Hyperbolic Sparsemax Kernel — Exact-Rational Demo

What this runs

Every quantity below is computed in exact rational arithmetic (BigInt numerator/denominator) — no floating point touches the kernel, honoring the SHD-CCP determinism rule. Decimals are shown only for reading. This instantiates assumptions (A1)–(A4) of the proof.

Spinors built as $q_i=g_i^2/N(g_i)$ for Lipschitz integer quaternions $g_i$ ⇒ exact rational unit quaternions on $S^3$. Lorentz lift $\tilde q_i=\tfrac1{w_i}(1,\mathbf m_i)$; $Q_h(i,j)=\langle\tilde q_i,\tilde q_j\rangle_{\mathbb L}^2-1$.

Panel A — Executable sparsemax row (Axiom I)

Transition row $K_{i\cdot}=[\,z_{i\to j}-\tau_i\,]_+$

Pick a source voxel $i=(a,b,c)$. Logits $z_{i\to j}=-\sigma\beta_i Q_h(i,j)-\lambda A_{ij}$ with $A_{ij}=0$ forward (increasing coord), $A_{ij}=A_0 s$ reverse. The reverse edge is suppressed by the torsional valve; sparsemax then collapses support to the survivors.

a: b: c:

Panel B — Torsional pump current (whole lattice)

Base-referenced forward current $J_0=\sum_{(i,j)\in E^+}\big(\pi^0_iK_{ij}-\pi^0_jK_{ji}\big)$

The full $64\times64$ kernel is built as a symmetric rational base $W_{ij}=\tfrac{1}{1+\sigma_b Q_h(i,j)}$, reversible walk $P^0=W/d$, then the rational valve $\rho_{ij}=\tfrac1{1+\lambda A_{ij}}$ on reverse edges (rejected mass → diagonal). Proof T5b predicts $J_0=\sum_{E^+}\tfrac{W_{ij}}{D}(1-\rho)>0$.

References

kernel_spec.md · Hyperbolic_Softmax_Proof.md · Hyperbolic_Softmax_Design.md · TTMPT