Manuscript IX-A@1 · IX-A.tex · section · 55aa6f6d

Coefficient hierarchy and dependency ledger

Coefficient hierarchy and dependency ledger · lines 1734–1828

IX-A@1#eq:four

Let a proposed relational functional have bare coefficients

F_{f} = F_{f}^{(0)} + κ B_{f} + κ^{2} D_{f} + \dots .

If the relevant retarded products exist, its first interacting coefficient is

B_{f} + R_{1} (V_{1}; F_{f}^{(0)}),

and its second is

D_{f} + R_{1} (V_{1}; B_{f}) + R_{1} (V_{2}; F_{f}^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; F_{f}^{(0)}) .

This is coefficient bookkeeping, not a construction of the maps.

{}1.2

Coefficient or output	Field degree	Boundary obligation	Status in IX-A
$F_{f}^{(0)} = \int f ϕ$	1	None	Bulk element
$B_{f}$	2	$q_{f d ϕ}$ and its scalar-direction descendants	Admitted
$J (a)$	1	Advanced Cauchy trace of $a$	Admitted
$D_{η} B_{f}$	1	$q_{f d σ}$	Admitted
$D_{η^{'}} D_{η} B_{f}$	0	None	Admitted
Finite pointwise words	Unbounded	Leibniz combinations of admitted traces	Admitted; separately continuous multiplication
Selected old two-factor Wick coefficients	0–4	Propositions prop:knownoutputs and prop:quadratic	Located in the carrier
Reflected mixed triple	0 after full contraction	Joint graph extension and two associations	Not constructed
$D_{f}$	Typically 3	Nested Cauchy kernels and their descendants	Not automatically admitted
$R_{1} (V_{1}; B_{f})$	Depends on contraction	Time-ordered extensions and contacts	Not constructed
$R_{1} (V_{2}; F_{f}^{(0)})$	Depends on vertex	Second-order vertex and retarded map	Not constructed
$R_{2} (V_{1}, V_{1}; F_{f}^{(0)})$	Depends on contractions	Two-vertex and three-insertion products	Not constructed

The order ledger is

(coupling order, ℏ -order, insertion count, field degree) .

A truncation in $ℏ$ does not bound the number of pointwise factors or the field degree.

The membership of $B_{f}$ gives no theorem admitting $D_{f}$ . Likewise, the selected old Wick coefficients do not imply that a retarded product with $B_{f}$ exists.

Displayed formulas in this passage

IX-A@1#eq:four · line 1747
$D_{f} + R_{1} (V_{1}; B_{f}) + R_{1} (V_{2}; F_{f}^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; F_{f}^{(0)}) .$
unlabeled · line 1738
$F_{f} = F_{f}^{(0)} + κ B_{f} + κ^{2} D_{f} + \dots .$
unlabeled · line 1743
$B_{f} + R_{1} (V_{1}; F_{f}^{(0)}),$
unlabeled · line 1817
$(coupling order, ℏ -order, insertion count, field degree) .$