# Subdivision: norm, boundary and generator gate v0.1

Date: 2026-10-14. Reading edition0.11. Development — no registry entry. Physical promotion **0**. Constructor-team AI verification only; independent human review pending. The new metric branch is an authored conditional compatibility construction, not a recovered UD constitutive law.

## 1. Sources and contract

The original fixed-complex substrate is the real signed simplex chain carrier, with boundary B, unit pairing and A=Bᵀ−B. Source R31 defines ordinary oriented barycentric subdivision S; R33 defines its bounded scalar/residual completion; R34 proves the strict Euclidean unit-vertex/generator obstruction. C0173 and C0243 distinguish a changed metric with its correct adjoint from a basis change or a fixed-metric operator modification. These source statements retain their dates, conditional premises and provenance. Their exact existing public-catalog UTF-8 text fields are snapshotted and hashed in the release; original release/PDF bytes are not implied by those text hashes. No imported archival programme was executed.

The supplied historical ledger v0.39, Q165/Q169–Q179, declares Euclidean squared chain-amplitude support, fixed-complex currents and separately typed simplex stocks. It does not select a cross-event metric. Q157–Q160 leave recursive support/source/sink semantics open. The attached CRL subdivision document is a governance proposal; constructor-team checks do not satisfy its non-constructor promotion gate. The unavailable Gap2 image was not used.

The user proposed a universal sqrt2 partition/coupling/saturation constant. This gate instead identifies the actual source map and separately tests: **SᵀHS=G**, **B′S=SB**, and **A′H S=SA_G**, where A_G=B†G−B and A′H=B′†H−B′. Pairings, rates and readout are declared; none is physical time or energy. Passing norm preservation does not imply either remaining condition.

## 2. Recovered map and the original metric

On a filled tetrahedron Δ³, old simplices are increasing vertex tuples. New barycentric vertices are the15 nonempty original faces. A new oriented simplex is a strictly nested flag F0⊂…⊂Fk. The subdivided f-vector is (15,50,60,24), so source and target chain dimensions are15 and149.

For an oriented old simplex [v0,…,vk], define ordinary subdivision

S[v0,…,vk]=Σπ sign(π)[{vπ0}⊂{vπ0,vπ1}⊂…⊂{vπ0,…,vπk}].

Each old degree-k basis simplex has (k+1)! signed descendants. Distinct columns have disjoint descendant supports in that grade. Consequently

**Theorem SG.1:** B′S=SB and SᵀS=⊕k (k+1)! I; the Gram factors are **1,2,6,24**.

The boundary identity follows by signed flag-face cancellation. The finite replay regenerates both boundaries directly by alternating deletion and every subdivision column by permutations. It verifies all matrix entries exactly. In the source Euclidean metric, S therefore fails norm preservation on every nonzero positive-degree basis chain. It preserves the vertex norm, but on each old vertex e_i the squared generator defect is

||(A′S−SA)e_i||²=7.

Old vertex incidence has3 edges; the new singleton vertex has7 incident flag edges. R34's no-go for simultaneous Euclidean native contraction, unit singleton coefficient and unchanged unit-incidence generator remains valid.

## 3. Why scalar and factorial normalization do not close the gate

**Bounded source branch:** T=S/√24 is a chain map and a contraction. Its degree squared norm factors are (1,2,6,24)/24. A separate residual map D=(I−TᵀT)½ gives the isometry c↦(Tc,Dc); it retains rather than removes the missing native norm. This does not make T itself an isometry. It attenuates singleton coefficients by1/√24 and has vertex generator defect squared7/24. R33's funded two-event ledger and R34's interface obstruction remain unchanged.

**Degree-by-degree amplitude normalization:** U_k=S_k/√((k+1)!) is an isometry in unit target coordinates but fails the chain equation because adjacent degrees use different factors. Already on one edge the squared boundary defect is3−2√2. No single universal sqrt2 factor normalizes all four grades.

**Naive metric alternative:** retain S and B′ but use Hdiag,k=I/(k+1)!. Then SᵀHdiag S=I and B′S=SB. Nevertheless A′Hdiag S≠SA. On a vertex the degree1 raising adjoint is multiplied by2. The generator defect has Euclidean squared norm22 and Hdiag squared norm11. The exact parallel check also finds positive-degree adjoint-defect ranks4,6,3. Thus a simple norm compensation is insufficient.

| Transfer/pairing | Native norm | Boundary | Generator | Unit singleton coefficient |
|---|---|---|---|---|
| S, target unit metric | fails | exact | fails | preserved |
| S/√24, unit native metric | contractive; residual completion isometric | exact | fails | attenuated |
| U_k=S_k/√((k+1)!), unit metric | exact | fails | not admitted | preserved in degree0 |
| S, Hdiag,k=I/(k+1)! | exact | exact | fails | preserved |
| S, retraction/complement metric below | exact conditional | exact | exact conditional | preserved |

These comparisons use different declared pairings. Their equal scalar norms do not identify them with Q165's unchanged149-coordinate Euclidean support.

## 4. A canonical affine chain retraction

The new conditional branch starts with a geometry-derived barycenter map on barycentric vertices. Map each face F to b_F=(1/|F|)Σv∈F e_v in the original formal vertex vector space. On a flag [F0,…,Fk], take the exterior product b_F0∧…∧b_Fk. Expand it in the original oriented degree-k simplex basis. Its coefficient in [i0,…,ik] is the determinant of the corresponding barycenter-coordinate minor. This defines R_k without reading any desired clock spectrum or physical numerical target.

**Theorem SG.2:** R B′=B R and RS=I. The first identity follows because each barycenter coefficient column sums to1, so the exterior extension commutes with boundary contraction. Every descendant of an old k-simplex maps back to its oriented source with coefficient sign(π)/(k+1)!; summing its (k+1)! signed descendants gives the second identity.

This R is an authored coarse retraction, not a source-selected event readout. It is natural under vertex relabeling, unlike selecting a preferred last vertex. The generated check verifies all24 signed S4 permutations on S,R and the reference metric below. Such symmetry consistency does not select the metric as native.

Set Q=I−SR. Then Q²=Q, RQ=0, QS=0, and B′Q=QB′. Every fine chain decomposes uniquely as x=S(Rx)+Qx. Thus im S and ker R are complementary boundary-invariant subspaces.

## 5. Exact conditional metric completion

Let G be any declared positive old-chain pairing and W any positive fine-space pairing. With degree-block pairings when grading is to remain explicit, define

H=Rᵀ G R+Qᵀ W Q.

**Theorem SG.3:** H is positive definite; SᵀHS=G; and A′H S=S A_G. No invertible149×149 numerical approximation is required for the proof.

Proof. The fine squared norm is ||Rx||²G+||Qx||²W. If it vanishes, Rx=Qx=0, hence x=S(Rx)+Qx=0. Also HS=RᵀG and SᵀHS=G. Using the chain retraction,

B′ᵀHS=B′ᵀRᵀG=RᵀBᵀG=HS G⁻¹BᵀG.

Since H is invertible, B′†H S=S B†G. Subtracting the boundary identity proves the generator equation. Therefore e^(s A′H) S=S e^(s A_G) for every internal s, with unchanged rate; no sqrt2 clock multiplier appears.

Equivalently, in normalized coordinates E=H½ S G⁻½ is a complete isometry of the old15-dimensional carrier into the fine149-dimensional carrier and Â′E=EÂ. This is exact preservation on the embedded coarse subspace, **not** unitary equivalence of the entire unequal-dimensional operators. The134 complementary fine coordinates remain a separate autonomous sector because ker R and im S are orthogonal boundary subcomplexes in H, hence reducing subspaces for A′H.

On Δ³ take G=I and use W=I as one reference witness. The root replay additionally checks W=2I and positive degree weights(1/2,2,3,4). All satisfy the three equations. W is unselected; its restriction to ker R is what matters. The existence of many admissible metrics is an underdetermination statement, not a promotion of any one member.

## 6. Complement, readout and support costs

The complement dimensions by degree are(11,44,56,23), total134. Its boundary ranks are(11,33,23), so the complementary complex is acyclic. For the reference G=I, the old coarse sector retains its one harmonic vertex line and seven signed frequency pairs at2 in the declared old rate unit. General positive G retains its own correctly declared old generator instead. No full fine spectrum, minimal physical implementation or default n10 clock embedding is claimed. A larger S4 carrier has different multiplicities; the prior single15-chain three-band theorem is not a classification of this149-chain.

The exact transfer S sends each old unit vertex basis vector to its singleton with coefficient1. An arbitrary old vertex amplitude c_i is retained as c_i. With reference G=I, that singleton still has norm1 under the new metric. This evades R34 only by changing the post-event pairing and its adjoint; it does not refute the Euclidean7>3 obstruction.

The new H is generally **nondiagonal within degree**. Individual simplex squares c_sigma² are therefore not additive contributions to xᵀHx. Cross terms cannot be relabeled as independent nonnegative local stocks. Q170's original local incidence-current/control-volume rule and Q171/Q172's separate simplex stocks do not automatically carry over. A local metric/source/readout law must be derived separately. This construction is not a diagonal volume metric, a free energy supply or a physical norm identification.

S changes the carrier dimension. Its isometry has an orthogonal completion after metrics are normalized and enough fine complementary coordinates are supplied. A full implementation still needs the134 extra coordinates prepared in the admitted complementary state, a declared transfer gate/clock and observation law, and preservation of outgoing records. The algebraic image condition Qx=0 describes the ideal prepared coarse image; it is not a demonstrated initialization device. A full fine input with Qx≠0 retains that additional information. Any operation setting it to zero requires an explicit export or other accounted resource; it is not a free reset.

Here134 is a **direct-sum chain-coordinate dimension**, not134 qubits, a tensor-factor donor bank or a minimal physical resource count. The ideal complement condition is zero amplitude in those directions. The fully retained linear identification (c,z)↦Sc+z, z∈ker R, has squared norm cᵀGc+zᵀWz and is bijective. It transports rather than erases z. For a declared old observation O, the conditional coarse readout OR satisfies OR(Sc+z)=Oc; source/measurement implementation remains open. Raw singleton-coordinate readout need not ignore arbitrary z.

A finite iteration can repeat this algebraic metric construction with newly declared G,S,R,W at each refinement. That statement alone does not supply new carriers, source-owned metrics, event occurrence, a uniform event spacing, protected export, a physical clock or indefinite funded growth. Existing finite-window/reset restrictions remain in force.

## 7. What survives from the sqrt2 proposal

For the specifically rotated rectangle-halving map F(r)=2/r, one-step similarity requires r=√2. But F(F(r))=r, so departure from√2 does not itself prove cumulative drift or instability. In isotropic d-dimensional volume halving the linear factor is2^(−1/d). Orthogonal equal-amplitude normalization over N alternatives is1/√N. On this actual subdivision the grade factors are√1,√2,√6,√24; sqrt2 is the edge case, not a universal folding/coupling constant.

The carrier spectrum statement Q175 is separately typed: the unit-metric glued K+ support generator has frequencies√2 and√5. It is not the refinement normalization above. Neither norm preservation nor these spectral coincidences provides a kinetic/potential coupling, CHSH observable algebra or feedback-instability threshold. The simple family α[[0,−1],[1,0]] preserves Euclidean norm for every real α and supplies a counterexample to inferring a universal stability ceiling from skewness alone.

## 8. Verification and revised frontier

Root Fraction replay generates the15/149 bases, alternating-deletion boundaries, subdivision permutations and barycenter minors. It checks both chain maps, RS=I, all Gram entries, original and naive-metric vertex defects, and three positive complement-metric cases. The weighted adjoint equation is verified as B′ᵀHS=HSBᵀ; the positive split-norm proof makes it equivalent to adjoint intertwining. A separate fresh implementation passes332 exact checks, including all24 signed actions and naive-defect ranks. Source/scope review checks that the metric change preserves previous no-go scopes. These are constructor-team AI checks, not independent human review.

The next lawful frontier is source selection of the coarse readout/retraction, fine metric/complement law and measurable local support; or a declared interface defect/current within the original source metric. Exact compatibility is now feasible under an explicit changed metric, but the mechanism selecting it is **OPEN**. No physical or registry promotion, universal sqrt2 law, finite-resource exception or global proof closure follows.