Hyperbolic Softmax — Owl Academy · Experimental Systems

Premise

A transition kernel for the SHD-CCP $4\times4\times4$ ($64$-voxel) packet that scores in hyperbolic geometry, runs a torsional Markov pump (broken detailed balance), and stays exactly rational so it survives the SHD-CCP handshake. The deterministic, on-chain normalizer is sparsemax ($\alpha$-entmax, $\alpha=2$); the word softmax is reserved for the analytic Axiom-II limit.

Why a vanilla softmax is the wrong shape

Transcendental ⇒ breaks the handshake. $e^{x}$ is float-dependent, so nodes disagree and fail torusWinding(mantle) % 7 == core → entropic collapse.
Statistical ⇒ violates the substrate. The breathing substrate demands exact topological resolution, not a dense probabilistic smear.
Reversible ⇒ cannot pump. A symmetric similarity is detailed-balanced; the pump needs $J>0$.
Euclidean & scalar ⇒ Derrick-unstable. A flat scalar logit field collapses or diffuses; no escape valve.

Resolution: a normalizer that is hyperbolic (geometry), asymmetric (pump), rational (determinism), and $w$-stabilized (Derrick).

Pipeline

 q_i ∈ S³∩ℚ⁴ ──Φ──► Lorentz lift  q̃_i = (1/w_i)(1, m_i) ∈ ℍ³            [T1]
                              │
       neighbor q̃_j           ▼
                    Q_h(i,j) = ⟨q̃_i, q̃_j⟩_L² − 1  ∈ ℚ≥0               [T2]
                              │
  z_{i→j} = −β_i·Q_h(i,j) − λ·A_ij      (A_ij = A₀·s·χ_ij, rational)     [valve]
                              │
            sparsemax:  K_ij = [ z_{i→j} − τ_i ]_+ ,  Σ_j K_ij = 1       [T3]  (exact ℚ)
                              │           ╲  Gibbs limit  K ∝ e^{z}       [Axiom II, ℝ]
       irrational ε_ij ──► Octet(8)  (never re-enters the kernel)        [T4]
                              ▼
   torsional pump  K(t) ─► π, J(t) ─► B(t₇) > B(t₅),  Θ₁₀ ≡ Θ₀           [T5–T7]

Locked decisions

Element	Lock
Model	Lorentz/hyperboloid; $S^3$ spinor $q_i$ kept distinct from lift $\tilde q_i=\tfrac1{w_i}(1,\mathbf m_i)$ (closed rational form).
Normalizer	sparsemax ($\alpha=2$) executable; softmax/Gibbs analytic only.
Valve	Analytic $e^{-\lambda A_{ij}}$; executable rational $\rho_{ij}=\tfrac{1}{1+\lambda A_{ij}}$.
12↔64	3 arms = axes $x,y,z$; 4 phases = coordinate positions; 6 signed directions lifted by chirality into 12 channels.
Residuals	Octet(8) quarantine; never perturb the 64-node rational kernel.
Temperature	$\beta_i=\tfrac{1-w_i}{w_i}$ on a Derrick-safe annulus $w\in[w_{\min},w_{\max}]\subset(0,1)$.

Verified (exact rational demo)

The simulation instantiates the locked spec with BigInt rationals:

64 spinors exactly on $S^3$ $w\in[7/29,\,40/41]\subset(0,1)$ all 64 rows $\sum=1$ exactly reverse edge → 0 (valve) $J_0\approx0.172>0$ direct = formula (bit-exact)

Artifacts

◈

kernel_spec.md

Frozen invariants: state set, rational lift, $Q_h$, valve, sparsemax row, parity & octet rules.

⊢

Hyperbolic_Softmax_Proof.md

Theorems T1–T7 + Main Theorem: lift, quadrance, sparsemax, determinism, broken balance & current, stability, phase reset.

✦

Hyperbolic_Softmax_Design.md

The design narrative — conflict, pipeline, locked decisions, handshake/axiom conformance, open items.

▶

sim/ — interactive demo

Exact-rational 4×4×4: compute a sparsemax row and the pump current $J_0$ live in the browser.

Upstream law: TTMPT formal proof.