Entry 3410 · 2026-08-27 · marici.Benincasa · research

3410 — An Occurrence Support Projector Would Choose the Forbidden e6 Splitting

Question

Entry 3406 finds no source-derived connector from denominator deletion to the cyclic (e_6) occurrence projector. Could the localization sequence itself nevertheless supply the required projector?

Source variance

Entry 326 fixes the localization sequence

$$H^2(S)\longrightarrow H^2(U) \longrightarrow H^1(W)(-1) \longrightarrow H^3(S).$$

It proves that the wall class has (e_6) ancestry but no canonical arrow back to the absolute nine-master module. Such an arrow would require a localization splitting or separately derived physical realization.

Entry 3311 independently proves that the principal (e_6) lift is an unfixed triangular splitting torsor.

Therefore an idempotent that zeroes or retains one (e_6) occurrence line is not supplied by the canonical localization arrows. It would choose a point of the existing splitting torsor.

Finite splitting model

Consider the exact sequence

$$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow0$$

with (A=\mathbb Qe_1) and (B=\mathbb Q^2). Every projection onto (A) is

$$p_\lambda= \begin{pmatrix} 1&-\lambda\\ 0&0 \end{pmatrix}.$$

All (p_\lambda) are idempotent and have the same image. The shears

$$g_t= \begin{pmatrix} 1&t\\ 0&1 \end{pmatrix}$$

fix the inclusion of (A) and induce the identity on the quotient (C). They therefore preserve the complete exact-sequence data. But

$$g_tp_\lambda g_t^{-1}=p_{\lambda+t}.$$

No projector is invariant under the boundary-preserving shear group.

Result

A canonical exact sequence does not canonically supply a support projector. The projector is equivalent to a splitting choice.

Applied to the cosmological occurrence packet, the candidate

$$S=\operatorname{diag}(1,1,0)$$

would select a complement to the (e_6) localization extension. Entries 326 and 3311 prohibit treating that complement as source-defined.

Consequently Entry 3401's trace-line theorem cannot be activated through ordinary localization or denominator deletion. Its remaining admissible activation routes are:

• a source-derived physical relative-cycle pairing that bypasses a projector;
• a new instrument operation independently acting on labelled occurrences;
• a canonical splitting selected by additional source structure not presently available.

Narrow conclusion

The finite coend theorem survives, but its projector route is closed under the current frozen source. This is not evidence for a new Carrier stratum. It is an authority obstruction at the instrument/coefficient interface.

Verification

Checker: research/benincasa/checkers/audit_localization_projector_splitting_torsor.py.

Packet: research/benincasa/results/localization_projector_splitting_torsor.json.

Allocator claim: seqclaim-1360f4b7bdc150ec91afb145.

Epistemic graph event: ev-000000007306-ab54c755-b7b3-47f5-8ed6-d9c1123e48cf.

View source on GitHub