# Refinement tower: source and scope review

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

## Sources inspected

Current published SG01 (`UD_SUBDIVISION_THREE_CONDITION_GATE_v0.1.md`) and MS01 (`UD_METRIC_OBSERVABLE_SELECTION_GATE_v0.1.md`); their published R33, R34 and C0559 extracted-text snapshots; Site AGENTS.md; the supplied historical coefficient ledger v0.39 Q157–Q179; the supplied CRL-2 subdivision governance proposal v1.0. The unavailable Gap2 image is not evidence. Catalog text snapshots do not establish recovery of the original archival release bytes. The historical ledger is not the expanded current registry. No archived scientific programme was executed for this review.

## Main finding

The proposed two-step tower test closes a mathematical consistency question inside the candidate metric branch. It does not derive the source law selecting that branch. Applying the one-step unit residual calibration at each step and applying fresh unit calibration to the composite complement produce different complete fine metrics, although both preserve the same old norm, chain boundary and generator. A grouping-independent extension therefore must transport the intermediate detail pairing instead of resetting its calibration to the Euclidean pairing of the final fine coordinate basis.

This is a limitation of an additional tower requirement, not a retraction of MS.4. MS.4 establishes unique one-step selection after its stated premises; it never asserts invariance under regrouping multiple refinements. The tower metric remains conditional on the affine retraction, the selected refinement history and the detail pairings. Algebraic coherence cannot turn those premises into native UD selection.

## Exact composition and mismatch

Let S10,R10 and S21,R21 be the retained chain sections and retractions, R10 S10=I and R21 S21=I. Define Q10=I−S10 R10 and Q21=I−S21 R21. The composite pair is S20=S21 S10 and R20=R10 R21, with R20 S20=I. Its complement projector obeys

Q20=I−S20 R20=Q21+S21 Q10 R21.

Thus every final chain x has a unique retained decomposition

x=S20 c+S21 z1+z2,

where c=R20 x, z1=Q10 R21 x∈ker R10 and z2=Q21 x∈ker R21. This is a direct-sum decomposition, not automatically a Euclidean-orthogonal one.

Starting with coarse pairing G and applying the MS.4 unit residual rule twice gives

H1=R10ᵀG R10+Q10ᵀQ10,
H2,seq=R21ᵀH1 R21+Q21ᵀQ21
       =R20ᵀG R20+(Q10 R21)ᵀ(Q10 R21)+Q21ᵀQ21.

Its norm is cᵀGc+||z1||²+||z2||². Applying the same fresh unit rule once to the composite pair gives

H2,fresh=R20ᵀG R20+Q20ᵀQ20,

with norm cᵀGc+||S21 z1+z2||². These are different declarations about retained details. Cross terms in the second formula cannot be omitted by assuming a Euclidean orthogonal decomposition that has not been established.

For a pure first-level detail x=S21 z1, z1∈ker R10, and z1 of chain degree k, the ordinary subdivision Gram identity gives

xᵀH2,seq x=||z1||²,
xᵀH2,fresh x=(k+1)! ||z1||².

There are nonzero first-level complement directions in every degree. Degree 0 gives no discrepancy, while degrees 1,2,3 give factors 2,6,24. An explicit edge witness is z1 equal to the sum of the two flag edges [{0}⊂{0,1}] and [{1}⊂{0,1}]. Their affine retractions cancel, ||z1||²=2, and the second-level fresh metric gives 4 instead of the sequential metric's 2.

Both complete metrics remain positive and admissible under the SG.3 construction for the composite chain retraction. Neither disagreement nor covariance selects the native one. They agree on the embedded coarse image S20 C0. Their generator intertwining with the original coarse operator is the same, but their complete fine pairings and potentially their detailed fine dynamics must not be identified.

## Conditional repair and its exact scope

To recover the sequential metric as a direct composite construction, use on ker R20 the induced detail pairing

W20=(Q10 R21)ᵀ(Q10 R21)+Q21ᵀQ21.

Its restriction is positive: both decoded detail components vanishing forces a vector in ker R20 to vanish. In the ambient fine space W20 is only semidefinite and annihilates S20; do not call it a positive-definite full-space matrix. A positive-definite ambient extension can be supplied without changing its complement restriction, for example H2,seq itself. Only the complement restriction enters SG.3.

The recovered rule is: retain and transport each declared detail pairing; compose retractions and sections; evaluate the old coarse quadratic observable after the composite retraction. This rule is associative under regrouping the same retained sequence. It does not imply equal metrics for genuinely different refinement histories, alternative retractions, changed detail calibrations, or distinct terminal carriers. Those require additional comparison maps and constitutive premises.

This repair is target-independent and mathematically reviewable, but remains a candidate constitutive contract. C0559 explicitly leaves local-first versus coarse-first nonlinear observation open; it does not authorize interpreting these tower details as a physical anisotropy field, renormalization flow or source law. R33's bounded residual construction is a different declared Euclidean/attenuated branch and remains intact. R34's fixed Euclidean strict-interface obstruction also remains intact.

## Observable composition and retained receipts

For a supplied old positive observable D_U, the complement-blind coarse observable is

K20=R20ᵀD_U R20=R21ᵀ(R10ᵀD_U R10)R21.

This associative pullback is independent of whether one uses H2,seq or H2,fresh. It proves consistency of a specified coarse-first observation, not uniqueness against local-first or complement-sensitive observables. Readout blindness does not erase either detail sector.

When G=I and the level detail pairings are unit Euclidean, the sequential total splits into the positive coarse contribution and the two separately retained positive detail contributions. If a complete quadratic support receipt is reported, both detail contributions must remain in its denominator; renormalizing coarse outcomes alone would be conditional observation/postselection. These mathematical receipts do not define a laboratory detector, quantum measurement channel, Born law or energy.

## Finite resources and limits

The ordinary barycentric tower has f-vectors (4,6,4,1), (15,50,60,24), and (149,796,1224,576), hence dimensions 15→149→2745. The two detail sectors have dimensions 134 and 2596; their direct sum has 2730 directions. By degree the sectors are (11,44,56,23) and (134,746,1164,552). A fully retained terminal state has 15 coarse directions plus those 2730 detail directions.

These counts are real direct-sum chain-coordinate dimensions, not qubits, tensor-factor donor banks, implementation minima or a reset-resource estimate. Refinement data maps and metric algebra do not supply fresh prepared coordinates, an event trigger, physical time, reset donors or protected outgoing records. Pure coarse preparation imposes zero detail amplitude; setting an arbitrary input to that condition deletes information unless an explicitly accounted export or other operation retains it. Existing finite-window joint-carrier and clock limits remain in force.

No factor here supplies a universal sqrt2 law. The exact degree factors remain 1,2,6,24. Source Q175's sqrt2 carrier frequency is a separately typed spectral fact.

## Publication language and governance

Use: “One-step unit detail calibration is exact conditional, but is not invariant under regrouping the same refinement maps. A transported level-detail pairing gives an associative conditional tower construction; the source still does not select its constitutive premises or measurement implementation.”

Avoid: “The source now selects the metric,” “canonical physical tower metric,” “unit complement is universally scale invariant,” “134/2596 qubits,” “coarse blindness erases details,” or “grouping coherence proves covariance across every alternative refinement.”

The governance attachment remains a proposal and its development lane permits analysis without a registry conclusion. Constructor-team checks do not meet the non-constructor review plus custodian promotion procedure. This review establishes no CRL advancement, physical promotion or global closure.