Shunyaya Symbolic Mathematical Hardware – Instant Convincer: Goals, Knobs & Identities (3.1–3.5)

3.1 — Goals and philosophy

  • Be convincing in minutes. Calculator/spreadsheet checks, nothing exotic.
  • Zero ambiguity. Fixed knobs, explicit formulas, single tolerance.
  • Same everywhere. Software and silicon share the exact semantics; collapse parity phi((m,a)) = m always holds.

3.2 — Deterministic knobs (freeze before testing)

  • Numerical. eps_a = 1e-6, eps_w = 1e-12, denom_soft_min = 1e-9.
  • Streaming weight. gamma = 1 so w_i := |m_i|^gamma unless stated.
  • Policy. division_policy in {strict, soft, meadow} (tests below are policy-agnostic unless noted).
  • Bands. Thresholds for A++/A+/A0/A-/A-- (not used in numeric checks).
  • Tolerance. tol = 1e-12 for float64 calculators/spreadsheets.

3.3 — Test E1: Clamp safety (edges never explode)

  • Inputs. a_in = +0.9999999, eps_a = 1e-6.
  • Clamp. a = clamp(a_in, -0.999999, +0.999999) = +0.999999.
  • Map. u = atanh(a) = 7.254328619247669 (finite).
  • Mirror. a_in = -0.9999999 -> a = -0.999999, u = -7.254328619247669.
  • Verdict. PASS (finite u after clamp).

3.4 — Test E2: Multiply identity (closed-form = rapidity)

  • Inputs. a1 = +0.8, a2 = +0.2.
  • Rapidity path. atanh(0.8)=1.0986122886681098, atanh(0.2)=0.2027325540540822, sum = 1.3013448427221919, tanh(sum) = 0.8620689655172413.
  • Closed form. (a1+a2)/(1+a1*a2) = 1.0/1.16 = 0.8620689655172413.
  • Verdict. PASS (diff = 0 within tol).

3.5 — Test E3: Divide identity (closed-form = rapidity)

  • Inputs. a_f = +0.8, a_g = +0.2.
  • Rapidity path. atanh(0.8) - atanh(0.2) = 0.8958797346140276, tanh(...) = 0.7142857142857144.
  • Closed form. (a_f - a_g) / (1 - a_f*a_g) = 0.6/0.84 = 0.7142857142857144.
  • Verdict. PASS (diff = 0 within tol).

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Disclaimer
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.