Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f

What can be entered in the catalogue

A finite Enter catalogue of supplied evaluations › What can be entered in the catalogue · lines 2039–2101

A wave kernel is generally a distribution. Even at a generic spatial pair it contains delta fronts in time. A purported “kernel value” on such a front is not a number. We therefore use smeared entries, or values restricted to a declared smooth part of a kernel.

Choose compact smooth sections $a, f$ whose pair supports satisfy the transfer contract, and define

Q_{cone} = ⟨ a, G_{+}^{F} f ⟩, Q_{flat} = ⟨ dev_{*} a, G_{+}^{flat} dev_{*} f ⟩ .

These are finite complex numbers, and Theorem thm:transfer proves their equality. Choose also one fixed generic spatial pair and define

C_{dif} (x, x^{'}) = - \frac{Φ _{β} ( 0 , ϑ )}{4 π ^{2} α r r ^{'}}, κ_{L} = \frac{2 e ^{- γ_{E}}}{L} .

A possible finite catalogue is:

Entry	Mathematical target	Required scope metadata
C1	$Q_{cone}$	$α, β, m, B, a, f$ , frame, support window
C2	$Q_{flat}$	Same tests and normalization as C1
C3	$C_{dif} (x, x^{'})$	Generic pair, angular lift, front convention
C4	$κ_{L}$ , and optionally $m^{2} - κ_{L}^{2}$	Boundary scale $L$ , units, mass

For a complex entry, its real and imaginary parts are reported separately.

A report word contains a finite rational approximation or enclosure, its error convention, and a status: for example ANALYTIC, SUPPLIED_NUMERIC, or UNCERTIFIED. The existence of an enclosure is not inferred merely by printing a decimal. An installed symbolic formula is also a finite description, not an exact finite-bit encoding of an arbitrary real number.

The number $κ_{L}$ is the magnitude of a transverse negative-energy scale. It is a positivity certificate for the mass-shifted operator only when accompanied by the comparison $m \geq κ_{L}$ .

Displayed formulas in this passage

unlabeled · line 2049
$Q_{cone} = ⟨ a, G_{+}^{F} f ⟩, Q_{flat} = ⟨ dev_{*} a, G_{+}^{flat} dev_{*} f ⟩ .$
unlabeled · line 2059
$C_{dif} (x, x^{'}) = - \frac{Φ _{β} ( 0 , ϑ )}{4 π ^{2} α r r ^{'}}, κ_{L} = \frac{2 e ^{- γ_{E}}}{L} .$