F176 · CF176

Centered Gram spectrum

Status: EXACT

Expression: spec(γ/L_c²) = {−(x+y+z), x+y−z, x−y+z, −x+y+z}

Definitions

• γ is centered Gram matrix; x,y,z are opposite-pair coordinates; spec denotes eigenvalues.

Explanation

Lists the eigenvalues governing centered Xyz causal integrity.

Cross-references: CF175, CF177, CF180, CF181

Source page: F9

F177 · CF177

Xyz determinant factorization

Status: EXACT

Expression: −det(γ/L_c²) = (x+y−z)(x−y+z)(−x+y+z)

Definitions

• γ is centered Gram matrix; x,y,z are pair coordinates; det is determinant.

Explanation

Factors the centered Gram determinant into the three causal-integrity margins.

Cross-references: CF175, CF176, CF178, CF181

Source page: F9

F178 · CF178

Xyz Law / Heron form

Status: EXACT CENTERED GEOMETRY

Expression: −det(γ/L_c²) = 16A_tri²

Definitions

• A_tri is triangle area; x,y,z are centered pair coordinates; γ is centered Gram matrix.

Explanation

Identifies the determinant with the Heron-form squared area of the associated triangle.

Cross-references: CF177, CF181, CF182

Source page: F9

F179 · CF179

Centered deviation coordinates

Status: EXACT PARAMETRIZATION

Expression: x=4/3+α; y=4/3+β; z=4/3+γ; α+β+γ=0

Definitions

• x,y,z are pair coordinates; α,β,γ are zero-sum deviations.

Explanation

Centers the pair coordinates around the neutral value 4/3.

Cross-references: CF174, CF177, CF178, CF180

Source page: F9

F180 · CF180

Centered Xyz Gram spectrum

Status: EXACT

Expression: spec(γ)=L_c²{−4, 4/3−2α, 4/3−2β, 4/3−2γ}

Definitions

• γ is centered Gram matrix; α,β,γ are centered deviations; L_c is native length.

Explanation

Rewrites the centered Gram spectrum in deviation coordinates.

Cross-references: CF176, CF179, CF181, CF182

Source page: F10

F181 · CF181

Centered Xyz degeneracy boundary

Status: EXACT

Expression: detγ=0 ⇔ α=2/3 or β=2/3 or γ=2/3

Definitions

• γ is centered Gram matrix; α,β,γ are deviation coordinates.

Explanation

Locates the three rank-loss boundaries of centered Xyz geometry.

Cross-references: CF177, CF180, CF182, CF204

Source page: F10

F182 · CF182

General common-mode Xyz determinant

Status: EXACT GIVEN x=s+α

Expression: detγ = −3s(s−2α)(s−2β)(s−2γ)L_c⁸

Definitions

• s is common mode; α,β,γ are deviations; γ is centered Gram matrix; L_c is native length.

Explanation

Factors the determinant for a general common mode and isolates the three causal margins.

Cross-references: CF179, CF181, CF183, CF204

Source page: F10

F183 · CF183

Celal/permutation invariance of centered degeneracy

Status: EXACT

Expression: det(ΓᵀγΓ)=detγ

Definitions

• Γ is a permutation or Celal frame map; γ is centered Gram matrix.

Explanation

Shows that centered degeneracy is unchanged by the permitted branch permutation.

Cross-references: CF085, CF181, CF182

Source page: F10

F184 · CF184

Schwarzschild principal tidal pattern used in Xyz test

Status: CONDITIONAL STRONG-FIELD INPUT

Expression: {ε_r,ε_t,ε_t}={r_s/r, r_s/(2r), −r_s/(2r)}

Definitions

• ε_r and ε_t are radial and tangential tidal eigenvalues; r_s is Schwarzschild radius; r is radius.

Explanation

Supplies the radial/tangential tidal eigenvalue pattern used to test Xyz geometry.

Cross-references: CF166, CF185, CF188

Source page: F10

F185 · CF185

Schwarzschild-Xyz radius

Status: CONDITIONAL COROLLARY

Expression: r_Xyz = (3β/2)r_sL_c²

Definitions

• r_Xyz is Xyz radius; β is normalization; r_s is Schwarzschild radius; L_c is causal length.

Explanation

Defines the radius at which the Schwarzschild tidal pattern is compared with the Xyz scale.

Cross-references: CF168, CF184, CF186

Source page: F10

F186 · CF186

Route-A / Route-B-Xyz radii

Status: CONDITIONAL; β AMBIGUITY

Expression: r_Xyz,A=(r_s/β)^(1/3); r_Xyz,B=r_s(L_c/4)^(1/3)

Definitions

• r_Xyz,A and r_Xyz,B are candidate radii; r_s is Schwarzschild radius; β is route coefficient; L_c is causal length.

Explanation

Lists the two radius scalings generated by the competing normalization routes.

Cross-references: CF168, CF185, CF187

Source page: F10

F187 · CF187

Mass-independent Xyz curvature threshold

Status: CONDITIONAL COROLLARY

Expression: K_Xyz=[16/(3β²)]L_c⁻⁴ ⇒ 12/L_c⁴ (A), 192/L_c⁴ (B)

Definitions

• K_Xyz is threshold curvature; β is route coefficient; L_c is causal length; A and B label routes.

Explanation

Gives the route-dependent curvature threshold after eliminating source mass.

Cross-references: CF168, CF185, CF186, CF190

Source page: F10

F188 · CF188

Kerr invariant principal tidal amplitude

Status: EXACT KERR GEOMETRY / INPUT

Expression: ψ₂=−m_0/(r−ia cosθ)³; 2|ψ₂|=r_s/(r²+ a²cos²θ)³ᐟ²

Definitions

• ψ₂ is Kerr Weyl scalar; m₀ is mass parameter; r is radius; a is spin; θ is polar angle; r_s is Schwarzschild radius.

Explanation

Supplies the Kerr principal curvature amplitude used in the rotating extension.

Cross-references: CF184, CF189, CF190

Source page: F10

F189 · CF189

Kerr Xyz candidate surface

Status: CONDITIONAL INVARIANT EXTENSION

Expression: r²+a²cos²θ=R_Xyz²; R_Xyz=[(3β/2)r_sL_c²]¹ᐟ³

Definitions

• R_Xyz is candidate surface radius; r,a,θ are Kerr coordinates; β is normalization; r_s is Schwarzschild radius.

Explanation

Extends the spherical Xyz surface condition to the Kerr invariant radius.

Cross-references: CF185, CF186, CF188, CF190

Source page: F10

F190 · CF190

Kerr invariant curvature at Xyz boundary

Status: CONDITIONAL COROLLARY

Expression: W=48|ψ₂|²=12r_s²/Σ³; W_Xyz=[16/(3β²)]L_c⁻⁴

Definitions

• W is Weyl curvature invariant; ψ₂ is Kerr scalar; Σ=r²+a²cos²θ; β is normalization.

Explanation

Evaluates the Kerr invariant at the candidate Xyz boundary.

Cross-references: CF187, CF188, CF189

Source page: F10

F191 · CF191

Spherical traceless tidal density-contrast law

Status: EXACT WITHIN WEAK SPHERICAL BRANCH

Expression: ε_r=(2A/3)(ρ−ρ̄); ε_t=−(A/3)(ρ−ρ̄), A=C_Σc³T_c²/ℏ

Definitions

• ε_r,ε_t are radial/tangential tidal strains; ρ,ρ̄ are local and mean densities; A is source coefficient.

Explanation

Relates the traceless spherical tidal pattern to density contrast in the weak branch.

Cross-references: CF067, CF184, CF192, CF193

Source page: F10

F192 · CF192

Spherical Xyz density-contrast threshold

Status: CONDITIONAL SOURCE-TO-SHAPE COROLLARY

Expression: |ρ̄−ρ|_Xyz = ℏ/(βC_Σc⁵T_c⁴)

Definitions

• ρ̄−ρ is density contrast; β is normalization; C_Σ is coefficient; c is light speed; T_c is causal time.

Explanation

Converts the Xyz curvature condition into a spherical density-contrast threshold.

Cross-references: CF032, CF167, CF191, CF193

Source page: F11

F193 · CF193

Dimensionless contrast loading and threshold

Status: EXACT GIVEN DEFINITIONS / β OPEN

Expression: A_Δ≡(2/3)|ρ̄−ρ|c⁵T_c⁴/ℏ; A_Δ,Xyz=2/(3βC_Σ)

Definitions

• A_Δ is contrast loading; ρ̄−ρ is density contrast; β and C_Σ are normalization coefficients.

Explanation

Defines dimensionless density-contrast loading and its Xyz threshold.

Cross-references: CF032, CF168, CF192

Source page: F11

F194 · CF194

Horizon-Xyz crossover

Status: CONDITIONAL

Expression: r_s^ν=(3β/2)L_c; r_Xyz=r_s

Definitions

• r_s is Schwarzschild radius; r_s^ν is model crossover scale; β is normalization; L_c is causal length.

Explanation

States the conditional radius at which the Xyz scale coincides with a Schwarzschild scale.

Cross-references: CF185, CF195, CF196

Source page: F11

F195 · CF195

Conditional black-hole crossover mass

Status: CONDITIONAL; CALIBRATED BRANCH

Expression: M_X=(L_c/2G_G)[(3β/2)^(1/3)]

Definitions

• M_X is crossover mass; L_c is causal length; G_G is model coupling; β is normalization.

Explanation

Converts the crossover radius into a mass using the calibrated model coupling.

Cross-references: CF151, CF194, CF196

Source page: F11

F196 · CF196

Density/horizon form of same Xyz crossover

Status: EXACT ALGEBRA GIVEN CF194-CF195

Expression: ρ̄_X = 3M_X/(4πr_s³) = ℏ/(βC_Σc⁵T_c⁴) = Δρ_Xyz

Definitions

• ρ̄_X is crossover mean density; M_X is crossover mass; r_s is Schwarzschild radius; Δρ_Xyz is threshold contrast.

Explanation

Shows the equivalent density, horizon, and Xyz threshold forms of the crossover.

Cross-references: CF192, CF194, CF195

Source page: F11

F197 · CF197

Mercury back-inversion for T_c

Status: OBSERVATIONAL INVERSION / CONSISTENCY

Expression: T_c = [625π²ℏG_G³a(1−e²)Δϖ/(18M_Sun)]¹ᐟ²

Definitions

• T_c is inferred causal time; G_G is coupling; a is semimajor axis; e is eccentricity; Δϖ is perihelion advance.

Explanation

Inverts the Mercury expression to infer the intrinsic time from orbital observables.

Cross-references: CF038, CF151, CF153, CF198

Source page: F11

F198 · CF198

K_M reconstruction from K_GD and T_c

Status: EXACT ALGEBRA GIVEN RETAINED K_M LAW

Expression: K_M=(25π³/12)ℏK_GD/T_c²

Definitions

• K_M is source modulus; K_GD is interface constant; ℏ is action; T_c is causal time.

Explanation

Reconstructs the source modulus from the retained interface constant and intrinsic time.

Cross-references: CF034, CF151, CF197, CF201

Source page: F11

F199 · CF199

Two-cloud measured effective gravitational coupling

Status: EXPERIMENTAL RECONSTRUCTION

Expression: G_eff = Δa_mod/[2M_mod(r_s⁻²−r_f⁻²)]

Definitions

• G_eff is measured coupling; Δa_mod is acceleration difference; M_mod is source mass; r_s,r_f are source and far radii.

Explanation

Defines an experimental two-radius reconstruction of effective coupling from differential acceleration.

Cross-references: CF151, CF200, CF201

Source page: F11

F200 · CF200

Laboratory T_c reconstruction

Status: EXPERIMENTAL RECONSTRUCTION FORM

Expression: T_c = [15625π²ℏK_GD⁵Δa_mod/(96M_mod(r_s⁻²−r_f⁻²))]¹ᐟ²

Definitions

• T_c is reconstructed causal time; K_GD is interface constant; Δa_mod and M_mod are measured source data; r_s,r_f are radii.

Explanation

Combines the two-cloud reconstruction with the retained modulus law to infer intrinsic time.

Cross-references: CF198, CF199, CF201

Source page: F11