research: AM completion cones and chamber transport

sonnet/README.md

@@ -84,6 +84,7 @@ deferred until its next oracle/evidence gate is affordable.
 | [`am-conformal-chart-normal-forms/`](am-conformal-chart-normal-forms/) | T0/T1; Phase 1 mechanism calibrated | exact Riccati lift, Möbius covariance, scalar-gauge invariance, cubic no-go, and eight-axis Pareto accounting; no discovery or economy theorem | [Phase-1 results](am-conformal-chart-normal-forms/02-phase1-riccati-results.md); next freeze a blind low-height grammar and run bounded recovery |
 | [`effective-scale-carrier-ladder/`](effective-scale-carrier-ladder/) | T1 / NARROW | finite syntax decision eliminates a surreal runtime; symbolic height is a real C2 obstruction, but semantic evaluation and C3/C4 separation remain open | [results](effective-scale-carrier-ladder/01-results.md), [compiler](../workstreams/carrier_ladder/compiler/), and [commit--reveal audit](../workstreams/carrier_ladder/redteam/); next implement an effective normalized hyperiteration fragment, not a general surreal runtime |
 | [`am-power-weight-compiler/`](am-power-weight-compiler/) | T1 / EXPAND | 12/12 frozen public cases and nonce-protected held-out pass with exact replay; sparse distant readout shows a 2-weight versus 33-weight window advantage, but no general performance or surreal claim | [results](am-power-weight-compiler/01-results.md), [compiler](../workstreams/am_weight_compiler/), and [commit--reveal audit](../workstreams/am_weight_compiler/HELD_OUT_REVEAL.json); next freeze a separate AM goal front-end and transport corpus |
+| [`am-completion-cones/`](am-completion-cones/) | T1 / CHAMBERS | separates power degree, exponential character, and M-weight; proves finite observer slices, cofinal observer heights, bounded-shift continuity, and the bidirectional pointed-cone no-go; the exact two-ray calibration returns 7/4 and replays A/M and resonance transport | [initial theorems](am-completion-cones/01-initial-cone-theorems.md), [calibration](am-completion-cones/02-two-generator-calibration.md), and [disposition](am-completion-cones/03-disposition.md); next study gluing across cones sharing a face |
 | [`s6-complex-arithmetic-tower/`](s6-complex-arithmetic-tower/) | T0 initialization | auditable two-question research contract only; neither the manuscript nor an arithmetic interface is verified | [problem frontier](s6-complex-arithmetic-tower/00-problem-frontier.md); next archive/checksum the source and reproduce its matrix/topology certificate |
 | [`boltzmann-bbgky-h-theorem/`](boltzmann-bbgky-h-theorem/) | Phase 1I charted fibre-response calibration passed | 47 exact certificates through Phases 1C and 1E–1I; target Lyapunov decrease need not survive continued fibre response; contrast/odds expose a chart-relative dynamics/composition tradeoff; no continuum response theorem, fibre objectification, or generic entropy claim | [Phase 1I result](boltzmann-bbgky-h-theorem/18-phase1i-charted-fibre-calculus-results.md); next run the separate finite collision-covector and continuum fibre-response gates |
 

sonnet/am-completion-cones/00-problem-frontier.md

@@ -0,0 +1,194 @@
+# AM completion cones and bigraded chamber transport
+
+Status: T0 mathematical contract for issue
+[#148](https://github.com/mountain/process-geometry/issues/148).
+
+## 1. Why this question follows from the first compiler
+
+The completed AM power--weight compiler in PR #147 established an exact
+rank-one observer calculus. It did **not** yet construct the completion of the
+entire native AM algebra.
+
+Write
+
+\[
+X_{\nu,\kappa}
+=a^\nu e^{\kappa v},
+\qquad
+\kappa=w-\nu.
+\]
+
+Then
+
+\[
+X_{\nu,\kappa}X_{\mu,\lambda}
+=X_{\nu+\mu,\kappa+\lambda},
+\]
+
+\[
+A X_{\nu,\kappa}
+=\nu X_{\nu-1,\kappa},
+\qquad
+M X_{\nu,\kappa}
+=(\nu+\kappa)X_{\nu,\kappa}.
+\]
+
+The intrinsic degree group is therefore
+
+\[
+G=\mathbb Z\oplus\Lambda,
+\]
+
+with coordinates `(power degree, exponential character)`.  The
+`M`-weight is the derived linear functional
+
+\[
+w(\nu,\kappa)=\nu+\kappa.
+\]
+
+The first compiler's single weight coordinate is an abstract rank-one
+completion coordinate.  It can be embedded along a declared homogeneous ray
+of `G`, but the compiler does not yet record that embedding.  Its completion
+evidence must therefore be read as **ray-local**.
+
+This distinction is mathematical, not an implementation detail.
+
+## 2. The completion problem
+
+Let `K` be a commutative exact `Q`-algebra and let
+
+\[
+C\subset G_{\mathbb R}
+\]
+
+be a rational cone.  Put
+
+\[
+S=C\cap G.
+\]
+
+The candidate completed monoid algebra is
+
+\[
+K[[S]]
+=\left\{
+\sum_{g\in S}c_gX^g
+\right\}.
+\]
+
+This notation is legitimate only if every requested product coefficient has
+finitely many contributing decompositions.  A sufficient structure is:
+
+1. `C` is finitely generated and rational;
+2. `C` is pointed, so `C` contains no nonzero line;
+3. an observer height
+   \(h:G_{\mathbb R}\to\mathbb R\) is strictly positive on
+   \(C\setminus\{0\}\).
+
+The observer then sees only
+
+\[
+S_{\le N}^{h}
+=\{g\in S:h(g)\le N\}.
+\]
+
+The first theorem target is that this set is finite.
+
+## 3. Algebra versus chambers
+
+The cone algebra `K[[S]]` is only the zero chamber.  A finite Laurent shift
+produces a translated support sector
+
+\[
+\mathcal H_{g_0,C}
+=K[[g_0+S]].
+\]
+
+These sectors are modules rather than copies of one untyped algebra:
+
+\[
+\mathcal H_{g_0,C}\,
+\mathcal H_{g_1,C}
+\longrightarrow
+\mathcal H_{g_0+g_1,C}.
+\]
+
+The AM generators have exact degree behaviour
+
+\[
+A:
+\mathcal H_{g_0,C}
+\longrightarrow
+\mathcal H_{g_0-e_\nu,C},
+\]
+
+\[
+M:
+\mathcal H_{g_0,C}
+\longrightarrow
+\mathcal H_{g_0,C},
+\]
+
+where \(e_\nu=(1,0)\).
+
+Away from resonance, an `A`-primitive has the opposite transport
+
+\[
+P_A:
+\mathcal H_{g_0,C}
+\longrightarrow
+\mathcal H_{g_0+e_\nu,C}.
+\]
+
+At \(\nu=-1\), the coefficient \((\nu+1)^{-1}\) does not exist and the
+typed `log-a` extension is forced.  This is a codimension-one resonance wall
+inside the chamber system, not a reason to insert `log(a)` into the base
+algebra.
+
+## 4. Frozen questions
+
+The research must answer:
+
+1. Is a pointed rational cone enough for a useful completed AM calculus?
+2. Are `A` and its primitive correctly understood as chamber transports?
+3. Do different positive observer heights define the same completion but
+   different finite costs?
+4. Can exact `exp` and `log1p` be defined on the positive augmentation ideal
+   without enlarging the coefficient algebra beyond the frozen domain?
+5. Which support obstruction, if any, genuinely forces Hahn, transseries, or
+   surreal structure?
+
+## 5. Controls
+
+Positive controls:
+
+- recover the PR #147 rank-one evaluator as a ray `S=N rho`;
+- use a two-generator pointed cone and prove finite coefficient convolution;
+- compare two positive heights on the same cone;
+- replay `A`, `M`, ordinary primitive transport, and `log-a` resonance;
+- compute one mixed two-ray completed coefficient exactly.
+
+Negative controls:
+
+- reject a nonpointed cone;
+- reject a height that vanishes on a nonzero support ray;
+- exhibit the failure of one pointed cone to be translation-invariant in both
+  power directions;
+- reject a general degree-zero exponential whose constant coefficient would
+  require an undeclared exponential closure;
+- keep symbolic-height iteration outside the fragment.
+
+## 6. Claim ceiling
+
+This work may justify finitely generated rational bigraded completion and
+typed chamber transport.  It may not claim:
+
+- a general Hahn field or transseries algebra;
+- a surreal runtime;
+- symbolic-height hyperiteration;
+- a full multivariable AM function theory;
+- a complexity or performance theorem;
+- Core, Theory Map, or Public API promotion.
+
+The required disposition is exactly one of `CONE`, `CHAMBERS`, `RAY-ONLY`, or
+`HIGHER-SUPPORT`.