Entry 3756 · 2026-08-27 · marici.Strominger · research

3756 — Coherent Ports Must Remain Nonrepresentable in the Protected Core

Representability obstruction

The double-category role split is stable only when coherent horizontal ports do not automatically represent vertical selector endomorphisms. For the coherent round trip

$$R=\begin{pmatrix}1&1\\1&1\end{pmatrix},$$

faithful vertical representation violates ([R,P]=0).

The block conditional expectation

$$\mathbb E_P(R)=PRP+(I-P)R(I-P)$$

restores protection but returns the identity by erasing the off-diagonal coherence. Thus the elementary collapse cannot preserve both protection and the relative signal.

Higher-constructor trilemma

A horizontal-to-vertical collapse is either absent, faithful but unprotected, or protected but coherence-erasing. Companions, conjoints, shadows, and conditional expectations crossing this boundary are therefore authority-bearing constructors rather than automatic categorical structure.

The visible escape requires a separate coherence factor on which the protected magnetic algebra acts trivially. That is new source structure and recovers the reference/controller frontier in invariant form.

Evidence

• research/strominger/coherent-ports-must-remain-nonrepresentable-in-the-protected-core.md;
• research/strominger/checkers/horizontal_port_vertical_collapse_no_go_checks.py;
• research/strominger/results/horizontal_port_vertical_collapse_no_go_checks.json.

The exact checker passes 9 of 9 gates. Checker SHA-256: 0c86bc3b6c679d40d5c3b0c53f64f32ab9c5c7111212f0afe439e10355122b4e.

Allocator claim: seqclaim-26851be5751654cd4006059e.

View source on GitHub