Theorem: Em03-F exterior scalar-source release

Under the stated stationary field input, scattering normalization, and effective exterior-source specification:

1. the compact source interaction has the exact unitary propagator and common time-ordering phase (eq:propagator)–(eq:UF);

2. its source-to-field map and all source matrix units are calculated, with the off-diagonal identities (eq:offdiagonalzero);

3. identified outgoing scalar packet restrictions are normal displaced Gaussian states with amplitudes (eq:alpha);

4. a compact exterior separable source profile and outgoing packet have a proved nonzero coupling;

5. the bounded sine-Weyl effect has the exact source-sign gap (eq:singap), while selected joint number probabilities are identical for opposite labels;

6. the packet radiation channel is entanglement breaking, with the exact optimal deterministic recovery values (eq:recovery);

7. the source-displaced states satisfy the common moment bound (eq:Csrc), and the compact local scalar probe gives (eq:localcertificate);

8. source-profile, observable, complete-readout, and independent-trial errors have the separate proved allowances of Theorem thm:profile, (eq:obsledger), and Theorem thm:observed;

9. spacelike field-observation invariance, retarded future response, and retained-source coherence are calculated;

10. excess outgoing scalar energy is finite and has the switching-work ledger of Theorem thm:energy and Proposition prop:work.

The construction is restricted to a massless s-wave source and full-shell access. The local approximation may enlarge its coupling region and requires increasing probe preparation effort. Independent repeated preparations are retained as an additional statistical hypothesis.

The source law is an effective exterior specification. No microscopic quantum-gravity derivation, black-hole interior information escape, coherent local field transfer, Page curve, backreaction, mass-loss equation, or complete evaporation theorem is established.

Proof. Theorem thm:unitary proves (i). Theorem thm:fieldchannel proves (ii). Proposition prop:displacement, Lemma lem:gaussian, and Theorem thm:packetnormal prove (iii). Theorem thm:nonzero proves (iv). Theorems thm:witness and thm:counts prove (v). Theorems thm:EB and thm:recovery prove (vi). Proposition prop:sourceclass, Theorem thm:packetbridge, and Theorem thm:local prove (vii). The cited profile and record arguments prove (viii). Theorems thm:causality and thm:backaction prove (ix). The energy and work arguments prove (x).

Every comparison retains its own domain and metric. The model contains neither a gravitational source derivation nor the interior, coherent-access, or evaporation dynamics required for the excluded physical conclusions.