research: AMP closure and ensemble carrier gate

sonnet/README.md

@@ -180,6 +180,16 @@ a correlation-fibre action, regular cocycle, future adequacy, or effective
 compression theorem.  The next work separates a finite collision-covector gate
 from a continuum Deng-calibrated fibre-response gate; neither has yet passed.
 
+## Research-local calibration — AMP closure and ensemble carrier gate
+
+[`amp-ensemble-carrier-gate/`](amp-ensemble-carrier-gate/) separates the
+finite M/P affine subsystem from the infinite A/M/P Lie closure.  It proves an
+exact logarithmic normal form, compiles homogeneous repeated ensembles without
+Cartesian state enumeration, and shows that Addition opens a completed scale
+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.
+
 ## Research-local calibration — the \(S^6\) complex structure claim
 
 [`s6-complex-arithmetic-tower/`](s6-complex-arithmetic-tower/) studies a

sonnet/amp-ensemble-carrier-gate/00-problem-frontier.md

@@ -0,0 +1,110 @@
+# Problem frontier: AMP closure, ensembles, and carrier necessity
+
+Status: frozen research contract for issue
+[#150](https://github.com/mountain/process-geometry/issues/150).
+
+## 1. Question
+
+On the positive real line, or on a complex history lift with a chosen branch
+of `Log`, consider
+
+\[
+A_t(x)=x+t,
+\qquad
+M_s(x)=e^s x,
+\qquad
+P_r(x)=\exp(e^r\Log x).
+\]
+
+The motivating hypothesis is that Addition describes local accumulation,
+Multiplication describes rescaling or independent assembly, and Power
+objectifies repeated same-kind assembly.  The hypothesis earns mathematical
+credit only if it changes closure, representation, or a frozen computation.
+
+The first gate asks:
+
+> What is the smallest task-sufficient carrier for finite and iterated A/M/P
+> histories, and can a Power-aware presentation compile an ensemble task
+> without materializing its full Cartesian state space?
+
+`AMP` is research-local terminology here.  It is not asserted to be the name
+of an established mathematical field or one three-dimensional Lie group.
+
+## 2. Frozen task family
+
+The positive control uses a finite weighted state space `Omega` with partition
+value
+
+\[
+Z_0=\sum_{\omega\in\Omega}w(\omega)>0.
+\]
+
+A homogeneous assembly stage is
+
+\[
+Z_{k+1}=e^{b_k}Z_k^{n_k},
+\qquad n_k\in\mathbb N_{>0}.
+\]
+
+Its declared observer asks only for total partition value, logarithmic
+partition value, and the number of base replicas.  It does not ask for named
+microstates, correlations, marginals, or an interacting Hamiltonian.
+
+The negative control inserts Addition in the state chart.  In the logarithmic
+observer `y=log x`, this produces
+
+\[
+y\longmapsto \log(e^{\alpha y+\beta}+t),
+\]
+
+which must either remain in the finite M/P carrier or exhibit an exact carrier
+upgrade witness.
+
+## 3. Evidence firewall
+
+Separate all of the following:
+
+- a finite M/P word from the full A/M/P closure;
+- integer replica count from arbitrary real or complex powering;
+- fixed iteration height from symbolic or ordinal height;
+- exact total-partition observation from reconstruction of the ensemble;
+- a finite observer truncation from the full completed series;
+- membership in a large ambient field from an effective implementation;
+- compilation cost, certificate storage, replay cost, and output size.
+
+Matrices and polynomials may verify local identities.  They receive no credit
+as the ontology of the process rank.
+
+## 4. Acceptance and kill conditions
+
+The phase passes only if it supplies:
+
+1. exact adjacent conjugation and Lie-bracket laws;
+2. a proof that the three infinitesimal generators do or do not close;
+3. an explicit infinite closure witness if they do not;
+4. an exact M/P normal form and replayable ensemble certificate;
+5. a negative control that leaves the finite normal form;
+6. a minimum-carrier disposition including a surreal necessity verdict.
+
+Narrow or stop if the apparent advantage is only:
+
+- relabelling repeated multiplication as Power without a reusable operation;
+- hiding full state enumeration in an oracle;
+- using noninteger powers as literal replica counts;
+- reporting `No` containment as an algorithm;
+- choosing Conway simplicity solely to force a surreal answer;
+- discarding interaction or correlation data that the observer actually asks
+  to recover.
+
+## 5. Claim ceiling
+
+This Sonnet does not claim a complexity-class separation, a solution of the
+three-dimensional Ising model, an exact renormalization theorem, a canonical
+thermodynamic surreal limit, or a general surreal runtime.
+
+```text
+Epistemic maturity: T1 exact finite results + T0 continuation
+Engineering status: Sonnet-local Python certificate
+Mathematical Core: unchanged
+Experimental/Public API: none
+```

sonnet/amp-ensemble-carrier-gate/01-closure-theorems.md

@@ -0,0 +1,226 @@
+# AMP closure theorems
+
+Status: exact on the declared positive-real or chosen-`Log` chart.
+
+## 1. The finite M/P subsystem
+
+Write
+
+\[
+\Phi_{a,b}(x)=e^b x^a,
+\qquad a>0.
+\]
+
+In the coordinate `y=log x`, this is the affine map
+
+\[
+y\longmapsto ay+b.
+\]
+
+**Proposition 1.1.**  The M/P histories close in the two-parameter family
+`Phi`:
+
+\[
+\Phi_{a_2,b_2}\circ\Phi_{a_1,b_1}
+=
+\Phi_{a_2a_1,\,a_2b_1+b_2}.
+\]
+
+The Multiplication flow is `Phi_(1,s)` and the Power flow is
+`Phi_(e^r,0)`.  Hence
+
+\[
+P_rM_sP_r^{-1}=M_{e^r s}.
+\]
+
+This is the orientation-preserving affine group of the logarithmic line.  A
+finite chronological word compiles into the two fields `(a,b)`, and replay on
+the declared observer costs one multiplication and one addition.
+
+For one repeated map `Phi_(a,b)`, the `N`-fold iterate is
+
+\[
+\Phi_{a,b}^{\circ N}
+=
+\begin{cases}
+\Phi_{a^N,\,b(a^N-1)/(a-1)},&a\ne1,\\
+\Phi_{1,\,Nb},&a=1.
+\end{cases}
+\]
+
+Symbolic natural height therefore does not by itself force hyperseries or
+surreal numbers for this subsystem.
+
+## 2. The full infinitesimal closure
+
+The generators are
+
+\[
+A=\partial_x,
+\qquad
+M=x\partial_x,
+\qquad
+P=x\log x\,\partial_x.
+\]
+
+With
+
+\[
+[f\partial_x,g\partial_x]=(fg'-gf')\partial_x,
+\]
+
+the first relations are
+
+\[
+[A,M]=A,
+\qquad
+[M,P]=M,
+\qquad
+[A,P]=(1+\log x)\partial_x.
+\]
+
+The last field is not a constant linear combination of `A`, `M`, and `P`.
+Thus the three-generator span is not a Lie algebra.
+
+For integers `m,p` and nonnegative integers `n,q`, define
+
+\[
+V_{m,n}=x^m(\log x)^n\partial_x.
+\]
+
+**Theorem 2.1.**  Their bracket is
+
+\[
+[V_{m,n},V_{p,q}]
+=x^{m+p-1}
+\left((p-m)(\log x)^{n+q}
+ +(q-n)(\log x)^{n+q-1}\right)\partial_x,
+\]
+
+where a zero coefficient removes the formally negative logarithmic degree.
+
+This follows by differentiating the two coefficient functions and collecting
+the two powers of `log x`.
+
+**Corollary 2.2.**  The Lie algebra generated by `A`, `M`, and `P` is
+infinite-dimensional.
+
+**Proof.**  Since
+
+\[
+V_{0,1}=[A,P]-A,
+\]
+
+we obtain
+
+\[
+[A,V_{0,1}]=V_{-1,0}.
+\]
+
+Repeated bracketing gives
+
+\[
+\operatorname{ad}_A^{k-1}(V_{-1,0})
+=(-1)^{k-1}(k-1)!V_{-k,0},
+\qquad k\ge1.
+\]
+
+The Laurent monomials `x^(-k)` are linearly independent.  Therefore the
+closure contains an infinite independent family.  QED.
+
+This is the first strict mathematical reason not to model AMP as three
+coordinates on an ordinary finite-dimensional manifold.
+
+## 3. Addition viewed by the M/P observer
+
+Set `q=e^(-y)=1/x`.  A state translation becomes
+
+\[
+\log(e^y+t)
+=y+\log(1+tq)
+=y+\sum_{k\ge1}\frac{(-1)^{k+1}}{k}t^kq^k.
+\]
+
+Formally this is an element of the completed positive ray `K[[q]]`; analytically
+the displayed Taylor equality holds for `|tq|<1`.  Its second derivative in
+`y` is generically nonzero, so no affine M/P pair `(a,b)` represents it.
+
+A finite observer through degree `N` sees only `N` coefficients.  Power sends
+`q` to `q^a`; rational `a` may enlarge the exponent lattice, while finitely
+many declared positive exponents still admit a pointed, locally finite
+support cone.  Arbitrary branches, infinite scale accumulation, or unbounded
+iteration require a separate carrier gate.
+
+## 4. The `3n` process frame is anchored, not a `3n`-manifold
+
+For physical coordinates `x_1,...,x_n`, introduce formal local generator
+labels
+
+\[
+e_{A_i},\qquad e_{M_i},\qquad e_{P_i}.
+\]
+
+They define a rank-`3n` generating bundle with anchor
+
+\[
+\rho(e_{A_i})=\partial_{x_i},
+\qquad
+\rho(e_{M_i})=x_i\partial_{x_i},
+\qquad
+\rho(e_{P_i})=x_i\log(x_i)\partial_{x_i}.
+\]
+
+**Proposition 4.1.**  At every point of the positive chart, the anchor has
+rank `n`, not `3n`.
+
+**Proof.**  Its image contains every `partial_(x_i)` through `e_(A_i)`, so the
+rank is at least `n`.  All three generators for index `i` are scalar multiples
+of the same tangent vector, with exact kernel relations
+
+\[
+\rho(e_{M_i}-x_i e_{A_i})=0,
+\qquad
+\rho(e_{P_i}-\log(x_i)e_{M_i})=0.
+\]
+
+Thus the image has rank at most `n` and the local kernel has dimension `2n`.
+QED.
+
+The `3n` description is nevertheless useful if it retains the generator
+grade, legal compositions, and history.  For an observable `f`, the process
+signature
+
+\[
+\bigl(A_i f,M_i f,P_i f\bigr)_{i=1}^n
+\]
+
+is a structured family of probes, not `3n` independent tangent coordinates.
+In `y_i=log x_i`, for example,
+
+\[
+A_i=e^{-y_i}\partial_{y_i},
+\qquad
+M_i=\partial_{y_i},
+\qquad
+P_i=y_i\partial_{y_i}.
+\]
+
+A good chart can therefore expose drift, scale response, and scale-of-scale
+response directly.  The new information comes from their transformation and
+composition laws, not from pretending that the anchored values are
+independent.  Because the brackets also escape the finite `3n` span, this
+finite generating bundle is not yet a closed Lie algebroid; its closure needs
+the completed, filtered fibre described above.
+
+## 5. Structural interpretation
+
+The first AMP picture is not a bigger matrix algebra:
+
+- M/P is one finite affine chart after logarithmic observation;
+- A/P interaction opens a completed scale fibre over that chart;
+- finite observers take finite quotients of the completion;
+- chart transport, support admissibility, and branch data are part of the
+  object.
+
+The matrices used to replay affine composition and the polynomials used to
+replay a finite truncation are local calculation engines for this structure.