Manuscript Em06-B@1 · Em06-B-v1.tex · section · 61e38859

Convention, support, and evidence audit

Convention, support, and evidence audit · lines 2065–2163

{ {}1.14

Item	Audited calculation	Support or convention
Item	Audited calculation	Support or convention
Density variation	Variation of (eq:action) gives $P_{m} + κ Q_{χ h}$ ; expansion gives (eq:Q-expanded).	Signature $+ - - -$ ; $□ = \partial_{t}^{2} - Δ$ ; trace–mass term retained.
Scalar response	$- G_{m}^{R} Q_{χ h} ϕ_{0}$ .	Compact source separated from $Σ_{0}$ ; retarded sign fixed in (eq:Green).
Coordinate response	$□ w = C (h)$ , zero $w, \partial_{t} w$ at $t = 0$ .	Unswitched; coordinate-time data; localized estimate uses $h$ only on $E$ .
Compact test	Green transposition and two compact integrations by parts give (eq:q).	Coefficient factors are compact; no time-truncated distribution is substituted for a smooth test.
Gauge identity	$C (K ξ) = □ ξ$ , $Q_{K ξ} = (\partial \cdot ξ) P_{m} - [P_{m}, X_{ξ}]$ , then (eq:test-gauge).	$ξ$ compact in the common plateau and away from $Σ_{0}$ .
Classical pairing	Fourier integration gives $o = 2 Re ⟨ k_{q}, z ⟩$ .	Unitary spatial Fourier transform; inner product linear in its second argument.
Quantum fields	Number-form bound, analytic vectors, Weyl identities.	$a (v)$ antilinear; $Φ (v) = a (v) + a^{†} (v)$ ; closures and form moments explicitly specified.
Ordering phase	Repeated Weyl multiplication gives $- ϵ \sum_{i < j} t_{i} t_{j} Im ⟨ v_{i}, v_{j} ⟩$ .	Factors ordered by increasing index; no noncommuting joint spectral law.

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ledger. Analytical derivations are supplied in this text. No symbolic expansion script, numerical witness evaluation, validated floating-point enclosure, compilation log, second-reader audit, or machine-checking report is supplied or claimed. Bibliographic searches did not provide accessible full texts for the additional VII-I comparison. Accordingly, its summary-only labels are not promoted to premises. This limitation does not leave a mathematical gap in B1–B5, whose required formulas and estimates are proved here.

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bibliography: References