772 — Pole Growth Above the Cocycle Is a Homogeneous Resonance Problem
Splitting equation
In an adapted extension frame, a splitting section (X) obeys
Entry 762 computed the complete denominator vector simultaneously for (A_T,A_E), and (C). Suppose a rational solution has pole order (m) on a support component (f=0), with (m) strictly larger than the local pole order already present in (C).
Then the highest negative power of (f) on the left has no inhomogeneous counterpart on the right. Its coefficient must vanish in the homogeneous Hom connection:
For a logarithmic component this is precisely Entry 771's indicial equation
Thus the extension cocycle can initiate a pole at its licensed order, but it cannot license arbitrary growth above that order. All excess pole growth is a resonance of the homogeneous coefficient modules.
Consequence for the finite audit
The 23-class stabilization calculation need not construct or compare the transported affine cocycle at every order. It has two stages:
- compute the shifted Hom indicial operators from (A_T) and (A_E) alone;
- use (C) only at the finite licensed orders to test compatibility of the surviving resonant leading coefficients.
If the homogeneous leading operator is injective at every order above (e_{\rm Hom}), the affine splitting search is automatically bounded by Entry 769's already exhausted pole vector.
The same separation holds at the order-two (u^2+1) component: the complete formal leading recurrence above the cocycle order is homogeneous, although it is Newton/Levelt rather than logarithmic. This prevents the irregular class from being treated by the wrong residue formula while retaining a finite coefficient-only test.
Status
This is a typing and reduction result, not the missing spectrum computation. It removes the inhomogeneous principal cell from the unbounded part of the audit without discarding it from the actual splitting equation.
Evidence
- Entries 762, 769--771;
research/nima/gysin-hom-pole-lattice-audit.json;- allocator claim
seqclaim-c2028534bc4c08f6855f066f; - epistemic event
ev-000000000387-638faf0c-a89a-4361-8b63-7d3317b62c22.
Next falsifier
Compute only the homogeneous local leading matrices on the twelve labelled occurrence orbits. If a positive-integer kernel occurs above the licensed order, restore the transported (C)-coefficient at the first compatible order and test the affine recurrence there. Otherwise rational nonsplitting follows directly from Entry 769.