# Hermitian-First Path

## Purpose

This note records the Hermitian-first admissibility path for `j3_oc`.

Its core question is:

> what if the first admissibility cut is simply to keep only Hermitian or reality-compatible ambient Jordan data?

---

## Core rule

Start from
$$
X \in J3(C \otimes O)
$$
and keep only candidates satisfying a Hermitian or equivalent reality-compatible condition:
$$
X = X^\dagger.
$$

In the current branch language, this is the first step of:

`X -> X_H -> X_{H,r} -> X_adm`

where `X_H` is the first admissible representative.

---

## Why it is attractive

- it is the cleanest first control on the complexified ambient arena
- it immediately separates ambient room from candidate physical data
- it does not smuggle in `u`, `SU(3)`, or reduced `Spin(2,3)` structure too early
- it gives the branch a conservative baseline

---

## What it buys

- a disciplined first admissibility cut before more interpretive choices enter
- a stable starting point for later rank/orbit refinement
- a clear way to ask whether complexification still earns anything after the first reality-compatible restriction
- a useful baseline against which stronger admissibility paths can be measured

---

## Where it is weak

- by itself it does not determine `u`
- by itself it does not tell us whether `SU(3)` survives in the right way
- it risks collapsing back toward a picture in which the complexified ambient language is large but the physically relevant structure is still effectively real
- it may function as cleanup only, not as branch-defining work

---

## Success condition

Hermitian-first succeeds only if it becomes a real first stage in a stronger path.

More concretely, it should:

- make later rank/orbit refinement cleaner
- constrain later `u` and `SU(3)` work rather than leaving them totally free
- support a more disciplined comparison against `J3(O)`

---

## Failure condition

Hermitian-first fails as a stand-alone path if:

- it does no more than broad cleanup
- all physically important structure is inserted only after the Hermitian cut
- it leaves no better handle on `u`, `SU(3)`, reduced-sector selection, or observable viability

In that case it remains useful as a baseline, but not as the main admissibility story.

---

## Best current verdict

Hermitian-first is the safest baseline path, but not a likely winner by itself.

The most disciplined reading is:

- keep it as stage one
- do not ask it to carry the whole bridge

---

## Relation to the main branch

This path aligns with the current default in:

- [0b - ambient-to-observable reduction.md](../0b%20-%20ambient-to-observable%20reduction.md)

but this note exists so that the path can be judged separately rather than hidden inside the default sequence.