Shunyaya Symbolic Mathematical Hardware – Fixed-Point & Invariance Guards (3.17–3.19)

3.17 — Test E10: Fixed-point parity (Q-format sanity)

  • Goal. Prove fixed-point Q implementations preserve bands and collapse within budget.
  • Profiles. a in Q1.15 or Q5.11; u in Q2.30 or Q5.27; apply the same to {U,W} as needed.
  • Vectors. Include adversarial a = ±(1 - eps_a), near-zero m, alternating signs.
  • Checks.
    |a_fp - a_dp| <= 2*ulp(a_fp) and phi_fp == phi_dp within magnitude tolerance;
    streaming parity: a_out_fp == a_out_dp within the Q-budget.
  • Verdict. PASS/WARN/FAIL per envelope; record the chosen Q formats and tolerances in the manifest.

3.18 — Test E11: Backfill/shuffle invariance (U/W)

  • Goal. Assert batch == stream == shuffled under {U,W} across random permutations.
  • Procedure. Fix a multiset of (a,w); compare outputs over N >= 50 random shuffles.
  • Acceptance. All deltas <= tol for float or within the declared Q-budget for fixed-point.
  • Verdict. PASS if every permutation is within bound.

3.19 — Test E12: Near-zero division guard (M2 + policy)

  • Goal. Show bounded alignment and declared magnitude behavior as denominators approach zero.
  • Procedure.
    Choose (m_f,a_f), (m_g,a_g) with |m_g| -> 0.
    Alignment: verify |a_div| < 1 and finite with a_div = (a_f - a_g) / (1 - a_f*a_g).
    Magnitude: follow the declared policy:
    strict: classical divide with explicit zero handling;
    soft: m_out = m_f / max(|m_g|, denom_soft_min) with sign;
    meadow: m_out = 0 if m_g = 0.
  • Verdict. PASS if alignment stays bounded and m_out matches the policy.

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