III.28 Earlier compact derivations retained as alternatives
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The following derivations retain valid mathematical arguments from the original manuscript. Their hypotheses and notation agree with the corrected finite core. The earlier abbreviated argument for error correction is replaced by the complete normalized and scaled recovery proofs above, and its substantive conclusions are included there.
III.28.1 Duhamel proof of leakage from a fixed packet
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Let \(H=H^*\) on a finite-dimensional complex Hilbert space, \(P=P^*=P^2\), \(Q=I-P\), and \(t\in\mathbb R\).
Proof : Alternative proof of the commutator leakage bound
Because \(QP=0\),
The Duhamel identity gives
Both unitary factors have norm one, so taking norms and using the integration interval's length \(|t|\) yields \(\|Qe^{-itH}P\|\le |t|\,\|[H,P]\|\).
III.28.2 Proof of projected curvature with exterior forms
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Let \(P\) be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by \(\nabla\omega=P\,d\omega\). Thus \(\nabla^2\) denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields.
Proof : Alternative proof of the curvature formula
For a section \(s\) with \(Ps=s\),
Since \(ds=d(Ps)=(dP)s+Pds\) and \(P(dP)P=0\), the surviving term is
III.28.3 Restricted low-mode proof
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Let \(h\ne0\), \(x=kh/2\), and \(0\le x\le1\).
Proof : Alternative proof in the original low-mode regime
Since \(0\le\sin x\le x\), the error is nonnegative. On \([0,1]\),
Therefore
Hence \(x^2-\sin^2x\le x^4/3\). Multiplication by \(4/h^2\) gives