# Different refinement histories and detail excitation: source/scope review

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

## Sources inspected and provenance

Site AGENTS.md; current atlas extracted catalog text for C0644 (NRF-1 quadratic refinement theorem), C0559 (observable ordering), C0656 (actual geometric Pachner evolution and topology-history arrow), R30 (native spatial-refinement source gate), SG01 (subdivision compatibility), MS01 (metric/observable conditional selector), and RT01 (refinement-tower calibration). The complete SG01, MS01 and RT01 published theorem/review records were inspected. The attached historical ledger v0.39 was checked at Q157–Q160 and Q165–Q179. The attached CRL-2 subdivision governance proposal was read in full. Catalog text snapshots are not recovery of original archival package bytes. No imported historical scientific program was executed. The unavailable `Gap2_FINAL_Finite_Genesis.png` was not inspected and supplies no evidence.

## 1. A genuine second refinement geometry exists, but its full interface remains conditional

C0656 describes actual simplicial mutation, including a legal 1→4 cone construction around a registered latent fifth vertex. Starting from one filled tetrahedron, its complete downward closure has f-vector (5,10,10,4), full chain dimension 29. Ordinary barycentric subdivision has f-vector (15,50,60,24), dimension 149. These are different refinement geometries, not different parentheses around the same tower maps.

C0644 already supplies an inherited-edge NRF-1 inclusion I1, child charts Fi, projected cycle maps Li, and a rational edge retraction satisfying R1 I1=I. These data support the stated edge-cycle and nonlinear-observable calculations. They do not by themselves select a full graded, oriented subdivision section Sstar with a globally affine exterior retraction Rstar and the metric Hstar=RstarᵀG Rstar+QstarᵀQstar. A new full graded construction can be mathematically checked, but must be labeled conditional rather than imported silently from the historical edge map.

In particular, C0656's fifth seat is combinatorial; the new vertex's equal-barycentric amplitude readout is an additional affine/covariance premise. C0644's child chart identifying the center with an omitted parent vertex is not the same thing as that global affine readout. The choice must not be treated as already sourced merely because both use a center-star geometry.

With a supplied full pair Sstar,Rstar, the star detail dimension is 29−15=14, whereas barycentric detail dimension is 149−15=134. No bijective full-state isometry identifies these unequal-dimensional carriers. Coarse comparison through their common 15-dimensional source is possible; it does not identify or erase their different detail sectors. A map Sbar Rstar annihilates all star detail and therefore cannot be called a full norm-preserving history transfer. Retaining both detail sectors in a larger carrier is an additional resource/interface contract, not a canonical consequence of the old coarse dynamics.

C0559/C0644 explicitly leave observable order and cross-refinement-branch covariance open. Exact coarse-first pullback of a supplied D_U is therefore a consistency test of a selected readout, not an adoption of the local-first anisotropy law or its 5/16,3/16 coefficients.

## 2. Exact inheritance prevents excitation of initially blank detail

Let V be the old real chain space with positive metric G; F the fine space with positive metric H. Let S:V→F satisfy SᵀHS=G. Define its metric-adjoint retraction R=G⁻¹SᵀH and Q=I−SR. Then RS=I and F=im S ⊕_H ker R.

Suppose a real generator A is H-skew, AᵀH+HA=0, and AS=SA0. The image im S is invariant by the intertwining identity. For z∈ker R and every c∈V,

⟨Sc,Az⟩_H=−⟨ASc,z⟩_H=−⟨SA0c,z⟩_H=0.

Thus ker R is also invariant. Both sectors reduce A; equivalently RA=A0R and QA=AQ. For fixed A, xdot=Ax decomposes as cdot=A0c and zdot=Ad z, where c=Rx, z=Qx and Ad is the restriction to ker R. Consequently z(0)=0 implies z(t)=0 at all times of the evolution. This is a universal linear algebra theorem under its premises, not merely a finite numerical observation about one Hodge operator.

The conclusion also holds for sufficiently regular time-dependent H-skew generators A(t) with the same fixed H,S,R and A(t)S=SA0(t). A separately added H-skew K(t) satisfying K(t)S=0 leaves the coarse image invariant and cannot create detail from a coarse-only state. Time-dependent metrics, changing sections, state jumps, non-skew terms or generator defects require new accounting and lie outside this fixed-pairing theorem.

## 3. What a genuine excitation law must change

For xdot=Ax+f with exact inheritance, the split equations are

cdot=A0c+Rf,  zdot=Ad z+Qf.

Thus detail can be seeded by a nonzero initial z, a forcing component Qf, a state jump with Qxplus≠0, or a violation of the full-generator intertwining identity. A supplied norm-preserving donor coupling may exchange norm with a retained external carrier, but does not create norm freely or identify it with energy.

In split coordinates with pairing diag(G,W), an H-skew coarse/detail coupling has block form

A = [[A0, −C†], [C, Ad]],

where C†=G⁻¹CᵀW. It gives zdot=Ad z+C c and cdot=A0c−C†z. Exact inheritance AS=SA0 requires C=0. Therefore autonomous blank-detail excitation and unchanged full-generator inheritance cannot both be demanded in this closed skew model. Omitting the feedback −C† while retaining C is not H-skew and conceals a norm source/sink unless explicitly supplied.

With forcing f, total support obeys d(xᵀHx)/dt=2xᵀHf. Detail support obeys d(zᵀWz)/dt=2zᵀWQf. These are receipts after a forcing law is supplied; neither selects that law. A small intertwiner defect can be quantified rather than ignored: D=QAS and zdot=Ad z+D c. For a constant H-skew A and initially coarse x0, the variation-of-constants estimate gives ||z(t)||_H≤t||D||_(G→H)||x0||_H. This is a sufficient leakage upper bound, not a lower bound or a physical error budget.

## 4. C4 launch funding is real historical source data, but not the new detail forcing

C0656's legal 1→4 move requires a registered latent five-seat descriptor and same-address C4 launch stock 3q². Its normalized q²=1 method fixture tracks N3 plus the separately addressed C4 stock, preserving its endpoint ledger. Pure low-side components with zero launch stock remain unchanged in that reported fixture. The ledger and event-frame B4 provenance are not interchangeable with the conditional fine-chain H norm, its detail component, a 14/134-direction state initialization, or physical energy.

C0656 contains no extraction map that converts C4 launch stock into Qf for the new S,R,H split; nor does it select a full generator coupling C, a transfer duration, or a cross-refinement measurement instrument. Its predictive open support source/sink and active-geometry arrow remain explicitly open. C0644 similarly leaves the proposed nonzero cut-cycle interaction law open. Historical Q157–Q160 corroborate that recursive ingress and predictive tagged sources remain unfinished, while Q165/Q169–Q179 define fixed-carrier stocks/currents instead of a new refinement interaction.

Therefore an actual detail seed law was **not recovered from the inspected sources**. This is a scoped source audit, not proof that no future source or unseen document can provide one. Geometry, exact metric compatibility, and inherited coarse observation alone do not generate seed amplitudes, source polarity, clock time, outgoing records, reset donors, or unlimited growth.

## 5. Publication and governance boundary

Permissible: “A genuine 1→4 star geometry and barycentric subdivision can be compared on their shared coarse carrier under declared full-chain interfaces. Their distinct retained details are not automatically equivalent. Exact H-skew generator inheritance makes a blank detail sector invariant; excitation requires an explicitly supplied seed, forcing, jump or generator defect. Existing C4 event funding is separately typed, and no conversion law into this new detail sector was found.”

Avoid: “The source now selects the refinement metric/history”; “C4 stock equals detail support”; “barycentric refinement autonomously creates new modes”; “a detail-only skew correction excites blank detail”; “coarse comparison is a full-state isometry”; “134/14 detail directions are qubits or tensor donors”; “norm preservation supplies energy”; or “constructor review closes CRL promotion.”

All new conditional premises remain candidate; native adoption, physical excitation/readout and non-constructor human review remain open. No registry entry, CRL advancement or physical promotion follows from this source review.