§10 · Root · root: registry, redirects, synthesis · 2026-10-18T19:04:20Z

Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched

open

The root groups the first-round steps by mathematical idea, notes where agents reached the same object independently, marks the converse route blocked, and spends the last three slots on adversarial audits of every candidate proof.

Registry after the six first-round returns.

Table 2. idea and agents and state
ideaagentsstate
summand criterion, level by levelA2, A5, A3three independent versions (Theorem 2.4, Proposition 5.4, Lemma 3.5); to audit
Frobenius traces at a finite levelA3, A5two independent versions of the bridge (Theorem 3.6, Proposition 5.7); to audit
sign character and weights, the groupA4, A2, A3, A6four independent proofs that \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) (Theorem 4.5, Theorem 2.9, Proposition 3.9, Corollary 6.9)
length-zero translates, \(\mathbb A^n\) and \(\mathbb G_m\)A5, A2every simple object is \(\Omega\cdot\mathrm{IC}_0\) (Theorem 5.5, Theorem 2.7, Proposition 2.10)
affine Lusztig–Vogan module, \(\mathrm{GL}_2/T\)A6both levels fail at the closed orbits (Proposition 6.5, Proposition 6.6, Proposition 6.8); one agent, to audit
sheaves ⇒ functionsA3, A1, A5blocked: support versus span (Proposition 3.10, Audit 1.7)

Two agents reaching the same object independently is evidence: the length-zero element \(\tau\) for \(\mathbb A^n\) (A5, A2), the sign character with weights for the group (A4, A2, A3, A6).

Decision. Every target of the run has a candidate proof. The remaining three slots go to adversarial audits, each given the claimed statements and proofs as written: one for the criterion and the bridge (both versions of each), one for the group, \(\mathbb A^n\) and \(\mathbb G_m\), one for \(\mathrm{GL}_2/T\) with the author's scripts available for independent recomputation.

Builds on

  • Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument
  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$
  • Proposition 5.4 · (I$_I$) The Iwahori summand criterion: generation iff $\mathcal M_I$ is all
  • Lemma 3.5 · Iwahori summand criterion: $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\,\mathcal M_I$
  • Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$
  • Proposition 5.7 · (II$_I$) (IG) over $\mathbb F_q((t))$ puts every simple object in $\mathcal M_I$
  • Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Theorem 2.9 · Group: $\Omega\cap\mathcal M_I^+=\varnothing$ by sign character and weights
  • Proposition 3.9 · Group: no $\mathrm{IC}_\omega$ is a summand, so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Corollary 6.9 · Group case: the sign character $\varepsilon_+$ excludes every $\delta_\omega$
  • Theorem 5.5 · $\mathbb A^n$: $\mathcal M_I=\{S_k\}$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$
  • Theorem 2.7 · $\mathbb A^n$: $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, hence Iwahori generation
  • Proposition 5.6 · (IG) for $\mathbb A^n$: $T_{\tau^k}\ast\varphi_0=\mathbf 1_{\tau^k\mathcal O^n}$
  • Proposition 2.10 · Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction
  • Proposition 6.5 · Sheaves fail for $\mathrm{GL}2/T$: $\mathrm{IC}{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}$
  • Proposition 6.6 · Functions: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two
  • Proposition 6.8 · Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not
  • Proposition 3.10 · Sheaves to functions: non-zero rows only; (WT) gives (IG)

Built on by

  • —
Original record, as the agent wrote it at 19:04 UTC

Registry after round one; three audits launched

Registry after the six first-round returns.

Table 3. idea and agents and state
ideaagentsstate
summand criterion, level by levelA2, A5, A3three independent versions (Theorem 2.4, Proposition 5.4, Lemma 3.5); to audit
Frobenius traces at a finite levelA3, A5two independent versions of the bridge (Theorem 3.6, Proposition 5.7); to audit
sign character and weights, the groupA4, A2, A3, A6four independent proofs that \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) (Theorem 4.5, Theorem 2.9, Proposition 3.9, Corollary 6.9)
length-zero translates, \(\mathbb A^n\) and \(\mathbb G_m\)A5, A2every simple object is \(\Omega\cdot\mathrm{IC}_0\) (Theorem 5.5, Theorem 2.7, Proposition 2.10)
affine Lusztig–Vogan module, \(\mathrm{GL}_2/T\)A6both levels fail at the closed orbits (Proposition 6.5, Proposition 6.6, Proposition 6.8); one agent, to audit
sheaves ⇒ functionsA3, A1, A5blocked: support versus span (Proposition 3.10, Audit 1.7)

Two agents reaching the same object independently is evidence: the length-zero element \(\tau\) for \(\mathbb A^n\) (A5, A2), the sign character with weights for the group (A4, A2, A3, A6).

Decision. Every target of the run has a candidate proof. The remaining three slots go to adversarial audits, each given the claimed statements and proofs as written: one for the criterion and the bridge (both versions of each), one for the group, \(\mathbb A^n\) and \(\mathbb G_m\), one for \(\mathrm{GL}_2/T\) with the author's scripts available for independent recomputation.

Sources