research: compile AMP polynomial and matrix layers

[mountain]

Closes #152.

Summary

• freeze a power-dominant benchmark f(x)=x^d+t and the normalized long-horizon observer
• derive the AMP polynomial-like escape coordinate on the exponential ray
• compile composition into an exact sparse nilpotent substitution matrix
• solve the finite conjugacy as the triangular linear system (dI-C)h=u
• add exact rational replay/residual certificates and independent research tests
• compare against both expanded symbolic iteration and a strong logarithmic recurrence

Result

The polynomial-like basis and matrix-like transport simplify the declared algorithm together:

• ordinary expanded support for d=2, N=100: 2^99+1 terms
• at observer order K=20: 55 sparse matrix entries versus 400 dense entries, 9 nonzero coordinate coefficients
• absolute error at y0=1.5: about 3.13e-14
• exact first omitted residual: 144/11 q^22

Disposition: EXPAND-NARROW.

This earns symbolic/storage compression, exact sparse compilation, replay, and compile-many reuse. It does not earn universal floating-point speedup: the strong binary64 log recurrence stops after nine effective corrections per query. The finite asymptotic chart also fails closed near y0=0, where increasing order can worsen the result.

The Böttcher, escape-rate, Koopman, and Carleman boundaries are cited explicitly; no classical novelty, global AMP solver, Core/API promotion, or complexity-class claim is made.

Validation

• python -m pytest -q tests/research/test_amp_polynomial_matrix_compiler.py tests/research/test_amp_ensemble_carrier_gate.py
  • 19 passed
• python -m pytest -q
  • 769 passed, 25 skipped, 1 existing deprecation warning
• git diff --check
• python -m compileall -q sonnet/amp-polynomial-matrix-compiler/amp_escape_compiler.py