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000001 | % Consolidated research snapshot; historical source-pinned records stay unchanged.
000002 | \section*{Research consolidation: source fields, dynamics and the remaining bridge (2026-09-09)}
000003 | \addcontentsline{toc}{section}{Research consolidation: source fields and dynamics (2026-09-09)}
000004 | \label{sec:research-20260909}
000005 | \noindent\textbf{Status and provenance.}
000006 | The research archive after Round~27 is now integrated, including the
000007 | Round~28--50 development chain and the subsequent named branches.
000008 | These are written, model-specific arguments accompanied by executable
000009 | controls, not newly formalized or independently reviewed theorems.
000010 | All physical T1--T8 gates and \texttt{TFPT.TOE.COMPLETE.01} remain \mO.
000011 | The source map, prerequisites and reproduction entry points are collected in
000012 | \path{experiments/theory-contracts/RESEARCH_2026-09-09.md}.
000013 | Historical records retain their original local/uncommitted labels and
000014 | hashes; the integration does not retroactively change their provenance.
000015 | 
000016 | \subsection*{1. A charged field from the original filled microscopic sea}
000017 | Fix the original complex, number-conserving QWZ strip at width eight,
000018 | mass one and holonomy sector $r=1$. Its one-particle space has dimension
000019 | $16N$ over $\mathbb C$; it is not the compiler's sixteen real Majoranas.
000020 | Write $P_N$ for its negative-energy projector of rank $8N$ and
000021 | $\Omega_N$ for the actually filled sea. Set
000022 | \[
000023 |  D_N=\frac{N}{2\pi}h_N,\qquad
000024 |  \mathcal H_N=d\Gamma(D_N)-\operatorname{Tr}(P_ND_N),\qquad
000025 |  Q_N=\mathcal N-8N.
000026 | \]
000027 | Here $\mathcal H_N\geq0$ on the full finite Fock space, including charged
000028 | sectors. For a smooth periodic function $g$ use the physical top row,
000029 | not an auxiliary charge register:
000030 | \[
000031 |  (R_Ng)(x,y)=N^{-1/2}e^{-i\pi x/(2N)}g(x/N)
000032 |        \delta_{y,7}\chi_+,\qquad \Psi_N(g)=a_N(R_Ng),
000033 |  \quad \chi_+=2^{-1/2}(1,1)^T.
000034 | \]
000035 | These fields obey exact CAR, have charge $-1$, and are supported on the
000036 | sampled support of $g$. Their adjoints have charge $+1$.
000037 | For $k=j-\tfrac14$, $p=2\pi k/N$, $\rho=1-\cos p$ and the normalized
000038 | top-row mode $r_{N,j}$, the original strip action gives
000039 | \[
000040 |  \|D_Nr_{N,j}\|\leq |k|,\qquad
000041 |  \|(D_N+k)r_{N,j}\|
000042 |  \leq\frac{\pi k^2}{N}+\frac{2\pi^2|k|^3}{3N^2}.
000043 | \]
000044 | Indeed $h(p)\chi_+$ on the top row has the two orthogonal components
000045 | $-\sin p\,\chi_+$ and $\rho\chi_-$; its norm is
000046 | $2|\sin(p/2)|$. This is a bound on the actual microscopic action,
000047 | not a fitted edge dispersion.
000048 | 
000049 | The geometric edge quasi-mode has amplitude proportional to
000050 | $\rho^{7-y}\chi_+$ and residual $\rho^8$. For
000051 | $|j|\leq M_N=\min(\lfloor N/8\rfloor,\lfloor\sqrt N\rfloor)$,
000052 | projection onto $P_N$ for
000053 | $j\geq1$ and onto $1-P_N$ for $j\leq0$ produces an isometry $J_N$ with
000054 | $P_NJ_N=J_NP_{\rm cur}$ exactly and
000055 | $\|J_Ne_j-r_{N,j}\|\leq4\rho_j$.
000056 | The discarded occupation of the raw top-row vector is at most
000057 | $9\rho_j^2$, hence $O(N^{-4})$ for each fixed label.
000058 | If $g=\sum_j f_je^{2\pi ijx}$, the locality-to-polarization comparison is
000059 | \[
000060 |  \|R_Ng-J_N\Pi_{M_N}f\|
000061 |  \leq4\rho_*\|\Pi_{M_N}f\|_2+
000062 |              \sum_{|j|>M_N}|f_j|\longrightarrow0.
000063 | \]
000064 | The last term is an $\ell^1$ alias tail; replacing it by an unqualified
000065 | $\ell^2$ tail would not justify sampled locality.
000066 | 
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.
000096 |