600-cell signal network

The marked planar reference starts from the unit circle and the opening $\varepsilon=\pi/648000$, with $\varphi=(1+\sqrt5)/2$. For scale $a>0$,

$C_{\mathrm{circ}}$ is the opposite semicircle, not the retained major arc. The selected ratio is

For a retained positive calibration $a_{\mathrm{ref}}$, $\chi(a)=1+a/a_{\mathrm{ref}}$ and $\Omega(v)=1+\lambda(v-1)$ satisfy $\chi(\lambda a)=\Omega(\chi(a))$ on positive scales and $v>1$. Zero scale is the degenerate template and the fixed scalar 1. Do not evaluate a length ratio as $0/0$.

Different objects

The 600-cell positions are 120 fixed vertices in $\mathbb R^4$, represented in $\mathbb Q(\sqrt5)$. A real network state is $v\in\mathbb R^{120}$. An integer state is $d\in\mathbb Z^{120}$. A signal entry is not a 4-dimensional vertex coordinate. Recovering a signal does not reconstruct an unrecorded orientation.

Declared mixing

$A$ is the adjacency matrix of the labeled graph: 120 vertices, 720 edges, degree 12. Adjacent vertices have squared distance $2-\varphi$. The checker stores doubled coordinates as integer pairs $(a,b)$ for $a+b\varphi$.

Each row and column of $P$ sums to one. The center has weight $13/25$ and each of twelve neighbors has weight $1/25$. For a degree-$r$ regular graph the same pattern is $((r+1)I+A)/(2r+1)$. The denominator 25 is a stated choice, not a constant forced by the planar curve.

Real update

Because $P$ is nonnegative and stochastic, $\|\mathcal F(v)-\mathcal F(w)\|_\infty\le\lambda\|v-w\|_\infty$. The unique fixed point is $\mathbf 1$. $M$ is nonsingular: if $Mx=0$ and $|x_i|$ is maximal, then $13|x_i|\le 12|x_i|$, so $x=0$. Exact finite real contraction is injective. Its limit alone does not name the initial state.

On constant data, $\mathcal F(t\mathbf 1)=\Omega(t)\mathbf 1$. For arbitrary real data, $\overline{\mathcal F(v)}=1+\lambda(\bar v-1)$. If $U$ preserves adjacency, $\mathcal F(Uv)=U\mathcal F(v)$. That is equivariance of the update, not symmetry of every input. The Laplacian eigenvalue $\nu=15$, multiplicity 16, has multiplier $2\lambda/5$.

The source manuscript is the 26 October 2026 assessment, revision 2.0, still unpublished. The author-reported reserved DOI 10.5281/zenodo.23688676 is not an issued publication, not a resolving deposit, and not a ninth source work.

Positive-simplex corollary

This is a written real-arithmetic derivation for 120 centered coordinates. It is not a TGA execution result, not a six-mark membership test, not a 45-byte codec, and not a training or physical-performance claim. It inherits no Lean declarations and no checker PASS.

For a strictly positive probability vector $p\in\mathbb R^{120}$, $\operatorname{clr}(p)=\log p-\operatorname{mean}(\log p)\,\mathbf 1$. On the zero-mean subspace, softmax and clr are inverse maps. Because $P$ preserves that subspace,

Then $\mathcal G_i(p)\propto p_i^{13\lambda/25}\prod_{j\sim i}p_j^{\lambda/25}$ and $\mathcal G^{-1}(p')=\operatorname{softmax}(\lambda^{-1}P^{-1}\operatorname{clr}(p'))$. Set $z=\operatorname{clr}(p)$, update $z'=\lambda Pz$, then apply the inverse chart. Expanding $P$ gives the product formula; the common mean-log factor cancels upon normalization. Invertibility follows from invertibility of $P$ on the zero-mean subspace. The full real signal also needs its mean, which evolves by scalar OMEGA. Softmax alone discards a common additive offset.