F76 · CF076

Dolu reciprocal-time invariant

Status: EXACT WITHIN CONSTRUCTION

Expression: T₊T₋ = T_c²

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

Differential reciprocal-time form

Status: EXACT WITHIN CONSTRUCTION

Expression: δln T₊ + δln T₋ = 0

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

Three-dimensional reciprocal time sectors

Status: CONDITIONAL

Expression: T₊ = T_ce⁺²ᵘ⁄³

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

Reciprocal volume sectors

Status: CONDITIONAL

Expression: V₊ = V_ce⁺²ᵘ

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

Spectral-volume equalization if shared-u bridge is adopted

Status: CONDITIONAL BRIDGE

Expression: 2V_cQ₊V₊ = Q₋V₋ = 8√2V_c

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

Jadvyga local reorientation

Status: DEFINITION

Expression: Kₓ(θ) = Λₓ(θ)Kₓ(0)

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

Lorentz-frame preservation

Status: EXACT

Expression: ΛₓᵀηΛₓ = η

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

Holonomy conjugation under local reorientation

Status: EXACT

Expression: W_f → ΛₓW_fΛₓ⁻¹

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

Celal invariant mismatch cost

Status: DEFINITION

Expression: C_xy(Π) = 1/2‖ΠᵀΠ_xy − Π‖²_F

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

Celal branch correspondence

Status: EXACT GEOMETRIC SELECTION

Expression: Π_Celal ∈ arg min_Π∈S₄ C_xy(Π)

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

Centered physical shape sector

Status: DEFINITION

Expression: q = (a,b,c,c,b,a), a+b+c = 0

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

Candidate reconnect relaxation law

Status: CONDITIONAL

Expression: T_c dq/dτ = −(I−P)q

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

Centered relaxation solution

Status: CONDITIONAL

Expression: q(τ) = q₀e⁻³ᵗ⁄²ᵀ_c

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

Candidate reconnect timescale

Status: CONDITIONAL

Expression: τ_rec = 2T_c/3

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

Lorentz bivector Hodge structure

Status: EXACT

Expression: J_*² = −I₆

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

Hodge compatibility with bivector metric

Status: EXACT

Expression: J_*ᵀΓJ_* = −Γ

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

Inter-cell branch-frame transporter

Status: CONDITIONAL/GEOMETRIC

Expression: Ω_yx = K_yΠ_yxKₓ⁻¹

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

Lorentz transport on matched frames

Status: EXACT WHEN GRAMS MATCH

Expression: Ω_yxᵀηΩ_yx = η

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

Four-component source-divergence candidate

Status: EXACT REPRESENTATION

Expression: C_b = (G₄⁻¹)ᵃ_cDₐH_cb

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

Capacity for four independent continuity equations

Status: EXACT REPRESENTATION

Expression: rank(H ↦ C_b) = 4

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

Weak scalar gradient-stress tensor

Status: DEFINITION

Expression: Tᵢⱼ^(M) = K_M(∂ᵢM∂ⱼM − 1/2δᵢⱼ|∇M|²)

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

Field-stress divergence; matter force is negative divergence

Status: EXACT

Expression: ∂ⱼTᵢⱼ^(M) = K_M(∂ᵢM)∇²M

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

Weak source-modulus form

Status: CONDITIONAL

Expression: K_M∇²M = ρc²

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

Six-channel strain hierarchy

Status: EXACT ALGEBRAIC

Expression: 3 reciprocal strains → 1 trace + 2 anisotropic

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

Massless-fabric/background-energy candidate

Status: OPEN HYPOTHESIS

Expression: ρ_f^rest = 0, u_f0 ≥ 0

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