G MODEL COMPLETE FORMULA REGISTER
Current register: CF001-CF222
Author: Gurbuz Dolu
Encoding: UTF-8

F1 / CF001
Title: Native causal length
Expression: 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
Title: Four future-directed null branches
Expression: 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
Title: Regular tetrahedral directional balance
Expression: 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
Title: Tetrahedral centroid balance
Expression: 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
Title: Null character of each branch
Expression: 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
Title: Four branches generate six unordered pairs
Expression: 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
Title: Pairwise causal bivector
Expression: 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
Title: Neutral pair-area invariant
Expression: |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
Title: Aggregate neutral six-bivector magnitude
Expression: 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
Title: Four-null Gram matrix
Expression: 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
Title: Four-null Gram spectrum
Expression: 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
Title: Non-degenerate local causal basis
Expression: 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
Title: Inverse Gram
Expression: 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
Title: Minkowski metric reconstruction
Expression: 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
Title: Branch directional derivative
Expression: 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
Title: Native d’Alembertian reconstruction
Expression: (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
Title: Boundary-of-boundary incidence identity
Expression: 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
Title: Exact six-edge Hodge decomposition
Expression: 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
Title: Gradient/circulation projectors
Expression: 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
Title: Complete six-channel decomposition
Expression: 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
Title: Historical normalized six-channel operator convention
Expression: 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
Title: Six-channel neutral spectrum
Expression: 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
Title: Native causal volume
Expression: 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
Title: Instantaneous causal-volume flux for dL/dt = c
Expression: 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
Title: Characteristic causal-volume flux
Expression: 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
Title: Intrinsic fabric energy scale
Expression: 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
Title: One intrinsic fabric update carries one action quantum
Expression: 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
Title: Native dimensional mass scale
Expression: 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
Title: Neutral pair-area scale
Expression: 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
Title: Candidate neutral area-stiffness closure
Expression: 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
Title: Candidate surface-tension scale
Expression: 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
Title: Retained causal-geometric coefficient
Expression: 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
Title: Microscopic coefficient plus spherical spreading
Expression: 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
Title: G Model Kutsal Interface Constant; not Big G
Expression: 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
Title: Exact algebraic interface identity
Expression: 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
Title: Continuum matter-spacetime coupling bridge
Expression: 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
Title: Big-G-calibrated fabric time
Expression: 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
Title: Reference value for C_Σ = 12/(5π)
Expression: 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
Title: Reference causal length
Expression: 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
Title: Neutral propagation normalization
Expression: 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
Title: Covariant local causal-cell energy
Expression: 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
Title: Stationary nonrelativistic limit
Expression: 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
Title: Accumulated matter action
Expression: 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
Title: Aggregation definition of matter-response time
Expression: 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
Title: Constant-energy Aggregation law
Expression: 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
Title: Dimensionless matter/fabric loading
Expression: 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
Title: Controlling Aggregation reciprocal relation
Expression: 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
Title: Stationary density loading
Expression: 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
Title: Equal-action/equal-scale condition
Expression: 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
Title: Additive matter-action accumulation rate
Expression: 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
Title: Independent energy contributions add in rate
Expression: 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
Title: Natural cell timelike direction
Expression: 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
Title: Matter-occupancy bookkeeping variable
Expression: 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
Title: Matter absent, neutral fabric
Expression: (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
Title: Matter present, deformed fabric
Expression: (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
Title: Matter absent locally, deformation persists
Expression: (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
Title: Neutral M; does not by itself imply D₀ = T_c
Expression: (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
Title: One-causal-depth column density
Expression: 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
Title: Projected-area mass additivity
Expression: ∫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
Title: Baseline fabric universality requirement
Expression: ∂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
Title: Operational deformation/clock relation
Expression: 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
Title: Local clock ratio
Expression: 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
Title: Weak-field potential identification
Expression: 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
Title: Weak-field acceleration
Expression: 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
Title: Orientation-balanced spherical mean
Expression: 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
Title: Discrete mean-defect source rule
Expression: 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
Title: Weak continuum source law
Expression: ∇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
Title: Big-G-calibrated weak source law
Expression: ∇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
Title: Poisson normalization
Expression: ∇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
Title: Reduced reciprocal quadratic spectrum
Expression: 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
Title: Mathematical minimum of Q₆
Expression: 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
Title: Minimum reciprocal spectral value
Expression: 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
Title: Reciprocal spectral-product invariant
Expression: 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
Title: Hyperbolic spectral invariant
Expression: 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
Title: Linear reciprocal spectral magnitude
Expression: 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
Title: Dolu reciprocal-time invariant
Expression: 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
Title: Differential reciprocal-time form
Expression: 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
Title: Three-dimensional reciprocal time sectors
Expression: 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
Title: Reciprocal volume sectors
Expression: 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
Title: Spectral-volume equalization if shared-u bridge is adopted
Expression: 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
Title: Jadvyga local reorientation
Expression: 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
Title: Lorentz-frame preservation
Expression: 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
Title: Holonomy conjugation under local reorientation
Expression: 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
Title: Celal invariant mismatch cost
Expression: 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
Title: Celal branch correspondence
Expression: 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
Title: Centered physical shape sector
Expression: 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
Title: Candidate reconnect relaxation law
Expression: 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
Title: Centered relaxation solution
Expression: 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
Title: Candidate reconnect timescale
Expression: 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
Title: Lorentz bivector Hodge structure
Expression: 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
Title: Hodge compatibility with bivector metric
Expression: 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
Title: Inter-cell branch-frame transporter
Expression: 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
Title: Lorentz transport on matched frames
Expression: 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
Title: Four-component source-divergence candidate
Expression: 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
Title: Capacity for four independent continuity equations
Expression: 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
Title: Weak scalar gradient-stress tensor
Expression: 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
Title: Field-stress divergence; matter force is negative divergence
Expression: ∂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
Title: Weak source-modulus form
Expression: 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
Title: Six-channel strain hierarchy
Expression: 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
Title: Massless-fabric/background-energy candidate
Expression: 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
Title: Branch encoding of metric perturbation
Expression: 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
Title: Branch encoding of stress-energy
Expression: 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
Title: Reconstruction of h_μν
Expression: 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
Title: Reconstruction of T_μν
Expression: 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
Title: Branch-space trace
Expression: 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
Title: Branch trace reversal
Expression: 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
Title: Harmonic condition in branch variables
Expression: (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
Title: Linearized Einstein operator
Expression: 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
Title: Identical conservation of linearized operator
Expression: 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
Title: Linearized gauge transformation
Expression: 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
Title: Big-G-calibrated linearized source equation
Expression: 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
Title: Two-function strong-field variable
Expression: 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
Title: Temporal metric factor
Expression: 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
Title: Spatial metric factor
Expression: 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
Title: Static spherical vacuum solution
Expression: 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
Title: Isotropic Schwarzschild exterior
Expression: 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
Title: Areal-radius transformation
Expression: 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
Title: Invariant areal-radius causal-support scalar
Expression: 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
Title: Horizon radius with empirical Big G
Expression: 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
Title: Schwarzschild Kretschmann scalar
Expression: 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
Title: Finite-cell abelian-type gauge transformation
Expression: 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
Title: Discrete face/loop field strength
Expression: 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
Title: Gauge invariance from BC = 0
Expression: 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
Title: Continuum closure
Expression: 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
Title: Minimal parity-even light action
Expression: 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
Title: Vacuum field equation
Expression: ∂_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
Title: Plane-wave closure form
Expression: 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
Title: Plane-wave co-closure
Expression: 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
Title: Nontrivial vacuum light is null
Expression: 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
Title: Vacuum light dispersion
Expression: 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
Title: Two physical transverse modes
Expression: 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
Title: Hodge helicity sectors
Expression: 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
Title: Helicity eigenvalue relation
Expression: 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
Title: Vertex conservation of edge-transfer current
Expression: 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
Title: Minimal source coupling
Expression: 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
Title: Discrete sourced light equation
Expression: 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
Title: Coulomb-form static field
Expression: 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
Title: Compact link variable
Expression: 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
Title: Compact U(1) integer representation weights
Expression: 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
Title: Native tetrahedral residual triality
Expression: 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
Title: Exact tetrahedral group-valued Bianchi closure
Expression: 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
Title: Formal BCH relation; additive curvature is the near-identity/small-curvature limit
Expression: 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
Title: Invariant G Model causal four-volume extension; not uniquely canonical
Expression: 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
Title: Dimensionless bivectors and six-space metric remove unit ambiguity
Expression: 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
Title: Dimensionally explicit candidate dual-area bivector; [B^dual]=L²
Expression: 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
Title: Discrete curvature has L⁻² units; contraction identity follows from definitions
Expression: 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
Title: Microscopic holonomy-curvature action with explicit unfixed normalization
Expression: 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
Title: Effective continuum normalization required by internal source matching
Expression: 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
Title: One-update matter action in constant-energy Aggregation
Expression: |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
Title: Continuum source equation matched by action normalization
Expression: 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
Title: G Model gravitational coupling used in Mercury benchmark
Expression: 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
Title: First post-Newtonian coefficients in the smooth metric-compatible vacuum closure
Expression: 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
Title: Mercury perihelion advance; M_Sun avoids conflict with deformation variable M
Expression: 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
Title: Mercury numerical benchmark
Expression: 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
Title: Post-7 Sep rationale for retained depth and Hodge-throughput factors
Expression: 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
Title: Centered mismatch definition
Expression: 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
Title: Causal re-equalization flow
Expression: 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
Title: Centered physical shape sector
Expression: 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
Title: Fixed-target centered tracking / recovery under CF157
Expression: 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
Title: Effective tracking operator
Expression: 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
Title: Tracking semigroup
Expression: 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
Title: Exact spectral projectors
Expression: 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
Title: Spectral tracking decomposition
Expression: 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
Title: Lyapunov stability of the effective tracking candidate
Expression: 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
Title: Aggregation-curvature bridge
Expression: 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
Title: Weak-static traceless tidal tensor
Expression: 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
Title: Tidal source-to-target shape direction
Expression: 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
Title: Two native target-normalization routes
Expression: 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
Title: Scalar loading cannot directly create centered shape
Expression: 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
Title: Layered Celal + effective tracking update
Expression: 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
Title: Moving-target tracking equation
Expression: 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
Title: Effective tracking-timescale to Aggregation ratio
Expression: 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
Title: Candidate rate-matching point; not a fundamental-physics threshold
Expression: 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
Title: Centered opposite-pair coordinates
Expression: 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
Title: Centered four-null Gram matrix
Expression: 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
Title: Centered Gram spectrum
Expression: 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
Title: Xyz determinant factorization
Expression: −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
Title: Xyz Law / Heron form
Expression: −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
Title: Centered deviation coordinates
Expression: 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
Title: Centered Xyz Gram spectrum
Expression: 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
Title: Centered Xyz degeneracy boundary
Expression: 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
Title: General common-mode Xyz determinant
Expression: 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
Title: Celal/permutation invariance of centered degeneracy
Expression: 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
Title: Schwarzschild principal tidal pattern used in Xyz test
Expression: {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
Title: Schwarzschild-Xyz radius
Expression: 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
Title: Route-A / Route-B-Xyz radii
Expression: 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
Title: Mass-independent Xyz curvature threshold
Expression: 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
Title: Kerr invariant principal tidal amplitude
Expression: 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
Title: Kerr Xyz candidate surface
Expression: 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
Title: Kerr invariant curvature at Xyz boundary
Expression: 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
Title: Spherical traceless tidal density-contrast law
Expression: 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
Title: Spherical Xyz density-contrast threshold
Expression: |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
Title: Dimensionless contrast loading and threshold
Expression: 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
Title: Horizon-Xyz crossover
Expression: 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
Title: Conditional black-hole crossover mass
Expression: 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
Title: Density/horizon form of same Xyz crossover
Expression: 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
Title: Mercury back-inversion for T_c
Expression: 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
Title: K_M reconstruction from K_GD and T_c
Expression: 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
Title: Two-cloud measured effective gravitational coupling
Expression: 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
Title: Laboratory T_c reconstruction
Expression: 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
Title: Laboratory K_M reconstruction
Expression: 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
Title: Hawking temperature at conditional Xyz crossover
Expression: 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
Title: Standard Schwarzschild lifetime evaluated at crossover
Expression: 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
Title: Xyz causal-integrity margin
Expression: 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
Title: Failed simple α-edge Xyz mode and branch dependence
Expression: 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
Title: Boundary-flow sign under the effective CF157 tracking candidate
Expression: 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
Title: General Xyz outward-crossing causal-integrity condition
Expression: 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
Title: Centered geometric mismatch norm
Expression: 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
Title: Celal Gram mismatch equals centered geometric mismatch up to fixed factor
Expression: (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
Title: Candidate post-Xyz transported mismatch / discontinuity bookkeeping
Expression: 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
Title: Minimum-discontinuity continuation selector
Expression: 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
Title: Observed fixed-target recovery-cost decay under CF157
Expression: 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
Title: Xyz 2+2 incidence cut selected by failed mode
Expression: 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
Title: Failed Xyz cut is pure gradient in six-channel decomposition
Expression: 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
Title: No unused eternal edge exists inside the one undirected four-vertex cell
Expression: |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
Title: Local candidate dual-area/action weight collapses at Xyz rank loss while 4-volume remains finite
Expression: 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
Title: Consolidated causal-integrity law; Celal may select among a separately derived admissible continuation set
Expression: 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
Title: Advection of a sharp material-support indicator with the matter flow
Expression: ∂_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
Title: Rigid-body matter velocity: translation plus axial rotation
Expression: 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
Title: Rotation is tangent to an ideal spherical matter boundary, including a tilted axis
Expression: (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
Title: Low-velocity dust source decomposition; rotation/tilt enters through momentum and stress
Expression: 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
Title: Ordinary weak matter-boundary matching without a delta-function surface layer
Expression: [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