F151 · CF151

G Model gravitational coupling used in Mercury benchmark

Status: CONDITIONAL; CALIBRATED WHEN Tᶜ FROM BIG G

Expression: G_G = [3/(5π²)]c⁵T_c²/ℏ

Definitions

• G_G is G Model coupling; c is light speed; T_c is causal time; ℏ is action quantum.

Explanation

Defines the model coupling used by the orbital benchmark, with its numerical time calibration conditional.

Cross-references: CF036, CF037, CF153, CF154

Source page: F7

F152 · CF152

First post-Newtonian coefficients in the smooth metric-compatible vacuum closure

Status: CONDITIONAL CONTINUUM CONSEQUENCE

Expression: β_G = 1, γ_G = 1

Definitions

• β_G and γ_G are post-Newtonian coefficients in the smooth metric closure.

Explanation

Records the retained first post-Newtonian coefficients of the smooth vacuum branch.

Cross-references: CF116, CF151, CF153

Source page: F7

F153 · CF153

Mercury perihelion advance; M_Sun avoids conflict with deformation variable M

Status: CONDITIONAL FORM; Tᶜ CALIBRATION MAY CONTAIN BIG G

Expression: Δϖ_G = [18/(5π)]c³T_c²M_Sun/[ℏa(1−e²)]

Definitions

• Δϖ_G is perihelion advance; M_Sun is solar mass; a is semimajor axis; e is eccentricity; T_c is causal time.

Explanation

Gives the retained Mercury advance in model variables and flags its calibration dependence.

Cross-references: CF119, CF151, CF152, CF154

Source page: F7

F154 · CF154

Mercury numerical benchmark

Status: CALIBRATED COMPATIBILITY

Expression: T_c = 2.18657×10⁻⁴³ s ⇒ Δϖ_G = 42.9826 arcsec/Julian century

Definitions

• T_c is calibrated causal time; Δϖ_G is predicted perihelion advance; arcsec is angular arcsecond.

Explanation

Reports the numerical Mercury comparison after inserting the calibrated intrinsic time.

Cross-references: CF038, CF151, CF153

Source page: F7

F155 · CF155

Post-7 Sep rationale for retained depth and Hodge-throughput factors

Status: CONDITIONAL PHYSICAL-SELECTION RATIONALE

Expression: w(x)=x ⇒ ∫₀¹4x³w(x)dx=4/5; ⟨|cosφ|+|sinφ|⟩=4/π

Definitions

• w(x) is a radial weight; x is normalized radius; φ is angular coordinate.

Explanation

Records the weighting and angular averages used as a physical-selection rationale for retained geometric factors.

Cross-references: CF032, CF148

Source page: F7

F156 · CF156

Centered mismatch definition

Status: DEFINITION

Expression: e = q − q_eq

Definitions

• e is mismatch; q is current centered shape; q_eq is target or equilibrium shape.

Explanation

Defines the deviation of the current shape from its selected target.

Cross-references: CF086, CF157, CF158

Source page: F8

F157 · CF157

Causal re-equalization flow

Status: STRUCTURAL EFFECTIVE STRUCTURAL CANDIDATE

Expression: T_c dq/dτ = −(I₆−P)e

Definitions

• q is shape; τ is update time; P is target projector; e is mismatch; I₆ is channel identity.

Explanation

Provides the effective re-equalization flow retained for the special tracking construction.

Cross-references: CF087, CF156, CF158, CF159, CF161

Source page: F8

F158 · CF158

Centered physical shape sector

Status: EXACT IN CENTERED SECTOR

Expression: q=(α,β,γ,γ,β,α), α+β+γ=0; Pq=−q/2

Definitions

• q is centered six-channel shape; α,β,γ are paired shape coordinates; P is target projector.

Explanation

Specifies the paired centered sector and its target-projector eigenvalue.

Cross-references: CF086, CF156, CF157, CF162, CF167

Source page: F8

F159 · CF159

Fixed-target centered tracking / recovery under CF157

Status: EXACT GIVEN CF157

Expression: e(τ)=e(0)e^{−3τ/(2T_c)}; τ_rec=2T_c/3

Definitions

• e is mismatch; τ is tracking time; T_c is causal time; τ_rec is recovery timescale.

Explanation

Solves the fixed-target mismatch flow and identifies its decay timescale.

Cross-references: CF157, CF156, CF160, CF161

Source page: F8

F160 · CF160

Effective tracking operator

Status: DEFINITION

Expression: R(t)=exp[−(I₆−P)t]

Definitions

• R(t) is tracking operator; P is target projector; I₆ is six-channel identity.

Explanation

Defines the operator exponential implementing the effective centered tracking flow.

Cross-references: CF157, CF159, CF161

Source page: F8

F161 · CF161

Tracking semigroup

Status: EXACT GIVEN CF157

Expression: R(δ₁)R(δ₂)=R(δ₁+δ₂)

Definitions

• R is effective tracking operator; δ₁ and δ₂ are successive tracking intervals.

Explanation

The fixed-target recovery operators compose as a one-parameter semigroup.

Cross-references: CF159, CF160

Source page: F8

F162 · CF162

Exact spectral projectors

Status: EXACT

Expression: E₁=(2/3)P+ (1/3)P̄; E₀=I+P−2P²; E_minus=(4/3)(P²−P)

Definitions

• P is target projector; P̄ is its complementary sector; E₁,E₀,E_minus are spectral operators.

Explanation

Lists the retained projector decomposition of the centered tracking operator.

Cross-references: CF158, CF160, CF163

Source page: F8

F163 · CF163

Spectral tracking decomposition

Status: EXACT GIVEN CF157

Expression: R=E₀+e^{−δ}E₁+e^{−3δ/2}E_minus

Definitions

• R is tracking operator; δ is normalized time; E₀,E₁,E_minus are spectral projectors.

Explanation

Expresses recovery as decay in the nonzero spectral sectors while retaining the neutral sector.

Cross-references: CF159, CF160, CF162, CF164

Source page: F8

F164 · CF164

Lyapunov stability of the effective tracking candidate

Status: EXACT GIVEN CF157

Expression: dV_L/dτ = −(1/T_c)eᵀ(I−P)e ≤ 0

Definitions

• V_L is Lyapunov function; e is mismatch; P is target projector; T_c is causal time.

Explanation

Shows non-increasing mismatch energy for the effective candidate flow.

Cross-references: CF156, CF157, CF163

Source page: F8

F165 · CF165

Aggregation-curvature bridge

Status: EXACT WITHIN COMBINED BRANCH

Expression: L_c²∇²M = (3C_Σ/2)A_cell = [18/(5π)]T_c/D₀

Definitions

• L_c is causal length; M is deformation; C_Σ is coefficient; A_cell is loading; D₀ is aggregation time.

Explanation

Connects deformation curvature to cell loading and the aggregation-time ratio.

Cross-references: CF032, CF046, CF047, CF067

Source page: F8

F166 · CF166

Weak-static traceless tidal tensor

Status: DEFINITION

Expression: E_ij = ∂_i∂_jM − (1/6)δ_ij∇²M

Definitions

• E_ij is traceless tidal tensor; M is deformation; δ_ij is spatial identity; ∇² is Laplacian.

Explanation

Removes the scalar trace from the Hessian of the weak deformation field.

Cross-references: CF064, CF096, CF167

Source page: F8

F167 · CF167

Tidal source-to-target shape direction

Status: CONDITIONAL STRONG PASS; ρ OPEN

Expression: q_eq = L_c²(ε₁, ε₂, ε₃, ε₃, ε₂, ε₁)

Definitions

• q_eq is target shape; ε_i are eigenvalues of the tidal tensor; L_c is causal length; ρ is density contrast.

Explanation

Maps the ordered tidal eigenvalues into the paired target channel direction; its source weighting remains open.

Cross-references: CF158, CF166, CF168, CF192

Source page: F8

F168 · CF168

Two native target-normalization routes

Status: OPEN NORMALIZATION AUDIT

Expression: β_A=2/3; β_B=1/6; β_A/β_B=4

Definitions

• β_A and β_B are candidate normalization coefficients for two target routes.

Explanation

Records two retained normalization choices and their factor-four ratio without selecting one.

Cross-references: CF167, CF173, CF187

Source page: F9

F169 · CF169

Scalar loading cannot directly create centered shape

Status: EXACT NO-GO BY SYMMETRY

Expression: E_minus(A_cell 1_6)=0

Definitions

• E_minus is anisotropic spectral projector; A_cell1₆ is uniform scalar loading.

Explanation

Uniform scalar loading lies outside the centered anisotropic target sector.

Cross-references: CF046, CF162, CF167

Source page: F9

F170 · CF170

Layered Celal + effective tracking update

Status: CANDIDATE CF157 UPDATE; DEPENDS ON CF157

Expression: q_{n+1}=q_eq,n+1+R(δ_n)(q_n−q_eq,n+1)

Definitions

• q_n is current shape; q_eq,n+1 is next target; R is recovery operator; δ_n is update interval.

Explanation

Updates toward a changing target by applying the fixed-target recovery operator at each layer.

Cross-references: CF157, CF159, CF160, CF167

Source page: F9

F171 · CF171

Moving-target tracking equation

Status: EXACT GIVEN CF157

Expression: de/dτ + [3/(2T_c)]e = −dq_eq/dτ

Definitions

• e is target mismatch; q_eq is moving target; τ is time; T_c is causal time.

Explanation

Adds the target-motion forcing term to the fixed-target mismatch decay equation.

Cross-references: CF157, CF159, CF170

Source page: F9

F172 · CF172

Effective tracking-timescale to Aggregation ratio

Status: EXACT GIVEN COMBINED CANDIDATES

Expression: R_rate ≡ τ_rec/D₀ = (2/3)A_cell

Definitions

• R_rate is timescale ratio; τ_rec is recovery time; D₀ is aggregation time; A_cell is loading.

Explanation

Relates recovery speed to aggregation through the intrinsic loading ratio.

Cross-references: CF047, CF049, CF089, CF159

Source page: F9

F173 · CF173

Candidate rate-matching point; not a fundamental-physics threshold

Status: CONDITIONAL COMBINED-CANDIDATE RESULT

Expression: R_rate=1 ⇔ A_cell=3/2 ⇔ D₀=2T_c/3

Definitions

• R_rate is rate ratio; A_cell is loading; D₀ is aggregation time; T_c is intrinsic time.

Explanation

Identifies the candidate point where recovery and aggregation rates coincide, without promoting it to a universal threshold.

Cross-references: CF047, CF089, CF172

Source page: F9

F174 · CF174

Centered opposite-pair coordinates

Status: EXACT CENTERED GEOMETRY

Expression: A₁₂=A₃₄=xL_c²; A₁₃=A₂₄=yL_c²; A₁₄=A₂₃=zL_c²; x+y+z=4

Definitions

• A_ij are opposite pair areas; x,y,z are normalized pair coordinates; L_c is native length.

Explanation

Parametrizes the three opposite pair-area classes in the centered tetrahedral geometry.

Cross-references: CF175, CF176, CF177, CF179

Source page: F9

F175 · CF175

Centered four-null Gram matrix

Status: EXACT GIVEN CF174

Expression: γ/L_c² = [[0,x,y],[x,0,z],[y,z,0]]

Definitions

• γ is centered three-coordinate Gram matrix; x,y,z are pair coordinates; L_c is native length.

Explanation

Gives the centered Gram matrix in the opposite-pair coordinates.

Cross-references: CF174, CF176, CF180

Source page: F9