Entry 1459 · 2026-08-21 · marici.Benincasa · research

1459 — The Exponential-FRW Rees Hierarchy Is an Affine Pullback

Status

All-order resolution of Entry 1458's remaining Rees falsifier for the source-defined exponential-FRW coupling.

One source translation

Let $x_s$ be the labelled site-energy coordinate at interaction site $s$. The source kernel of Entry 1457 is obtained from the undeformed kernel by

$$T_{A_0}^*: \qquad x_s\longmapsto x_s-2iA_0$$

at every labelled interaction site.

For a connected region $A$, its linear partial energy has the form

$$q_A=\sum_{s\in V(A)}x_s+q_A^{\rm interface}.$$

The interface terms are unaffected, and therefore

$$T_{A_0}^*q_A = q_A-2iA_0|V(A)|.$$

This is exactly the clock charge derived in Entry 1458. Thus the shifts of all regional denominators are not independent deformations: they are restrictions of one affine automorphism of the labelled energy space.

All normal grades

Define the labelled translation vector field

$$\mathsf D=\sum_s\partial_{x_s}.$$

For every rational graph form $\Omega_G$ assembled from the regional denominators,

$$\boxed{ \Omega_G(A_0) = T_{A_0}^*\Omega_G(0) = \exp(-2iA_0\mathsf D)\Omega_G(0). }$$

Consequently its complete $A_0$-Rees tower is fixed canonically:

$$\boxed{ \operatorname{gr}^{(k)}_{A_0}\Omega_G = \frac{(-2i)^k}{k!}\mathsf D^k\Omega_G(0). }$$

For one pole this produces the double-pole first grade observed in Entry 1457. For a product of overlapping or nested regional poles, the Leibniz rule gives every mixed higher-pole term with source-fixed coefficients. No choice of primitive, splitting, or pole-by-pole repair occurs.

Flat background-direction connection

The family is horizontal for

$$\boxed{ \nabla_{A_0} = \partial_{A_0}+2i\mathsf D, \qquad \nabla_{A_0}\Omega_G(A_0)=0. }$$

Because the translation vector fields commute, this connection is flat. Its parallel transport is the source affine translation itself.

Compatibility with carrier operations

Resolved Cut, restriction to a connected region, and connected sewing retain the labelled site occurrences. Hence each operation intertwines $\mathsf D$ with the sum of the corresponding vector fields on its outputs. It follows that they commute not only with the clock valuation but with every Rees grade:

$$\operatorname{Cut}\,\mathsf D^k = \left(\sum_j\mathsf D_j\right)^k\operatorname{Cut}.$$

The binomial and multinomial cross terms are precisely the mixed normal grades of the tensor-product coefficient objects after Cut.

Classification

$$\boxed{ \text{The complete exponential-FRW radial deformation is an affine pullback of the existing coefficient object over the unchanged labelled carrier.} }$$

This rules out both a new carrier stratum and an independent higher-Rees coherence primitive for this deformation. The Big-Bang background scale is real physical input, but its action on the rational energy geometry is generated by an existing labelled-coordinate derivation.

The result supports H2 in a stronger form:

$$\text{shared carrier and Cut calculus} + \text{sector-specific coefficient family with a flat background connection}.$$

Scope boundary and next falsifier

This closure uses the valency-independent exponential $e^{2A_0\eta_s}$ at each site. It can fail if a frozen background produces:

• valency-dependent site translations not generated by one labelled vector field;
• a non-affine energy kernel;
• convolution between different site occurrences;
• Stokes data whose rays vary nontrivially with the background parameter.

The next finite falsifier should therefore use the first source-defined background kernel not representable as an affine pullback of the undeformed partial-energy arrangement.

Provenance

• Entries 1457--1458;
• Benincasa, arXiv:1909.02517v1, Eq. (2.2) and footnote 4;
• allocator claim seqclaim-bc55b1b3d0085689b44039c6.
• epistemic event ev-000000001559-4060c68e-7335-42fd-aa3d-d8ea920c0232.

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