F201 · CF201

Laboratory K_M reconstruction

Status: EXPERIMENTAL RECONSTRUCTION FORM

Expression: K_M = M_s(r_s⁻²−r_f⁻²)/[1250π⁹K_GD⁴Δa_mod]

Definitions

• K_M is source modulus; M_s is source mass; r_s,r_f are radii; K_GD is interface constant; Δa_mod is measured acceleration.

Explanation

Provides the corresponding direct laboratory reconstruction of source modulus.

Cross-references: CF098, CF199, CF200

Source page: F11

F202 · CF202

Hawking temperature at conditional Xyz crossover

Status: SEMICLASSICAL COMPARISON ONLY

Expression: T_H,X = ℏ/[4πk_BT_C√(3β/2)]

Definitions

• T_H,X is Hawking temperature; ℏ is action; k_B is Boltzmann constant; T_C is crossover time; β is normalization.

Explanation

Compares the conditional Xyz crossover with a semiclassical Hawking temperature scale.

Cross-references: CF194, CF195, CF203

Source page: F11

F203 · CF203

Standard Schwarzschild lifetime evaluated at crossover

Status: SEMICLASSICAL COMPARISON ONLY

Expression: T_H,X = (3200π³/3)(3β/2)³ᐟ²T_c

Definitions

• T_H,X is comparison temperature; β is normalization; T_c is intrinsic causal time.

Explanation

Re-expresses the crossover Hawking comparison solely in the intrinsic-time parameterization.

Cross-references: CF202, CF194

Source page: F11

F204 · CF204

Xyz causal-integrity margin

Status: DEFINITION GIVEN CF182

Expression: χ_Xyz ≡ min{s−2α, s−2β, s−2γ_d}

Definitions

• χ_Xyz is causal-integrity margin; s is common mode; α,β,γ_d are deviation coordinates.

Explanation

Defines the smallest distance to any of the three determinant rank-loss boundaries.

Cross-references: CF181, CF182, CF205, CF207

Source page: F12

F205 · CF205

Failed simple α-edge Xyz mode and branch dependence

Status: EXACT AT SIMPLE α EDGE

Expression: u_x=(1,1,−1,−1), γu_x=0; k₁+k₂−k₃−k₄=0 at α=s/2

Definitions

• u_x is candidate null mode; γ is centered Gram matrix; k_i are branch vectors; α,s are shape coordinates.

Explanation

Identifies a null mode at a simple edge and shows the corresponding branch-sum relation.

Cross-references: CF181, CF204, CF206

Source page: F12

F206 · CF206

Boundary-flow sign under the effective CF157 tracking candidate

Status: EXACT GIVEN CF157; NOT GENERAL TRIGGER

Expression: λ_α=s−2α; (dλ_α/dτ)_CF157=(3/T_c)(s/2−α_eq) at α=s/2; dλ/dτ=0

Definitions

• λ_α is α-edge margin; α_eq is target coordinate; τ is time; T_c is causal time.

Explanation

Evaluates the effective candidate flow at the α-edge and explicitly does not promote it to a general trigger.

Cross-references: CF157, CF204, CF205, CF207

Source page: F12

F207 · CF207

General Xyz outward-crossing causal-integrity condition

Status: GENERAL GEOMETRIC / KINEMATIC NECESSARY CONDITION; DYNAMICS OPEN

Expression: χ_Xyz=0 and D_τχ_Xyz<0 ⇒ smooth ordinary 1+3 continuation cannot remain in retained phase unchanged

Definitions

• χ_Xyz is integrity margin; D_τ is material derivative; τ is evolution time.

Explanation

States the necessary geometric condition for an outward crossing while leaving the dynamical mechanism open.

Cross-references: CF204, CF206, CF208, CF217

Source page: F12

F208 · CF208

Centered geometric mismatch norm

Status: DEFINITION

Expression: Δq=(Δα,Δβ,Δγ,Δβ,Δα); V_geom=(1/2)||Δq||²

Definitions

• Δq is centered shape mismatch; Δα,Δβ,Δγ are coordinate differences; V_geom is mismatch energy.

Explanation

Defines a permutation-symmetric mismatch measure in the centered shape sector.

Cross-references: CF156, CF157, CF209, CF211

Source page: F12

F209 · CF209

Celal Gram mismatch equals centered geometric mismatch up to fixed factor

Status: EXACT IN CENTERED FIXED-COMMON-MODE SECTOR

Expression: (1/2)||Δγ||_F² = (1/2)L_c⁴V_geom

Definitions

• Δγ is Gram-matrix mismatch; L_c is causal length; V_geom is geometric mismatch.

Explanation

Relates the matrix Frobenius mismatch to the coordinate mismatch in the fixed common-mode sector.

Cross-references: CF175, CF182, CF208, CF210

Source page: F12

F210 · CF210

Candidate post-Xyz transported mismatch / discontinuity bookkeeping

Status: DEFINITION; PHYSICAL MOVE SET OPEN

Expression: e_r=U_(n,r)q_eq,r; C_r=(1/2)‖r−U_(n,r)U_(n,r)ᵀ‖_F²

Definitions

• e_r is transported mismatch; U_(n,r) is transport basis; q_eq,r is target; C_r is discontinuity cost.

Explanation

Defines candidate transported mismatch and a projector-based discontinuity cost after an Xyz transition.

Cross-references: CF085, CF167, CF208, CF211

Source page: F12

F211 · CF211

Minimum-discontinuity continuation selector

Status: STRONG RATIONALE; admissible R_r remains OPEN

Expression: r_* ∈ arg min_r(C_r+R_r), R_r admissible

Definitions

• r_* is selected continuation; C_r is mismatch cost; R_r is admissibility penalty.

Explanation

Selects the continuation minimizing discontinuity plus an admissibility penalty, without specifying that penalty fully.

Cross-references: CF210, CF217

Source page: F12

F212 · CF212

Observed fixed-target recovery-cost decay under CF157

Status: EXACT GIVEN CF157; EFFECTIVE TRACKING CANDIDATE ONLY

Expression: V_L(τ)=V_L(0)e^(−3τ/T_c)

Definitions

• V_L is recovery cost; τ is time; T_c is causal time.

Explanation

States the exponential decay of the fixed-target recovery cost in the retained effective candidate.

Cross-references: CF157, CF159, CF164

Source page: F12

F213 · CF213

Xyz 2+2 incidence cut selected by failed mode

Status: EXACT UP TO EDGE ORIENTATION / ORDERING

Expression: c_X=BᵀP_Xu_x, c_X∈(0,±2,±2,±2,±2,0)

Definitions

• c_X is edge cut vector; B is incidence; P_X is Xyz projector; u_x is failed mode.

Explanation

Constructs the discrete 2+2 edge cut associated with the failed simple-edge mode.

Cross-references: CF017, CF205, CF214, CF215

Source page: F12

F214 · CF214

Failed Xyz cut is pure gradient in six-channel decomposition

Status: EXACT

Expression: P_grad c_X ≠ c_X; P_circ c_X = c_X

Definitions

• P_grad and P_circ are gradient/circulation projectors; c_X is Xyz edge cut.

Explanation

Classifies the selected cut entirely in the circulation sector rather than the gradient sector.

Cross-references: CF019, CF020, CF213

Source page: F12

F215 · CF215

No unused eternal edge exists inside the one undirected four-vertex cell

Status: EXACT COMBINATORIAL NO-GO

Expression: |E(K₄)| = C(4,2) = 6

Definitions

• E(K₄) is the edge set of the complete four-vertex graph; C(4,2) counts its six edges.

Explanation

The complete four-vertex cell already uses all six unordered branch pairs.

Cross-references: CF006, CF018, CF213

Source page: F12

F216 · CF216

Local candidate dual-area/action weight collapses at Xyz rank loss while 4-volume remains finite

Status: EXACT CONSEQUENCE WITHIN CF143-CF147 CANDIDATE ACTION

Expression: detγ_X=0 ⇒ V₄,x→0 ⇒ B^dual_X→0

Definitions

• γ_X is degenerate Xyz Gram; V₄,x is four-volume; B^dual_X is candidate dual area.

Explanation

Propagates Gram rank loss through the candidate determinant-weighted dual area, without claiming physical collapse of volume.

Cross-references: CF143, CF145, CF146, CF147, CF181

Source page: F12

F217 · CF217

Consolidated causal-integrity law; Celal may select among a separately derived admissible continuation set

Status: EXACT BOUNDARY + CONDITIONAL SELECTOR; ORDINARY TRANSPORT AND POST-Xyz MOVE SET OPEN

Expression: χ_Xyz>0: ordinary evolution may remain on existing causal relations; χ_Xyz=0,D_τχ_Xyz≥0: retained phase may remain admissible; χ_Xyz=0,D_τχ_Xyz<0: ordinary smooth continuation is inadmissible and an admissible intervention is required

Definitions

• χ_Xyz is integrity margin; D_τ is its material derivative; Celal is the branch-selection procedure.

Explanation

Summarizes interior, tangent-boundary, and outward-crossing cases while preserving the open continuation dynamics.

Cross-references: CF085, CF204, CF207, CF210, CF211

Source page: F12

F218 · CF218

Advection of a sharp material-support indicator with the matter flow

Status: EXACT KINEMATICS FOR MATERIAL SUPPORT

Expression: ∂_tχ_m + v_m·∇χ_m = 0

Definitions

• χ_m is material-support indicator; v_m is material velocity; t is time; ∇ is spatial gradient.

Explanation

Advects a sharp matter-support field with the material velocity without making it a graph-topology boundary.

Cross-references: CF053, CF219, CF222

Source page: F13

F219 · CF219

Rigid-body matter velocity: translation plus axial rotation

Status: EXACT NONRELATIVISTIC RIGID-BODY KINEMATICS

Expression: v_m(x,t)=V(t)+ω(t)×[x−X(t)]

Definitions

• v_m is matter velocity; V is translation velocity; ω is angular velocity; X is center; x is position.

Explanation

Decomposes rigid-body material motion into translation and rotation.

Cross-references: CF218, CF220, CF221

Source page: F13

F220 · CF220

Rotation is tangent to an ideal spherical matter boundary, including a tilted axis

Status: EXACT GEOMETRY

Expression: (ω×r)·n = 0 for n ∥ r

Definitions

• ω is angular velocity; r is radial position; n is spherical outward normal.

Explanation

Shows that rigid rotation has no normal component at an ideal spherical boundary.

Cross-references: CF219, CF221

Source page: F13

F221 · CF221

Low-velocity dust source decomposition; rotation/tilt enters through momentum and stress

Status: STANDARD NONRELATIVISTIC SOURCE LIMIT / INPUT

Expression: T^{00}≈ρc²; T^{0i}≈ρcv^i; T^{ij}≈ρv^iv^j

Definitions

• T^{μν} is stress-energy; ρ is density; v^i is material velocity; c is light speed.

Explanation

Provides the nonrelativistic stress-energy components used as the matter-layer source input.

Cross-references: CF041, CF042, CF102, CF218, CF219

Source page: F13

F222 · CF222

Ordinary weak matter-boundary matching without a delta-function surface layer

Status: EXACT WITHIN RETAINED WEAK SOURCE EQUATION AND NO-SURFACE-LAYER ASSUMPTION

Expression: [M]_S=0; [∂_nM]_S=0; [∇²M]_S=C_Σ(c³T_c²/ℏ)ρ|_S

Definitions

• [·]_S denotes the matter-boundary value; n is boundary normal; M is deformation; C_Σ is retained coefficient; ρ is boundary density.

Explanation

Matches deformation, normal derivative, and weak-source curvature at the material boundary while excluding a delta-function surface source.

Cross-references: CF032, CF053, CF067, CF218, CF220

Source page: F13