F126 · CF126

Vacuum field equation

Status: CONDITIONAL CONSEQUENCE

Expression: ∂_μF^{μν} = 0

Definitions

• F^{μν} is the raised light-sector field strength; ∂_μ is the spacetime derivative; ν is a spacetime index.

Explanation

The source-free Euler-Lagrange equation follows from the proposed light action in vacuum.

Cross-references: CF124, CF125, CF127, CF129

Source page: F6

F127 · CF127

Plane-wave closure form

Status: EXACT CONSEQUENCE

Expression: F = q ∧ a

Definitions

• F is plane-wave field strength; q is wave covector; a is polarization one-form; ∧ is exterior product.

Explanation

Represents a closed plane-wave field as the wedge of its wave covector and polarization.

Cross-references: CF126, CF128, CF129, CF132

Source page: F6

F128 · CF128

Plane-wave co-closure

Status: EXACT

Expression: q_μF^{μν} = q²a^ν − (q·a)q^ν

Definitions

• q is wave covector; F is plane-wave field; a is polarization; q² and q·a use the Lorentz metric.

Explanation

Contracts the plane-wave field with its wave covector and exposes the transverse condition.

Cross-references: CF127, CF129, CF130

Source page: F6

F129 · CF129

Nontrivial vacuum light is null

Status: EXACT WITHIN CANDIDATE DYNAMICS

Expression: q² = 0

Definitions

• q is the wave covector; q² is its Lorentz norm.

Explanation

Requires a nontrivial transverse vacuum wave to have a null wave covector.

Cross-references: CF126, CF128, CF130, CF131

Source page: F6

F130 · CF130

Vacuum light dispersion

Status: EXACT WITHIN CANDIDATE DYNAMICS

Expression: ω² = c²k²

Definitions

• ω is angular frequency; k is spatial wave number; c is light speed.

Explanation

Writes the null-wave condition as the standard vacuum dispersion relation.

Cross-references: CF129, CF131, CF137

Source page: F6

F131 · CF131

Two physical transverse modes

Status: EXACT WITHIN CANDIDATE DYNAMICS

Expression: dim(q⊥/span{q}) = 2

Definitions

• q⊥ is the Lorentz-orthogonal polarization space; span{q} is the gauge direction.

Explanation

After quotienting the gauge direction, two independent transverse polarization modes remain.

Cross-references: CF127, CF129, CF132, CF133

Source page: F6

F132 · CF132

Hodge helicity sectors

Status: DEFINITION

Expression: F_± = F ∓ iJ_*F

Definitions

• F_± are helicity sectors; F is field strength; J_* is the bivector Hodge operator; i²=−1.

Explanation

Splits the field into the two complex Hodge-helicity sectors.

Cross-references: CF090, CF127, CF133

Source page: F6

F133 · CF133

Helicity eigenvalue relation

Status: EXACT

Expression: J_*F_± = ±iF_±

Definitions

• J_* is Hodge star; F_± are helicity fields; i is the imaginary unit.

Explanation

Shows that each helicity sector is an eigenvector of the Hodge operator.

Cross-references: CF090, CF132

Source page: F6

F134 · CF134

Vertex conservation of edge-transfer current

Status: EXACT

Expression: BJ = 0

Definitions

• B is vertex-edge incidence; J is edge-transfer current.

Explanation

Imposes vanishing net current at every vertex in the discrete cell.

Cross-references: CF017, CF121, CF135

Source page: F6

F135 · CF135

Minimal source coupling

Status: CONDITIONAL IDENTIFICATION

Expression: S_int = −g_AJᵀA

Definitions

• S_int is interaction action; g_A is coupling; J is edge current; A is edge potential.

Explanation

Couples the discrete current linearly to the edge potential.

Cross-references: CF121, CF125, CF134, CF136

Source page: F6

F136 · CF136

Discrete sourced light equation

Status: CONDITIONAL

Expression: κ_γCCᵀJ = g_AJ

Definitions

• κ_γ is light stiffness; C is edge-face incidence; J is edge current; g_A is source coupling.

Explanation

Gives the sourced discrete light equation from the quadratic action and current coupling.

Cross-references: CF122, CF125, CF135, CF137

Source page: F6

F137 · CF137

Coulomb-form static field

Status: CONDITIONAL

Expression: E(r) = g_AQ/(4πκ_γr²)

Definitions

• E(r) is radial static field; g_A is source coupling; Q is source charge; κ_γ is stiffness; r is radius.

Explanation

Provides the inverse-square static solution for a point source in the candidate light sector.

Cross-references: CF136, CF138

Source page: F6

F138 · CF138

Compact link variable

Status: CONDITIONAL U(1) COMPLETION

Expression: U_ij = e^{i a_ij}

Definitions

• U_ij is a compact link; a_ij is a real link phase; i is the imaginary unit.

Explanation

Exponentiates the link phase to make the discrete variable compact and U(1)-valued.

Cross-references: CF139, CF140

Source page: F6

F139 · CF139

Compact U(1) integer representation weights

Status: EXACT GROUP THEORY

Expression: ψ_i → g_i^nψ_i, n ∈ ℤ

Definitions

• ψ_i is a site field; g_i is a U(1) group element; n is an integer representation weight.

Explanation

Records the integer-weight action of compact U(1) on site fields.

Cross-references: CF138, CF140

Source page: F6

F140 · CF140

Native tetrahedral residual triality

Status: EXACT GROUP THEORY

Expression: A₄/V₄ ≅ ℤ₃

Definitions

• A₄ is the tetrahedral rotation group; V₄ is its Klein-four normal subgroup; ℤ₃ is the quotient.

Explanation

Identifies the residual three-class quotient of the tetrahedral rotation group.

Cross-references: CF006, CF138, CF139

Source page: F6

F141 · CF141

Exact tetrahedral group-valued Bianchi closure

Status: EXACT ALGEBRAIC

Expression: W₀₁₂W₀₂₃W₀₃₁(Ω₀₁W₁₂₃Ω₁₀⁻¹) = I

Definitions

• W_ijk are face holonomies; Ω₀₁ transports between cells; I is the group identity.

Explanation

States exact ordered holonomy closure around the tetrahedral boundary.

Cross-references: CF083, CF142, CF146

Source page: F7

F142 · CF142

Formal BCH relation; additive curvature is the near-identity/small-curvature limit

Status: LOCAL / SMALL-CURVATURE BCH

Expression: log(eᴷ¹eᴷ²⋯) = ΣK_i + 1/2Σ[K_i,K_j] + O(K³)

Definitions

• K_i are near-identity Lie-algebra generators; [·,·] is a commutator; O(K³) is higher order.

Explanation

Relates group-valued holonomy products to additive curvature only in the local small-curvature limit.

Cross-references: CF141, CF146, CF147

Source page: F7

F143 · CF143

Invariant G Model causal four-volume extension; not uniquely canonical

Status: CANDIDATE INVARIANT EXTENSION

Expression: V₄,x = (2/3)L_c⁴[|detγ_x|/|detγ₀|]¹ᐟ²

Definitions

• V₄,x is causal four-volume at x; L_c is native length; γ_x and γ₀ are local and neutral metrics.

Explanation

Proposes a determinant-weighted four-volume while explicitly retaining non-uniqueness.

Cross-references: CF023, CF144, CF145

Source page: F7

F144 · CF144

Dimensionless bivectors and six-space metric remove unit ambiguity

Status: DEFINITION / DIMENSIONAL CLEANUP

Expression: Σ̂_A = Σ_A/L_c²; Γ̂_AB = ⟨Σ̂_A,Σ̂_B⟩

Definitions

• Σ_A is a bivector; Σ̂_A is dimensionless; Γ̂_AB is the six-space metric; L_c is native length.

Explanation

Normalizes bivectors by the native area so six-space contractions are dimensionless.

Cross-references: CF007, CF091, CF143, CF145

Source page: F7

F145 · CF145

Dimensionally explicit candidate dual-area bivector; [B^dual]=L²

Status: CANDIDATE DEFINITION; UNIQUENESS OPEN

Expression: B^dual_B = −2(V₄,x/L_c²)Γ̂_ABJ_*Σ̂_A

Definitions

• B^dual is candidate dual bivector; V₄,x is four-volume; Γ̂ is six-metric; J_* is Hodge star; Σ̂ is normalized bivector.

Explanation

Defines a dimensionally explicit dual-area candidate but leaves uniqueness open.

Cross-references: CF090, CF143, CF144, CF146

Source page: F7

F146 · CF146

Discrete curvature has L⁻² units; contraction identity follows from definitions

Status: IDENTITY EXACT GIVEN DEFINITIONS; R_disc→R OPEN

Expression: R^disc_x = (2/L_c²)Γ̂^AB⟨Σ̂_A,K_B⟩; Σ_B⟨B^dual,J_*K⟩ = V₄,xR^disc_x

Definitions

• R^disc_x is discrete scalar curvature; K_B is curvature bivector; Γ̂ is six-metric; V₄,x is four-volume.

Explanation

Defines discrete curvature and its dual-area contraction, while leaving the continuum limit open.

Cross-references: CF141, CF143, CF145, CF147

Source page: F7

F147 · CF147

Microscopic holonomy-curvature action with explicit unfixed normalization

Status: CANDIDATE ACTION

Expression: S_micro^G/ℏ = ζ_microΣ⟨B^dual/L_c²,(J_*+βI)K(W)⟩_Γ

Definitions

• S_micro^G is microscopic gravitational action; ζ_micro is normalization; β is parity parameter; K(W) is holonomy curvature.

Explanation

Proposes a microscopic holonomy-curvature action with normalization still to be fixed.

Cross-references: CF142, CF145, CF146, CF148

Source page: F7

F148 · CF148

Effective continuum normalization required by internal source matching

Status: DERIVED MATCHING; MICRO FACTORS/UNIQUENESS OPEN

Expression: ζ_eff ≡ ζ_microN_disc = 1/(4C_Σ) = 5π/48 ≈ 0.327249

Definitions

• ζ_eff is effective normalization; ζ_micro is microscopic normalization; N_disc is discrete count; C_Σ is retained coefficient.

Explanation

Matches the microscopic normalization to the continuum source coefficient, without claiming microscopic uniqueness.

Cross-references: CF032, CF067, CF147, CF150

Source page: F7

F149 · CF149

One-update matter action in constant-energy Aggregation

Status: EXACT WITHIN AGGREGATION DEFINITIONS

Expression: |ΔS_m| = E_cellT_c = ℏA_cell

Definitions

• ΔS_m is one-update matter action; E_cell is cell energy; T_c is causal time; A_cell is loading.

Explanation

Relates one intrinsic-time matter update to both cell energy and dimensionless loading.

Cross-references: CF027, CF041, CF046, CF050

Source page: F7

F150 · CF150

Continuum source equation matched by action normalization

Status: CONDITIONAL STRUCTURAL

Expression: E^G_ab = R_ab − (1/2)γ_abR = 2C_Σ(cT_c²/ℏ)H_ab

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