F51 · CF051

Independent energy contributions add in rate

Status: EXACT

Expression: Γ_{m,tot} = Σ_i Γ_{m,i}

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

Status: EXACT

Expression: t^μ = K^μ/√(−K²), K^μ = Σ_a k_a^μ

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

Status: BOOKKEEPING

Expression: χ_m ∈ {0,1}

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

Status: STATE

Expression: (χ_m,M) = (0,0)

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

Status: STATE

Expression: (χ_m,M) = (1,M ≠ 0)

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

Status: STATE

Expression: (χ_m,M) = (0,M ≠ 0)

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

Status: SPECIAL STATE

Expression: (χ_m,M) = (1,0)

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

Status: DEFINITION

Expression: Σ_col = ρL_c

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

Status: CONDITIONAL GEOMETRY

Expression: ∫Σ_col dA = M_source

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

Status: POSTULATE

Expression: ∂κ_C/∂m = ∂κ_C/∂ρ = ∂κ_C/∂Ω = 0

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

Status: CONDITIONAL

Expression: M = ln(dτ/dt)

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

Status: CONDITIONAL

Expression: dτ/dt = e^M

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

Status: CONDITIONAL

Expression: Φ = c²δM

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

Status: CONDITIONAL

Expression: g = −c²∇M

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

Status: EXACT EXPANSION

Expression: Ā[M] = M + (L_c²/6)∇²M + (L_c⁴/120)∇⁴M + ⋯

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

Status: CONDITIONAL

Expression: Ā[M] − M = (C_Σ/4)A_cell

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

Status: CONDITIONAL

Expression: ∇²M = C_Σ(c³T_c²/ℏ)ρ

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

Status: CALIBRATED COMPATIBILITY

Expression: ∇²M = (4πG/c²)ρ

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

Status: CALIBRATED COMPATIBILITY

Expression: ∇²Φ = 4πGρ

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

Status: EXACT WITHIN REDUCED MODEL

Expression: Q₆(M) = 16e²M + 8e⁻²M

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

Status: EXACT

Expression: M_* = −(1/4)ln 2

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

Status: EXACT

Expression: Q₆,min = 16√2

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

Status: EXACT

Expression: Q₊Q₋ = 128

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

Status: EXACT

Expression: S₆² − D₆² = 512

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

Status: EXACT IN STATED SPECTRUM

Expression: Tr|K(M)| = 8 cosh M

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