The vertex code and its two-dimensional image
=============================================

Hamiltonian and transfer matrix
--------------------------------

Take a circle with L vertices and a qubit at each vertex. For p=0 the
unaugmented complex has no A stabilizers. With only a transverse source,

.. math::

   H_z=-\frac K2\sum_j(1+Z_jZ_{j+1}),\qquad H_x=-g\sum_jX_j,
   \qquad H=H_z+H_x.

The source-free code space is spanned by the two aligned Z product states.
The trace in this package includes the whole Hilbert space. Let
:math:`D=e^{-\delta H_z}`, :math:`U=e^{-\delta H_x}` and Q=DU, with
delta=1/(TM). Inserting Z-basis states between factors gives

.. math::

   Z_M=C\sum_\sigma\exp\left[
       \kappa_x\sum_{j,t}\sigma_{j,t}\sigma_{j+1,t}
       +\kappa_\tau\sum_{j,t}\sigma_{j,t}\sigma_{j,t+1}\right],

.. math::

   \kappa_x=\frac{\delta K}{2},\qquad
   \kappa_\tau=-\frac12\log\tanh(\delta g),\qquad
   C=e^{\beta KL/2}
     [\cosh(\delta g)\sinh(\delta g)]^{LM/2}.

Both directions are periodic for the ordinary trace. This is an exact
finite-M representation of Q, including the normalization C. VertexModel
uses L,M>=3 to avoid coincident-bond conventions.

The isotropic one-parameter family
-----------------------------------

At a chosen delta, set

.. math::

   K=\frac{2\kappa}{\delta},\qquad
   g=-\frac{1}{2\delta}\log\tanh\kappa.

Then kappa_x=kappa_tau=kappa and the normalized measure is the ordinary
zero-field square-lattice Ising measure. Its exact bulk critical coupling is
:math:`\kappa_c=\tfrac12\log(1+\sqrt2)`; see :doc:`../references`.
For a square spacetime torus L=M=n with delta=1, this path uses T=1/n.
Changing n along that path changes the quantum temperature. It is not a
Trotter refinement at fixed quantum H and T. A fixed-H refinement instead
holds K,g,T fixed and increases M.

Wilson points and moments
--------------------------

Every collection of points is a Wilson 0-cycle. A same-slice quantum product
:math:`\prod_j Z_j^{\nu_j}` maps to the corresponding spin product.
For diagonal products D_nu_t on different discrete time slices, the insertion is

.. math::

   \frac{\operatorname{Tr}(D_{\nu_0}Q D_{\nu_1}Q\cdots D_{\nu_{M-1}}Q)}
        {\operatorname{Tr}Q^M}.

This fixes the Euclidean multi-time meaning of a general VertexObservable
point pattern. It is not a real-time correlation function.

For ``average=True`` the configuration value is the average of all translations
of that product. A translated joint pattern and a product of two translated
averages are different observables:

.. math::

   \frac1{LM}\sum_x A_xB_{x+r}\ne
   \left(\frac1{LM}\sum_x A_x\right)
   \left(\frac1{LM}\sum_y B_y\right)

in general. For a fixed-separation connected correlation, declare the joint
point pattern explicitly and subtract the product of the two ensemble means
after pooling. Repeated point insertions cancel modulo two.

The spacetime magnetization is :math:`m=(LM)^{-1}\sum_x\sigma_x`.
Its powers sum products across all discrete times and positions. The Binder
functional :math:`U_4=1-\langle m^4\rangle/(3\langle m^2\rangle^2)`
uses pooled moments. A finite torus has a rounded crossover; comparisons
between sizes can provide finite-size critical indicators, with statistical,
grid-interpolation and finite-size effects assessed separately.

Boundary twists: mathematical context
--------------------------------------

The current vertex executor samples the periodic-periodic model and Wilson
point products. The following twist identities define useful terminology but
do not advertise a twist API.

Let Z_ab be the spin partition sum with a reversed spatial seam if a=1 and a
reversed temporal seam if b=1. The closed global-X operator
:math:`\mathcal T=\prod_jX_j` imposes temporal antiperiodicity:

.. math::

   \frac{\operatorname{Tr}(\mathcal T Q^M)}{\operatorname{Tr}Q^M}
      =\frac{Z_{01}}{Z_{00}}.

A repeated magnetic Euclidean insertion
:math:`U_m=\exp[-\delta K Z_{L-1}Z_0]` reverses the spatial bond on every
slice. Thus :math:`\operatorname{Tr}[(U_m Q)^M]=C Z_{10}`. Both insertions
give the corresponding combined seam choice at zero longitudinal source.
U_m is a transfer-weight insertion, not a single same-time Pauli operator.

An expectation in a twisted background has its own denominator Z_ab.
The four backgrounds are not equally weighted physical sectors of the
ordinary periodic trace. For reweighting, reverse the desired bond signs
explicitly. Reversing a seam twice composes modulo two; squaring its original
reweighting factor is not the identity operation on the action. Practical
twist-ratio sampling and its variance control require additional methods.