Manuscript Em01-F@3 · Em01-F-v3.tex · theorem · 225e59af

Theorem: Exact combined positive-scale formula

Exact localized crossed-line and tensor regrouping › Combined expression for analysis · lines 2800–2860 · proof included or linked

Em01-F@3#thm:combined, Em01-F@3#eq:combinedpairing, Em01-F@3#eq:combined-seeds

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For all sufficiently small positive scales,

t_{δ}^{app} (Φ) = λ^{3} ℏ^{2} I, J \sum ⟨ K_{ret}^{I, J}, U_{δ}^{I, J} [Φ]⟩ .

Equivalently,

t_{δ}^{app} (Φ) = λ^{3} ℏ^{2} I, J \sum {I_{RR} (P_{47}^{I, J} [U_{δ}^{I, J} [Φ] \circ κ_{47}]) + I_{RR} (P_{5, +}^{I, J} [U_{δ}^{I, J} [Φ] \circ κ_{5, +}]) + I_{RR} (P_{5, -}^{I, J} [U_{δ}^{I, J} [Φ] \circ κ_{5, -}]) + I_{resp} (P_{supp}^{I, J} [U_{δ}^{I, J} [Φ] \circ κ_{resp}])} .

This formula equals the original 42-entry ledger on every compact joint test. It has not removed any insertion row or vertex term.

Proof. Use (eq:localized-smoothing) in the complete vertex and then apply Lemma lem:tensorsimplify. The factor $i$ in the selected family multiplies the factor $i$ in the localized crossed array. Consequently

t_{δ}^{app} = - λ^{3} ℏ^{2} V (K_{ret}, N d_{δ}) .

Transposing the output derivative gives the positive kinetic term and the negative mass term in (eq:Ucombined). This proves (eq:combinedpairing), including its overall sign.

Apply Theorem thm:seed to each row and combine the four response seed actions by linearity. This gives (eq:combined-seeds). All preceding steps are equalities with smooth crossed coefficients at positive scale. The ledger expansion is the same complete vertex before these algebraic regroupings, so the two expressions agree on arbitrary joint tests.

Displayed formulas in this passage

Em01-F@3#eq:combinedpairing · line 2803
$t_{δ}^{app} (Φ) = λ^{3} ℏ^{2} I, J \sum ⟨ K_{ret}^{I, J}, U_{δ}^{I, J} [Φ]⟩ .$
Em01-F@3#eq:combined-seeds · line 2811
$t_{δ}^{app} (Φ) = λ^{3} ℏ^{2} I, J \sum {I_{RR} (P_{47}^{I, J} [U_{δ}^{I, J} [Φ] \circ κ_{47}]) + I_{RR} (P_{5, +}^{I, J} [U_{δ}^{I, J} [Φ] \circ κ_{5, +}]) + I_{RR} (P_{5, -}^{I, J} [U_{δ}^{I, J} [Φ] \circ κ_{5, -}]) + I_{resp} (P_{supp}^{I, J} [U_{δ}^{I, J} [Φ] \circ κ_{resp}])} .$
unlabeled · line 2844
$t_{δ}^{app} = - λ^{3} ℏ^{2} V (K_{ret}, N d_{δ}) .$

Theorem: Exact combined positive-scale formula

Displayed formulas in this passage

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