# Verdict: do orbit standards generate without finite orbit closures?

Origin: Toan, 2026-10-20 13:43, on `proof_graphs/spherical-generation/nodes/filtration.html`, main proposition (b).

| statement | status | steps |
|---|---|---|
| I. For \(\mathbb A^n\): \(\operatorname{Thick}\{\Delta_m\}=\{F:F_0=0\}\ne\mathcal C\) | proved (in-run audit) | A1-2, A1-3, A2-3, A2-5 |
| II. \(\mathcal C=\operatorname{Thick}(\{\Delta_m\}\cup\{\underline k_{t^{M_0}\mathcal O^n}\})\) | proved (in-run audit) | A1-4, A2-7 |
| III. Piece standards generate (Theorem G); orbit standards iff \((\star)\) | proved (in-run audit) | A1-5–A1-8, A2-8, A2-11, A2-12 |

## Proposition I: orbit standards miss the origin (proved)
**Assumes.** \(X=\mathbb A^n\), \(G=\mathrm{GL}_n\), \(\mathcal C\) = objects pulled back from a level \(Y_{N,M}\) (Proposition IV).
**Statement.** \(\operatorname{Thick}_{\mathcal C}\{\Delta_m:m\in\mathbb Z\}=\ker i_0^*=\{F\in\mathcal C:F_0=0\}\). In particular \(\mathrm{IC}_0=\underline k_{\mathcal O^n}\notin\operatorname{Thick}\{\Delta_m\}\).
**Proof.**
1. \(i_0^*\) (stalk at the origin, computed at any level) is exact on \(\mathcal C\); its kernel is thick [[A1-2]].
2. \(i_0^*\Delta_m=0\), \(i_0^*\underline k_{t^M\mathcal O^n}=k\) [[A1-3]].
3. On a piece, one stalk determines the object (from (L3)) [[A2-3]].
4. If \(F_0=0\), level-wise dévissage on \(Y_N\) leaves only shell terms; the tail term vanishes by 3 [[A2-5]].

## Proposition II: one lattice repairs it (proved)
**Statement.** For any \(M_0\), \(\mathcal C=\operatorname{Thick}(\{\Delta_m\}_{m\in\mathbb Z}\cup\{\underline k_{t^{M_0}\mathcal O^n}\})\). In \(\operatorname{Ind}\mathcal C\), \(\varinjlim_M\underline k_{t^M\mathcal O^n}\) is right-orthogonal to all \(\Delta_m\).
**Proof.** Theorem G at each level; the triangle \(\Delta_m\to\underline k_{t^m\mathcal O^n}\to\underline k_{t^{m+1}\mathcal O^n}\) moves between lattices [[A1-4]]; the Ind witness via the closed adjunction [[A2-7]].

## Proposition III: piece standards generate; criterion for orbits (proved)
**Assumes.** A level system with finitely many \(L^+G\)-stable pieces per level, (L0) objects of level \(\alpha\) supported on \(Y_\alpha\), (L0′) exact stalks, (L1)–(L3) of Proposition IV; for the criterion also directedness (D).
**Statement.** (a) \(\mathcal C=\operatorname{Thick}\{\Delta_P: P \text{ a piece of some level}\}\). (b) \(\Delta_O\in\mathcal C\) iff the orbit \(O\) is a piece of some level ("visible"). (c) Orbit standards generate iff \((\star)\): every piece is a finite union of visible orbits. (d) \((\star)\iff\) H1(a)\(_{\rm fin}\) + H1(b) + every orbit visible. For \(\mathbb A^n\) H1(a)\(_{\rm fin}\) fails at every shell and visibility fails at \(\{0\}\).
**Proof.** [[A1-5]] with (L0) [[A2-8]]; [[A1-6]]; [[A1-7]]; [[A1-8]], scoped by [[A2-11]].

## Gaps
- Whether H1(a)\(_{\rm fin}\) alone forces \((\star)\) for natural placid presentations (it does not for the bare axioms, [[A2-12]]).
- Whether H1(a)\(_{\rm fin}\) + visibility imply H1(b).

## Proposed edit (filtration.html, after Proposition (b))
"Without H1(a) this fails: for \(\mathbb A^n\) the orbit standards span exactly the objects with zero stalk at \(0\) (Proposition IV); what always generates are the standards of the pieces of the finite-type levels, e.g. \(\{\Delta_m\}\cup\{\mathrm{IC}_0\}\)."