Manuscript IX-D@3 · IX-D-v3.tex · section · 3ad4586e

D7: dependency and non-identification ledger

D7: dependency and non-identification ledger · lines 1607–1706

The objects below are not interchangeable. The assertion is that no identification needed by this construction has been supplied, not that special numerical coincidences are universally impossible.

{}1.14

Object	Scope and missing identification
Object	Scope and missing identification
IX-Free-2's $⋆_{c}$	A truncated free product on the declared tensor domain $D_{decl}$ . It is not $T_{2}$ , not $R_{1} (V_{1}; B_{f})$ , and not the complete free-product domain of an unspecified general IX-D construction.
IX-Free's two-edge outputs	Admitted nonconstant quadratic functionals with fixed flat free contractions. Their availability does not supply a time-ordering map.
Triangle $U_{c}$	A ternary numerical joint-test distribution, now the constant projection of the constructed free word. It does not select a causal subtraction or a source contact.
V's external $R_{1, 1}$	A switched external-metric response with its stated smooth comparison test $+ Q u_{f}$ . Its extension statement concerns that object, not the reflected triangle.
V's oscillator coefficient	A switched bilinear in a finite-mode representation. No map identifies it with $U_{c}$ or a sharp causal carrier coefficient.
V's Newton–wave cubic	A distinct Wick form with spatial Newton-transfer singularities, not a substitute for the reflected graph or its causal extension.
VI's relational coefficient	A first-order dressing/forced-response comparison with the copied switch witness. It supplies no sharp retarded map.
VII-A's plateau intertwiner	A restricted classical identity retaining switch derivatives, not an established quantum Ward identity or an identification with $U_{c}$ .
VII-E/F and VIII certificates	Certificate or geometric data in separate tracks. No map into the present polynomial carrier or into $U_{c}$ is supplied.
Source and packet data	No supplied map identifies $j [f]$ , conical packets, slicing events, or source tokens with $B_{f}, W_{g}$ , the partial contractions, or $U_{c}$ .

The complete V, VI, VII-A, VII-E/F, and VIII sources are not provided here. Their entries record the stated scope boundaries, not independently certified theorems used in an unconditional causal construction.

The updated dependency chain is

IX-Free-2 partial product on D_{decl} ⇓ still required time-ordered extension on a named domain ⇓ still required retarded subtraction with causal support ⇓ separate additional gates G2 and interacting maps .

For a general causal construction, any required free-domain enlargement is an additional obligation alongside the time-ordering step. The old degree-four stop is no longer the live stop for this word.

Displayed formulas in this passage

unlabeled · line 1690
$IX-Free-2 partial product on D_{decl} ⇓ still required time-ordered extension on a named domain ⇓ still required retarded subtraction with causal support ⇓ separate additional gates G2 and interacting maps .$