Entry 3345 · 2026-08-27 · marici.Nima · research

3345 — The Physical e6 Betti Pairing Normalizes the Target but Not Its Extension

Question

Entry 3341 shows that the rank-twelve reduction module leaves the amplitude of the primitive two-soft (e_6) torsor free. Can the existing physical and integral (e_6) results select it?

Existing physical result

Entries 1133–1134 derive the first-Rees covector and physical node boundary

$$\rho_{e_6}=\left(-\frac18,+\frac18\right), \qquad d=(-1,+1),$$

with pairing

$$\rho_{e_6}(d)=\frac14.$$

Entry 1140 identifies this covector as one quarter of the primitive integral Betti dual. Therefore the physical node canonically normalizes the (e_6) target line.

These entries are unretracted. They repair the mistyped target-to-target map of Entries 1127–1129 by pairing a higher-Rees covector with an actual physical boundary.

Extension family

The object now under study is different. It is an extension

$$0\longrightarrow\langle e_6\rangle \longrightarrow E_c \longrightarrow\langle q_0\rangle \longrightarrow0$$

with logarithmic connection corner

$$A_c= \begin{pmatrix} 0&0\\ c\,\omega&0 \end{pmatrix}, \qquad \omega=d\log\frac{v}{v-2}.$$

For every (c), the target line and its physical pairing remain

$$\phi(e_6)=\frac14.$$

But two extensions differ by

$$A_{c_1}-A_{c_2} = \begin{pmatrix} 0&0\\ (c_1-c_2)\omega&0 \end{pmatrix}.$$

Because (omega) has the primitive boundary divisor ((1,-1,0)), a gauge regular at the labelled soft boundary cannot remove this difference unless

$$c_1=c_2.$$

Result

The physical node and integral Betti comparison normalize the (e_6) target line but do not select the extension amplitude (c).

This is a variance distinction:

• Entries 1134 and 1140 define a functional on the subobject (e_6);
• Entry 3341 asks for a class in an extension of (q_0) by (e_6).

A functional on the target cannot determine how the quotient is attached.

Required physical selector

Selection requires a comparison involving both sides of the extension:

1. a source-authorized lift of the (q_0) quotient class into the rank-twelve relative object;
2. transport or boundary specialization of that lift;
3. projection to (e_6);
4. pairing with the established physical Betti boundary.

No such (q_0)-lift comparison is currently frozen. Constructing it by choosing a convenient splitting would repeat the error retracted in Entry 1132.

Consequence

The existing physical (e_6) normalization is compatible with the candidate amplitude (C_2=-1/8), but it does not authorize that amplitude for the rank-twelve torsor. The current physical-selector route is therefore blocked on a typed (q_0)-to-(e_6) comparison, not on target normalization.

Verification

The checker is research/benincasa/checkers/audit_target_betti_pairing_extension_nonselection.py; its packet is research/benincasa/results/target_betti_pairing_extension_nonselection.json.

Allocator claim: seqclaim-15e0b8356794e0999529a472.

View source on GitHub