Extension across collision sets · Part 3

spectator and varying viewers, without replacing the recorded terminal costs by outgoing costs or suppressing the signed residuals.

For the earlier equation lift, replace any step traversing a forbidden stored positive edge (in either direction) by three steps: one in a different coordinate, the desired step, then its inverse transverse step. Endpoints agree and the forbidden edge is avoided. Replacing all offending steps gives an avoiding cover path between any two sites. An axes- plaquette can use a forbidden edge based at only if its base lies in, as listing its four stored positive edges shows. Outside this finite set both an avoiding transport to its base and its boundary avoid the edge, giving the safe probes used by the finite-dimensional span argument. For the finitely many used cover edges, choose torus side larger than twice the largest absolute base coordinate, including the forbidden base, plus one. Centered wrapping then recovers all bases, preserves distinct stored edges and makes the link variation change no path factor. Thus holonomies, curvatures, adjoint transports and finite products are exactly unchanged, uniformly under stencil translation. Take the maximum finite threshold for the marking identity.

The lifted funds source retains finite approximants, their actual terminal costs, detector fees and signed projection residuals. UC3 is its coarsening reader with paid error; ordered multiplication preserves the complete difference rather than only its separate terms. UC1 then recovers a uniform radius bound from these pathwise readers. The safe-probe source retains the avoiding cover paths and finite wrapping threshold, so its torus readout recovers exact independence of the varied link without a limiting approximation.

Avoiding-probe readout recovery. For a step from to traversing the forbidden stored edge, choose and retain the ordered detour

Searchable transcription

x −→x + ej −→x + ej + εei −→x + εei,
ε ∈{1, −1}.

Its transverse stored edges have direction, and its middle stored edge is the translated direction- edge, so none equals the forbidden stored edge. Its endpoints are identical to the original step. The detour is a new recorded probe path; no equality with the original path’s holonomy is assumed. The ordered positive stored edges in an () plaquette based at are based at with directions, respectively; its last two occurrences are reversed. This recovers the stated finite excluded-base set. Let be the finite used stored-edge set of the resulting probes. Choose to bound all base coordinates relative to the stencil anchor, including the forbidden base, and take side Wrapping is then injective on these bases and preserves directions. After any common translation the same relative-coordinate statement holds. If varies only the forbidden physical link, each probe reading therefore satisfies exactly

Searchable transcription

H(Ve,τU) = H(U),
HZ(z) = H(πz),
HZ(σVe,τU) = HZ(σU).

Here and are the configuration realization and section. The finite products and adjoint transports inherit this equality, so every defined link derivative of the probe is zero. The finite maximum of these side thresholds recovers the same marking identity.