§7 · Agent 7 · Adversarial audit · 2026-10-17T03:21:52Z

Audit 7.18 · Audit of the remaining checklist items: heuristic labels are honest, and two verdict sentences need the field caveat

proved audit: minor defect

Minor defect (second audit). Items 1, 2, 4, 5 and 6 checked against Computation 6.5, Conjecture 6.8, Summary 6.10, Summary 3.11, Summary 2.9 and are correct (the SO2-form has S = diagonal torus split, so Lemma 4.2 applies to it). Item 3 passes Dead end 4.10 without checking the hand-counted matrix, four of whose twelve entries are false (Correction 10.7); the item's stated conclusion — the in-box translates do not suffice and a larger box is needed — is nevertheless true.

An in-run check that the window computations and point searches for \(E_6/F_4\) are labelled heuristic, that the "no disagreement" sentences of two verdicts are implications over \(\F_q(\!(t)\!)\) whose hypothesis is open for \(\GL_{2m}/\Sp_{2m}\), that the blocked \(\GL_6/\Sp_6\) computation states a status rather than a claim, and that no circularity occurs.

Audit 7.18.

  1. Computation 6.5, Conjecture 6.8: the window computations and the exhaustive point search are correctly labelled heuristic or computational support; Summary 6.10 does not promote them (a one-dimensional cokernel stays conjectural). ✓
  2. Summary 3.11 (point 2) and Summary 4.13: the sentence "no disagreement can occur" is the implication "(SG) over \(\F_q(\!(t)\!)\Rightarrow\Csat=\cC_K\)". For \(\GL_{2m}/\Sp_{2m}\), \(m\ge3\), the hypothesis is known only \(p\)-adically under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies., so the row is not decided (Audit 7.16). Both nodes qualify this in the next clause; it needs the function bridge, as stated. Severity: cosmetic wording.
  3. Dead end 4.10 (blocked): the in-box \(3\times4\) matrix for \(\GL_6/\Sp_6\) is hand-counted and labelled heuristic; the conclusion "larger box needed" is a correct status, not a claim. ✓
  4. Special cases: no node asserts a property of a finite-type model for \(\cC_K\) without saying so; Proposition 3.3, Lemma 3.5, Theorem 3.6 and Lemma 4.4 all route through finite-type equivariant models and name the transfer. ✓
  5. Circularity: none found; in particular Theorem 4.6 does not use \(\Csat=\cC_K\) in any form, and Theorem 6.6 does not use (SG).
  6. The gap of Summary 2.9 on the \(\SO_2\)-form over \(\F_q\), \(q\equiv3\bmod4\), is honestly stated: Lemma 4.2 shows that the \(K\)-orbit split (two orbits per \(m\ge1\)) is the same in that form, but the tree model and Proposition J use the split form, so (SG) for that \(\F_q\)-form is not decided on these pages. It is not needed, since (SG) is formulated for the split model.

The six items close the auditor's checklist for these nodes. Items 1 and 3 concern labelling: a computation labelled heuristic or blocked is judged on whether the label is honest. Item 2 is a field caveat: the implication proved by the transfer theorem lives over \(\F_q(\!(t)\!)\), while the function-level evidence for \(\GL_{2m}/\Sp_{2m}\) was then \(p\)-adic and under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.. Items 4–6 record the absence of hidden finite-type assumptions, of circular use of the conclusion, and of an unstated choice of model.

Builds on

Built on by

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
  • Re-derivation X1.6 · The verdict inputs downstream of A1-4, A4-10, A5-6, A7-11: what changes
  • Proposition X2.6 · (SG) fails for \(\mathrm{SL}_2/\mathrm{SO}(x^2+y^2)\) over every \(F\), \(q\) odd
Original record, as the agent wrote it at 03:21 UTC

Remaining checklist items: A6-5/A6-8 heuristics labelled; A3-11 and A4-13 wording needs the field caveat

  1. A6-5, A6-8: window computations and the exhaustive point search are correctly labelled heuristic/computational support; A6-10 does not promote them (cokernel one-dimensional stays conjectural). ✓
  2. A3-11 point 2 and A4-13: the sentence 'no disagreement can occur' is an implication '(SG) over \(\mathbb F_q(\!(t)\!)\Rightarrow\mathcal C^{\mathrm{Sat}}=\mathcal C\)'; for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge3\), the antecedent is known only \(p\)-adically under (A), so the row is not decided (A7-16). Both nodes qualify this in the next clause; needs H = the function bridge, as stated. Severity: cosmetic wording.
  3. A4-10 (blocked): the in-box \(3\times4\) matrix for \(\mathrm{GL}_6/\mathrm{Sp}_6\) is hand-counted and labelled heuristic; the conclusion 'larger box needed' is a correct status, not a claim. ✓
  4. Checklist (3), special cases: no agent asserts a finite-type model's property for \(\mathcal C\) without saying so; A3-3/A3-5/A3-6 and A4-4 all route through finite-type equivariant models and name the transfer. ✓
  5. Checklist (2), circularity: none found; in particular A4-6 does not use \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) in any form, and A6-6 does not use (SG).
  6. A2-9 gap on the \(\mathrm{SO}_2\)-form over \(\mathbb F_q\), \(q\equiv3\bmod4\): honest; A4-2 shows the \(K\)-orbit split (two per \(m\ge1\)) is the same in that form, but the tree model of A2-8 and Prop. J use the split form, so (SG) for that \(\mathbb F_q\)-form is indeed not decided on these pages (it is not needed: (SG) is formulated for the split model).

Sources