§5 · Agent 5 · Functions without (A1)–(A5) · 2026-10-17T02:06:59Z

Strategy 5.1 · Strategy: triangular generation from explicit orbit geometry, without Theorem E

proved audit: minor defect

Minor defect (second audit). The plan is sound and is carried out in the later nodes. One imprecision: it pairs the row \(\lambda^*(b)\) with the reverse dominance order inside Proposition B, whose input (b) asks for finite lower sets. Reverse dominance on all labels of a fixed degree has infinite lower sets (e.g. \((k,-k)\ge_{\mathrm{dom}}(0,0)\) for every \(k\ge0\)), so Proposition B does not apply to it as announced. The execution in Theorem 5.7 works in finite slices \(S_{d,c}\) instead.

A plan for proving \((\mathrm{SG})\) unconditionally, over every non-archimedean local field, for the spheres, \(\GL_n/(\GL_p\times\GL_q)\) and the symplectic family: classify the \(K\)-orbits over \(\cO\), bound supports by lattice algebra, and apply Proposition B; where dominant lifts are missing, use a different Hecke row and the reverse dominance order.

Conventions. \(F\) is any non-archimedean local field (either characteristic, any residue cardinality \(q\)), \(K=G(\cO)\), \(\operatorname{vol}K=1\), \((f\ast\varphi)(x)=\sum_{gK}f(g)\varphi(g^{-1}x)\), \(T_\lambda=\one_{Kt^\lambda K}\) and \(\varphi_0=\one_{X(\cO)}\), as on the classical page.

What Proposition B needs. Proposition B (triangular generation) uses only three inputs: (a) a set of compact open \(K\)-orbits whose indicators span \(C_K\); (b) a partial order on them with finite lower sets; (c) for each orbit \(O_\mu\) an element \(f_\mu\in\cH_K\) such that \(f_\mu\ast\varphi_0\) is supported in the lower set of \(\mu\) and is nonzero on \(O_\mu\). None of these needs \(\operatorname{char}F=0\), Borel–Serre, the Cartan decomposition, or Sakellaridis's hypotheses (A1)–(A5).

Plan. For each family, (a) comes from a direct classification of the \(K\)-orbits over \(\cO\), (b) from a lattice-theoretic support bound, and (c) from Lemma A(2): \(T_\lambda\ast\varphi_0\) takes nonnegative integer values and is at least \(1\) at \(t^\lambda x_0\).

Where coverage fails. For \(\GL_{2m}/\Sp_{2m}\) some labels have no dominant lift. For such a label \(b\) the new input is a different Hecke row, \(\lambda^*(b):=(b_1,\dots,b_m,0,\dots,0)\), which is not a lift of \(b\), together with the reverse dominance order; a module-theoretic inequality (Lemma 5.6) gives the triangularity.

Every result below is proved in this track unless it is marked as cited.

Builds on

Built on by

Lemma 5.2 · Spheres: the \(\Spin_{2n}(\cO)\)-orbits on \(Q(F)\) are the pole-order shells, for every local field
Theorem 5.4 · \(\GL_n/(\GL_p\times\GL_q)\), pair model: orbits, lifts, a componentwise support bound and \((\mathrm{SG})\) over every \(F\)
Computation 5.5 · \(\GL_4/\Sp_4\): lattice counts and a Hecke element for the uncovered label \((1,1)\)
Proposition 5.8 · The support bound of H5(a) at function level over every \(F\), and the same-points question

Original record, as the agent wrote it at 02:06 UTC

Strategy: Proposition B from explicit orbit geometry, no use of Theorem E

Conventions as on classical.html: \(F\) any non-archimedean local field (both characteristics, any \(q\)), \(K=G(\mathcal O)\), \(\mathrm{vol}K=1\), \((f\ast\varphi)(x)=\sum_{gK}f(g)\varphi(g^{-1}x)\), \(T_\lambda=\mathbf 1_{Kt^\lambda K}\), \(\varphi_0=\mathbf 1_{X(\mathcal O)}\).

Proposition B (classical.html §2) needs only: (a) a set of compact open \(K\)-orbits whose indicators span \(C_K\); (b) a partial order with finite lower sets; (c) for each orbit an \(f_\mu\in\mathcal H_K\) with support bound and nonzero top value. None of these needs \(\operatorname{char}F=0\), Borel–Serre, the Cartan decomposition, or (A1)–(A5). For each family I prove (a) by a direct orbit classification over \(\mathcal O\), (b) by a lattice-theoretic support bound, (c) by Lemma A(2) (\(T_\lambda\) has nonnegative integer values and value \(\ge1\) at \(t^\lambda x_0\)).

Where coverage fails (\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\)) the new input is a different Hecke row: for a label \(b\) use \(\lambda^*(b):=(b_1,\dots,b_m,0,\dots,0)\) (not a lift of \(b\)) and the REVERSE dominance order; a module-theoretic inequality (node 6) gives the triangularity.

All labels below are ones I prove; "cited" marks a standard input with the reference I rely on.

Sources

classical.html#Proposition B
classical.html#Lemma A

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866