TFPT scientific source excerpt
Original repository path: tex-artefacts/toe_round7_quantum_tensor_curvature.tex
Original file SHA-256: 2427c7bec1c1fb5f38a579d2871c6174f03956bfbba07bdfbb0c77e8e151bc15
Inclusive original lines: 19-131
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000019 | \fbox{\parbox{0.92\linewidth}{
000020 | \textbf{Exact scope.}  This note quantizes only the free, quadratic, zero-mean
000021 | curvature target whose classical constraint quotient was established at
000022 | repository commit \texttt{4417f1ba8da3d5dc622c942efcbfc46ebaae546d}.
000023 | It constructs neither a microscopic TFPT parent nor finite-dimensional quantum
000024 | cells.  It proves no nonlinear constraint algebra, state selection mechanism,
000025 | universal matter coupling, interacting graviton theory, T7 closure, TOE claim,
000026 | or RH statement.  Strict lattice light-cone support remains false.  The
000027 | microcausality theorem below is a distributional statement about the
000028 | $a\to0$ limit of this free target.
000029 | }}
000030 | \end{center}
000031 | 
000032 | \paragraph{Assumptions and preregistered obligations.}
000033 | 
000034 | Let a periodic cubic lattice have $V=L^3$ cells and spacing $a>0$.  The six
000035 | orthonormal symmetric-tensor coordinates are ordered as
000036 | $(11,22,33,23,13,12)$, with the off-diagonal basis tensors normalized by
000037 | $1/\sqrt2$.  Component $A$ lives at $x+a m_A/2$, where
000038 | \[
000039 |  m_A=(000,000,000,011,101,110).
000040 | \]
000041 | We use the physical-position Fourier phase
000042 | $\exp[-\ii k\cdot(x+a m_A/2)]$.  The staggered derivative symbol is
000043 | \[
000044 |  \ii\kappa_i^{(a)}(k),\qquad
000045 |  \kappa_i^{(a)}(k)=\frac{2}{a}\sin\frac{ak_i}{2},\qquad
000046 |  r_a(k)=|\kappa^{(a)}(k)|.
000047 | \]
000048 | It vanishes only at the homogeneous momentum in the first Brillouin zone.
000049 | The complete homogeneous canonical block is removed as an additional
000050 | zero-mean prescription.  It is not declared gauge.
000051 | 
000052 | The proof was preregistered against the following obligations.
000053 | \begin{enumerate}[label=\textbf{O\arabic*.},leftmargin=2.5em]
000054 | \item Prove, for every nonzero real momentum and not from samples, that the
000055 | curvature commutator operator is $r^4P_{\TT}$, is positive, has rank two, and
000056 | is transverse and traceless.
000057 | \item Construct the Weyl quotient, prove the Heisenberg inequality by an
000058 | explicit factorization, and identify the finite-volume GNS representation and
000059 | ground state.  A finite matrix approximation to the CCR is inadmissible.
000060 | \item Treat $k$ and $-k$, self-conjugate momenta, and half-grid Bloch signs
000061 | without choosing a globally inconsistent polarization frame.
000062 | \item Prove convergence of local states as $L\to\infty$ at fixed $a$.
000063 | \item Either prove an $a\to0$ tempered-distribution limit and its curvature
000064 | microcausality, or leave that claim open.  Finite grids may only be regression
000065 | tests for the analytic proof.
000066 | \end{enumerate}
000067 | 
000068 | The input from the classical target is the canonical reduced Hamiltonian and
000069 | the already-proved staggered identities.  In one momentum fibre set
000070 | \[
000071 |  Bq=q\kappa,\quad t=(1,1,1,0,0,0)^T,\quad
000072 |  v=B^T\kappa,\quad s=v-r^2t,
000073 | \]
000074 | \[
000075 |  K_p=I-\frac12tt^T,\qquad
000076 |  K_q=r^2(I-tt^T)-2B^TB+tv^T+vt^T.                 \tag{1}
000077 | \]
000078 | The four local constraints are the scalar $s^Tq=0$ and the three momentum
000079 | constraints $Bp=0$.  For $r>0$ they are independent and first class.  The
000080 | curvature variables are
000081 | \[
000082 |  E=K_qq,\qquad F=K_qK_pp,                           \tag{2}
000083 | \]
000084 | and the classical result supplies
000085 | \[
000086 |  K_qB^T=0,\quad K_ps=B^T\kappa,\quad K_qt=s,
000087 | \]
000088 | \[
000089 |  K_q^2=r^2K_q+ss^T,\qquad
000090 |  K_qK_pK_q=r^2K_q+\frac12ss^T.                    \tag{3}
000091 | \]
000092 | All matrices in (1)--(3) are real in the physical-position trivialization.
000093 | 
000094 | \paragraph{The exact spin-two commutator projector.}
000095 | 
000096 | For $\kappa\ne0$, let $P=I_3-\kappa\kappa^T/r^2$.  The orthogonal
000097 | spin-two projector on symmetric tensors is
000098 | \[
000099 |  (P_{\TT}h)_{ij}=P_{ia}P_{jb}h_{ab}
000100 |  -\frac12P_{ij}P_{ab}h_{ab}.                       \tag{4}
000101 | \]
000102 | It has rank two, $P_{\TT}^2=P_{\TT}=P_{\TT}^T$, and its range is
000103 | $\{h:h\kappa=0,\ \Tr h=0\}$.
000104 | 
000105 | \begin{theorem}[Polynomial curvature projector]
000106 | For every $\kappa\ne0$,
000107 | \[
000108 |  C:=K_qK_pK_q=r^4P_{\TT}.                          \tag{5}
000109 | \]
000110 | Consequently
000111 | \[
000112 |  C^T=C\ge0,\quad C^2=r^4C,\quad \Tr C=2r^4,
000113 | \quad \operatorname{rank}C=2,
000114 | \quad CB^T=0,\quad Ct=0.                           \tag{6}
000115 | \]
000116 | Although (4) has inverse powers of $r$, $C$ itself is a homogeneous
000117 | degree-four polynomial in $\kappa$.
000118 | \end{theorem}
000119 | 
000120 | \begin{proof}
000121 | The TT space is $\Ker B\cap t^\perp$.  If $h$ is TT, (1) gives
000122 | $K_qh=r^2h$ and $K_ph=h$, hence $Ch=r^4h$.  Equations (3) give
000123 | $CB^T=0$.  They also give
000124 | $Ct=K_qK_ps=K_qB^T\kappa=0$.  The scalar and momentum constraints are
000125 | independent at $r>0$, so
000126 | $\Ran B^T+\operatorname{span}\{t\}$ is four-dimensional and is the
000127 | orthogonal complement of the two-dimensional TT space.  Thus $C$ is $r^4$
000128 | on TT and zero on its orthogonal complement, proving (5).  The properties in
000129 | (6) follow immediately.  Finally, $C=K_qK_pK_q$ is polynomial; equivalently
000130 | the two factors of $r^{-2}$ implicit in (4) cancel against $r^4$.  A direct
000131 | component multiplication of (4) is included in the executable checker.