From abc886c4534f22c84f030e6dcba8e116910342c7 Mon Sep 17 00:00:00 2001 From: Mingli Yuan Date: Thu, 27 Aug 2026 16:28:01 +0800 Subject: [PATCH 1/2] research: derive AM completion chamber theorems --- sonnet/README.md | 1 + .../00-problem-frontier.md | 194 ++++++++++++++ .../01-initial-cone-theorems.md | 236 ++++++++++++++++++ .../02-two-generator-calibration.md | 223 +++++++++++++++++ sonnet/am-completion-cones/03-disposition.md | 111 ++++++++ 5 files changed, 765 insertions(+) create mode 100644 sonnet/am-completion-cones/00-problem-frontier.md create mode 100644 sonnet/am-completion-cones/01-initial-cone-theorems.md create mode 100644 sonnet/am-completion-cones/02-two-generator-calibration.md create mode 100644 sonnet/am-completion-cones/03-disposition.md diff --git a/sonnet/README.md b/sonnet/README.md index a72307e6..8070b781 100644 --- a/sonnet/README.md +++ b/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 | diff --git a/sonnet/am-completion-cones/00-problem-frontier.md b/sonnet/am-completion-cones/00-problem-frontier.md new file mode 100644 index 00000000..b8b10b6d --- /dev/null +++ b/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 the exact coefficient 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`. diff --git a/sonnet/am-completion-cones/01-initial-cone-theorems.md b/sonnet/am-completion-cones/01-initial-cone-theorems.md new file mode 100644 index 00000000..4f7400e1 --- /dev/null +++ b/sonnet/am-completion-cones/01-initial-cone-theorems.md @@ -0,0 +1,236 @@ +# Initial theorems for AM completion cones + +Status: T1 exact deductions inside the frozen rational-polyhedral scope. + +## 1. Setup + +Let + +\[ +G=\mathbb Z\oplus\Lambda +\] + +be a rank-two lattice after clearing the denominator of the rational +rank-one character lattice `Lambda`. Let + +\[ +X^g=X_{\nu,\kappa}=a^\nu e^{\kappa v}, +\qquad +g=(\nu,\kappa). +\] + +Fix a finitely generated rational cone + +\[ +C=\mathbb R_{\ge0}s_1+\cdots+\mathbb R_{\ge0}s_r +\] + +and its affine monoid `S=C cap G`. A height `h` is admissible when + +\[ +h(s_i)>0 +\] + +for every nonzero generator. + +## 2. Finite-observer theorem + +**Theorem 2.1.** If `C` is pointed and rational polyhedral and `h` is +admissible, then for every real `N` the observer set + +\[ +S^h_{\le N}=\{g\in S:h(g)\le N\} +\] + +is finite. + +**Proof.** Because `h` is strictly positive on the compact section of `C` +obtained by intersecting it with a unit sphere, there is a constant `c>0` +such that + +\[ +h(x)\ge c\lVert x\rVert +\qquad (x\in C). +\] + +Thus `h(x)<=N` implies `||x||<=N/c`. A lattice has only finitely many points +in a bounded set. Hence the observer set is finite. \(\square\) + +**Corollary 2.2.** For any `g in S`, the set of decompositions + +\[ +\{(p,q)\in S^2:p+q=g\} +\] + +is finite. + +Indeed, admissibility gives + +\[ +0\le h(p),h(q)\le h(g), +\] + +so both entries lie in a finite observer set. + +Therefore the coefficient + +\[ +[X^g](fg) +=\sum_{p+q=g} [X^p]f\,[X^q]g +\] + +is a finite exact sum even when `f` and `g` have infinite support in `S`. + +## 3. Completed exp and log + +Let + +\[ +\mathfrak m +=\left\{f\in K[[S]]:[X^0]f=0\right\}. +\] + +For `u in m`, define + +\[ +\exp(u)=\sum_{n\ge0}\frac{u^n}{n!}, +\qquad +\log(1+u)=\sum_{n\ge1}\frac{(-1)^{n+1}}n u^n. +\] + +**Theorem 3.1.** Every target coefficient of these two expressions is a +finite exact sum. + +**Proof.** Let + +\[ +\epsilon +=\min\{h(s):s\in S\setminus\{0\}\}>0. +\] + +The minimum exists because Theorem 2.1 makes every bounded observer slice +finite, while strict positivity excludes zero. + +Every nonconstant monomial in `u^n` has height at least `n epsilon`. +Consequently only + +\[ +n\le h(g)/\epsilon +\] + +can contribute to the coefficient at target `g`. Each product coefficient is +finite by Corollary 2.2. \(\square\) + +This theorem deliberately assumes zero constant term. If a general +degree-zero coefficient `c_0(a)` is allowed inside an exponential, then +`exp(c_0(a))` need not belong to the frozen coefficient algebra. The rational +constant case handled by PR #147 is a declared narrow extension, not evidence +for arbitrary degree-zero exponential closure. + +## 4. Observer-height cofinality + +The observer height is not unique. + +**Theorem 4.1.** Let `h_1` and `h_2` both be admissible on the same finitely +generated cone. Then their filtrations are cofinal: there exist constants +`0 an interior observer chart may change computational economy without +> changing the completed carrier. + +## 5. Bidirectional invariant-cone no-go + +Let `e_nu=(1,0)`. + +**Proposition 5.1.** No pointed cone `C` containing zero can satisfy both + +\[ +C+e_\nu\subseteq C +\qquad\text{and}\qquad +C-e_\nu\subseteq C. +\] + +**Proof.** Applying the two inclusions to zero gives + +\[ +e_\nu\in C, +\qquad +-e_\nu\in C. +\] + +Thus `C` contains the nonzero line `R e_nu`, contradicting pointedness. +\(\square\) + +This no-go concerns literal closure, not continuity. It does not imply that +differentiation or primitive transport is unavailable. + +## 6. Bounded-shift continuity + +For a translated chamber + +\[ +\mathcal H_{g_0,C}=K[[g_0+S]], +\] + +the AM operators satisfy + +\[ +A:\mathcal H_{g_0,C}\to\mathcal H_{g_0-e_\nu,C}, +\qquad +M:\mathcal H_{g_0,C}\to\mathcal H_{g_0,C}. +\] + +**Proposition 6.1.** These maps are continuous for every admissible height. + +**Proof.** `M` preserves degree. `A` translates every degree by the fixed +vector `-e_nu`; therefore it translates every height by the fixed scalar +`-h(e_nu)`. Given an output horizon, increasing the input horizon by that +fixed amount suffices. \(\square\) + +Away from the resonance wall `nu=-1`, the termwise primitive + +\[ +P_A X_{\nu,\kappa} +=\frac1{\nu+1}X_{\nu+1,\kappa} +\] + +is the opposite continuous chamber transport. On the wall, its codomain +must be enlarged by a typed `log-a` sector. + +## 7. Initial disposition + +The exact deductions already rule out a naive `CONE` interpretation in which +one pointed support cone is literally invariant under both power directions. +They do **not** force higher support such as surreal or Hahn structure. + +The leading disposition is therefore + +\[ +\boxed{\texttt{CHAMBERS}} +\] + +subject to the frozen two-generator and paired-observer calibrations. + +The emerging object is a chambered completed algebra: + +- pointed cones provide locally finite multiplication and completion; +- interior heights are observer charts on one completion; +- `A` and ordinary primitives transport between translated sectors; +- resonance walls force typed extensions; +- stronger support orders enter only if a later frozen task defeats every + admissible rational-polyhedral chamber system. diff --git a/sonnet/am-completion-cones/02-two-generator-calibration.md b/sonnet/am-completion-cones/02-two-generator-calibration.md new file mode 100644 index 00000000..bc4c34f5 --- /dev/null +++ b/sonnet/am-completion-cones/02-two-generator-calibration.md @@ -0,0 +1,223 @@ +# Two-generator AM cone calibration + +Status: exact calibration of the initial cone theorems. + +## 1. The cone + +Take + +\[ +G=\mathbb Z^2, +\qquad +r=(1,0), +\qquad +s=(0,1), +\] + +and the pointed cone + +\[ +C=\mathbb R_{\ge0}r+\mathbb R_{\ge0}s. +\] + +Write + +\[ +X=X^r=a, +\qquad +Y=X^s=e^v. +\] + +Then + +\[ +K[[C\cap G]]=K[[X,Y]]. +\] + +The native AM operators are + +\[ +A(X^pY^q)=pX^{p-1}Y^q, +\] + +\[ +M(X^pY^q)=(p+q)X^pY^q. +\] + +Thus `A` is differentiation in the power direction, while the exponential +character direction remains fixed. + +## 2. A genuinely two-ray completed expression + +Consider + +\[ +F(X,Y)=\exp(X+Y+XY). +\] + +Because the argument has zero constant term and support in the positive cone, +`F` is defined in the completed cone algebra. Factorization gives + +\[ +F=\exp(X)\exp(Y)\exp(XY). +\] + +Therefore + +\[ +[X^pY^q]F +=\sum_{k=0}^{\min(p,q)} +\frac1{(p-k)!(q-k)!k!}. +\] + +At target bidegree `(2,2)`, + +\[ +[X^2Y^2]F +=\frac1{2!2!} ++1 ++\frac1{2!} +=\boxed{\frac74}. +\] + +This coefficient cannot be interpreted as a calculation on one unspecified +rank-one ray: it receives contributions from the two generators and their +mixed degree. + +## 3. Exact `A` transport + +Differentiating the completed expression gives + +\[ +AF=(1+Y)F. +\] + +The coefficient transported from source degree `(2,2)` to target degree +`(1,2)` is + +\[ +[X^1Y^2]AF +=2[X^2Y^2]F +=\frac72. +\] + +The product side gives independently + +\[ +[X^1Y^2](1+Y)F +=[X^1Y^2]F+[X^1Y^1]F +=\frac32+2 +=\frac72. +\] + +This is exact chamber transport by `-r`; it is not an endomorphism preserving +the zero cone sector term by term. + +## 4. Exact `M` action + +Since `M` measures total AM weight, + +\[ +MF=(X+Y+2XY)F. +\] + +At `(2,2)`, the eigenvalue calculation gives + +\[ +[X^2Y^2]MF +=4\cdot\frac74 +=7. +\] + +The product calculation agrees: + +\[ +[X^2Y^2](X+Y+2XY)F +=\frac32+\frac32+2\cdot2 +=7. +\] + +## 5. Paired observer heights + +Two admissible heights are + +\[ +h_1(p,q)=p+q, +\qquad +h_2(p,q)=2p+q. +\] + +Both define the same formal completion `K[[X,Y]]`. They assign different +horizons to the same target: + +\[ +h_1(2,2)=4, +\qquad +h_2(2,2)=6. +\] + +Their scalar bounded slices contain respectively + +\[ +\#\{(p,q)\in\mathbb N^2:p+q\le4\}=15, +\] + +\[ +\#\{(p,q)\in\mathbb N^2:2p+q\le6\}=16 +\] + +lattice degrees. The exact componentwise down-set of `(2,2)` contains only +nine degrees. + +Hence three notions must remain distinct: + +1. the completed carrier, which is unchanged; +2. the scalar observer height, which defines a cofinal topology; +3. the exact target down-set, which may give a sharper dependency slice. + +This is the first exact instance in which observer charts preserve semantics +but alter computational economy. + +## 6. Primitive chambers and resonance + +On the zero chamber, every monomial has `p>=0`, so the ordinary primitive + +\[ +P_A(X^pY^q) +=\frac1{p+1}X^{p+1}Y^q +\] + +maps `K[[C cap G]]` into the translated chamber `X K[[X,Y]]` and satisfies + +\[ +A P_A F=F. +\] + +After one negative chamber shift, the boundary contains + +\[ +X^{-1}Y^q. +\] + +Its primitive is + +\[ +\log(X)Y^q, +\] + +with the declared positive-real witness `X=a>0`. The whole boundary +`p=-1`, not a single isolated scalar, is therefore a resonance wall. + +## 7. What this calibration establishes + +The two-ray example verifies: + +- locally finite completed multiplication; +- exact mixed exp coefficients; +- `A` as degree-shifting chamber transport; +- `M` as an endomorphism of each chamber; +- semantic invariance under paired admissible heights; +- a genuine distinction between height windows and exact target down-sets; +- ordinary primitive transport and the codimension-one `log-a` wall. + +No stronger support order is needed for any of these operations. diff --git a/sonnet/am-completion-cones/03-disposition.md b/sonnet/am-completion-cones/03-disposition.md new file mode 100644 index 00000000..730458d2 --- /dev/null +++ b/sonnet/am-completion-cones/03-disposition.md @@ -0,0 +1,111 @@ +# Disposition: `CHAMBERS` + +Status: mathematical disposition for the initial scope of issue #148. + +## Decision + +The initial rational-polyhedral AM completion problem receives the disposition + +\[ +\boxed{\texttt{CHAMBERS}}. +\] + +The correct bounded object is not one pointed cone invariant under all power +translations. It is a chambered system of cone-bounded completions. + +For a pointed rational cone `C` and affine monoid `S=C cap G`, define sectors + +\[ +\mathcal H_{g_0,C}=K[[g_0+S]]. +\] + +Their products and AM actions are typed by degree: + +\[ +\mathcal H_{g_0,C}\mathcal H_{g_1,C} +\longrightarrow +\mathcal H_{g_0+g_1,C}, +\] + +\[ +A:\mathcal H_{g_0,C} +\longrightarrow +\mathcal H_{g_0-e_\nu,C}, +\] + +\[ +M:\mathcal H_{g_0,C} +\longrightarrow +\mathcal H_{g_0,C}, +\] + +\[ +P_A:\mathcal H_{g_0,C} +\dashrightarrow +\mathcal H_{g_0+e_\nu,C}. +\] + +The dashed arrow records the resonance wall `nu=-1`, where the codomain must +include a typed `log-a` sector. + +## Why not `CONE` + +A fixed pointed cone cannot be invariant under both `+e_nu` and `-e_nu`. +Demanding literal closure under differentiation and primitives would insert a +line into the cone and destroy the finite-observer property. + +The operators remain continuous because their degree shifts are bounded. The +correct repair is typed chamber transport, not abandonment of completion. + +## Why not `RAY-ONLY` + +The two-generator calibration computes + +\[ +[X^2Y^2]\exp(X+Y+XY)=\frac74 +\] + +and exactly replays both `A` and `M`. Thus the rank-one compiler's mathematics +extends coherently to a genuine rational bigrading. + +## Why not `HIGHER-SUPPORT` + +Pointed rational-polyhedral cones already guarantee: + +- finite bounded observer slices; +- finite coefficient convolution; +- well-defined completed `exp` and `log1p` on the augmentation ideal; +- cofinality of all interior observer heights; +- continuous fixed-degree AM transport. + +No frozen obstruction currently requires Hahn, transseries, hyperseries, or +surreal support. Such objects remain candidates only if later tasks require +non-polyhedral well-ordered supports, unbounded rank raising, or symbolic +iteration heights. + +## Consequence for the programme + +The arithmetic geometry has acquired a more precise form: + +> AM completion is a chambered, observer-filtered geometry over the +> `(power, character)` lattice. Observer heights choose finite views; AM +> operators transport between views; resonance walls force typed extensions. + +This is stronger than the first compiler result and narrower than a general +transseries claim. It also explains why an observer with finite reach does +not need the whole ensemble: the declared cone and height make every visible +slice finite, while the completed carrier is recovered as their inverse +limit. + +## Next mathematical gate + +The next gate should study cone changes rather than parser syntax: + +1. define transport between two different pointed cones sharing a face; +2. determine when their completed sectors glue along the common chamber; +3. identify the cocycle or obstruction carried by successive chart changes; +4. test whether this gluing recovers the AM affine action independently of a + chosen cone. + +That is the first place where completion, fibering, observer charts, and the +earlier ensemble discussion meet in one exact construction. From f464adf54cca63946230369c24fdb702084552e6 Mon Sep 17 00:00:00 2001 From: Mingli Yuan Date: Thu, 27 Aug 2026 16:29:52 +0800 Subject: [PATCH 2/2] research: close AM completion proof gaps --- .../00-problem-frontier.md | 2 +- .../01-initial-cone-theorems.md | 32 +++++++++++++++++-- .../02-two-generator-calibration.md | 3 +- sonnet/am-completion-cones/03-disposition.md | 3 +- 4 files changed, 35 insertions(+), 5 deletions(-) diff --git a/sonnet/am-completion-cones/00-problem-frontier.md b/sonnet/am-completion-cones/00-problem-frontier.md index b8b10b6d..1fd593ca 100644 --- a/sonnet/am-completion-cones/00-problem-frontier.md +++ b/sonnet/am-completion-cones/00-problem-frontier.md @@ -55,7 +55,7 @@ This distinction is mathematical, not an implementation detail. ## 2. The completion problem -Let `K` be the exact coefficient algebra and let +Let `K` be a commutative exact `Q`-algebra and let \[ C\subset G_{\mathbb R} diff --git a/sonnet/am-completion-cones/01-initial-cone-theorems.md b/sonnet/am-completion-cones/01-initial-cone-theorems.md index 4f7400e1..b23ab931 100644 --- a/sonnet/am-completion-cones/01-initial-cone-theorems.md +++ b/sonnet/am-completion-cones/01-initial-cone-theorems.md @@ -11,7 +11,8 @@ G=\mathbb Z\oplus\Lambda \] be a rank-two lattice after clearing the denominator of the rational -rank-one character lattice `Lambda`. Let +rank-one character lattice `Lambda`. Let `K` be a commutative `Q`-algebra +and let \[ X^g=X_{\nu,\kappa}=a^\nu e^{\kappa v}, @@ -25,7 +26,7 @@ Fix a finitely generated rational cone C=\mathbb R_{\ge0}s_1+\cdots+\mathbb R_{\ge0}s_r \] -and its affine monoid `S=C cap G`. A height `h` is admissible when +and its affine monoid \(S=C\cap G\). A height `h` is admissible when \[ h(s_i)>0 @@ -148,6 +149,33 @@ continuous ratio `h_2/h_1` has a positive minimum and a finite maximum. same convergent formal objects, but generally assign different finite observer horizons and different dependency costs. +For an admissible height, define the observer ideal + +\[ +F_h^{>N} +=\left\{ +f\in K[[S]]:[X^g]f=0\text{ whenever }h(g)\le N +\right\}. +\] + +It is an ideal because every degree in `S` has nonnegative height. + +**Theorem 4.3.** The completed cone algebra is the inverse limit of its +finite observer quotients: + +\[ +K[[S]] +\cong +\varprojlim_N K[[S]]/F_h^{>N}. +\] + +**Proof.** Each quotient retains only the finite set +`S^h_{le N}` from Theorem 2.1. A formal series determines a compatible +family of these finite restrictions. Conversely, compatibility assigns one +coefficient to every degree of `S`, and hence determines a unique formal +series. The two constructions are inverse algebra homomorphisms. +\(\square\) + This is the precise form of a useful geometric principle: > an interior observer chart may change computational economy without diff --git a/sonnet/am-completion-cones/02-two-generator-calibration.md b/sonnet/am-completion-cones/02-two-generator-calibration.md index bc4c34f5..64e40b52 100644 --- a/sonnet/am-completion-cones/02-two-generator-calibration.md +++ b/sonnet/am-completion-cones/02-two-generator-calibration.md @@ -187,7 +187,8 @@ P_A(X^pY^q) =\frac1{p+1}X^{p+1}Y^q \] -maps `K[[C cap G]]` into the translated chamber `X K[[X,Y]]` and satisfies +maps \(K[[C\cap G]]\) into the translated chamber \(XK[[X,Y]]\) and +satisfies \[ A P_A F=F. diff --git a/sonnet/am-completion-cones/03-disposition.md b/sonnet/am-completion-cones/03-disposition.md index 730458d2..6bcaad81 100644 --- a/sonnet/am-completion-cones/03-disposition.md +++ b/sonnet/am-completion-cones/03-disposition.md @@ -13,7 +13,8 @@ The initial rational-polyhedral AM completion problem receives the disposition The correct bounded object is not one pointed cone invariant under all power translations. It is a chambered system of cone-bounded completions. -For a pointed rational cone `C` and affine monoid `S=C cap G`, define sectors +For a pointed rational cone `C` and affine monoid \(S=C\cap G\), define +sectors \[ \mathcal H_{g_0,C}=K[[g_0+S]].