Shunyaya Symbolic Mathematical Hardware – Two-Lane Fundamentals (1.1–1.3)

1.1 — The pair on purpose

  • Type. x := (m,a) where m is the classical magnitude and a in (-1,+1) is a bounded confidence lane that always travels with m.
  • Interpretation. Act on m; use a for visibility, bands, QA, advisory scheduling, and audits.
  • Collapse parity (non-negotiable). phi((m,a)) = m. Classical outputs remain identical.

1.2 — Safety rails (always on)

  • Clamp before combine. a := clamp(a, -1+eps_a, +1-eps_a) with default eps_a = 1e-6.
  • No explosions at the edges. Clamping guarantees atanh(a) is finite and every path stays bounded.
  • Re-clamp on output. After any combine, reapply the clamp so a remains inside (-1,+1).

1.3 — Combine once, everywhere (tiny kernel)

  • Rapidity map. u := atanh(a); combine confidence with simple ops in u (add, subtract, scale).
  • Inverse map. a := tanh(u); then re-clamp.
  • Closed forms for products/ratios (M2 policy; exact).
    • Multiply lane. a_mul = (a1 + a2) / (1 + a1*a2)
    • Divide lane. a_div = (a_f - a_g) / (1 - a_f*a_g)
      These match rapidity add/sub exactly, since atanh(a_mul) = atanh(a1) + atanh(a2) and atanh(a_div) = atanh(a_f) - atanh(a_g).
  • Scale (magnitude multiply by scalar k). If k > 0, lane unchanged; if k < 0, lane flips sign a' = sign(k)*a. If k = 0, m' = 0 and a' = a is allowed (policy may display (0,+1)); clamp as usual.

Minimal kernel (ready to paste; ASCII inline)

  • EPS_A = 1e-6
  • def clamp_a(a): return max(-1.0+EPS_A, min(+1.0-EPS_A, a))
  • def a_mul(a1,a2): a1,a2 = clamp_a(a1),clamp_a(a2); num = a1 + a2; den = 1.0 + a1*a2; return clamp_a(num/den)
  • def a_div(af,ag): af,ag = clamp_a(af),clamp_a(ag); num = af - ag; den = 1.0 - af*ag; return clamp_a(num/den)

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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.