Research: AM completion cones and bigraded chamber transport

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Motivation

PR #147 proves exact observer-directed coefficient semantics on a frozen rank-one rational ray, but it also exposes the next mathematical boundary.

For

$$\Phi_{\nu,w}=a^\nu e^{(w-\nu)v},$$

write the exponential character as

$$\kappa=w-\nu.$$

The native AM algebra is naturally bigraded by $(\nu,\kappa)$, while the first compiler uses one abstract positive weight coordinate. That implementation is therefore evidence for a completed ray or one-parameter slice; it is not yet a construction of the full bigraded AM completion.

The next task is mathematical, not performance optimization.

Core questions

1. Which coordinate controls completion: power degree $\nu$, character $\kappa$, $M$-weight $w=\nu+\kappa$, or an observer-chosen positive functional on the bigrading?
2. Which pointed support cones make multiplication locally finite and target coefficients decidable?
3. How do the operators

$$A:(\nu,\kappa)\mapsto(\nu-1,\kappa),\qquad M\Phi_{\nu,\kappa}=(\nu+\kappa)\Phi_{\nu,\kappa}$$

act continuously across such completions?
4. Can differentiation and primitive formation live in one cone, or must they transport between adjacent chambers?
5. At what precise obstruction would Hahn, transseries, hyperseries, or surreal support become necessary?

Initial propositions to formalize

P1 — ray-scope theorem

State precisely that the #147 compiler evaluates completed expressions supported on one finitely generated rational rank-one monoid, with finite Laurent shifts, rather than the entire $(\nu,\kappa)$ lattice.

P2 — bidirectional cone no-go

A pointed cone cannot be invariant under both translations by $(1,0)$ and $(-1,0)$. Hence a single well-founded support cone cannot be literally closed under both unrestricted $A$-differentiation and inverse-$A$ primitive transport. Continuity and closure must be distinguished.

P3 — bounded-shift continuity

For a positive observer height $h$, an operator of fixed bigraded degree $d$ maps $F^N$ to $F^{N+h(d)}$ and is continuous even when it does not preserve the positive ideal.

P4 — finite-observer theorem

For a pointed finitely generated rational cone $C$ and a height $h$ strictly positive on its nonzero generators, every bounded observer slice

$${g\in C\cap G:h(g)\le N}$$

is finite. Therefore coefficient convolution and completed exp/log readouts have finite dependency slices at every declared horizon.

P5 — chamber transport

Determine whether $A$, ordinary primitives, and the two resonant extensions should be typed as maps between cone completions/chambers, rather than endomorphisms of one completed algebra.

Frozen mathematical controls

• one ray recovered as the research: completed AM power-weight compiler #147 special case;
• a two-generator cone with exact paired coefficient extraction;
• two observer heights on the same support giving transport-compatible readouts;
• $A$ as a bounded degree shift;
• ordinary primitive chamber transport;
• typed log-a resonance at $\nu=-1$;
• a cone with an infinite bounded slice rejected;
• a nonpointed cone rejected;
• an attempted bidirectional invariant cone produces the frozen no-go witness.

Claim ceiling

This issue may establish exact theorems for finitely generated rational bigraded cones and observer-directed completions. It does not authorize a general Hahn/transseries/surreal completion, multivariable AM function theory, complexity breakthrough, Core promotion, or Public API change.

Success criterion

Publish exactly one disposition:

• CONE: one admissible cone completion supports the required AM calculus;
• CHAMBERS: differentiation/primitive calculus requires typed transport among completions;
• RAY-ONLY: the evidence does not yet justify a reusable bigraded completion;
• HIGHER-SUPPORT: a frozen obstruction genuinely forces a stronger support order.

The default expectation is CHAMBERS, but the research must be allowed to refute it.