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libm-scalar-math retrofit sweep — rc40 (math.sqrt) + rc41 residue (math.{sin,cos,atan2} / math.pi)

2026-06-05 (srmech v0.7.0rc40). The §22 discipline closeout: when srmech itself ships the scalar primitive, route the scalar fallback through srmech's own A–N cascade, not libm. Now that srmech.amsc.rational.{sqrt,hypot} (rc35) and rational.{sin,cos,tan,atan,atan2} (rc33) + the pi_cascade exist, the remaining math.* scalar calls in shipped srmech are routable. This is the sibling of the rc32 abs()-sweep and the rc33 numpy-math-sweep.

Full AST audit (26 real math.* routable references across 8 files)

A precise AST walk (real ast.Call / ast.Attribute nodes, ignoring docstring/comment text) found 26 routable references. Note: rational.py, pi_cascade.py and trigonometry.py have zero real calls — the float-trig cascades are genuinely libm-free (the grep hits there are all docstrings).

family count sites
math.sqrt 12 laplacian.py ×5 (Jacobi), bell.py ×2, octonion.py ×1, sm.py ×3, hypercomplex_dft.py ×1
math.sin / math.cos / math.atan2 12 kepler.py ×7, compose.py ×3, hypercomplex_dft.py ×2
math.pi 2 hypercomplex_dft.py ×1, form_function_rotation.py ×1

rc40 — math.sqrt routed (12 sites → rational.sqrt)

All 12 sites have provably non-negative arguments (sm.py guards >0; octonion sum_sq ≥ 0; the Jacobi rotations are √(1+x²) ≥ 1 and 1/√d under if d > 0), so rational.sqrt (which raises on negatives, matching nothing libm-silent here) is a safe machine-ε drop-in.

  • amsc/laplacian.py — the 5 cyclic-Jacobi sites (the off-diagonal norm + the rotation t/c angles). Class L now visibly composes Class N: the eigensolver's leaf root is the Class-N rational sqrt. Verified Jacobi vs numpy.linalg.eigvalsh = 8.9e-16.
  • qm/bell.py — the Tsirelson bound 2√2 and 1/√2 constants (bit-exact 0.0).
  • qm/octonion.py — octonion_norm (bit-exact 0.0).
  • qm/sm.py — Higgs vev √(μ²/2λ), Z-mass √(g²+g'²), Yukawa 1/√2 (bit-exact 0.0). (sm.py already imported rational as _srn.)
  • amsc/cascade/hypercomplex_dft.py — the 1/√3 normalisation constant (bit-exact 0.0).

import math was swapped for the rational import in the four modules that used math only for sqrt (laplacian/bell/octonion/sm); hypercomplex_dft keeps it (its cos/sin/pi are rc41 residue).

Ratchet: test_no_math_sqrt_hypot_anywhere_in_srmech (AST) — math.sqrt / math.hypot calls only go DOWN to zero, never up. cmath.sqrt (genuine complex root, e.g. the rc39 eigvals 2×2 shift discriminant — no rational peer) and np.sqrt on a bulk array (no scalar-cascade peer for the vectorised op) are NOT flagged.

rc41 — math.{sin,cos,atan2} + math.pi routed (14 sites)

Done. A different primitive family from rc40's sqrt, with its own machine-ε anchor considerations (rc33 had to bump the trig float anchor 10**12 → 10**15), and kepler.py's Kepler-equation solver is an iterative Newton loop where trig accuracy gates convergence — so this was verified machine-ε before routing: sin 7.8e-16, cos 6.7e-16, atan2 4.4e-16 (all quadrants, multiple periods); the cascade-π float (pi_cascade_digits(30) → float) is bit-exact 0.0 vs math.pi.

  • amsc/kepler.py ×7 — cos/sin/atan2 in pin_slot + the Kepler Newton solver + the equation-of-centre series → rational.{cos,sin,atan2}. pin_slot bit-exact 0.0 vs libm; Kepler residual |E − e·sinE − M| = 4.4e-16. import math dropped (kepler used math only for trig).
  • amsc/cascade/compose.py ×3 — sin in the Kuramoto-coupling DSL worked examples → rational.sin. import math KEPT (math.fsum is an exact numerical sum with no transcendental peer).
  • amsc/cascade/hypercomplex_dft.py ×2 + math.pi ×1 — the exp(μθ) twiddle cos/sin + the 2π factor → rational.{cos,sin} + a module-level cascade-π float. import math dropped.
  • signal_processing/form_function_rotation.py ×1 — math.pi in the fundamental-mode eigenvalue → cascade-π (its trig was already _srn.cos/sin). import math dropped.

Ratchet: test_no_math_trig_pi_anywhere_in_srmech (AST) — no math.{sin,cos,tan,asin,acos,atan,atan2,exp,pi,tau} reference (call OR bare constant) anywhere in shipped srmech. math.{gcd,isqrt,fsum} are exact integer / numerical helpers and are NOT flagged.

The libm-scalar-math audit is now fully swept (12 sqrt @ rc40 + 14 trig/π @ rc41 = all 26 routable references). The only math.* remaining in shipped srmech are gcd ×2, isqrt ×1, fsum ×1 — exact integer / numerical, no libm transcendental.

Out of scope (legitimately stays)

  • cmath.sqrt — complex root; no rational complex-sqrt primitive. Scientific tier (the rc39 eigvals shift discriminant).
  • np.sqrt(array) / np.cos(array) etc. on bulk arrays — vectorised math with no scalar-cascade peer; the container/bulk carve (elementwise_transcendental is the explicit scientific-tier vectorised op). Replacing with a Python loop of rational.* would be a perf catastrophe and is NOT what the discipline asks.
  • **0.5 scalar power-as-root — not surveyed here; a future micro-sweep if any appear (none flagged by the routable-math.* audit).