Entry 3362 · 2026-08-27 · marici.Benincasa · research

3362 — The Raw Second-Rees Arrow Cannot Realize the Relational Residue

Question

Entry 3358 constructs the primitive Leray–soft relational cell

$$R_{fb}=d_f\otimes d_b.$$

The frozen four-stratum reduction also displays the raw coefficient

$$[E^{-2}e_6],\nabla_E\Omega_{111}=\frac18.$$

Can this coefficient define the missing comparison

$$\kappa: \mathbb Q\langle R_{fb}\rangle \longrightarrow R_{q_0,e_6}?$$

Tempting numerical composition

Let $\eta_f$ be the primitive integral dual to the vertical Leray boundary, so $\eta_f(d_f)=1$. Contracting the vertical leg gives

$$(\eta_f\otimes1)(R_{fb})=d_b.$$

Multiplication by $1/8$, together with the Koszul orientation sign, would produce

$$-\frac18d_b,$$

which is exactly the candidate residue vector $C_2(1,-1)$.

This numerical match is not sufficient because the proposed coefficient must be invariant under the admitted Rees gauge.

Rees-gauge audit

Entries 292–293 derive the shear

$$\widehat\Omega_{111} =\Omega_{111}+\frac{h}{E}e_6.$$

If the raw double-pole coefficient is $b$, then differentiation of the shear changes it by

$$b\longmapsto b-h.$$

For

$$b=\frac18, \qquad h=\frac18,$$

the double pole vanishes. Other admitted values of $h$ produce other coefficients. Thus the raw $1/8$ is a coordinate of a Rees presentation, not a natural transformation on the relational residue.

Result

The fifth-tower relational cell survives this test, but the obvious four-stratum arrow does not realize it canonically in the $q_0/e_6$ extension.

The distinction is now:

• $R_{fb}$: primitive universal composability residue;
• raw $E^{-2}e_6/8$: removable Rees-presentation coefficient;
• $R_{q_0,e_6}$: intrinsic logarithmic extension line;
• $\kappa$: still missing comparison between the first and third objects.

The equality of the raw coefficient with the magnitude of $C_2$ is compatibility evidence only. Using it as $\kappa$ would make the fifth-tower comparison gauge-dependent.

Revised finite falsifier

Compute the mixed vertical/horizontal map only after passing to the Rees-normalized logarithmic complex. The candidate comparison must:

1. commute with both signed boundary operators;
2. be invariant under $\Omega_{111}\mapsto\Omega_{111}+h e_6/E$;
3. land in the intrinsic logarithmic extension costalk;
4. retain the rank-one relational cell before any instrument pairing.

If every such map vanishes, the current coefficient complex does not realize the fifth-tower residue. If a nonzero map survives but depends on $h$, it is presentation data. Only a gauge-independent map establishes $\kappa$.

Verification

The checker is research/benincasa/checkers/audit_raw_rees_adapter_no_go.py; its packet is research/benincasa/results/raw_rees_adapter_no_go.json.

Allocator claim: seqclaim-ad93976bfad99bc7dabe2d85.

Epistemic graph event: ev-000000007208-c00c9151-a15a-47ef-b7b6-5444d1b5ba33.

View source on GitHub