Manuscript VIII-A@3 · VIII-A-v3.tex · section · 2398a747

Notation and proof dependencies

Notation and proof dependencies · lines 2748–2834

Symbol	Meaning
Symbol	Meaning
$Π$	Spinless packet $(p, μ, W, u, A, H_{Π})$ .
$Ein_{0} (g)$	Evaluated realization-dependent matching data, not a continuum Einstein equation.
$Cert (g)$	Mathematical witnesses and verification of all applicable realization and matching obligations.
$Sig (g)$	Persistent marked particle/peripheral signature.
$A = (C, a)$	Common affine reference frame.
$A_{f}$	Adapted face chart, distinct from the vertices $A_{i}$ .
$J (μ), H_{Π}$	Rest-frame affine rotation and its framed conjugate.
$n_{g f}, N (γ)$	Overlap seam index and net oriented winding number.
$A_{src}, ω_{ρ}$	Named surviving source algebra and expectation functional.
$ν_{G}, ν_{src}$	Separate incidence and coupling permissions.
$F_{s}, Σ_{s}$	Explicit fixed-cone graph and its slice.
$ρ (x)$	Sectorwise spatial function $ξ + ζ$ , not a density operator in the fixed-cone formulas.
$P_{s}$	Slice intersection with the fixed singular worldline.
$A_{i}, D_{i}, B_{i}, C_{i}$	Cone-labelled vertices with sectorwise developed copies.
$V_{i}, U_{i}^{\pm}, Q_{i}$	Persistent core, born core, and horizontal collar-side faces.
$e_{i}$	Persistent abstract edge label with different incoming and outgoing embedded segments.
$∥ \cdot ∥_{2}, ∥ \cdot ∥_{row}$	Euclidean spectral and maximum-row-sum coordinate matrix norms in the rotation rejection test.
$d_{⋄}$	Half-diamond channel distance; used only in the finite recovery safeguards.

The proof dependencies are:

affine conventions ⟶ spinless matching and canonical transitions, explicit graph dominance regions ⟶ realization, metric tables, and unique event, realization + canonical transitions + marked winding ⟶ complete fixed-cone certificate, fixed presentation ⟶ A1, equal complete inputs + deterministic specification ⟶ A2, declared edge-path presentation + regular relators + primitive peripheral word ⟶ A3, explicit rotation and translation residuals ⟶ A4 .

The instrument, spectator, deferred, and recovery identities are separately proved finite-dimensional statements. No unavailable companion theorem, Hamiltonian flow, field product, or source constructor is a premise.

Displayed formulas in this passage

unlabeled · line 2807
$affine conventions ⟶ spinless matching and canonical transitions, explicit graph dominance regions ⟶ realization, metric tables, and unique event, realization + canonical transitions + marked winding ⟶ complete fixed-cone certificate, fixed presentation ⟶ A1, equal complete inputs + deterministic specification ⟶ A2, declared edge-path presentation + regular relators + primitive peripheral word ⟶ A3, explicit rotation and translation residuals ⟶ A4 .$