Lattice modes displayed by the Quantum Gravity Lab
16 September 2026

Scope and observable
The page displays fractional edge-length perturbations of the free,
linearized Euclidean Kuhn Regge symbol, continued with p4=i omega.
Spatial positions are a FIXED reference lattice. Color is signed normalized
edge strain, not a displacement of those vertices or an embedding of the
curved metric in Euclidean 3-space. The overall weak-field amplitude is
arbitrary. The displayed edge strains depend on the gauge specified below.
The two-dimensional mode space modulo vertex displacements is physical;
individual edge colors are not gauge-invariant curvature measurements.

Construction from the paper's exact factorization
Use a=c=1, p_j=k n_j, omega=2 asinh(sqrt(sum_j sin^2(p_j/2))), p4=i omega.
Let z_i=exp(i p_i), q_i=2 sin(p_i/2), f_i=z_i-1, t_i=1-z_i^-1.
Then f_i=i q_i exp(i p_i/2), t_i=i q_i exp(-i p_i/2).
Choose real orthonormal a,b perpendicular to the spatial q vector.
Take H+=(aa^T-bb^T) or Hx=(ab^T+ba^T), padded with zero time components.
Both have trace zero and H q=0. Define the paper's trace-reversed coordinates

  y_ii=H_ii,
  y_ij=exp(i(p_i+p_j)/2) H_ij   (i != j).

Its constraint obeys, identically,

  (C y)_i = f_i y_ii + sum_(j != i) t_j y_ij
          = i exp(i p_i/2) sum_j q_j H_ij = 0.

The scalar branch L=0 and the paper's Wsharp R W=-(L V-Csharp C)/2
therefore give R(Wy)=0. Apply the exact edge map

  X_i=z_i (beta y)_i,
  X_ij=2 y_ij-z_i z_j sum_(r not in {i,j}) y_rr,
  beta=I-J/2.

As the diagonal trace is zero here, beta y=y. For the four eliminated
body diagonals restore X_B=2 M_BA X_A, since M_BB=-I/2 exactly. Set the
decoupled hypercube diagonal amplitude to zero. This produces a full
15-component null vector in the vertex phase convention of the paper.
The implementation evaluates every edge equation at each parameter update
and rejects a result with relative residual above 1e-10.

Why these are physical modes rather than vertex-displacement modes
For a nonzero spatial momentum, omega>0 and f4,t4 are nonzero. In y
coordinates a vertex displacement xi has, up to an overall sign,

  y_ii=2 t_i xi_i - sum_r t_r xi_r,
  y_ij=f_j xi_i+f_i xi_j.

Our modes have yi4=0, y44=0 and sum_(i=1..3) yii=0. The off-diagonal
time conditions imply xi_i=-f_i xi4/f4. On L=-sum_i f_i t_i=0 this gives
sum_r t_r xi_r=2 t4 xi4 and sum_(i=1..3) yii=-4 t4 xi4.
The trace condition forces xi4=0, hence every xi_i=0. Thus our two
independent tensors intersect the gauge subspace only at zero and represent
the two propagating modes. At k=0 this argument does not apply; the page
explicitly labels the pattern as a static uniform-strain limit.

What the colors mean
For edge D starting at integer spatial vertex x, the linear length strain is

  delta ell_D / ell_D = epsilon Re[(X_D / (2 |D|^2)) exp(i p.x-i omega t)].

Each polarization is divided by its largest spatial strain amplitude,
so the displayed scale runs from -1 to +1. epsilon is arbitrary and small;
the color scale does not set a physical wave amplitude. Blue denotes
contraction and orange extension. All seven nonzero spatial edge classes
in {0,1}^3 are drawn, including face and body diagonals. The '+' and 'cross'
labels name this specified basis, approaching ordinary continuum TT
polarizations as k tends to zero. They need not be transverse to ordinary
p at finite momentum: the constraint uses lattice differences q and phases.
The spatial amplitudes pair with their complex conjugates under reversal
of spatial momentum and frequency, so taking the real part gives a real
spatial strain field. The verification script checks this pairing too.

Wave-vector and group-velocity arrows
The white arrow is n=p/|p|, the direction normal to phase fronts. The blue
arrow has components vg_j=sin(p_j)/(2 sqrt(S(1+S))), with
S=sum_j sin^2(p_j/2). Its length represents |vg|/c. It can point in a
different direction from n; it vanishes at the spatial-zone corner.
c is the model's long-wavelength speed, not a separately derived photon
dispersion law. Time is normalized by a/c. At k=0 the group speed is
reported as its directional limit c.

Verification and source
verify_modes.py independently assembles the MIDPOINT symbol directly from
the rational coupling table and converts the displayed VERTEX amplitudes
using exp(-i p.D/2). It checks both modes, non-gauge independence, zero and
zone limits, and rejects corrupted fields. Run python3 verify_modes.py.
The reference receipt is reference/ModeChecks.json. The derivation above
is mathematical reasoning using the paper's verified factorization, not a
new Lean formalization or a derivation of nonlinear Lorentzian evolution.
Source paper: Lattice Graviton Dispersion on the Kuhn Triangulation,
Washburn and Beltracchi, proposed 15 September 2026 revision, exact branch
and axis-time sections. The original paper programs remain unchanged.
