nx_isqrt.nx

nx_isqrt.nx -- integer square root, shared helper. Per cardinal feedback-15 DRY-through-shared-libraries: the same digit-by-digit binary-sqrt routine appeared inlined in nx_henyey_greenstein.nx and nx_worley_noise.nx within hours of each other. Third caller would have triggered "rule of three"; factoring at two callers + planned third (nx_distance_field_2d, nx_curl_noise magnitude, nx_atmosphere_scatter optical-depth) is the right time. Algorithm: digit-by-digit binary square root (Wikipedia "Methods of computing square roots", section "Binary numeral system (base 2)"). Returns floor(sqrt(n)) for n >= 0; refuses (returns 0) for n < 0. O(log n) iterations. No division, no multiplication except shifts -- safe on every Nishi target without depending on the host's idiv. Q10 wrapper: nx_isqrt_q10(x_q10) computes sqrt(x) where x is Q10- scaled, returning the result also in Q10. Implementation: sqrt(x) = sqrt(x_q10 / Q) = sqrt(x_q10) / sqrt(Q) = sqrt(x_q10) / sqrt(Q). To recover Q10 scaling: sqrt(x_q10) * Q / sqrt(Q) = sqrt(x_q10) * sqrt(Q) = sqrt(x_q10 * Q) -- this is the standard Q-to-Q sqrt trick. For Q = 1024, sqrt(Q) = 32, so we can alternatively compute nx_isqrt(x_q10) * 32 -- but the multiply- then-sqrt form preserves more precision for small x_q10 values. Loss audit: floor() in the integer-sqrt step truncates the fractional bit; max relative error 1 / floor(sqrt(n)) at the smallest representable values, asymptotically 0% as n grows. genealogy_id: wikipedia_binary_isqrt + abacus_long_division_sqrt lineage_id: nx_isqrt_digit_by_digit_binary nx_safety_envelope: intended_use: "Integer square root (digit-by-digit binary) + Q10 wrapper -- shared math primitive for substrate kernels (noise, distance fields, attenuation, atmospheric scattering)" sil_target: SIL2 (math primitive; correctness affects every consumer)