Native admission, literal preparation and the realized sector

Native admission, literal preparation and the realized sector

The pre-sectoral source signature consists of a dependency preorder on source vertices, depositionpotential tokens, a marked subsort, finite supports, lineage, invariant data and intrinsic boundary data. A marked support contains a strict dependency pair. Strong maps preserve these structures and are injective on marked supports. No vertex, token or dependency is by definition a spacetime point, gauge variable or coupling.

Proposition 6.1 (An admitted antecedent). The walking marked dependency, with vertices and one inherited marked tokensupported on, is admitted by the initial witness

clause of the source-admission construction. Its literal preparation is

Searchable transcription

Y⋆= Jreal(X⋆, m⋆) = (X⋆, m⋆, Sm⋆, ⪯m⋆, lm⋆, Invm⋆, Bndm⋆).

This supplies a native antecedent for the realization below without introducing a physical label into its source signature.

Proof. The walking marked dependency is one of the specified initial witnesses. Its support is finite and nonempty; the displayed pair is strict, and its lineage, invariant and boundary data are inherited from that witness. The least isomorphism-saturated admission closure contains every initial witness, hence contains Restricting the inherited data to its marked support gives precisely the literal preparation tuple. Neither this admission step nor restriction supplies any physical coordinate or law.

The realized Wilson sector is a relation

Searchable transcription

KYM(Y⋆) ⊆Oraw
YM × DYM.

A raw tuple retains the supplied group and representation, lattice and orientation, cutoffand period, coupling, frames and links, local functions, supports and all subsequent auxiliary construction data. The discrepancy column retains equation and comparison residuals, including nonzero residuals. Thus admission is not defined by requiring a successful final quantum theory. The supplied construction specifies the physical law and proves its consequences; the present lift proves recovery of that construction on this realized branch. A single marked occurrence supports this whole realized tuple. It is not a map from the two abstract vertices onto every physical lattice endpoint.

For a target configuration, a recovery witness consists of this antecedent, its literal preparation, a realized history, and an allowed comparison isomorphism. The corresponding lift groupoid is Lift( YM Comp YM

Here the arrows and retained data are those of the specified realized branch. The explicit section below proves nonempty recovery fibres for every finite link configuration. It does not classify every possible native antecedent of the same target. Every later observable and limit uses the same realized history and its declared refinements; successful outputs from incompatible histories are not combined.