research: compile AMP polynomial and matrix layers

sonnet/README.md

@@ -190,6 +190,14 @@ tail outside the finite carrier.  The AMP line earns `EXPAND`; every frozen
 workload eliminates a surreal runtime, so the overall gate remains `NARROW`
 until an interacting residual yields a measured computation advantage.
 
+[`amp-polynomial-matrix-compiler/`](amp-polynomial-matrix-compiler/) performs
+that first interacting-residual gate on `x -> x^d+t`.  The AMP exponential-ray
+basis turns composition into an exact sparse nilpotent matrix and the
+long-horizon coordinate into a triangular linear solve.  It eliminates
+exponential symbolic support and supplies exact residual certificates, while a
+strong logarithmic recurrence prevents any claim of universal numerical
+speedup.  The result is `EXPAND-NARROW` and remains Sonnet-local.
+
 ## Research-local calibration — the \(S^6\) complex structure claim
 
 [`s6-complex-arithmetic-tower/`](s6-complex-arithmetic-tower/) studies a

sonnet/amp-polynomial-matrix-compiler/00-problem-frontier.md

@@ -0,0 +1,112 @@
+# Problem frontier: do AMP polynomials and matrices simplify an algorithm?
+
+Status: frozen contract for issue
+[#152](https://github.com/mountain/process-geometry/issues/152).
+
+## 1. The question
+
+Issue #150 identified the finite AMP polynomial-like family
+
+\[
+\sum_{\gamma,n}^{\mathrm{finite}}
+c_{\gamma,n}x^\gamma(\log x)^n
+\]
+
+and the sparse operator matrices induced by `A`, `M`, and `P` on the degree
+lattice `(gamma,n)`.  Algebraic closure alone does not establish algorithmic
+value.  This phase asks whether the representation reduces a frozen task under
+same-information baselines.
+
+## 2. Frozen power-dominant task
+
+Take
+
+\[
+f(x)=x^d+t,
+\qquad d\ge2,
+\qquad t>0,
+\]
+
+on a declared positive chart near infinity.  Put
+
+\[
+y=\log x,
+\qquad
+q=e^{-y}.
+\]
+
+Then
+
+\[
+F(y)=dy+\log(1+tq^d),
+\qquad
+g(q)=e^{-F(y)}=\frac{q^d}{1+tq^d}.
+\]
+
+The main observer is the normalized long-horizon quantity
+
+\[
+G_N(y)=d^{-N}F^{\circ N}(y)
+\]
+
+and its limit when the asymptotic coordinate converges.  This is deliberately
+narrower than reconstructing the full iterate or orbit.
+
+## 3. Same-information baselines
+
+Three paths receive separate ledgers.
+
+1. **Expanded symbolic baseline:** form `f^[N](x)` as one ordinary expanded
+   polynomial.
+2. **Strong numerical baseline:** update `F(y)` directly in the logarithmic
+   chart, accumulate the normalized correction, and stop when floating-point
+   correction is zero.  It is forbidden to expand the polynomial.
+3. **AMP compiler:** compile a finite polynomial-like escape coordinate once
+   from a sparse substitution matrix, then evaluate it for the declared
+   observer.
+
+The expanded baseline measures symbolic support only.  It may not be used as
+the sole numerical competitor.
+
+## 4. Metrics
+
+- exact expanded support count;
+- observer order and nonzero polynomial-like terms;
+- dense versus sparse matrix entries;
+- exact compilation and residual certificate;
+- strong-baseline executed steps;
+- compile-once/evaluate-many online work;
+- numerical error across observer orders;
+- chart failure outside the asymptotic domain;
+- decoder and output scope.
+
+## 5. Acceptance and kill conditions
+
+The representation earns algorithmic credit only if:
+
+- the basis makes the declared transport exactly sparse;
+- the matrix construction changes the solve, not only its notation;
+- the result is replayable without hidden symbolic expansion;
+- a strong recurrence baseline is reported;
+- compilation, online work, and output precision are separated.
+
+Narrow or stop a claim if:
+
+- an ordinary Taylor or monomial basis is relabelled AMP without a support
+  advantage;
+- a dense generic eigensolver replaces an available triangular solve;
+- symbolic expansion is treated as the only numerical baseline;
+- the finite truncation is evaluated outside its chart without a refusal;
+- one scalar long-time observer is presented as full-orbit reconstruction;
+- a classical Böttcher/Koopman result is claimed as new.
+
+## 6. Claim ceiling
+
+This phase does not claim a new Böttcher theorem, generic Koopman solver,
+complexity-class improvement, Ising solver, or Public API.
+
+```text
+Epistemic maturity: T1 exact finite compiler + bounded numerical calibration
+Engineering status: Sonnet-local Python
+Mathematical Core: unchanged
+```

sonnet/amp-polynomial-matrix-compiler/01-sparse-compiler-theorems.md

@@ -0,0 +1,180 @@
+# Sparse AMP compiler theorems
+
+Status: exact over rational `t>0` at every fixed observer order.
+
+## 1. Polynomial-like observer basis
+
+On the `y=log x` chart, use
+
+\[
+q^k=e^{-ky}=x^{-k},
+\qquad 1\le k\le K.
+\]
+
+The compiled coordinate has the finite AMP form
+
+\[
+H_K(y)=y+\sum_{k=1}^K h_kq^k.
+\]
+
+It is a finite slice of the exponential-logarithmic algebra: the affine `y`
+term records the Power direction, and the `q` ray records the completed
+Addition residual around the power-dominant chart.
+
+## 2. Matrix-like composition
+
+Let `C_K` be composition by
+
+\[
+g(q)=\frac{q^d}{1+tq^d}
+\]
+
+on the truncated ray.  Its entries are
+
+\[
+(C_K)_{r,k}=[q^r]g(q)^k.
+\]
+
+Using the negative-binomial expansion,
+
+\[
+g(q)^k
+=q^{dk}(1+tq^d)^{-k}
+=\sum_{j\ge0}
+(-1)^j\binom{k+j-1}{j}t^j q^{d(k+j)}.
+\]
+
+Therefore
+
+\[
+(C_K)_{d(k+j),k}
+=(-1)^j\binom{k+j-1}{j}t^j,
+\]
+
+and every other entry is zero.
+
+**Proposition 2.1.**  `C_K` is nilpotent.  In particular,
+
+\[
+C_K^r=0
+\quad\text{whenever}\quad
+d^r>K.
+\]
+
+**Proof.**  One application sends a monomial of degree `k` to degrees at
+least `dk`.  After `r` applications, every surviving degree is at least
+`d^r k`.  No positive degree survives the observer cutoff when `d^r>K`.
+QED.
+
+This nilpotence is the finite-observer version of power-driven scale escape.
+
+## 3. The linear eigenproblem
+
+Write
+
+\[
+u(q)=\log(1+tq^d)
+=\sum_{j\ge1}\frac{(-1)^{j+1}}{j}t^jq^{dj}.
+\]
+
+Composition gives
+
+\[
+H_K(F(y))
+=dy+u(q)+\sum_{k=1}^K h_k g(q)^k.
+\]
+
+Thus the truncated conjugacy condition
+
+\[
+H_K\circ F=dH_K+O(q^{K+1})
+\]
+
+is exactly
+
+\[
+(dI-C_K)h=u_{\le K}.
+\]
+
+**Theorem 3.1.**  This equation has a unique rational solution for rational
+`t`.
+
+**Proof.**  `C_K` strictly raises degree, so `dI-C_K` is triangular with
+nonzero diagonal `d`.  Equivalently, nilpotence gives the finite inverse
+
+\[
+(dI-C_K)^{-1}
+=\frac1d\sum_{r\ge0}^{\mathrm{finite}}
+\left(\frac{C_K}{d}\right)^r.
+\]
+
+All entries remain rational.  QED.
+
+For `d=2`, `t=1`, and `K=10`, the compiler obtains
+
+\[
+H_{10}(y)
+=y+\frac12q^2-\frac13q^6+\frac58q^8-\frac9{10}q^{10}.
+\]
+
+The finite eigenrelation replays exactly.  Its first omitted residual is
+
+\[
+2q^{12}.
+\]
+
+## 4. Long-horizon observer
+
+The exact, untruncated coordinate is the logarithm of a Böttcher coordinate
+and satisfies
+
+\[
+H(F(y))=dH(y).
+\]
+
+Consequently,
+
+\[
+d^{-N}H(F^{\circ N}(y))=H(y).
+\]
+
+When the correction `H(z)-z` vanishes along the escaping orbit,
+
+\[
+\lim_{N\to\infty}d^{-N}F^{\circ N}(y)=H(y).
+\]
+
+The finite compiler approximates this observer directly, without constructing
+the degree-`d^N` iterate.
+
+## 5. Classical boundary
+
+This construction meets established mathematics:
+
+- Böttcher coordinates conjugate a degree-`d` polynomial near infinity to
+  `z -> z^d`;
+- the Green/escape-rate function is the logarithmic long-horizon observer;
+- the Koopman operator acts linearly on observables by composition;
+- Carleman-style methods represent nonlinear composition in an infinite
+  function basis.
+
+The AMP-specific question is narrower: does the arithmetic chart select the
+right sparse dictionary and typed completion automatically?  The present
+example gives one positive calibration, not a novelty claim about these
+classical structures.
+
+Primary references used as boundaries:
+
+1. B. O. Koopman,
+   [“Hamiltonian Systems and Transformation in Hilbert Space”](https://www.pnas.org/doi/10.1073/pnas.17.5.315),
+   1931.
+2. C. Favre and T. Gauthier,
+   [“The arithmetic of polynomial dynamical pairs”](https://arxiv.org/abs/2004.13801),
+   including formal Böttcher expansions at infinity.
+3. L. DeMarco, K. Lindsey,
+   [“Convergence properties of the Gronwall area formula for quadratic Julia sets”](https://arxiv.org/abs/1405.1933),
+   including coefficient-level numerical use of Böttcher maps.
+4. M. J. Colbrook,
+   [structure-preserving finite approximations of Koopman operators](https://arxiv.org/abs/2209.02244),
+   a distinct data-driven setting that reinforces the need to state the
+   dictionary, truncation, and convergence target.