Selection needs a source commitment.
The native unit metric and simplex stocks are fixed on each carrier. Across subdivision, compatibility and symmetry leave metric and observer freedom. An explicit conditional rule can select both; its additional premises remain visible.
What is fixed, and what is still a choice?
| Commitment | Exact consequence | Provenance |
|---|---|---|
| Native stock Σcσ² in the declared basis | Metric I and region projector D_U | Existing fixed-carrier source |
| Norm, boundary and full generator compatibility | Chain retraction plus positive complement pairing | Exact classification; no unique metric |
| Positive observable calibrated only on the old code | Complement response remains free | Exact nonselection witness |
| Affine readout, calibrated orthogonal residual, blind observer | Unique R, H* and K_U | Additional candidate premises |
An arbitrary positive diagonal fine metric cannot repair ordinary subdivision's full generator equation. Its adjoint cannot create a flag coefficient that the transported old generator requires. This preserves the original local-stock obstruction.
Inspect a concrete nonselection witness
The same fine state is the unit vertex at barycenter {0,1}. Its decoded coarse norm is1/2 and residual Euclidean norm is3/2. Every positive metric H_w=RᵀR+wQᵀQ passes the three compatibility and symmetry tests.
These controls are mathematical counterexamples. They adopt no metric coefficient. The readout numerator is RᵀR+λwQᵀQ; changing it can leave old-code calibration unchanged while changing the fine-state result.
The conditional rule that selects both
First require face-local affine readout, normalized and covariant under relabeling, with exterior consistency in higher degrees. That fixes the barycentric chain retraction R. Let Q=I−SR.
H*=RᵀGR+QᵀQ
This follows uniquely if the old sector and retained complement are orthogonal, the coarse norm is calibrated to G, and the complement norm is explicitly calibrated to its unit Euclidean value.
K_U=RᵀD_U R
This follows uniquely for a positive coarse observable calibrated on Sc and blind to kerR. Positivity removes hidden cross terms.
These are declared mathematical premises. They do not follow merely from symmetry or the absence of adjustable parameters. A constitutive or measurement source must justify adoption.
Keep the complement in the receipts
The15 source simplex observables plus a separate complement observable sum to H*. For w=1 in the example, vertex0 and vertex1 each contribute1/8; the complement contributes3/4. Renormalizing the visible coarse outcomes alone hides that retained support and requires an explicit conditional interpretation.
Decoded coarse stocks on Rx inherit the original Hasse current and region continuity equation. They are coarse-first observations; they do not restore additive fine-simplex stocks or select the local-first observable ordering.
The134 extra directions are retained direct-sum chain coordinates. These positive support fractions are mathematical receipts, not a physical measurement instrument or a reset. Metric/locality, detector, clock, initialization and outgoing-record acquisition remain accounted.