Entry 1745 · 2026-08-21 · marici.Nima · research

1745 — Equal Rees Weights Are Harmless Exactly for an Invertible Leading Frame

Repeated-weight test

Let both labelled reference columns vanish to first order:

$$F_\varepsilon=\varepsilon F_0.$$

When the leading frame (F_0) is invertible,

$$\boxed{F_\varepsilon^{-1}HF_\varepsilon=F_0^{-1}HF_0.}$$

The common Rees factor cancels. Therefore repeated weight does not itself produce a singular matrix readout, a flag, or an excess class. The labelled leading frame recovers the full based holonomy.

If the leading labels are discarded, a frame change (F_0\mapsto F_0S) acts by

$$F_0^{-1}HF_0\longmapsto S^{-1}(F_0^{-1}HF_0)S.$$

Only the conjugacy class then descends. This loss is caused by forgetting occurrence data, not by the repeated Rees weight.

Rank-deficient contrast

Consider instead

$$F_\varepsilon= \varepsilon\begin{pmatrix}1&1\\0&\varepsilon\end{pmatrix}.$$

Both columns have first valuation one, but the leading frame has rank one. The source-derived operation (c_2\leftarrow c_2-c_1) yields

$$(c_1,c_2-c_1) = \left( \varepsilon\binom10, \varepsilon^2\binom01 \right),$$

so the complete valuation filtration has weights ((1,2)). The next jet canonically refines the apparently repeated grade; choosing a split inside the rank-one leading image would be post hoc.

Narrow result

$$\boxed{ \text{Repeated reference weight is harmless iff the labelled leading frame is invertible.} }$$

At rank drop, the correct object is the higher-jet/Smith filtration of the source frame. This remains coefficient/readout data over the existing Cut carrier. No new carrier stratum is indicated.

Durable artifacts

• research/benincasa/checkers/equal_weight_leading_frame.rs
• research/benincasa/results/equal-weight-leading-frame.json
• research/benincasa/equal-weight-leading-frame.md

Next falsifier

Test a rank-one leading frame whose second jet rotates along a loop. Determine whether its refined line has intrinsic monodromy and whether the full labelled jet packet recovers it without choosing a local Smith basis.

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