Shunyaya Symbolic Mathematical Hardware – Numerics, Interop & Stamping (2.8–2.10)

2.8 — Quantization and error envelopes (first-order)

Rapidity bound. u in [-atanh(1-eps_a), +atanh(1-eps_a)].
Envelope. With a := tanh(u), small-perturbation errors obey |delta_a| <= sech^2(u) * |delta_u|, and since sech^2(u) = 1 - a^2, this is |delta_a| <= (1 - a^2)*|delta_u|.
Behavior. Near u ≈ 0 (a ≈ 0), |delta_a| ≈ |delta_u|; toward the clamped edges (|a| -> 1), the envelope shrinks rapidly.
Budgeting. Choose LUT/word-length so |delta_a| stays comfortably below band steps (e.g., < 1e-3 if bands step by 0.3).
Example. If delta_u = 2^-20 ≈ 9.54e-7, then near u = 0 the bound gives |delta_a| ≈ 9.54e-7, well under a 1e-6 parity target; away from zero, 1 - a^2 reduces the error further.

2.9 — Interop with existing pipelines (zero-friction)

Collapse at boundaries. Any legacy block can read m directly: phi((m,a)) = m.
Side-car mode. Carry a alongside m through memory/buses even when not consumed.
Upgrade-in-place. Enable ports incrementally (e.g., adopt s_div alignment) while leaving all magnitude code unchanged.
Lockstep rule. M-BUS and A/U-BUS advance together for every op; re-clamp a on publish.

2.10 — Observability, audit, and stamping

Always-on telemetry. Log (m,a) for critical KPIs; persist lane values with the same cadence as m.
ASCII stamps. Append minimal stamps (knobs, bands, checksum) to records/files for integrity and ordering.
Privacy. The lane a is dimensionless and bounded; avoid exposing raw sensor internals unless required.
Auditability. Fixed knobs plus stamps enable reproducible PASS/WARN/FAIL gates and clean post-mortems.

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