# Metric and observable selection: source/scope review

Date: 2026-10-14. Constructor-team AI source review only. Development — no registry entry. Physical promotion 0. Independent human review pending.

## Sources actually inspected

Current public atlas catalog text for C0243 (incidence, metric and source-selection audit), C0173 (geometric inner-product interface), C0176 (chain semantics and weighted receipts), C0559 (refinement and observable ordering), and SG01 (subdivision compatibility v0.1); attached historical ledger v0.39 Q165–Q179; attached governance proposal CRL-2 subdivision v1.0; Site AGENTS.md. Catalog text is an extracted text source, not recovery of original package bytes. The unavailable Gap2 image supplies no evidence.

## Main finding

The present source commitments select the native fixed-complex Euclidean pairing and squared-coordinate stocks. They do not uniquely select a metric or observation ordering across ordinary barycentric subdivision. Exact three-way compatibility is insufficient: for fixed affine retraction R and Q=I−SR, every w>0 gives H_w=RᵀGR+wQᵀQ with the same old norm, boundary and generator intertwiners. R and Q transform covariantly under all relabelings, so this ambiguity persists under signed S4 covariance. The source-selection result is therefore an exact underdetermination statement. A conditional constitutive extension can be uniquely solved after its extra axioms are declared, but its selection axioms are not derived from the existing source.

## Source fidelity

* Ledger Q165 declares squared amplitudes in the orthonormal oriented-simplex basis: G=I on the admitted fixed carrier.
* Q169 retains the unit-incidence global skew Hodge–Dirac operator on unique simplex coordinates. With this coordinate basis and that original support convention unchanged, the native fine carrier also has H=I.
* Q170 defines a selected Hasse control volume U by S(U)=∑σ∈U cσ²=cᵀD_Uc, with D_U its diagonal coordinate indicator. It does not choose U, a cross-event coarse retraction, or a new fine quadratic observable.
* Q171/Q172 distinguish cell-core and face-reservoir stocks. Q177–Q179 allocate a fixed-complex simplex stock equally among incident top cells. That ownership prescription does not provide a cross-refinement nonlinear observable-ordering law.
* C0176 proves that the unchanged unit-incidence generator can conserve a diagonal positive additive stock only when weights are constant on each connected Hasse component. A reference unit weight fixes I. It explicitly does not exclude nondiagonal conserved quadratic forms.
* C0173 gives two geometry-defined, positive, covariant metric alternatives with different dimensionless spectra. Correct adjoints and regular-baseline recovery do not select the metric. Its source interpretation and physical norm/energy remain unselected.
* C0243 selects native h=1 by preserving the originally fixed entries and Euclidean pairing in its tested family. Transporting a changed coordinate basis and metric creates no new dynamics. Minimizing an operator norm or purity is an extra law, even when its optimizer has no adjustable parameter.
* C0559 contrasts local-tensor-first, carrier-average-first and exact-chain-retraction-first prescriptions. The source explicitly leaves observable ordering open. Support-stock ownership does not select the anisotropy ordering.

Consequently, H*=RᵀGR+QᵀQ cannot be called the recovered original fine support metric. If used, it is a changed-pairing candidate branch with a correspondingly changed adjoint. The original Euclidean obstruction remains true.

## Conditional selector, stated honestly

The following is a target-independent candidate constitution, not an existing source-derived physical law.

1. Declare the old simplex basis, old positive pairing G, ordinary subdivision S, and selected old stock region U before evaluating target spectra or data.
2. For each barycentric vertex F, require its coarse readout to be affine, supported only on vertices of F, and invariant under every permutation of those vertices. Affine coefficients sum to one. Thus every coefficient is exactly 1/|F|, and R_0(F)=|F|⁻¹∑v∈F e_v. Require exterior consistency on oriented flags to select every R_k by determinant minors. These locality/affinity/exterior axioms are new selector premises; relabeling covariance alone is weaker.
3. Require the full fine chain to split as x=S(Rx)+Qx, with no deletion of the residual. Require the two sectors to be orthogonal, the coarse-sector norm to be (Rx)ᵀG(Rx), and the residual z∈ker R to have precisely the unit Euclidean norm in the declared fine simplex coordinate basis. These requirements uniquely select H*=RᵀGR+QᵀQ. Replacing the last calibration by any positive W produces the already admitted H_W. Unit calibration is a substantive new axiom, not a consequence of G=I on the original carrier.
4. Require the coarse-region observable to ignore residual inputs entirely and agree with the declared old stock on every Sc. These requirements select q_U(x)=(Rx)ᵀD_U(Rx), with matrix K_U=RᵀD_UR when G=I. For a separately declared weighted old stock D_U must be defined accordingly; it cannot silently be replaced by the unweighted indicator.

Uniqueness in step 3 follows by polarization on the direct decomposition im S⊕ker R: specified restrictions and zero cross pairing determine the entire bilinear form. Uniqueness in step 4 follows because residual blindness removes both residual quadratic terms and coarse–residual cross terms, leaving the already calibrated form on Rx.

This conditional selector is falsifiable and reviewable: changing the residual calibration, violating face locality, or choosing local-first fine stock changes the result. Its consistency is not evidence selecting it as native UD physics.

## Observable ambiguity and ordering

For any λ≥0, K_{U,λ}=RᵀD_UR+λQᵀQ is positive semidefinite and satisfies SᵀK_{U,λ}S=D_U. Thus all coarse-image calibration tests agree for every λ; they do not determine residual response. For any nonzero z∈ker R, zᵀK_{U,λ}z=λ||z||². Requiring residual blindness uniquely sets λ=0 within this family and, more generally, annihilates the complement and its cross terms. That requirement must be specified before solving.

Under H_W, R is the metric-adjoint of S: R=G⁻¹SᵀH_W. Because the two subcomplexes reduce the weighted generator, R A′_H=A_G R. Hence the reconstructed coarse amplitudes y=Rx obey the original coarse flow even for arbitrary residual input. With G=I, q_U obeys the old Q170 cut-current balance evaluated on y. This is a coarse observable made after retraction, not fine-simplex local stock additivity. The corresponding K_U is generally dense within degree and includes cross terms in the fine coefficients. The residual is retained by Qx even though the coarse detector is blind to it; detector blindness does not erase or reset it.

For full coarse support U, K_U=RᵀR. Under the unit-residual candidate metric H*=K_U+QᵀQ, total support divides into nonnegative coarse support and nonnegative retained residual support. The fine-simplex squares do not supply these terms independently. A source-selected physical detector or local realization remains open.

## Diagonal-metric obstruction scope

A positive diagonal fine metric cannot intertwine the weighted Hodge–Dirac generator with ordinary S while retaining the old unit-metric basis, even though a degree-factorial diagonal metric can preserve its norm. For an original vertex i, the raising adjoint gives a nonzero coefficient on every fine incidence [{i}⊂F] with i∈F. On the other hand, S applied to the old generator's edge component contains only singleton-to-two-vertex flags [{j}⊂{i,j}] with signs. For |F|≥3, the former coefficient is ±h_{ {i} }/h_{ [{i}⊂F] }, strictly nonzero under positivity, and the latter is zero. Grade separation prevents cancellation by other pieces. Therefore no positive diagonal H passes the generator condition. This proof should explicitly assume the ordinary S, old G=I, fixed fine boundary, finite positive diagonal H, and weighted-adjoint generator; it does not classify every modified operator or transfer map.

The contradiction reinforces the semantic gate: requiring unchanged independent fine-simplex additive stocks and all three compatibility equations is inconsistent in this specific carrier. A nondiagonal extension resolves compatibility by changing that stock convention, not by proving it equivalent.

## Governance and publication language

Use: “The existing source fixes I and D_U within a specified fixed carrier; cross-refinement selection remains underdetermined. Under separately declared affine coarse-readout, residual-calibration and detector-blindness axioms, the unique conditional construction is R,H*,K_U.”

Avoid: “The source derives H*,” “canonical physical metric,” “parameter-free therefore native,” “coarse stock is the sum of fine simplex stocks,” or “constructor review closes the independent promotion gate.” Any theorem of uniqueness must list its additional selector premises next to the result.

The attached CRL document is a governance proposal. It permits development analysis with no registry entry and requires non-constructor review plus custodian ruling for promotion. The historical ledger is not the current expanded registry. This review supplies constructor-team scope checking only; it does not close human review, observable implementation, physical energy identification, detector calibration, laboratory clock, or funded-reset gates.