F1 · CF001

Native causal length

Status: DEFINITION

Expression: L_c = cT_c

Definitions

• L_c is the native causal length; c is the speed of light; T_c is intrinsic causal time.

Explanation

Defines the length obtained when the causal time scale is propagated at c.

Cross-references: CF002, CF023, CF039

Source page: F1

F2 · CF002

Four future-directed null branches

Status: MODEL GEOMETRY

Expression: kₐ = L_c(1, nₐ)

Definitions

• kₐ is branch a; L_c is native length; nₐ is one of four unit spatial branch directions.

Explanation

Encodes each future-directed null branch as a temporal component and a tetrahedral spatial direction.

Cross-references: CF001, CF003, CF004, CF005

Source page: F1

F3 · CF003

Regular tetrahedral directional balance

Status: EXACT

Expression: nₐ·nᵦ = −1/3 (a ≠ b)

Definitions

• nₐ and nᵦ are distinct unit branch directions; a,b ∈ {1,2,3,4}.

Explanation

The four unit directions have the pairwise dot product required by a regular tetrahedron.

Cross-references: CF002, CF004, CF010

Source page: F1

F4 · CF004

Tetrahedral centroid balance

Status: EXACT

Expression: Σₐ₌₁⁴ nₐ = 0

Definitions

• nₐ are the four tetrahedral unit directions; Σ denotes the sum over all branches.

Explanation

The four spatial branch directions have a vanishing centroid.

Cross-references: CF002, CF003, CF010

Source page: F1

F5 · CF005

Null character of each branch

Status: EXACT

Expression: kₐ² = 0

Definitions

• kₐ is a branch four-vector; the square uses the model Lorentz metric.

Explanation

Each branch is null, so its temporal and spatial parts balance in the causal metric.

Cross-references: CF002, CF010, CF014

Source page: F1

F6 · CF006

Four branches generate six unordered pairs

Status: EXACT

Expression: C(4,2) = 6

Definitions

• 4 is the number of branches; C(4,2) counts unordered two-branch pairs.

Explanation

Counts the six pair channels carried by four causal branches.

Cross-references: CF007, CF018, CF019, CF020

Source page: F1

F7 · CF007

Pairwise causal bivector

Status: DEFINITION

Expression: Σₐᵦ = kₐ ∧ kᵦ

Definitions

• Σₐᵦ is the bivector for a branch pair; kₐ,kᵦ are branch four-vectors; ∧ is the exterior product.

Explanation

Defines each pair channel as the exterior product of its two null branches.

Cross-references: CF002, CF006, CF008, CF009

Source page: F1

F8 · CF008

Neutral pair-area invariant

Status: EXACT

Expression: |kₐ ∧ kᵦ| = 3/2 L_c²

Definitions

• kₐ,kᵦ are distinct branches; L_c is native length; |·| is bivector magnitude.

Explanation

Every neutral branch pair has the same causal area magnitude.

Cross-references: CF001, CF007, CF009, CF029

Source page: F1

F9 · CF009

Aggregate neutral six-bivector magnitude

Status: EXACT

Expression: AΣ,0 = 6(4L_c²/3) = 8L_c²

Definitions

• AΣ,0 is aggregate neutral pair area; the factor 6 counts pairs; L_c is native length.

Explanation

Sums the six pair-area magnitudes into the neutral aggregate area scale.

Cross-references: CF006, CF008, CF023

Source page: F1

F10 · CF010

Four-null Gram matrix

Status: EXACT

Expression: Gₐₐ = 0, Gₐᵦ = −1/3 (a ≠ b)

Definitions

• G is the four-branch Gram matrix; a,b index the four null branches.

Explanation

Records the diagonal and off-diagonal inner products of the four-null basis.

Cross-references: CF003, CF005, CF011, CF012, CF013

Source page: F1

F11 · CF011

Four-null Gram spectrum

Status: EXACT

Expression: spec(G₄) = {−4/3, 4/3, 4/3, 4/3}L_c²

Definitions

• G₄ is the four-null Gram matrix; spec denotes its eigenvalue multiset; L_c is native length.

Explanation

Gives the one longitudinal and three degenerate transverse Gram eigenvalues.

Cross-references: CF010, CF012, CF013

Source page: F1

F12 · CF012

Non-degenerate local causal basis

Status: EXACT

Expression: det G₄ = −256/27 L_c⁸

Definitions

• G₄ is the four-null Gram matrix; det is its determinant; L_c is native length.

Explanation

The nonzero determinant establishes that the four branch directions form a non-degenerate local basis.

Cross-references: CF010, CF011, CF014

Source page: F1

F13 · CF013

Inverse Gram

Status: EXACT

Expression: G₄⁻¹ = L_c⁻²(3/4 I₄ − 1/4 11ᵀ)

Definitions

• G₄⁻¹ is the inverse Gram matrix; I₄ is the 4×4 identity; 1 is the all-ones vector.

Explanation

Provides the explicit inverse needed to reconstruct the Minkowski metric from branch data.

Cross-references: CF010, CF012, CF014

Source page: F1

F14 · CF014

Minkowski metric reconstruction

Status: EXACT

Expression: K G₄⁻¹ Kᵀ = η

Definitions

• K contains the four branch vectors; G₄⁻¹ is the inverse Gram matrix; η is the Minkowski metric.

Explanation

Reconstructs the Minkowski metric by changing from branch coordinates to spacetime coordinates.

Cross-references: CF005, CF010, CF013, CF015

Source page: F1

F15 · CF015

Branch directional derivative

Status: DEFINITION

Expression: Dₐ = kₐ^μ∂_μ

Definitions

• Dₐ is the derivative along branch a; kₐ^μ is its four-vector; ∂_μ is the spacetime derivative.

Explanation

Defines differentiation along each causal branch.

Cross-references: CF002, CF014, CF016

Source page: F1

F16 · CF016

Native d’Alembertian reconstruction

Status: EXACT

Expression: (G₄⁻¹)ᵃᵇDₐDᵦ = □

Definitions

• Dₐ are branch derivatives; G₄⁻¹ is the inverse branch Gram; □ is the Minkowski d’Alembertian.

Explanation

Combines branch derivatives with the inverse Gram to recover the wave operator.

Cross-references: CF013, CF014, CF015, CF067

Source page: F1

F17 · CF017

Boundary-of-boundary incidence identity

Status: EXACT

Expression: BC = 0

Definitions

• B is the vertex-to-edge incidence matrix; C is the edge-to-face incidence matrix.

Explanation

The incidence composition vanishes because a boundary has no boundary.

Cross-references: CF018, CF019, CF020, CF123

Source page: F1

F18 · CF018

Exact six-edge Hodge decomposition

Status: EXACT

Expression: BᵀB + CCᵀ = 4I₆

Definitions

• B,C are incidence matrices; I₆ is the identity on six edge channels.

Explanation

Relates vertex and face incidence operators on the six tetrahedral edge channels.

Cross-references: CF006, CF017, CF019, CF020

Source page: F1

F19 · CF019

Gradient/circulation projectors

Status: EXACT

Expression: Pgrad = 1/4 BᵀB, Pcirc = 1/4 CCᵀ

Definitions

• Pgrad and Pcirc are six-channel projectors; B,C are incidence matrices.

Explanation

Normalizes the gradient and circulation components of edge-channel data.

Cross-references: CF017, CF018, CF020, CF021, CF022

Source page: F1

F20 · CF020

Complete six-channel decomposition

Status: EXACT

Expression: Pgrad + Pcirc = I₆

Definitions

• Pgrad and Pcirc are complementary projectors; I₆ is the six-channel identity.

Explanation

States that the gradient and circulation sectors exhaust the six pair channels.

Cross-references: CF018, CF019, CF022

Source page: F1

F21 · CF021

Historical normalized six-channel operator convention

Status: RETAINED

Expression: K₀ = BᵀB − 2I₆

Definitions

• K₀ is the retained normalized edge operator; B is incidence; I₆ is the six-channel identity.

Explanation

Records the normalized operator convention retained for the neutral channel spectrum.

Cross-references: CF018, CF019, CF022

Source page: F1

F22 · CF022

Six-channel neutral spectrum

Status: EXACT

Expression: spec(K₀) = {4,0,0,0,−2,−2}

Definitions

• K₀ is the normalized six-channel operator; spec denotes its eigenvalue multiset.

Explanation

Lists the three gradient-zero modes and the two negative circulation eigenvalues.

Cross-references: CF019, CF020, CF021

Source page: F1

F23 · CF023

Native causal volume

Status: DEFINITION

Expression: V_c = (2/3)L_c³ = (√2/12)(cT_c)³

Definitions

• V_c is native causal volume; L_c is native length; c and T_c are the causal speed and time.

Explanation

Defines the four-null causal cell volume in equivalent native-length and causal-time forms.

Cross-references: CF001, CF008, CF024, CF025

Source page: F1

F24 · CF024

Instantaneous causal-volume flux for dL/dt = c

Status: CONDITIONAL

Expression: Γᵥ = dV/dt = 2cL_c²

Definitions

• Γᵥ is volume flux; V is causal volume; L_c is native length; c is propagation speed.

Explanation

Differentiates the native volume while assuming radial causal growth dL/dt = c.

Cross-references: CF023, CF025, CF040

Source page: F1

F25 · CF025

Characteristic causal-volume flux

Status: CONDITIONAL

Expression: Γ_c = 2c³T_c²

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