation:** Lists the eigenvalues governing centered Xyz causal integrity. **Cross-references:** CF175, CF177, CF180, CF181 **Source page:** F9 ## F177 / CF177 — Xyz determinant factorization **Expression** ```text −det(γ/L_c²) = (x+y−z)(x−y+z)(−x+y+z) ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF175, CF176, CF178, CF181 **Source page:** F9 ## F178 / CF178 — Xyz Law / Heron form **Expression** ```text −det(γ/L_c²) = 16A_tri² ``` **Status:** EXACT CENTERED GEOMETRY **Definitions** - 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. **Cross-references:** CF177, CF181, CF182 **Source page:** F9 ## F179 / CF179 — Centered deviation coordinates **Expression** ```text x=4/3+α; y=4/3+β; z=4/3+γ; α+β+γ=0 ``` **Status:** EXACT PARAMETRIZATION **Definitions** - x,y,z are pair coordinates; α,β,γ are zero-sum deviations. **Explanation:** Centers the pair coordinates around the neutral value 4/3. **Cross-references:** CF174, CF177, CF178, CF180 **Source page:** F9 ## F180 / CF180 — Centered Xyz Gram spectrum **Expression** ```text spec(γ)=L_c²{−4, 4/3−2α, 4/3−2β, 4/3−2γ} ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF176, CF179, CF181, CF182 **Source page:** F10 ## F181 / CF181 — Centered Xyz degeneracy boundary **Expression** ```text detγ=0 ⇔ α=2/3 or β=2/3 or γ=2/3 ``` **Status:** EXACT **Definitions** - γ is centered Gram matrix; α,β,γ are deviation coordinates. **Explanation:** Locates the three rank-loss boundaries of centered Xyz geometry. **Cross-references:** CF177, CF180, CF182, CF204 **Source page:** F10 ## F182 / CF182 — General common-mode Xyz determinant **Expression** ```text detγ = −3s(s−2α)(s−2β)(s−2γ)L_c⁸ ``` **Status:** EXACT GIVEN x=s+α **Definitions** - 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. **Cross-references:** CF179, CF181, CF183, CF204 **Source page:** F10 ## F183 / CF183 — Celal/permutation invariance of centered degeneracy **Expression** ```text det(ΓᵀγΓ)=detγ ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF085, CF181, CF182 **Source page:** F10 ## F184 / CF184 — Schwarzschild principal tidal pattern used in Xyz test **Expression** ```text {ε_r,ε_t,ε_t}={r_s/r, r_s/(2r), −r_s/(2r)} ``` **Status:** CONDITIONAL STRONG-FIELD INPUT **Definitions** - 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. **Cross-references:** CF166, CF185, CF188 **Source page:** F10 ## F185 / CF185 — Schwarzschild-Xyz radius **Expression** ```text r_Xyz = (3β/2)r_sL_c² ``` **Status:** CONDITIONAL COROLLARY **Definitions** - 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. **Cross-references:** CF168, CF184, CF186 **Source page:** F10 ## F186 / CF186 — Route-A / Route-B-Xyz radii **Expression** ```text r_Xyz,A=(r_s/β)^(1/3); r_Xyz,B=r_s(L_c/4)^(1/3) ``` **Status:** CONDITIONAL; β AMBIGUITY **Definitions** - 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. **Cross-references:** CF168, CF185, CF187 **Source page:** F10 ## F187 / CF187 — Mass-independent Xyz curvature threshold **Expression** ```text K_Xyz=[16/(3β²)]L_c⁻⁴ ⇒ 12/L_c⁴ (A), 192/L_c⁴ (B) ``` **Status:** CONDITIONAL COROLLARY **Definitions** - 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. **Cross-references:** CF168, CF185, CF186, CF190 **Source page:** F10 ## F188 / CF188 — Kerr invariant principal tidal amplitude **Expression** ```text ψ₂=−m_0/(r−ia cosθ)³; 2|ψ₂|=r_s/(r²+ a²cos²θ)³ᐟ² ``` **Status:** EXACT KERR GEOMETRY / INPUT **Definitions** - 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. **Cross-references:** CF184, CF189, CF190 **Source page:** F10 ## F189 / CF189 — Kerr Xyz candidate surface **Expression** ```text r²+a²cos²θ=R_Xyz²; R_Xyz=[(3β/2)r_sL_c²]¹ᐟ³ ``` **Status:** CONDITIONAL INVARIANT EXTENSION **Definitions** - 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. **Cross-references:** CF185, CF186, CF188, CF190 **Source page:** F10 ## F190 / CF190 — Kerr invariant curvature at Xyz boundary **Expression** ```text W=48|ψ₂|²=12r_s²/Σ³; W_Xyz=[16/(3β²)]L_c⁻⁴ ``` **Status:** CONDITIONAL COROLLARY **Definitions** - 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. **Cross-references:** CF187, CF188, CF189 **Source page:** F10 ## F191 / CF191 — Spherical traceless tidal density-contrast law **Expression** ```text ε_r=(2A/3)(ρ−ρ̄); ε_t=−(A/3)(ρ−ρ̄), A=C_Σc³T_c²/ℏ ``` **Status:** EXACT WITHIN WEAK SPHERICAL BRANCH **Definitions** - 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. **Cross-references:** CF067, CF184, CF192, CF193 **Source page:** F10 ## F192 / CF192 — Spherical Xyz density-contrast threshold **Expression** ```text |ρ̄−ρ|_Xyz = ℏ/(βC_Σc⁵T_c⁴) ``` **Status:** CONDITIONAL SOURCE-TO-SHAPE COROLLARY **Definitions** - 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. **Cross-references:** CF032, CF167, CF191, CF193 **Source page:** F11 ## F193 / CF193 — Dimensionless contrast loading and threshold **Expression** ```text A_Δ≡(2/3)|ρ̄−ρ|c⁵T_c⁴/ℏ; A_Δ,Xyz=2/(3βC_Σ) ``` **Status:** EXACT GIVEN DEFINITIONS / β OPEN **Definitions** - 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. **Cross-references:** CF032, CF168, CF192 **Source page:** F11 ## F194 / CF194 — Horizon-Xyz crossover **Expression** ```text r_s^ν=(3β/2)L_c; r_Xyz=r_s ``` **Status:** CONDITIONAL **Definitions** - 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. **Cross-references:** CF185, CF195, CF196 **Source page:** F11 ## F195 / CF195 — Conditional black-hole crossover mass **Expression** ```text M_X=(L_c/2G_G)[(3β/2)^(1/3)] ``` **Status:** CONDITIONAL; CALIBRATED BRANCH **Definitions** - 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. **Cross-references:** CF151, CF194, CF196 **Source page:** F11 ## F196 / CF196 — Density/horizon form of same Xyz crossover **Expression** ```text ρ̄_X = 3M_X/(4πr_s³) = ℏ/(βC_Σc⁵T_c⁴) = Δρ_Xyz ``` **Status:** EXACT ALGEBRA GIVEN CF194-CF195 **Definitions** - 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. **Cross-references:** CF192, CF194, CF195 **Source page:** F11 ## F197 / CF197 — Mercury back-inversion for T_c **Expression** ```text T_c = [625π²ℏG_G³a(1−e²)Δϖ/(18M_Sun)]¹ᐟ² ``` **Status:** OBSERVATIONAL INVERSION / CONSISTENCY **Definitions** - 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. **Cross-references:** CF038, CF151, CF153, CF198 **Source page:** F11 ## F198 / CF198 — K_M reconstruction from K_GD and T_c **Expression** ```text K_M=(25π³/12)ℏK_GD/T_c² ``` **Status:** EXACT ALGEBRA GIVEN RETAINED K_M LAW **Definitions** - 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. **Cross-references:** CF034, CF151, CF197, CF201 **Source page:** F11 ## F199 / CF199 — Two-cloud measured effective gravitational coupling **Expression** ```text G_eff = Δa_mod/[2M_mod(r_s⁻²−r_f⁻²)] ``` **Status:** EXPERIMENTAL RECONSTRUCTION **Definitions** - 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. **Cross-references:** CF151, CF200, CF201 **Source page:** F11 ## F200 / CF200 — Laboratory T_c reconstruction **Expression** ```text T_c = [15625π²ℏK_GD⁵Δa_mod/(96M_mod(r_s⁻²−r_f⁻²))]¹ᐟ² ``` **Status:** EXPERIMENTAL RECONSTRUCTION FORM **Definitions** - 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. **Cross-references:** CF198, CF199, CF201 **Source page:** F11 ## F201 / CF201 — Laboratory K_M reconstruction **Expression** ```text K_M = M_s(r_s⁻²−r_f⁻²)/[1250π⁹K_GD⁴Δa_mod] ``` **Status:** EXPERIMENTAL RECONSTRUCTION FORM **Definitions** - 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. **Cross-references:** CF098, CF199, CF200 **Source page:** F11 ## F202 / CF202 — Hawking temperature at conditional Xyz crossover **Expression** ```text T_H,X = ℏ/[4πk_BT_C√(3β/2)] ``` **Status:** SEMICLASSICAL COMPARISON ONLY **Definitions** - 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. **Cross-references:** CF194, CF195, CF203 **Source page:** F11 ## F203 / CF203 — Standard Schwarzschild lifetime evaluated at crossover **Expression** ```text T_H,X = (3200π³/3)(3β/2)³ᐟ²T_c ``` **Status:** SEMICLASSICAL COMPARISON ONLY **Definitions** - 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. **Cross-references:** CF202, CF194 **Source page:** F11 ## F204 / CF204 — Xyz causal-integrity margin **Expression** ```text χ_Xyz ≡ min{s−2α, s−2β, s−2γ_d} ``` **Status:** DEFINITION GIVEN CF182 **Definitions** - 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. **Cross-references:** CF181, CF182, CF205, CF207 **Source page:** F12 ## F205 / CF205 — Failed simple α-edge Xyz mode and branch dependence **Expression** ```text u_x=(1,1,−1,−1), γu_x=0; k₁+k₂−k₃−k₄=0 at α=s/2 ``` **Status:** EXACT AT SIMPLE α EDGE **Definitions** - 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. **Cross-references:** CF181, CF204, CF206 **Source page:** F12 ## F206 / CF206 — Boundary-flow sign under the effective CF157 tracking candidate **Expression** ```text λ_α=s−2α; (dλ_α/dτ)_CF157=(3/T_c)(s/2−α_eq) at α=s/2; dλ/dτ=0 ``` **Status:** EXACT GIVEN CF157; NOT GENERAL TRIGGER **Definitions** - 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. **Cross-references:** CF157, CF204, CF205, CF207 **Source page:** F12 ## F207 / CF207 — General Xyz outward-crossing causal-integrity condition **Expression** ```text χ_Xyz=0 and D_τχ_Xyz<0 ⇒ smooth ordinary 1+3 continuation cannot remain in retained phase unchanged ``` **Status:** GENERAL GEOMETRIC / KINEMATIC NECESSARY CONDITION; DYNAMICS OPEN **Definitions** - 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. **Cross-references:** CF204, CF206, CF208, CF217 **Source page:** F12 ## F208 / CF208 — Centered geometric mismatch norm **Expression** ```text Δq=(Δα,Δβ,Δγ,Δβ,Δα); V_geom=(1/2)||Δq||² ``` **Status:** DEFINITION **Definitions** - 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. **Cross-references:** CF156, CF157, CF209, CF211 **Source page:** F12 ## F209 / CF209 — Celal Gram mismatch equals centered geometric mismatch up to fixed factor **Expression** ```text (1/2)||Δγ||_F² = (1/2)L_c⁴V_geom ``` **Status:** EXACT IN CENTERED FIXED-COMMON-MODE SECTOR **Definitions** - 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. **Cross-references:** CF175, CF182, CF208, CF210 **Source page:** F12 ## F210 / CF210 — Candidate post-Xyz transported mismatch / discontinuity bookkeeping **Expression** ```text e_r=U_(n,r)q_eq,r; C_r=(1/2)‖r−U_(n,r)U_(n,r)ᵀ‖_F² ``` **Status:** DEFINITION; PHYSICAL MOVE SET OPEN **Definitions** - 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. **Cross-references:** CF085, CF167, CF208, CF211 **Source page:** F12 ## F211 / CF211 — Minimum-discontinuity continuation selector **Expression** ```text r_* ∈ arg min_r(C_r+R_r), R_r admissible ``` **Status:** STRONG RATIONALE; admissible R_r remains OPEN **Definitions** - 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. **Cross-references:** CF210, CF217 **Source page:** F12 ## F212 / CF212 — Observed fixed-target recovery-cost decay under CF157 **Expression** ```text V_L(τ)=V_L(0)e^(−3τ/T_c) ``` **Status:** EXACT GIVEN CF157; EFFECTIVE TRACKING CANDIDATE ONLY **Definitions** - 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. **Cross-references:** CF157, CF159, CF164 **Source page:** F12 ## F213 / CF213 — Xyz 2+2 incidence cut selected by failed mode **Expression** ```text c_X=BᵀP_Xu_x, c_X∈(0,±2,±2,±2,±2,0) ``` **Status:** EXACT UP TO EDGE ORIENTATION / ORDERING **Definitions** - 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. **Cross-references:** CF017, CF205, CF214, CF215 **Source page:** F12 ## F214 / CF214 — Failed Xyz cut is pure gradient in six-channel decomposition **Expression** ```text P_grad c_X ≠ c_X; P_circ c_X = c_X ``` **Status:** EXACT **Definitions** - 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. **Cross-references:** CF019, CF020, CF213 **Source page:** F12 ## F215 / CF215 — No unused eternal edge exists inside the one undirected four-vertex cell **Expression** ```text |E(K₄)| = C(4,2) = 6 ``` **Status:** EXACT COMBINATORIAL NO-GO **Definitions** - 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. **Cross-references:** CF006, CF018, CF213 **Source page:** F12 ## F216 / CF216 — Local candidate dual-area/action weight collapses at Xyz rank loss while 4-volume remains finite **Expression** ```text detγ_X=0 ⇒ V₄,x→0 ⇒ B^dual_X→0 ``` **Status:** EXACT CONSEQUENCE WITHIN CF143-CF147 CANDIDATE ACTION **Definitions** - 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. **Cross-references:** CF143, CF145, CF146, CF147, CF181 **Source page:** F12 ## F217 / CF217 — Consolidated causal-integrity law; Celal may select among a separately derived admissible continuation set **Expression** ```text 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 ``` **Status:** EXACT BOUNDARY + CONDITIONAL SELECTOR; ORDINARY TRANSPORT AND POST-Xyz MOVE SET OPEN **Definitions** - 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. **Cross-references:** CF085, CF204, CF207, CF210, CF211 **Source page:** F12 ## F218 / CF218 — Advection of a sharp material-support indicator with the matter flow **Expression** ```text ∂_tχ_m + v_m·∇χ_m = 0 ``` **Status:** EXACT KINEMATICS FOR MATERIAL SUPPORT **Definitions** - 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. **Cross-references:** CF053, CF219, CF222 **Source page:** F13 ## F219 / CF219 — Rigid-body matter velocity: translation plus axial rotation **Expression** ```text v_m(x,t)=V(t)+ω(t)×[x−X(t)] ``` **Status:** EXACT NONRELATIVISTIC RIGID-BODY KINEMATICS **Definitions** - 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. **Cross-references:** CF218, CF220, CF221 **Source page:** F13 ## F220 / CF220 — Rotation is tangent to an ideal spherical matter boundary, including a tilted axis **Expression** ```text (ω×r)·n = 0 for n ∥ r ``` **Status:** EXACT GEOMETRY **Definitions** - 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. **Cross-references:** CF219, CF221 **Source page:** F13 ## F221 / CF221 — Low-velocity dust source decomposition; rotation/tilt enters through momentum and stress **Expression** ```text T^{00}≈ρc²; T^{0i}≈ρcv^i; T^{ij}≈ρv^iv^j ``` **Status:** STANDARD NONRELATIVISTIC SOURCE LIMIT / INPUT **Definitions** - 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. **Cross-references:** CF041, CF042, CF102, CF218, CF219 **Source page:** F13 ## F222 / CF222 — Ordinary weak matter-boundary matching without a delta-function surface layer **Expression** ```text [M]_S=0; [∂_nM]_S=0; [∇²M]_S=C_Σ(c³T_c²/ℏ)ρ|_S ``` **Status:** EXACT WITHIN RETAINED WEAK SOURCE EQUATION AND NO-SURFACE-LAYER ASSUMPTION **Definitions** - [·]_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. **Cross-references:** CF032, CF053, CF067, CF218, CF220 **Source page:** F13