OpenAI claim summary for Math-2194

Arthur's Conjectures

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 014; 2026-10-05), the asserted result is that tame regular-unipotent, geometrically dense parameters admit nonzero constructible perverse Hecke eigensheaves with one Borel-level point for the specified simply connected groups and characteristic hypotheses; In manuscript 2 (family 014; 2026-10-05), the asserted result is that after a finite constant-field extension, the one-point tame eigensheaf construction for $\mathrm{SL}_n$, $p>n$, can retain the prescribed arithmetic Frobenius and full multi-leg eigenstructure; In manuscript 3 (family 014; 2026-10-05), the asserted result is that assuming the stated finite-level decomposition, each occurring cuspidal excursion parameter for a split semisimple function-field group has a global Arthur enhancement with one algebraic $\mathrm{SL}_2$ and a commuting Weil action, including inertia; In manuscript 4 (family 014; 2026-09-24), the asserted result is that cuspidal functions for split semisimple groups over function fields decompose by dual nilpotent orbits at every full finite level; for adjoint absolutely simple groups, one generic unramified component forces temperedness at every unramified place; In manuscript 5 (family 014; 2026-09-24), the asserted result is that the canonical unramified Arthur filtration on all finitely supported automorphic functions is rational under the restricted theory's characteristic hypotheses for split connected semisimple groups; In manuscript 6 (family 014; 2026-10-05), the asserted result is that for $\mathrm{SL}_n$ with $p>n$, dense tame parameters with arbitrary unipotent Jordan types at several marked points admit nonzero Borel-level perverse eigensheaves retaining tensor and fusion compatibility; In manuscript 7 (family 014; 2026-10-05), the asserted result is that globally generic cuspidal representations of split adjoint exceptional groups over function fields are tempered at ramified as well as unramified places, in every characteristic; In manuscript 8 (family 014; 2026-09-24), the asserted result is that restricted geometric Langlands has full spectral support in positive characteristic in the two stated characteristic regimes, with additional very-good and Weyl-order conditions over arbitrary algebraically closed fields.

Mathematical statement of the claimed result

Manuscript 1.

Let $k=\overline{\mathbb F}_q$ have characteristic $p$, $E=\overline{\mathbb Q}_{\ell}$ with $\ell\ne p$, and let $X_0/\mathbb F_q$ be smooth projective geometrically connected of genus at least two with rational point $x_0$. Set $X=(X_0)_k$, $x=(x_0)_k$ and $U=X\setminus\{x\}$. Let $G_0$ be split simple simply connected, with Borel $B_0$, and write $G,B$ for base changes. The characteristic hypotheses are: a nondegenerate invariant symmetric form on $\mathfrak g$ whose restriction to the central Lie algebra of each Levi is nondegenerate; the Chevalley-restriction isomorphism $k[\mathfrak m]^M\simeq k[\operatorname{Lie}(T_M)]^{W_M}$ for every Levi; semisimple Lie-algebra centralizers are Levi subgroups; and over every extension field each nilpotent Lie-algebra element lies in a parabolic unipotent-radical Lie algebra.

For a continuous Zariski-dense parameter $\rho:\pi_1^{\rm et}(U)\to\widehat G(L)$, $L/\mathbb Q_{\ell}$ finite, require trivial wild inertia and tame monodromy \[ \rho(\gamma)=\exp(t_{\ell}(\gamma)N),\qquad \dim Z_{\widehat G}(N)=\operatorname{rank}(\widehat G), \] with nilpotent $N$ and $t_{\ell}:I_x\to\mathbb Z_{\ell}$. On the stack $\mathcal A=\operatorname{Bun}_{G,B,x}(X)$, the claim gives a nonzero locally constructible perverse $E$-adic sheaf $M$ with nilpotent parabolic singular support and \[ \operatorname{Hecke}_{I,(V_i)}(M)\simeq M\boxtimes\mathop{\boxtimes}_{i\in I}(V_i)_{\rho} \quad\hbox{on }\mathcal A\times U^I \] for every finite set $I$ and representations $V_i$ of $\widehat G$. The eigenisomorphisms respect units, convolution, permutations and collision-diagonal fusion. No Frobenius descent of $\rho$ is assumed. The nilpotent cone consists of Higgs fields with nilpotent generic value and residue in the Borel unipotent radical.

Manuscript 2.

Let $n\ge2$, $\mathbb F_q$ have characteristic $p>n$, and $\ell\ne p$, $E=\overline{\mathbb Q}_{\ell}$. Let $X_0/\mathbb F_q$ be smooth projective geometrically connected of genus at least two, with rational point $x_0$, and $U_0=X_0\setminus\{x_0\}$. A continuous $\rho:\pi_1^{\rm et}(U_0)\to\operatorname{PGL}_n(L)$, $L/\mathbb Q_{\ell}$ finite, is assumed geometrically Zariski dense, trivial on wild inertia, and regularly unipotent on tame inertia: \[ \rho(\gamma)=\exp(t_{\ell}(\gamma)N),\qquad N\text{ regular nilpotent}. \] The claim gives $m\ge1$ and a Weil sheaf $M_0\ne0$ with $E$-coefficients on $\operatorname{Bun}_{\operatorname{SL}_n,B,x_0}(X_0)_{\mathbb F_{q^m}}$. After geometric pullback it is perverse and locally constructible, its singular support lies in the parabolic nilpotent locus, and \[ \operatorname{Hecke}_{I,(V_i)}(M_0)\simeq M_0\boxtimes\mathop{\boxtimes}_{i\in I}(V_i)_{\rho_m} \] for all finite leg sets and dual-group representations. These are Weil isomorphisms coherent for units, convolution, permutations and all fusion diagonals; $\rho_m$ is the restriction to the extended constant field.

Manuscript 3.

Let $X/\mathbb F_q$ be smooth projective geometrically connected, $G/\mathbb F_q$ split connected semisimple, $D$ an effective divisor, $U=X\setminus\operatorname{supp}D$ and $K_D=\ker(G(\mathcal O_{\mathbb A})\to G(\mathcal O_D))$. Fix $\ell\ne\operatorname{char}\mathbb F_q$, $k=\overline{\mathbb Q}_{\ell}$, an embedding $j:\overline{\mathbb Q}\to k$ and $a=j(\sqrt q)$. For an occurring character $\nu$ of the excursion algebra on cuspidal functions at full level $K_D$, write $\sigma_{\nu}:\pi_1(U)\to\widehat G(k)$ for its parameter and $\Gamma=W_U$ for the Weil subgroup.

The theorem assumes the companion finite-level decomposition: one nilpotent orbit $\mathcal O_{\nu}$ works at all good points, and each semisimple Frobenius class has the form $[\phi_{\mathcal O_{\nu}}(\operatorname{diag}(a^{d_x},a^{-d_x}))c_x]$, with $c_x$ commuting, algebraic and compact at every complex embedding. Under this assumption it claims a global algebraic $\phi_{\nu}:\operatorname{SL}_2\to\widehat G$ and continuous $\tau_{\nu}:\Gamma\to Z_{\widehat G}(\phi_{\nu}(\operatorname{SL}_2))(k)$, defined over a finite $\ell$-adic extension, with reductive Zariski closure, such that after one fixed conjugation \[ \sigma_{\nu}(w)=\tau_{\nu}(w)\phi_{\nu}(\operatorname{diag}(a^{\deg w},a^{-\deg w}))\quad(w\in\Gamma). \] The differential of $\phi_{\nu}$ sends the standard upper nilpotent matrix into $\mathcal O_{\nu}$. For every rational-form algebraic representation and closed $x\in U$, eigenvalues of $\tau_{\nu}(\operatorname{Frob}_x)$ are algebraic and have absolute value one at every complex embedding. The full centralizer may be disconnected. Ellipticity, packet classification and a multiplicity formula are not asserted.

Manuscript 4.

Let $F=\mathbb F_q(X)$ for a smooth projective geometrically connected curve, $G/\mathbb F_q$ split connected semisimple, and $D$ any effective divisor. Write $K_D=\ker(G(\mathcal O_{\mathbb A})\to G(\mathcal O_D))$ and $C_{D,\mathbb Q}$ for compactly supported rational cuspidal functions on $G(F)\backslash G(\mathbb A_F)/K_D$. For each dual-group nilpotent orbit $\mathcal O$, choose its Jacobson--Morozov $\phi_{\mathcal O}:\operatorname{SL}_2\to\widehat G$. Let $C_{D,\mathcal O,\mathbb Q}$ consist of functions whose spherical Hecke spectra, at every good point $x$, have semisimple class \[ [\phi_{\mathcal O}(\operatorname{diag}(q_x^{1/2},q_x^{-1/2}))c], \] where $c$ centralizes $\phi_{\mathcal O}$ and is algebraic and compact under every complex embedding. The same orbit is required at every good point. The claimed rational and $\ell$-adic decompositions are \[ C_{D,\mathbb Q}=\bigoplus_{\mathcal O} C_{D,\mathcal O,\mathbb Q},\qquad C_{D,\ell}=\bigoplus_{\mathcal O} C_{D,\mathcal O,\ell}, \] with $\ell\ne\operatorname{char}\mathbb F_q$ and scalar extensions from $\mathbb Q$. For split adjoint absolutely simple $G/F$, a cuspidal representation generic and unramified at one place is further claimed tempered at every unramified place. Ramified temperedness and a full multiplicity formula are not asserted here.

Manuscript 5.

Let $X/\mathbb F_q$ be smooth projective geometrically connected and $G$ split connected semisimple. On $H=G_{\overline{\mathbb F}_q}$ assume an invariant nondegenerate Lie-algebra form nondegenerate on all Levi centers, Chevalley restriction for every Levi, semisimple Lie-algebra centralizers being Levi, and every nilpotent element over any extension lying in a parabolic unipotent-radical Lie algebra. For $E=\overline{\mathbb Q}_{\ell}$, $\ell\ne\operatorname{char}\mathbb F_q$, let $S$ be bundle isomorphism classes over $\mathbb F_q$ and $\mathcal A_E=E^{(S)}$. For a closed invariant $Y$ in the pinned dual-group nilpotent cone, use the derived-Satake support category $\mathcal C_Y$ and Frobenius $\Phi$ to define \[ \mathcal F_Y\mathcal A_E=\operatorname{im}(H^0\operatorname{Tr}(\Phi,\mathcal C_Y)\to\mathcal A_E). \] The claim says this entire canonical filtration is rational: \[ E\otimes_{\mathbb Q}(\mathcal A_{\mathbb Q}\cap\mathcal F_Y\mathcal A_E) \xrightarrow{\sim}\mathcal F_Y\mathcal A_E \] for every such $Y$. The intersection is inside $\mathcal A_E$. Noncuspidal functions are included, and supported traces are defined by their image without assuming concentration in degree zero.

Manuscript 6.

Let $n\ge2$, $k=\overline{\mathbb F}_q$ of characteristic $p>n$, $\ell\ne p$, and $E=\overline{\mathbb Q}_{\ell}$. Take a smooth projective geometrically connected $X_0/\mathbb F_q$ of genus at least two with distinct rational $x_1,\ldots,x_s$, $s\ge2$, and set $X=(X_0)_k$, $D=\sum_ix_i$, $U=X\setminus|D|$. A continuous Zariski-dense $\rho:\pi_1^{\rm et}(U)\to\operatorname{PGL}_n(L)$, $L/\mathbb Q_{\ell}$ finite, is assumed trivial on wild inertia and has \[ \rho(\gamma)=\exp(t_{\ell,i}(\gamma)N_i)\quad(\gamma\in I_{x_i}), \] with any nilpotent $N_i$, including zero. On $\mathcal A=\operatorname{Bun}_{\operatorname{SL}_n,B,D}(X)$ the claim supplies a nonzero locally constructible perverse geometric sheaf $M$ with singular support in the generically nilpotent Higgs cone with Borel-unipotent residues, and \[ \operatorname{Hecke}_{I,(V_i)}(M)\simeq M\boxtimes\mathop{\boxtimes}_{i\in I}(V_i)_{\rho} \] for all finite leg sets and dual-group representations. Eigenmaps are coherent for units, convolution, permutations and every collision diagonal, with relative Satake normalization $E[d_{\lambda}](d_{\lambda}/2)$ and no extra moving-leg shift.

Manuscript 7.

Let $F$ be a global function field and $G/F$ split connected adjoint absolutely simple of one of the exceptional types $G_2,F_4,E_6,E_7,E_8$. For every complex globally generic cuspidal automorphic representation $\pi=\bigotimes'_v\pi_v$ of $G(\mathbb A_F)$, the claimed conclusion is \[ \pi_v\text{ is tempered for every place }v. \] Ramified places are included. The statement imposes no characteristic, ramification-depth or local-representation-type restriction.

Manuscript 8.

Let $k$ be algebraically closed of characteristic $p>0$, $\ell\ne p$, $E=\overline{\mathbb Q}_{\ell}$, $X/k$ smooth projective connected and $G/k$ connected reductive. Require the four common conditions: an invariant nondegenerate Lie-algebra form nondegenerate on every Levi center, Chevalley restriction for every Levi, semisimple Lie-algebra centralizers being Levi, and nilpotent elements over every extension lying in parabolic unipotent-radical Lie algebras. Regime A additionally requires $k=\overline{\mathbb F}_q$ and descent of $X$ and split $G$ to $\mathbb F_q$. Regime B instead permits arbitrary algebraically closed $k$, but requires $p$ very good for $G$ and $p\nmid|W_G|$. Very good excludes primes dividing $n+1$ for $A_n$, $2$ for $B,C,D$, $2,3$ for $G_2,F_4,E_6,E_7$, and $2,3,5$ for $E_8$.

Let $Y=\operatorname{LS}_{\widehat G}^{\rm restr}(X)$ be the restricted spectral prestack, parametrizing right-t-exact symmetric monoidal functors from dual-group representations to $\operatorname{QCoh}(T)\otimes_E\operatorname{QLisse}_E(X)$ on derived affine $T$. The claim says the previously occurring open-and-closed union $Y'$ equals all of $Y$, giving a $\operatorname{QCoh}(Y)$-linear equivalence \[ \mathbb L_G^{\rm restr}:\operatorname{Shv}_{\rm Nilp}(\operatorname{Bun}_G(X)) \xrightarrow{\sim}\operatorname{IndCoh}_{\rm Nilp}(Y). \] The automorphic and spectral categories each retain nilpotent singular support. This positive-characteristic result is restricted to the two stated regimes.

Outline of the argument

For manuscript 1 (family 014; 2026-10-05), the parameter is glued through a separating node to an unramified family. Two specializations and transverse Hecke cuts detect a nonzero flag-level object; unipotent monodromy on the enhancement torus permits descent to ordinary Borel level. For manuscript 2 (family 014; 2026-10-05), an arithmetic parameter is lifted to characteristic zero with its boundary action intact and then glued across a separating node. Equivariant nearby cycles and a compact conjugator preserve Frobenius throughout specialization and descent from the enhanced flag level. For manuscript 3 (family 014; 2026-10-05), shtuka cohomology supplies a finite-type cyclic support carrying full excursion data. Frobenius-compatible coherent models of Iwahori categories at several places identify an open-orbit condition, which assembles a single commuting Weil and $\mathrm{SL}_2$ pair.

For manuscript 4 (family 014; 2026-09-24), iwahori translation eigencharacters are analyzed before taking the Weyl quotient. Frobenius traces of central kernels and Springer tests bound the extreme Bernstein exponent, while trace identities synchronize the governing nilpotent orbit across places and coefficient embeddings. For manuscript 5 (family 014; 2026-09-24), a bilinear detector tests supported spectral local cohomology against finite evaluations. Orbit distributions, compact Weil lifts, and noncancellation separate the nilpotent supports, and rational descent then includes the noncuspidal functions. For manuscript 6 (family 014; 2026-10-05), a curve with several separating nodes realizes all marked types simultaneously. Nilpotent cotangent dimension bounds and transverse modifications control specialization, after which commuting unipotent torus actions remove all flag enhancements without destroying the object.

For manuscript 7 (family 014; 2026-10-05), the companion unramified result supplies global purity. Local functional equations compare semisimple parameters independently of monodromy; regularity of the adjoint local factor then identifies the monodromy orbit needed to apply the local temperedness criterion. For manuscript 8 (family 014; 2026-09-24), a hypothetical missing component is transported along a trait to a geometric specialization. Levi reduction and transverse Hecke modifications provide cotangent tests whose nonvanishing prevents that spectral summand from being annihilated.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 014; 2026-10-05), manuscript 2 (family 014; 2026-10-05), manuscript 3 (family 014; 2026-10-05), manuscript 4 (family 014; 2026-09-24), manuscript 5 (family 014; 2026-09-24), manuscript 6 (family 014; 2026-10-05), manuscript 7 (family 014; 2026-10-05), manuscript 8 (family 014; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

• OpenAI, Constructible tame Hecke eigensheaves in positive characteristic, 2026-10-05; citation key OAI:Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main, eq:main-eigen.
• OpenAI, Frobenius Structures on Tame Hecke Eigensheaves, 2026-10-05; citation key OAI:Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main, eq:regular-monodromy, eq:main-eigenmaps.
• OpenAI, Global Arthur Enhancements of Cuspidal Excursion Parameters, 2026-10-05; citation key OAI:Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main, intro:global-specialization.
• OpenAI, Ramanujan–Arthur Decompositions of Cuspidal Functions at Full Finite Level, 2026-09-24; citation key OAI:Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: main:decomposition, intro:decomposition.
• OpenAI, Rationality of the Canonical Unramified Arthur Filtration, 2026-09-24; citation key OAI:Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: intro:main, intro:main-map.
• OpenAI, Tame Hecke Eigensheaves with Several Marked Points, 2026-10-05; citation key OAI:Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: intro:main, intro:eigen.
• OpenAI, Temperedness at ramified places for globally generic exceptional groups, 2026-10-05; citation key OAI:Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, The Restricted Geometric Langlands Equivalence in Positive Characteristic, 2026-09-24; citation key OAI:The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:full-support.
• OpenAI catalogue entry.