Manuscript XVI-H@1 · XVI-H.tex · proposition · af895076

Proposition: Trisolution-preserving representative

Conditional audit of the inherited switch representative · lines 1489–1563 · proof included or linked

XVI-H@1#prop:switch, XVI-H@1#eq:tilde, XVI-H@1#eq:W-preserved

Let $u$ be a compact kernel in that inherited slot calculus, and put

P_{1} = □_{1}, P_{2} = □_{2}, P_{3} = P_{m, 3}, W_{u} = E_{0, 1} E_{0, 2} E_{m, 3} u .

Choose the switches with transition times covering the relevant compact support projections. Then

u = P_{1} P_{2} P_{3} [ζ_{1} ζ_{2} ζ_{3} W_{u}]

is compactly supported and satisfies

W_{u} = W_{u} .

The intermediate switched trisolution need not be compact. These conclusions alone are not an admission result.

Proof. For a compact input $v$ in one slot, the inherited Green identities give

ζ_{i} E_{i} v = E_{i}^{R} v - r_{i}, r_{i} = (1 - ζ_{i}) E_{i}^{R} v + ζ_{i} E_{i}^{A} v .

The causal support bounds and the opposite temporal truncations make $r_{i}$ compact: the retarded contribution is truncated above, and the advanced contribution below, while the source is compact.

Therefore

T_{i} v := P_{i} [ζ_{i} E_{i} v] = v - P_{i} r_{i}

is compact, and $E_{i} T_{i} v = E_{i} v$ because $E_{i} P_{i} r_{i} = 0$ . There is no additional minus sign in this equality.

Apply the three operations successively to compact inputs. The inherited distinct-slot commutations identify their composition with (eq:tilde). Equivalently, telescoping the three differences gives

u - u = P_{1} r_{1} + P_{2} r_{2} + P_{3} r_{3}

for compact corrections at the successive stages. Applying all three causal propagators annihilates each term, proving (eq:W-preserved).

Compactness follows already from the successive compact representatives. It can also be read from

P_{1} P_{2} P_{3} [ζ_{1} ζ_{2} ζ_{3} W_{u}] = [P_{1}, ζ_{1}] [P_{2}, ζ_{2}] [P_{3}, ζ_{3}] W_{u},

using the trisolution equations. Each commutator is supported in its slot's transition region, and the inherited causal support controls the spatial projections there.

Displayed formulas in this passage

XVI-H@1#eq:tilde · line 1500
$u = P_{1} P_{2} P_{3} [ζ_{1} ζ_{2} ζ_{3} W_{u}]$
XVI-H@1#eq:W-preserved · line 1507
$W_{u} = W_{u} .$
unlabeled · line 1493
$P_{1} = □_{1}, P_{2} = □_{2}, P_{3} = P_{m, 3}, W_{u} = E_{0, 1} E_{0, 2} E_{m, 3} u .$
unlabeled · line 1518
$ζ_{i} E_{i} v = E_{i}^{R} v - r_{i}, r_{i} = (1 - ζ_{i}) E_{i}^{R} v + ζ_{i} E_{i}^{A} v .$
unlabeled · line 1531
$T_{i} v := P_{i} [ζ_{i} E_{i} v] = v - P_{i} r_{i}$
unlabeled · line 1544
$u - u = P_{1} r_{1} + P_{2} r_{2} + P_{3} r_{3}$
unlabeled · line 1554
$P_{1} P_{2} P_{3} [ζ_{1} ζ_{2} ζ_{3} W_{u}] = [P_{1}, ζ_{1}] [P_{2}, ζ_{2}] [P_{3}, ζ_{3}] W_{u},$

Proposition: Trisolution-preserving representative

Displayed formulas in this passage

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