F101 · CF101

Branch encoding of metric perturbation

Status: EXACT REPRESENTATION

Expression: Q_ab = k_a^μ k_b^ν h_μν

Definitions

• Q_ab is branch metric data; k_a^μ,k_b^ν are branch vectors; h_μν is metric perturbation.

Explanation

Projects a spacetime metric perturbation into the four-branch basis.

Cross-references: CF014, CF103, CF105, CF107

Source page: F5

F102 · CF102

Branch encoding of stress-energy

Status: EXACT REPRESENTATION

Expression: H_ab = k_a^μ k_b^ν T_μν

Definitions

• H_ab is branch stress data; k_a^μ,k_b^ν are branch vectors; T_μν is stress-energy.

Explanation

Projects stress-energy into the same branch basis used for metric reconstruction.

Cross-references: CF094, CF101, CF104, CF111

Source page: F5

F103 · CF103

Reconstruction of h_μν

Status: EXACT

Expression: h = (K⁻¹)ᵀQK⁻¹

Definitions

• h is spacetime metric perturbation; K is branch-frame matrix; Q is branch metric data.

Explanation

Reconstructs the spacetime perturbation from branch metric components.

Cross-references: CF101, CF104, CF105

Source page: F5

F104 · CF104

Reconstruction of T_μν

Status: EXACT

Expression: T = (K⁻¹)ᵀHK⁻¹

Definitions

• T is spacetime stress-energy; K is branch-frame matrix; H is branch stress data.

Explanation

Reconstructs stress-energy from its branch-space encoding.

Cross-references: CF102, CF103, CF111

Source page: F5

F105 · CF105

Branch-space trace

Status: EXACT

Expression: Q = (G₄⁻¹)ᵃᵇQ_ab

Definitions

• Q is branch trace; G₄⁻¹ is inverse branch Gram; Q_ab is branch metric data.

Explanation

Contracts branch metric data with the inverse Gram.

Cross-references: CF010, CF013, CF101, CF106

Source page: F5

F106 · CF106

Branch trace reversal

Status: EXACT

Expression: Q̄_ab = Q_ab − 1/2(G₄)_abQ

Definitions

• Q̄_ab is trace-reversed branch metric data; Q_ab is branch data; Q is its trace; G₄ is Gram.

Explanation

Removes the branch trace in the standard trace-reversal combination.

Cross-references: CF101, CF105, CF107

Source page: F5

F107 · CF107

Harmonic condition in branch variables

Status: CONDITIONAL GAUGE

Expression: (G₄⁻¹)ᵃ_cDₐQ̄_cb = 0

Definitions

• Q̄_cb is trace-reversed branch perturbation; Dₐ is branch derivative; G₄⁻¹ is inverse Gram.

Explanation

Imposes the harmonic gauge condition after translating it to branch variables.

Cross-references: CF015, CF016, CF103, CF106

Source page: F5

F108 · CF108

Linearized Einstein operator

Status: EXACT UNDER CONTINUUM ASSUMPTIONS

Expression: E_ab^(1)[Q] = 1/2[−□Q_ab + DₐD_cQ_cb + D_bD_cQ_ca − DₐD_bQ − (G₄)_abD_cDᵈQ_cd]

Definitions

• E_ab^(1) is linearized Einstein operator; Q_ab is branch perturbation; Dₐ are branch derivatives; □ is d’Alembertian.

Explanation

Writes the linearized Einstein operator in branch variables before gauge simplification.

Cross-references: CF016, CF101, CF105, CF107

Source page: F5

F109 · CF109

Identical conservation of linearized operator

Status: EXACT

Expression: DᵃE_ab^(1) = 0

Definitions

• E_ab^(1) is the linearized Einstein operator; Dᵃ is the branch derivative with raised index.

Explanation

States the linearized Bianchi identity for the branch-space operator.

Cross-references: CF107, CF108, CF110

Source page: F5

F110 · CF110

Linearized gauge transformation

Status: CONDITIONAL CONTINUUM

Expression: δQ_ab = Dₐξ_b + D_bξₐ

Definitions

• δQ_ab is metric-data variation; ξ_a is infinitesimal branch gauge vector; Dₐ is branch derivative.

Explanation

Defines the infinitesimal gauge transformation of branch metric data.

Cross-references: CF101, CF107, CF108, CF109

Source page: F5

F111 · CF111

Big-G-calibrated linearized source equation

Status: CALIBRATED COMPATIBILITY

Expression: E_ab = (8πG/c⁴)H_ab

Definitions

• E_ab is linearized field tensor; G is calibrated coupling; c is light speed; H_ab is branch stress data.

Explanation

Matches the linearized branch source equation to the Big-G normalization.

Cross-references: CF068, CF102, CF108, CF109

Source page: F5

F112 · CF112

Two-function strong-field variable

Status: CONDITIONAL COMPLETION

Expression: ψ = 2 tanh(M/2)

Definitions

• ψ is bounded strong-field variable; M is deformation.

Explanation

Maps unbounded deformation M to a bounded variable for the strong-field completion.

Cross-references: CF061, CF113, CF114, CF115

Source page: F5

F113 · CF113

Temporal metric factor

Status: CONDITIONAL COMPLETION

Expression: T = e^M = (1+ψ/2)/(1−ψ/2)

Definitions

• T is temporal metric factor; M is deformation; ψ is bounded strong-field variable.

Explanation

Defines the temporal metric factor in both deformation and bounded-variable forms.

Cross-references: CF061, CF112, CF114

Source page: F5

F114 · CF114

Spatial metric factor

Status: CONDITIONAL COMPLETION

Expression: X = (1−ψ/2)²

Definitions

• X is spatial metric factor; ψ is bounded strong-field variable.

Explanation

Defines the companion spatial factor used by the isotropic strong-field metric.

Cross-references: CF112, CF113, CF115

Source page: F5

F115 · CF115

Static spherical vacuum solution

Status: EXACT WITHIN COMPLETION

Expression: ψ = −μ/r

Definitions

• ψ is bounded strong-field variable; μ is spherical mass-length parameter; r is isotropic radius.

Explanation

Uses the inverse-radius vacuum solution for the bounded strong-field variable.

Cross-references: CF112, CF116, CF117, CF118

Source page: F5

F116 · CF116

Isotropic Schwarzschild exterior

Status: EXACT WITHIN COMPLETION

Expression: ds² = (1−μ/(2r))²/(1+μ/(2r))² c²dt² + (1+μ/(2r))⁴(dr²+r²dΩ²)

Definitions

• ds² is spherical line element; μ is mass-length parameter; r is radius; dΩ² is unit-sphere metric.

Explanation

Gives the isotropic-coordinate spherical exterior produced by the two-function completion.

Cross-references: CF113, CF114, CF115, CF117, CF119

Source page: F5

F117 · CF117

Areal-radius transformation

Status: EXACT

Expression: R = r(1+μ/2r)²

Definitions

• R is areal radius; r is isotropic radius; μ is mass-length parameter.

Explanation

Converts the isotropic radius to the Schwarzschild areal radius.

Cross-references: CF115, CF116, CF118

Source page: F5

F118 · CF118

Invariant areal-radius causal-support scalar

Status: EXACT WITHIN SCHWARZSCHILD BRANCH

Expression: C_F = 1 − 2μ/R

Definitions

• C_F is causal-support scalar; μ is mass-length parameter; R is areal radius.

Explanation

Defines the invariant radial factor that vanishes at the spherical causal boundary.

Cross-references: CF117, CF119, CF120

Source page: F5

F119 · CF119

Horizon radius with empirical Big G

Status: CALIBRATED/GR COMPATIBILITY

Expression: R_h = 2μ = 2GM/c²

Definitions

• R_h is horizon radius; μ is mass-length parameter; G is empirical coupling; M is gravitating mass; c is light speed.

Explanation

Identifies the spherical horizon radius with the empirical Schwarzschild value.

Cross-references: CF116, CF118, CF120

Source page: F5

F120 · CF120

Schwarzschild Kretschmann scalar

Status: EXACT

Expression: K = 48μ²/R⁶

Definitions

• K is Kretschmann scalar; μ is mass-length parameter; R is areal radius.

Explanation

Gives the curvature invariant for the spherical vacuum branch.

Cross-references: CF117, CF118, CF184

Source page: F5

F121 · CF121

Finite-cell abelian-type gauge transformation

Status: EXACT KINEMATICS

Expression: A → A + Bᵀχ

Definitions

• A is edge potential; B is incidence matrix; χ is vertex gauge parameter.

Explanation

Defines the discrete gauge shift of an edge potential by a vertex coboundary.

Cross-references: CF017, CF122, CF123

Source page: F5

F122 · CF122

Discrete face/loop field strength

Status: DEFINITION

Expression: F = CᵀA

Definitions

• F is face/loop field strength; C is edge-to-face incidence; A is edge potential.

Explanation

Defines discrete curvature or field strength as the face incidence of the edge potential.

Cross-references: CF017, CF121, CF123

Source page: F5

F123 · CF123

Gauge invariance from BC = 0

Status: EXACT KINEMATICS

Expression: F′ = F

Definitions

• F is discrete field strength; F′ is its gauge-transformed value; B,C are incidence matrices.

Explanation

The field strength is unchanged because the boundary-of-boundary composition vanishes.

Cross-references: CF017, CF121, CF122

Source page: F5

F124 · CF124

Continuum closure

Status: EXACT KINEMATICS

Expression: F = dA, d²A = 0

Definitions

• A is a continuum gauge potential; F is its field strength; d is exterior derivative.

Explanation

States the continuum differential closure corresponding to the discrete incidence identity.

Cross-references: CF121, CF122, CF123

Source page: F5

F125 · CF125

Minimal parity-even light action

Status: CONDITIONAL COMPLETION

Expression: S_γ = −(κ_γ/4)∫F_μνF^μν d⁴x

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