Manuscript IX-E@1 · IX-E.tex · section · 8f8d5e7f

Inherited objects, rather than reconstructed sources

Prerequisites and entering state › Inherited objects, rather than reconstructed sources · lines 274–325

The following are imported premises. They are not proved anew in IX-E. The actual definitions of the sharp functional carrier, $C$ , $E_{0}$ , the locked kernel $K$ , and the old products belong to the prerequisite sources.

The labels C-I1–C-I9 are retained as inherited free-sector hypotheses. Their individual wording was not included in the supplied text; no purported verbatim reconstruction of those clauses is made here. The proofs below state the particular inherited properties they use explicitly.

Object	Entering status	IX-E use
$⋆_{c}$ on $D_{decl}$	Constructed in IX-Free-2, relative to its premises	Named free product, not time ordering
Both associations of $B_{f}, W_{g}, B_{f^{'}}$	Constructed through $ℏ^{3}$	Free data only
$ker p_{L} = ker p_{R} = {0}$	Evaluated	Degree-four descent remains closed
$C_{desc} = C_{glue} = R$	Evaluated	Not a causal normalization
F9a	Fixed-support families only	Not F9b or hypocontinuity
New $T_{2}, R_{1}, R_{2}$	Absent	Binary construction obligation
VII-C under its hypotheses (E1)–(E4)	Retained	Not repaired by $U_{c}$
G2	Untouched	Not opened by binary time ordering

The word “new” distinguishes the absent sharp extension from any old time-ordered product that may separately be supplied.