TFPT scientific source excerpt
Original repository path: tex-artefacts/toe_research_20260909.tex
Original file SHA-256: 361c3843779f78c66ff7e19b610152f38d5dc4f6623ed9017c9d515f1dc94433
Inclusive original lines: 67-95
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000067 | Second quantization relative to the filled sea maps the target vacuum,
000068 | particles and holes into the same source Fock space. With
000069 | $a_j=b_j$ for $j\leq0$ and $a_j=d_j^*$ for $j\geq1$, the target is
000070 | \[
000071 |  H=\sum_j |j-\tfrac14|\,n_j^{\rm exc},\qquad
000072 |  Q=\sum_{j\leq0}n_j^b-\sum_{j\geq1}n_j^d.
000073 | \]
000074 | The resulting contractions intertwine both field adjoints and charge in
000075 | the limit on the common finite-energy core. Orthogonality of momentum
000076 | residuals bounds the $k$-excitation generator error by $k$ times the
000077 | one-particle error, not by the $8N$ sea occupation. Since each excitation
000078 | costs at least $1/4$, $k\leq\lfloor4E\rfloor$ on an energy-$E$ core.
000079 | For $S_m=\sum_j|j-\tfrac14|^m|f_j|$ one also obtains the uniform bounds
000080 | \[
000081 |  \|\Psi_N(g)\|\leq S_0,\quad
000082 |  \|[\mathcal H_N,\Psi_N(g)]\|\leq S_1,\quad
000083 |  \text{time-transport error}\leq
000084 |  |t|\left(\frac{\pi S_2}{N}+\frac{2\pi^2 S_3}{3N^2}\right),
000085 | \]
000086 | with transported test function
000087 | $g_t(x)=e^{it/4}g(x-t/(2\pi))$ in the source convention.
000088 | All fixed smooth finite field words and their vacuum moments follow by
000089 | bounded-product convergence. The top-edge charge-sector minima are
000090 | \[
000091 |  E_{\rm top}(q)=q^2/2-q/4,\qquad q\in\mathbb Z.
000092 | \]
000093 | They are not minima over both edges of the whole cylinder.
000094 | The microscopic holonomy term has not been removed to fit a target.
000095 | Full proof and controls: \path{microscopic-charged-car-limit/} in the archive.