9.1 — Data type and invariants
• Pair type. x := (m, a) with a in (-1,+1).
• Collapse parity. phi((m,a)) = m.
• Clamp discipline. Always a := clamp(a, -1+eps_a, +1-eps_a) before any atanh(a), and re-clamp after any tanh(u).
• Streaming invariance. Batch, stream, and shuffled fusions match (via {U,W} accumulators).
• Canonical zero. Display (0,+1) when m == 0 (value lane unchanged; lane is bounded).
9.2 — Minimal API (language-agnostic pseudocode)
a_clamped := clamp_a(a, eps_a)
u := atanh(a_clamped)
a := tanh(u) ; inverse map
a_mul(a1, a2) :=
tanh( atanh(clamp_a(a1,eps_a)) + atanh(clamp_a(a2,eps_a)) )
a_div(af, ag) :=
tanh( atanh(clamp_a(af,eps_a)) – atanh(clamp_a(ag,eps_a)) )
uw_accum!(U, W, a, w) :=
( U + w * atanh(clamp_a(a,eps_a)), W + w )
uw_flush(U, W, eps_w) :=
tanh( U / max(W, eps_w) )
a_env(a_op, g_t, eps_a) :=
clamp_a( g_t * a_op, eps_a )
collapse((m,a)) := m ; phi((m,a)) = m
9.3 — One-minute quickstart (spreadsheet or REPL)
- Choose
eps_a = 1e-6,eps_w = 1e-12. - Paste three cells
a1=0.2,a2=0.5,a3=-0.3. - Compute
u_i = ATANH(a_i), thenU = SUM(u_i),W = 3,a_out = TANH( U / MAX(W, 1e-12) ). - Reorder rows; confirm
a_outdoes not change. - Check multiply identity:
A_MUL = TANH(ATANH(0.8)+ATANH(0.2))equals(0.8+0.2)/(1+0.8*0.2).
Spreadsheet cells (ready to paste).U_cell = SUM(ATANH(A2:A4))W_cell = COUNTA(A2:A4)A_OUT = TANH( U_cell / MAX(W_cell, 1e-12) )A_MUL = TANH(ATANH(0.8)+ATANH(0.2))A_MUL_CF = (0.8+0.2)/(1+0.8*0.2)
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The contents in the Shunyaya Symbolic Mathematical Hardware (SSMH) materials are research/observation material. They are not engineering advice, not a safety standard or certification, and not operational guidance. Do not use for safety-critical, medical, legal, or financial decisions. Use at your own discretion; no warranties are provided; results depend on correct implementation and inputs.
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