Manuscript IX-E-2@1 · IX-E-2.tex · section · e70d9077

Frozen domain

Release contract and fixed target › Frozen domain · lines 227–306

IX-E-2@1#eq:frozen-domain, IX-E-2@1#eq:domain-stability, IX-E-2@1#eq:old-intersection

Let $σ$ be complex-linear flip and $γ$ componentwise antilinear conjugation:

σ (F \otimes G) = G \otimes F, γ (F \otimes G) = F^{*} \otimes G^{*} .

Thus

d^{†} = σ γ d, σ^{2} = γ^{2} = id, σ γ = γ σ .

The inherited first-stage space and frozen target are

D_{first} D_{2} = (B \otimes W) + (W \otimes B), = span_{C} {σ^{a} γ^{b} d : d \in D_{first}, a, b \in {0, 1}}, O_{2} = {0} .

The spaces $B$ and $W$ are the inherited spaces of actual first-stage functionals. In particular, every $B_{f} \otimes W_{g}$ under consideration is retained.

IX-E proves the stability and containment statements

D_{2} \subset D_{cmp}, D_{cmp} = D_{decl} \cap σ (D_{decl}), σ D_{2} = γ D_{2} = D_{2}^{†} = D_{2} .

These are imported results, not new domain claims.

The zero old seed does not evaluate the actual old intersection:

O_{act} = D_{2} \cap D_{old}, D_{old} = P_{mc} \otimes P_{mc} .

No nonzero old time-ordered map is supplied here. Conjugation saturation cannot be ignored when auditing (eq:old-intersection).

{{0.92} Release box. The outcome is Unresolved on the fixed domain (eq:frozen-domain).

Witness-class result: the commutator vanishes exactly on every pair in IX-E's strongly separated witness class.

Fixed-domain result: no nonzero commutator on a certified retained two-way pair is established. Complete two-way coverage remains unfinished at the actual functional-support bridge.

No retained pair is deleted. No time-ordered replacement is introduced. No implication from witness-class vanishing to a constructed causal product is asserted. }

Displayed formulas in this passage

IX-E-2@1#eq:frozen-domain · line 246
$D_{first} D_{2} = (B \otimes W) + (W \otimes B), = span_{C} {σ^{a} γ^{b} d : d \in D_{first}, a, b \in {0, 1}}, O_{2} = {0} .$
unlabeled · line 231
$σ (F \otimes G) = G \otimes F, γ (F \otimes G) = F^{*} \otimes G^{*} .$
unlabeled · line 237
$d^{†} = σ γ d, σ^{2} = γ^{2} = id, σ γ = γ σ .$
IX-E-2@1#eq:domain-stability · line 263
$D_{2} \subset D_{cmp}, D_{cmp} = D_{decl} \cap σ (D_{decl}), σ D_{2} = γ D_{2} = D_{2}^{†} = D_{2} .$
IX-E-2@1#eq:old-intersection · line 274
$O_{act} = D_{2} \cap D_{old}, D_{old} = P_{mc} \otimes P_{mc} .$