\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=1.1in]{geometry} \usepackage{booktabs,array} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \hypersetup{pdftitle={The Hodge conjecture for Fermat fourfolds of odd degree at most 199}, pdfauthor={Rifat Jumagulov}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\PP}{\mathbb{P}} \newcommand{\Ds}{\mathfrak{D}} \newcommand{\fr}[1]{\langle #1\rangle} \newcommand{\V}{V} \DeclareMathOperator{\Gal}{Gal} \newcommand{\zt}{\zeta} \newcommand{\C}{\mathbb{C}} \newcommand{\Qbar}{\overline{\Q}} \title[Hodge classes on odd-degree Fermat fourfolds]{The Hodge conjecture for Fermat fourfolds of odd degree at most 199} \author{Rifat Jumagulov} \email{jum.rifm@gmail.com} \subjclass[2020]{Primary 14C30, 14C25; Secondary 14J70, 11T24} \keywords{Hodge conjecture, Fermat varieties, algebraic cycles, Jacobi sums, computer-assisted proof} \date{July 2026} \begin{document} \raggedbottom\vfuzz=3pt \begin{abstract} Let $X^4_m=\{x_0^m+\dots+x_5^m=0\}\subset\PP^5$ be the Fermat fourfold of degree $m$. We give a computer-assisted proof of the Hodge conjecture for $X^4_m$ for every odd $m\le199$: three geometric closure criteria are combined with an exhaustive machine census of the Hodge $(2,2)$-orbits, which is accompanied by a completeness proof and by re-verifiable per-orbit certificates. The three closure mechanisms for Hodge $(2,2)$-characters are as follows. First, if the character multiset splits into two zero-sum triples, then the corresponding rational Hodge block is transported from a $(1,1)$-substructure of a product of Fermat curves and is therefore algebraic. Second, algebraicity follows for characters that, after adjoining two vanishing pairs, decompose into an Aoki standard sextuple and a grade-$2$ Hodge quadruple. Third, the exceptional class at $m=33$ has a lift to level $66$ that is quasi-decomposable; its algebraicity then descends along the finite morphism $X^4_{66}\to X^4_{33}$. Together with the known decomposable, quasi-decomposable, and standard cases, these criteria cover all odd degrees $m\le199$. The census runs over the $89$ machine-examined levels $21\le m\le199$, $m\neq23$ (smaller odd degrees and $m=23$ are classical), classifies all $78{,}299$ Galois-orbit representatives, and isolates \emph{thirteen} orbits beyond decomposability, quasi-decomposability and Aoki's standard cycles: six close by the $*$-split criterion and the remaining \emph{seven} by the two-pair and level-lifted closures with level transport, leaving \emph{none}. Each classification carries a machine-checked witness --- positive for the classified orbits, re-established negative screenings for the terminal ones --- The enumeration is reproduced by two further implementations: an algorithmically independent census over all $89$ levels, and a direct brute-force enumeration --- every sorted zero-sum sextuple generated exhaustively --- through $m=143$, re-run live through $m=45$. Seven of the thirteen are gap classes outside Aoki's standard lattice calculus --- the $m=33$ witness with its inflations at $99,165$, the first $m=39$ orbit with $117,195$, and the $\Q(\zt_{21})$ orbit at $m=105$ --- and are new to the author's knowledge; for the other six algebraicity is also derivable from that calculus, the explicit presentations given here being the new content. Finally, an exact Jacobi-sum computation at $p=67$ shows that no cycle defined over $\Q(\zt_{33})$ projects nontrivially onto the exceptional $m=33$ block. More generally, for every finite extension of $\Q(\zt_{33})$ over which a certifying cycle is defined, every residue degree above $67$ is divisible by $6$. \end{abstract} \maketitle \section{Introduction and main results}\label{sec:intro} Let $X^4_m\subset\PP^5$ be the Fermat fourfold $x_0^m+\dots+x_5^m=0$, $G=\mu_m^6/\mu_m$, and for a character $a=(a_0,\dots,a_5)\in(\Z/m)^6$ with $\sum a_i\equiv0$, $a_i\neq0$, let $\V(a)\subset H^4_{\mathrm{prim}}$ be the ($1$-dimensional) eigenspace; $a$ is a Hodge $(2,2)$ character iff $\sum_i\fr{ta_i}=3m$ for all $t\in(\Z/m)^\times$, where $\fr{x}\in\{1,\dots,m-1\}$ is the residue (Shioda \cite{Shioda79}). The known algebraicity machinery is: (i)~the \emph{induced structure} (Shioda \cite{Shioda79}, Shioda--Katsura \cite{SK}), which transports algebraicity through decompositions---da Silva's conditions $(P1)/(P2)$ are quasi-decomposability in the semigroup $M_m$ \cite{daSilva}; and (ii)~\emph{standard cycles} (Aoki \cite{Aoki87}). da Silva \cite[Prop.~3.4 (= Prop.~3.6 of the arXiv version)]{daSilva} found the first Hodge class beyond both at $m=33$ and proposed a candidate cycle $[W]$ (Question~1). (His closures --- $m\le20$ [Thm.~2.7], $m$ prime or $m=4$ [Thm.~2.6], $m=p^2$ [Thm.~2.8], $m=21,27$ [Thm.~3.3], and fourfolds of degree $m\le100$ coprime to $6$ [Prop.~3.1], numbering of the published version \cite{daSilva} --- leave the composite non-square degrees with $3\mid m$, beyond his $21,27$, untouched; every class closed below lies there.) In summary: \begin{center}\footnotesize \begin{tabular}{ll>{\raggedright\arraybackslash}p{4.9cm}} \toprule mechanism & source & covers\\ \midrule induced structure & Shioda \cite{Shioda79}; Shioda--Katsura \cite{SK} & decomposable, quasi-decomposable\\ standard cycles; lattice calculus & Aoki \cite{Aoki87,Aoki00} & standard characters; $\nu\in S_m$; Thm.~0.1 degree forms\\ computational closures & da Silva \cite{daSilva} & $m\le20$; $m$ prime or $4$; $p^2$; $21,27$; $m\le100$ coprime to $6$\\ Theorems~\ref{thm:A}--\ref{thm:App} $+$ census & this paper & the residual odd orbits through $199$\\ \bottomrule \end{tabular} \end{center} We work in his semigroup formalism. The main result is the following. \begin{theorem}[main theorem]\label{thm:main} For every odd $m$ with $3\le m\le199$ the Hodge conjecture holds for the Fermat fourfold $X^4_m$. \end{theorem} \noindent It follows from the three closure criteria below (Theorems~\ref{thm:A}, \ref{thm:Aprime}, \ref{thm:App}), the level-transport lemmas, and the census (Theorem~\ref{thm:B}), which certifies that these criteria and the known machinery leave no orbit uncovered; odd degrees below the census range are classical (\cite[Thms.~2.6--2.7]{daSilva}: $m\le20$, and $m$ prime --- which covers $m=23$). What is genuinely new is delimited as follows: the level-lifted closure of the $m=33$ class (Theorem~\ref{thm:App}) is a new mechanism; the $*$-split criterion (Theorem~\ref{thm:A}) is a new criterion extracted from the classical Shioda--Katsura correspondence; the two-pair presentations (Theorem~\ref{thm:Aprime}) are new explicit applications of Aoki's calculus; and the census with its certificates is a new computational classification. Of the orbits they close, those at $m=33$ (with $99,165$), the first orbit at $m=39$ (with $117,195$), and the $\Q(\zt_{21})$ orbit at $m=105$ are gap classes --- outside the standard lattice calculus --- while the second $m=39$ orbit (with $117,195$), $m=45$ (with $135$) and the $\Q(\zt_7)$ orbit at $m=105$ are also derivable from \cite[Lem.~4.1(iii)]{Aoki00}, the contribution there being the explicit presentations. We prove: \medskip \noindent\emph{Notation.} Throughout $m\ge3$ ($m\le2$ gives a hyperplane or a smooth quadric, where the conjecture is classical). A multiset $a$ of nonzero residues with $\sum a_i\equiv0\pmod m$ has \emph{grade} $g$ if $\sum_i\fr{ta_i}=gm$ for \emph{every} $t\in(\Z/m)^\times$ (the constancy is the Hodge condition: grade-$3$ sextuples are the Hodge $(2,2)$ characters of $X^4_m$, grade-$2$ quadruples the $(1,1)$ characters of the surface $X^2_m$); $\sim$ is equality of multisets up to coordinate permutation, and the \emph{content} of a class $a=(a_0,\dots,a_5)$ is $\gcd(a_0,\dots,a_5,m)$ (content-$e$ classes are inflations from level $m/e$). $\mathrm{claim}_m(a)$ (or $\mathrm{claim}(a)$ when $m$ is clear) is Aoki's \textsc{claim}$(a)$: $\V(a)$ is spanned by algebraic cycle classes \cite[Introduction]{Aoki87} --- equivalently, some \emph{rational} algebraic cycle class has nonzero $\V(a)$-component, $\omega_a(Z)\neq0$. Three levels are kept distinct throughout: eigenlines are indexed by \emph{ordered} tuples ($\V(b)\neq\V(b')$ for distinct ordered characters even when they agree as multisets; coordinate permutations are realized by automorphisms of $X^4_m$, which is why claim is invariant under them, and it is likewise a Galois-orbit-level property); $M(a):=\bigoplus_b\V(b)$ is the rational Galois block, with rational form $M(a)_\Q:=M(a)\cap H^4(X^4_m,\Q)$ (the sum is over a full Galois orbit, so $M(a)_\Q\otimes\C=M(a)$; the same applies verbatim to the grade-$2$ blocks used below) --- the sum over the distinct \emph{ordered} conjugates $b=ta$, $t\in(\Z/m)^\times$, their number being the index of the ordered stabilizer $\{t:ta=a\text{ coordinatewise}\}$, \emph{not} the number of multiset classes; and the census classifies sorted representatives modulo both actions --- legitimate since claim is invariant under both, but never identified with an eigenline count. An unqualified \emph{class} always means a Galois orbit of characters (equivalently its sorted canonical representative); \emph{Hodge class} alone keeps its usual cohomological meaning. $\omega_a(Z)$ denotes the $\V(a)$-component of the cycle class of $Z$ (Aoki's averaged projector \cite[Introduction]{Aoki87}). $*$ is Aoki's juxtaposition --- concatenation of the entry tuples; characters of $X^r_m$ and $X^s_m$ concatenate to one of $X^{r+s+2}_m$ ($n=r+s+2$ in Aoki's indexing by cohomological degree), grades adding \cite[\S1]{Aoki87} --- and, for even $s$, $\Ds^s_m$ his set of $(s+2)$-entry characters that are coordinate permutations of $(a_0,-a_0,\dots,a_{s/2},-a_{s/2})$ ($s/2+1$ vanishing pairs), represented by linear cycles \cite[\S1, Thm.~1-1]{Aoki87}; only $\Ds^2_m$ is used below. $M_m$ is da Silva's additive semigroup of solutions $(x_1,\dots,x_{m-1};y)$ in non-negative integers $x_i$ (multiplicity vectors), $y>0$, of $\sum_i\fr{ti}\,x_i=my$ for all $t$ (multiplicities intrinsic), $M_m(g)$ its grade-$g$ part; a class is \emph{decomposable} if it splits in $M_m$ and \emph{quasi-decomposable} if it splits after adjoining one element of $M_m(1)$ --- da Silva's $(P1)/(P2)$ \cite[Def.~2.4]{daSilva}. $\#$ is his pairing: for $\beta=(b_0,\dots,b_{r+1})$, $\gamma=(c_0,\dots,c_{s+1})$ with nonzero entries and $b_{r+1}+c_{s+1}\equiv0$ (the set $U^{r,s}_m$), $\beta\#\gamma=(b_0,\dots,b_r,c_0,\dots,c_s)$ \cite[before Cor.~2.3]{daSilva}. $\sigma_{p,x}$ denotes the standard characters (Proposition~\ref{prop:D}). $u(a,p)$ is the scalar by which geometric Frobenius at a split prime $p$ acts on $\V(a)(n/2)$ --- the Tate-normalized Jacobi sum $j(a)/p^{n/2}$ (the Jacobi-sum count of diagonal hypersurfaces, \cite[pp.~500--502]{Weil49}; \cite{Weil52}) --- so a divisorial class has $u=1$ at split primes whose Frobenius fixes the class of a certifying divisor (the second $m=39$ class of \S\ref{sec:A} calibrates this in-text). Three symbols are kept apart: $u(a,p)$ above is the \emph{local} normalized scalar; $\chi_a$ is the associated \emph{global} Jacobi-sum Hecke character \emph{in its Tate-normalized form}: Weil's unnormalized character $J_a$ \cite{Weil52} has $|J_a(\mathfrak p)|=N\mathfrak p^{\,n/2}$ in every embedding ($n$ the cohomological degree of the ambient Fermat variety), so $J_a$ itself is \emph{not} of finite order, and $\chi_a:=J_a/N^{n/2}$ is its finite-order part (grade constancy makes the infinity type of $J_a$ the pure norm power $N^{n/2}$, which the normalization removes) --- every use of $\chi_a$ in this paper is of the normalized character, whose value at a split prime is the local scalar $u(a,p)$; it is multiplicative under juxtaposition, $\chi_{\alpha*\beta}=\chi_\alpha\chi_\beta$ (the normalizing powers add along the weights; the machine-verified instances are the displayed $u$-identities); and $\nu(a)\in\Z^{m-1}$ is Aoki's \emph{multiplicity vector}, additive under juxtaposition, $\nu(\alpha*\beta)=\nu(\alpha)+\nu(\beta)$ (Aoki writes $u$ for $\nu$). $S_m$ denotes Aoki's lattice: the subgroup of $\Z^{m-1}$ generated by the $\nu(\delta)$ over the vanishing pairs $\delta=\{k,m-k\}$ and the $\nu(\sigma_{p,x})$ over Aoki's standard elements for all primes $p\mid m$ --- for odd $p$ the AP family of Proposition~\ref{prop:D}, at even $m$ also Aoki's $p=2$ family $\sigma_{2,x}=(x,\,x+\frac m2,\,m-2x,\,\frac m2)$ \cite[\S2]{Aoki00} --- the ``standard calculus''; this generating family is verbatim the generator set of Aoki's lattice $S_m$ (Proposition~\ref{prop:ident}), with membership decided exactly by integer linear algebra (Hermite normal form), and $B^{4}_m$ the grade-$3$ Hodge-character set of the Fermat fourfold at level $m$; in lattice statements we freely identify $a$ with $\nu(a)$ ($a\in B^4_m\smallsetminus S_m$ means $\nu(a)\notin S_m$ --- a statement about the formal calculus, not about the character group). The witness $w=(1,4,16,22,25,31)$ below is, up to coordinate permutation, the Galois conjugate $7w$ of da Silva's displayed representative $(7,10,13,19,22,28)\sim7w$ of \cite[Prop.~3.4 (= Prop.~3.6 of the arXiv version)]{daSilva}. \medskip \noindent\emph{Symbols at a glance.} \begin{center}\small \begin{tabular}{l>{\raggedright\arraybackslash}p{10.4cm}} \toprule $a=(a_0,\dots,a_5)$ & an \emph{ordered} character; $\V(a)$ its one-dimensional eigenline\\ $M(a)$ & the rational Galois block $\bigoplus_{t}\V(ta)$ (sum over distinct ordered conjugates)\\ $\mathrm{claim}(a)$ & $\V(a)$ is spanned by algebraic cycle classes\\ grade $g$ & $\sum_i\fr{ta_i}=gm$ for every unit $t$ (the Hodge condition)\\ $B^4_m$ & the grade-$3$ Hodge characters of $X^4_m$\\ $M_m$, $M_m(g)$ & da Silva's semigroup of solutions and its grade-$g$ part\\ $\Ds^s_m$ & Aoki's all-vanishing-pairs classes (linear cycles); only $\Ds^2_m$ is used\\ $*$, $\#$ & juxtaposition of characters; da Silva's pairing on $U^{r,s}_m$\\ $\sigma_{p,x}$ & Aoki's standard characters (Proposition~\ref{prop:D})\\ $\nu(a)\in\Z^{m-1}$ & the multiplicity vector (additive: $\nu(\alpha*\beta)=\nu(\alpha)+\nu(\beta)$)\\ $S_m$ & Aoki's lattice, spanned by the pair and standard vectors (Prop.~\ref{prop:ident})\\ \emph{gap class} & a Hodge class with $\nu\notin S_m$ (nonzero in Aoki's gap group $A_m/S_m$)\\ $J_a$, $\chi_a$ & the unnormalized and the Tate-normalized Jacobi-sum Hecke characters\\ $u(a,p)$ & the local scalar $j(a)/p^{n/2}$ on $\V(a)(n/2)$, $n=\#\text{entries}-2$ ($n=4$ for sextuples, $n=2$ for grade-$2$ quadruples)\\ \bottomrule \end{tabular} \end{center} \medskip \begin{theorem}[$*$-split closure]\label{thm:A} Let $a$ be a Hodge $(2,2)$ character of $X^4_m$, $m$ arbitrary, whose $6$-multiset splits as two zero-sum triples $\beta'\uplus\gamma'$. Then the rational Hodge classes of $M(a)$ are algebraic: each is the image, under the algebraic joining-line correspondence of the Shioda--Katsura structure map, of a $\Q$-linear combination of divisor classes on $X^1_m\times X^1_m$ furnished by Lefschetz $(1,1)$ (no single ruled-surface representative is claimed). \end{theorem} In particular the classes at $m=39,117,195$ (two orbits each), none quasi-decomposable or standard, are closed; the orbit of $(1,7,16,22,34,37)$ is a certified gap family ($\nu\notin S_m$ at $39$ and at its inflations $117,195$); so is the $m=33$ witness of Theorem~\ref{thm:App}, at $33$ and at its inflations $99,165$ --- every one of these verdicts carries a modular kernel witness in the ancillary verifier, together with the doubling certificate $2\nu\in S_m$, not covered by the published mechanisms compared in \S\ref{sec:dossier} (to the author's knowledge), while the second orbit's $\nu$ lies in $S_{39}$ (an explicit six-generator $\pm1$ certificate), so its algebraicity is also derivable from the standard calculus (Lemma~4.1(iii) of \cite{Aoki00}) --- the closure certificates and the join-transport construction scheme are new for both. \begin{theorem}[two-pair standard closure]\label{thm:Aprime} Let $a$ be a Hodge $(2,2)$ character of $X^4_m$ admitting vanishing pairs $p_1=\{k,m-k\}$, $p_2=\{\ell,m-\ell\}$ with \[ a\uplus p_1\uplus p_2=S\uplus Q\qquad(\text{equality of }10\text{-element multisets}), \] where $S$ is an Aoki $5$-standard sextuple and $Q$ a grade-$2$ Hodge quadruple. Then the rational Hodge classes of $M(a)$ are algebraic. \end{theorem} This applies to---and, by exhaustive search, uniquely presents---the three remaining $5\mid m$ classes of the census: \begin{center}\small \begin{tabular}{llll} \toprule $m$ & $a$ & $p_1,p_2$ & $S\uplus Q$\\ \midrule $45$ & $(1,19,20,28,30,37)$ & $(5,40),(10,35)$ & $(1,10,19,28,37,40)\uplus(5,20,30,35)$\\ $105$ & $(3,24,50,66,85,87)$ & $(15,90),(45,60)$ & $(3,24,45,66,87,90)\uplus(15,50,60,85)$\\ $105$ & $(1,22,43,64,90,95)$ & $(5,100),(20,85)$ & $(1,22,43,64,85,100)\uplus(5,20,90,95)$\\ \bottomrule \end{tabular} \end{center} ---none quasi-decomposable, standard, or Theorem~\ref{thm:A}-splittable: closing, with the induced $m=135$ copy, the $p=5$-standard-block classes of the census. Of these, the $\Q(\zt_{21})$ orbit at $105$ is a certified gap class, not covered by the published mechanisms compared in \S\ref{sec:dossier} (to the author's knowledge); $m=45$, $135$ and the $\Q(\zt_7)$ orbit at $105$ have $\nu\in S_m$ (explicit $\pm1$ certificates, re-summed by the ancillary verifier), so for those three the algebraicity is also derivable from \cite[Lem.~4.1(iii)]{Aoki00} --- and, for $45$ and $135$, from the degree forms of \cite[Thm.~0.1(i)]{Aoki00} via the reduction opening its \S4 proof (quoted in Appendix~\ref{app:aoki}) --- the explicit two-pair presentations are the new content. \begin{theorem}[level-lifted closure; the $m=33$ witness]\label{thm:App} Let $w=(1,4,16,22,25,31)$, the unique non-quasi-decomposable non-standard Hodge class of $X^4_{33}$ (uniqueness is the census's, used for Theorems~\ref{thm:B} and~\ref{thm:C}; the proof below uses only the displayed identities), and $a=2w\bmod 66$ its induced copy on $X^4_{66}$. Then \[ a\uplus\{1,65\}\uplus\{25,41\}=Q\uplus S,\qquad Q=(1,25,44,62),\ S=(2,8,32,41,50,65), \] with $Q$ a grade-$2$ Hodge quadruple and $S$ quasi-decomposable at level $66$ via $S\uplus\{33,33\}=(2,32,33,65)\uplus(8,33,41,50)$, the self-paired element $(33,33)\in M_{66}(1)$ being its unique quasi-witness. Hence $\V(a)$ is algebraic (Lefschetz $+$ da Silva $+$ Aoki~1-4, as in Theorem~\ref{thm:Aprime}), and $\V(w)$ is algebraic by descent along $\pi\colon X^4_{66}\to X^4_{33}$, $(x_i)\mapsto(x_i^2)$ (Lemma~\ref{lem:descent}). \end{theorem} \begin{theorem}[completion of the odd-degree census; $m\le199$]\label{thm:B} For every odd $m\le199$, every grade-$3$ Hodge $(2,2)$ class of $X^4_m$ is algebraic: the classes beyond $\text{decomposable}\cup\text{quasi}\cup\text{standard}$ are exhausted by Theorems~\ref{thm:A}, \ref{thm:Aprime}, \ref{thm:App} and level transport (the inflation/descent pair). In particular Theorem~\ref{thm:main} holds --- the Hodge conjecture is true \emph{in full} for every Fermat fourfold of odd degree $\le199$: in codimension $2$ the non-primitive summand of $H^4$ is spanned by the algebraic class $h^2$, so the statement follows from the primitive one; codimension $1$ is Lefschetz $(1,1)$; and codimension $3$ follows from codimension $1$ by hard Lefschetz --- $L^2\colon H^2(\Q)\xrightarrow{\ \sim\ }H^6(\Q)$ is an isomorphism of Hodge structures, so every $(3,3)$ Hodge class is $L^2$ of a $(1,1)$ Hodge class, and $L^2$ of a divisor class is algebraic. (Codimensions $0$ and $4$ are trivial: $H^0$ and $H^8$ are spanned by the classes of $X$ and of a point.) \end{theorem} \begin{theorem}[field obstruction; Question~1]\label{thm:C} Let $a_0=(7,10,13,19,22,28)=7w$ be da Silva's representative of the $m=33$ orbit. No algebraic cycle defined over $\Q(\zt_{33})$ --- a cycle of codimension $2$ with $\Q$-coefficients, $Z\in CH^2(X^4_{33,\Qbar})_\Q$ fixed by $G_{\Q(\zt_{33})}$ --- has nonzero projection onto the Galois orbit of $\V(a_0)$: \emph{geometric} Frobenius $\mathrm{Frob}^{\mathrm{geom}}_{\mathfrak p}$ at any prime $\mathfrak p$ above $67$ acts on every conjugate eigenline by a primitive sixth root of unity, of exact order $6$ --- an exact certificate $u(a_0,67)=1+\zt_3=\zt_6$ (for the pinned character choice of \S\ref{sec:C}) computed in $\Z[x]/\Phi_{33}$ with no floating point. (Under an alternative Jacobi-sum definition $j_{\mathrm{alt}}=-j$ the formula relating $j_{\mathrm{alt}}$ to geometric Frobenius acquires the compensating sign; the geometric eigenvalue remains $\zt_6^{\pm1}$.) Since $W$ --- da Silva's candidate cycle, defined over $\Q$ by explicit equations \cite[Question~1]{daSilva} --- and all its $\Q(\zt_{33})$-rational translates are fixed by $G_{\Q(\zt_{33})}$, their classes have zero projection: \emph{the answer to Question~1 is NO at $m=33$}. \end{theorem} \begin{corollary}[local residue-degree obstruction]\label{cor:resdeg} The obstruction is local and exact: if a cycle defined over a finite extension $L/\Q(\zt_{33})$ --- not assumed Galois over $\Q(\zt_{33})$; the constraint is per prime --- has nonzero projection onto the Galois orbit of $\V(a_0)$, then at every prime $\mathfrak q\mid\mathfrak p\mid67$ of $L$ one needs $u(a_0,67)^{f(\mathfrak q/\mathfrak p)}=1$, whence $6\mid f(\mathfrak q/\mathfrak p)$: every residue degree of $L$ above $67$ is divisible by $6$. (No hypothesis on the ramification of $L/\Q(\zt_{33})$ enters: the eigenline's character is unramified at $67$ over the base, so inertia at $\mathfrak q$ acts trivially and Frobenius acts through $u^{f(\mathfrak q/\mathfrak p)}$.) \end{corollary} \begin{remark} The same machinery, run at the closed levels, computes values of order $9$ at $m=45$ and of orders $42$ and $7$ at $m=105$ at sampled split primes --- computational observations, not part of any theorem here; they indicate that the cycles of Theorem~\ref{thm:Aprime} likewise cannot be $\Q(\zt_m)$-rational. \end{remark} \begin{proposition}[standard $=$ AP]\label{prop:D} Let $m$ be odd. Aoki's standard characters $\sigma_{p,x}$ for an odd prime $p\mid m$ --- defined for $x$ with $d/\gcd(x,d)>2$, $d=m/p$ \cite[\S1]{Aoki87} --- are exactly the padded arithmetic-progression characters $\{x,x+m/p,\dots,x+(p-1)m/p,-px\}$ with $px\not\equiv0\pmod m$, of grade $(p+1)/2$; grade-$3$ standard classes exist iff $5\mid m$ and $m>5$. \end{proposition} \medskip \noindent\emph{Computational verification and reproducibility (summary).} Theorem~\ref{thm:A}: proved (transport pinned to the Shioda--Katsura primary text). Theorem~\ref{thm:Aprime}: proved (a citation chain through Aoki's Theorems~1-1, 1-4(i),(ii), 2-1 \cite{Aoki87} $+$ Lefschetz $(1,1)$ $+$ an inflation lemma; all identities verified exactly, including $u(a,p)=u(S,p)u(Q,p)$). Theorem~\ref{thm:B}: machine-verified census, reproduced exactly by an algorithmically independent implementation (the four-tier certificate structure is laid out in \S\ref{sec:B}). Theorem~\ref{thm:C}: proved, exact certificate. Proposition~\ref{prop:D}: read off the primary construction. We give no explicit defining equations for a cycle on $X^4_{33}$ (Theorem~\ref{thm:App}'s certifying cycles are constructed at level $66$ and transported by the finite morphism). \section{Standard cycles are the AP family (Proposition~\ref{prop:D})}\label{sec:D} For odd $p\mid m$, $d=m/p$, and $x$ with $d/\gcd(x,d)>2$, the standard character is the multiset $\{x,x+d,\dots,x+(p-1)d,\,m-px\}$ of grade $(p+1)/2$ (Aoki \cite[\S1]{Aoki87}; da Silva \cite{daSilva}, Appendix). For \emph{odd} $m$ the admissibility hypothesis is equivalent to the visible one: $d$ is odd, so $d/\gcd(x,d)\le2$ forces $d\mid x$, i.e.\ $px\equiv0\pmod m$ --- exactly the degenerate case excluded in the statement. Grade $3$ forces $p=5$; at $m=5$ every candidate degenerates ($5x\equiv0$), whence ``$m>5$''. The identification is load-bearing for Theorem~\ref{thm:B}: it is what makes the census's standard classifier complete for odd $m$ (Lemma~\ref{lem:complete}). The padded arithmetic-progression family is exactly the witness family of the field-of-definition threshold of the companion papers: at the surface level in \cite{P1} (with explicit Hasse--Davenport witnesses), and in general even dimension in \emph{Galois-invariant ranks of middle algebraic cycles on Fermat varieties} (in preparation). \qed \begin{remark}[even $m$]\label{rem:evenstd} For even $m$ (used only in the even census of \S\ref{sec:even}) the admissibility hypothesis is strictly stronger than $px\not\equiv0$: it excludes $x\equiv d/2\pmod d$. Every excluded multiset contains a vanishing pair --- $(x+k_1d)+(x+k_2d)\equiv0\pmod m$ reduces to $1+2j+k_1+k_2\equiv0\pmod5$ for $x=d/2+jd$, solvable with $0\le k_11$ is the $g$-inflation of $\sigma_{5,x/g}$ at level $m/g$, where the gcd is $1$, so Lemma~\ref{lem:infl} and Thm.~2-1 cover every admissible block.) (3)~claim$(S*Q)$ is \cite[Thm.~1-4(i)]{Aoki87} ($r=4$, $s=2$, $n=8$). (4)~claim$(a)$ is \cite[Thm.~1-4(ii)]{Aoki87}: \emph{if there exists $\delta\in\Ds^s_m$ with claim$(\alpha*\delta)$, then claim$(\alpha)$}---applied with $\alpha=a$, $s=2$. \end{proof} \begin{lemma}[inflation]\label{lem:infl} Let $m=e\,m'$ and $\beta$ a character of level $m'$ (all entries nonzero mod $m'$, so all entries of $e\beta$ are nonzero mod $m$). Then $\mathrm{claim}_{m'}(\beta)\Rightarrow\mathrm{claim}_m(e\beta)$. \end{lemma} \begin{proof} $\pi\colon X^n_m\to X^n_{m'}$, $(x_i)\mapsto(x_i^e)$ is a morphism with $\pi(\zt x)=\zt^e\pi(x)$ for $\zt\in\mu_m$, so $\pi^*$ maps $\V(\beta)$ into $\V(e\beta)$ equivariantly (the entries of $e\beta$ are nonzero, so the target eigenline exists); $\pi$ finite surjective gives $\pi_*\pi^*=(\deg\pi)\,\mathrm{id}\neq0$ on $\Q$-cohomology, so $\pi^*\xi$ spans $\V(e\beta)$, and for a representing cycle $Z$, $\omega_{e\beta}(\pi^*Z)=\pi^*(\omega_\beta(Z))\neq0$. (Here $\pi^*Z$ is the pullback in $CH_\Q$: $\pi$ is a finite surjective morphism of smooth projective varieties, hence flat by miracle flatness, and flat pullback is compatible with cycle classes.) \end{proof} \begin{remark} da Silva's quasi-decomposability adjoins one pair and splits $4+4$; the two-pair $6+4$ pattern is outside the definition. The three presentations were found by exhaustive iterated-closure search and are unique per class; equivalently the three classes are sum-preserving two-entry surgeries of standard sextuples ($(10,40)\mapsto(20,30)$ at $45$; $(45,90)\mapsto(50,85)$, $(85,100)\mapsto(90,95)$ at $105$). The characters multiply exactly: $\chi_a=\chi_S\chi_Q$, i.e.\ $u(a,p)=u(S,p)\,u(Q,p)$ at every split prime. (Juxtaposition gives $\chi_a\chi_\delta=\chi_S\chi_Q$ for the adjoined $\delta=p_1*p_2$; the displayed form uses $\chi_\delta=\chi^{k+\ell}(-1)=1$ --- automatic for odd $m$, and at $m=66$ because $k+\ell=26$ is even.) The surviving $m=33$ class resists this genre exhaustively (no augmentation by up to five vanishing pairs admits a certifying partition; $5\nmid33$, so no standard blocks exist). \end{remark} \section{Level-lifted closure of the witness (Theorem~\ref{thm:App})}\label{sec:App} \begin{proof} All identities are exact (machine-verified; the ancillary re-derives each). The augmented multiset partitions as stated; $Q$ is grade-$2$ Hodge. For $S$: the pair $(33,33)$ is a legitimate element of $M_{66}(1)$ ($\fr{33t}+\fr{33t}=66$ for every odd $t$; multiplicities are permitted), and in da Silva's coordinates $S=\beta\#\gamma$ with $\beta=(2,32,65;33)$, $\gamma=(8,41,50;33)$ --- an element of his $U^{2,2}_{66}$, the distinguished entries $33,33$ summing to $0\bmod66$ (the definition preceding \cite[Cor.~2.3]{daSilva}); Corollary~2.3(b) with $(r,s)=(2,2)$ and Lefschetz~$(1,1)$ for $\beta,\gamma$ give claim$(S)$. The hypotheses of Corollary~2.3(b) are exactly: $r,s$ even, algebraicity of both factors, nonzero entries, and the matching condition --- each verified above; the self-paired value $b_3=c_3=33=m/2$ is admissible (Remark~\ref{rem:selfpair}). Then, exactly as in Theorem~\ref{thm:Aprime}: claim$(S*Q)$ by \cite[Thm.~1-4(i)]{Aoki87}; $S*Q\sim a*\delta$ with $\delta=(1,65,25,41)\in\Ds^2_{66}$; claim$(a)$ by \cite[Thm.~1-4(ii)]{Aoki87}. Descent is Lemma~\ref{lem:descent}. \end{proof} \begin{remark}[self-pair admissibility]\label{rem:selfpair} The hypotheses of the pairing statement used above --- membership in $U^{2,2}_{66}$ and Corollary~2.3(b) of \cite{daSilva} --- are: even $r,s$, nonzero entries, algebraicity of both factors, and the matching condition $b_3+c_3\equiv0$. Pairwise distinctness of the entries is not among them, and the underlying structure isomorphism \cite[Thm.~2.2]{daSilva} carries no nondegeneracy hypothesis on the characters; the self-paired distinguished value $b_3=c_3=33=m/2$ is therefore legitimate. This is the one point where the even level is essential: no self-pair exists at an odd level (Remark~\ref{rem:App}(iv)). \end{remark} \begin{lemma}[descent]\label{lem:descent} Let $m=e\,m'$ and $\beta$ a character of level $m'$ (equivalently: $e\beta$ has content $e$ and all entries nonzero mod $m$). Then $\mathrm{claim}_m(e\beta)\Rightarrow\mathrm{claim}_{m'}(\beta)$. \end{lemma} \begin{proof} $\pi\colon X^n_m\to X^n_{m'}$, $(x_i)\mapsto(x_i^e)$ is a finite surjective morphism ($\sum x_i^m=\sum(x_i^e)^{m'}$ identically) intertwining the group actions along the $e$-th-power surjection $G_m\twoheadrightarrow G_{m'}$; hence $\pi_*\V(e\beta)\subseteq\V(\beta)$ (again: the entries of $e\beta$ are nonzero, so $\V(e\beta)$ exists). Since $\pi_*\pi^*=(\deg\pi)\,\mathrm{id}\neq0$ and $\pi^*\V(\beta)=\V(e\beta)$ with both eigenlines $1$-dimensional, $\pi_*\colon\V(e\beta)\xrightarrow{\ \sim\ }\V(\beta)$. Proper pushforward carries algebraic cycle classes to algebraic cycle classes ($\mathrm{cl}\circ\pi_*=\pi_*\circ\mathrm{cl}$), and $\pi$ is finite of relative dimension zero, so there is no degree shift or Tate twist. \end{proof} \begin{remark}\label{rem:App} (i) With the inflation lemma (Lemma~\ref{lem:infl}), claim is level-transparent: $\mathrm{claim}_{m'}(\beta)\Leftrightarrow\mathrm{claim}_m(e\beta)$ --- every lift of an open class is an independent chance to close it. (ii) The mechanism is selective: the two-pair-at-the-double search does not fire for the induced copies of the $m=54$ classes at $108$ or of the $m=50$ class at $100$ (all of these close at their own levels at depth $3$--$4$; even-census receipts, \S\ref{sec:even}), and the beyond-machinery $m=66$ classes resist the single two-pair at level $66$ and its lift $132$ (they too close at depth $3$; even-census receipts, \S\ref{sec:even}). (iii) Consistency with the field obstruction (\S\ref{sec:C}): the constructed cycles (Lefschetz divisors on $X^2_{66}$, Aoki's linear spaces with $\varepsilon=\zt_{132}$, radicals) are not $\Q(\zt_{66})$-rational, and $\Q(\zt_{66})=\Q(\zt_{33})$ --- the obstruction forbids exactly the $\Q(\zt_{33})$-rational certificates, and this theorem produces non-rational ones (their precise minimal field of definition is not computed here). The local Frobenius values agree exactly: $u(2w@66,67)=\zt_6=u(w@33,67)$. (iv) The uniqueness of the quasi-witness $v=33$ shows the self-paired $m/2$ element --- which exists only at even levels --- is strictly load-bearing: this is why the exhaustive level-$33$ search (five augmentation pairs, all partition types) could not see the closure. \end{remark} \section{The census and its completeness (Theorem~\ref{thm:B})}\label{sec:B} For each odd $m$: enumerate grade-$3$ Hodge multisets (meet-in-the-middle on half-unit sum profiles); reduce to Galois orbits; classify decomposable $\to$ quasi $\to$ standard (Prop.~\ref{prop:D}) $\to$ $*$-split (Thm.~\ref{thm:A}) $\to$ survivors; the surviving orbits are then closed by Theorems~\ref{thm:Aprime}/\ref{thm:App} and level transport. The engine reproduces the Shioda/da Silva anchors inside the census range ($(P_m)$ at $m=21,27$; the $m=33$ witness) and was reproduced, on all $89$ odd levels of the census list ($21\le m\le199$, $m\neq23$; the omitted small levels and $m=23$ are classical \cite[Thms.~2.6--2.7]{daSilva}), by a second, algorithmically independent implementation (structured-array meet-in-the-middle), and, as a third method, by a direct brute-force enumeration --- every sorted zero-sum sextuple generated exhaustively, no meet-in-the-middle, no code shared with either implementation --- which reproduces the representative lists and their canonical hashes at every census level $m\le143$ (live in the runner for $m\le45$; the shipped receipt covers the rest). After Theorem~\ref{thm:A} (which closes the $*$-split classes at $39,117,195$), seven beyond-machinery orbits remain: four primitive ($m=33,45,105\times2$) and the induced copies at $99,135,165$; after Theorem~\ref{thm:Aprime} (which closes $45$ and both $105$, hence by Lemma~\ref{lem:infl} also $m=135$) and Theorem~\ref{thm:App} (which closes $33$, hence $99$ and $165$), nothing remains. Induced copies are tied to their primitives by the finite pull--push correspondences of Lemmas~\ref{lem:infl} and~\ref{lem:descent}, which transfer claim in both directions. \begin{lemma}[census completeness]\label{lem:complete} Fix $m$ and consider the shipped generator, classifiers, and canonizer. \begin{enumerate} \item[(i)] Every element of $M_m$ has even length $2g$, $g$ its grade: for nonzero entries $\fr{(m-t)x}=m-\fr{tx}$, so $\sum_i\fr{(m-t)a_i}=Lm-\sum_i\fr{ta_i}$ and grade-constancy forces $g=L/2$. In particular $M_m(1)$ consists exactly of the vanishing pairs $\{k,m-k\}$ (including the self-pair $k=m/2$ at even $m$), and a grade-$3$ sextuple is decomposable exactly when it contains a vanishing pair --- the complementary quadruple is then automatically grade-$2$ Hodge. \item[(ii)] The quasi-decomposability test is complete for da Silva's definition: for a pair-free sextuple $a$, the $8$-multiset $a\uplus\{k,m-k\}$ splits in $M_m$ only with part-lengths $(2,6)$ or $(4,4)$, by (i); in every $(2,6)$ split the pair is either $\{k,m-k\}$ itself or $\{a_i,m-a_i\}$ with $k\in\{a_i,m-a_i\}$, and in both cases the split is the trivial one (the pair part is the adjoined pair, the sextuple part is $a$ itself) --- excluded by the non-triviality clause of \cite[Def.~2.4]{daSilva}. Hence $a$ is quasi-decomposable iff some $a\uplus\{k,m-k\}$ splits into two grade-$2$ Hodge quadruples --- exactly the test performed. \item[(iii)] The meet-in-the-middle generator produces exactly the grade-$3$ Hodge sextuples: the join condition enforces $\sum_i\fr{ta_i}=3m$ for every $t$ in a half-unit system, and the complement identity of (i) extends it to all units; conversely every sorted sextuple is the join of its two sorted halves, each of which occurs in the enumeration of \emph{all} $3$-multisets of nonzero residues. \item[(iv)] Canonization retains the lexicographically least sorted conjugate --- a system of distinct representatives for the $(\Z/m)^\times$-orbits. \end{enumerate} Consequently the census verdicts (decomposable / quasi / standard / $*$-split / survivor) coincide with their definitions level by level; the standard classifier is complete for odd $m$ by Proposition~\ref{prop:D}. \end{lemma} \begin{proof} (i) is the two displayed identities; for the last claim, a vanishing pair contributes $m$ to every unit sum, so the complementary quadruple has constant grade $2$. (ii): the parts have even lengths summing to $8$ with grades $(1,3)$ or $(2,2)$; in a $(2,6)$ split the length-$2$ part is a vanishing pair inside $a\uplus\{k,m-k\}$, and since $a$ is pair-free it is $\{k,m-k\}$ (complement $=a$) or uses one adjoined entry, forcing $k\in\{a_i,m-a_i\}$ and complement $(a\smallsetminus\{a_i\})\uplus\{a_i\}=a$ (multiplicities included; the even-$m$ self-pair case is identical). (iii): if the joined profile sums to $3m$ on the half-units $t$, then at $m-t$ it sums to $6m-3m=3m$; the converse containment is the split of a sorted sextuple at position $3$. (iv): distinct orbits have distinct minima. The implementations are additionally calibrated against direct brute force (the shipped third-method enumeration, which reproduces representative counts and canonical hashes at every census level $m\le143$; the even-parity engine against brute force for $m\le14$) and reproduce the Shioda/da~Silva anchors above. The full $89$-level run --- per-level counts, classification tallies, survivor lists, and per-orbit closure witnesses --- ships as checksummed data artifacts with the ancillary files. \end{proof} \emph{What is certified, and how.} Four tiers, in decreasing logical strength. (1)~\emph{Per-orbit certificates}: for each of the $78{,}299$ orbit representatives across the $89$ levels, a stored witness of its classification (the vanishing pair / the quasi split / the standard parameter / the $*$-split triples), re-verified against the definitions on every run of the supplied runner; for the seven beyond-machinery orbits the runner re-establishes the \emph{negative} screenings live (no vanishing pair, no quasi witness at any $k$, not standard, no zero-sum split), so the terminal labels are re-checked certificates, not labels. (2)~The \emph{completeness lemma} above, which reduces ``no orbit is missed'' to the definitional correctness of the shipped generator and canonizer. (3)~The \emph{independent reimplementation}: an algorithmically independent second program regenerates the census from scratch and matches it level by level (its complete $89$-level log ships; the runner re-executes it live at anchor levels and regenerates a fresh sub-range against the per-level summary table of Appendix~\ref{app:census}). (4)~\emph{Checksummed receipts} of the historical campaign runs: a checksum certifies file integrity, never recomputation --- recomputation is what tiers (1)--(3) execute. A per-level summary table (\texttt{census\_level\_summaries.json}: representative count, tally by kind, survivor list, and the SHA-256 hash of the canonically sorted representative list) makes ``no missed orbit'' checkable level by level: any regeneration, by any implementation, must reproduce the count and the hash. \emph{The bound $199$.} The bound is a computational horizon, not a structural one: every closure mechanism used here is degree-uniform, and nothing in the method stops at $199$. What grows is the census cost --- on the reference machine (Apple M4 Max, Python 3.13) the engine takes $158$\,s at the single level $m=199$, the full two-engine $89$-level sweep of the pinned receipt took $\approx36$ minutes, and the complete independent rerun $\approx12$ minutes --- and $199$ was chosen to include, with margin, the largest levels at which new primitive phenomena appear in the odd census: the two-pair genre at $105$ (the last new primitive beyond-machinery orbit) and the $*$-split gap family through $195=5\cdot39$. The probes beyond the bound (next paragraph) found no new phenomenon. Additional single-engine censuses at the induced-risk levels beyond $199$ --- computational observations: these three levels sit outside the witness-tier receipts of Theorem~\ref{thm:B} --- found: at $231=33\cdot7$ and $297=33\cdot9$ one beyond-machinery orbit each, the induced copy of the $m=33$ class (content $7$, resp.\ $9$), closed by Theorem~\ref{thm:App} and transport; at the control level $273=3\cdot7\cdot13$, two $*$-split classes (closed by Theorem~\ref{thm:A}) and nothing else. They indicate that the conclusion of Theorem~\ref{thm:B} also holds at $m=231,273,297$; the theorem itself does not depend on these levels. \section{The field obstruction and Question~1 (Theorem~\ref{thm:C})}\label{sec:C} \emph{Normalization.} Fix a split prime $p\equiv1\bmod m$ and a multiplicative character $\chi\colon\F_p^\times\to\Qbar^\times$ of exact order $m$ (extended by $\chi(0)=0$). Set \[ S_0(a)=\sum_{\substack{v\in(\F_p^\times)^6\\ v_0+\dots+v_5=0}}\ \prod_{i}\chi^{a_i}(v_i), \qquad j(a)=\frac{S_0(a)}{p-1}\in\Z[\zt_m] \] (the scaling $v\mapsto\lambda v$ is free since $\sum a_i\equiv0$, so $S_0$ is exactly divisible by $p-1$; the certificate asserts this). By Weil's count of points of diagonal hypersurfaces \cite[pp.~500--502]{Weil49}, \cite{Weil52}, \[ \#X^4_m(\F_p)=1+p+p^2+p^3+p^4+\sum_a j(a), \] the sum over all $a=(a_0,\dots,a_5)$ with $a_i\neq0$ and $\sum a_i\equiv0$; the sign in front of the character sum is $+1$ in our even dimension, which is what fixes the normalization below. (The identity is verified exactly in the ancillary at $m=3$, $p=7$, where the left side is computed by brute force: $2801+889=3690$.) Consequently \emph{geometric} Frobenius at $p$ acts on the \'etale eigenspace $\V(a)\subset H^4_{\mathrm{prim}}$ by $j(a)$, with $|j(a)|=p^2$ in every embedding --- hence on $\V(a)(2)$ by $u(a,p)=j(a)/p^2$ (arithmetic Frobenius acts by the inverse scalar; every statement in this paper is about geometric Frobenius); the conjugate eigenlines carry $u(ta,p)$, computed by replacing $\chi$ with $\chi^t$. Under an inhomogeneous convention ($\sum v_i=1$) the Jacobi symbol changes by a sign that the point-count formula compensates; the certificate's asserted $(u')^3=-1$ pins the geometric order regardless. A $30$-digit numeric cross-check against the Gauss-sum product $\prod_ig(\chi^{a_i})/p^3$ corroborates the normalization (ancillary; corroboration only --- the certificate itself is exact). The certificate's character is pinned: $\chi$ is the character of $\F_{67}^\times$ with $\chi(g)=\zt_{33}$ on the smallest primitive root $g=2$ (the shipped code computes discrete logarithms base $2$); replacing $\chi$ by a conjugate character permutes the twenty conjugate values and can exchange $\zt_6\leftrightarrow\zt_6^{-1}$ at $t=1$ --- the convention-independent assertions are the exact order six and the two-value distribution. \begin{proof} Fix a prime $\ell\neq67$ and an embedding $\iota\colon\Qbar\hookrightarrow\Qbar_\ell$; cohomology in this proof is $\ell$-adic \'etale cohomology with $\Qbar_\ell$-coefficients (the eigenspaces correspond under comparison), and the algebraic integers $u(ta_0,67)\in\Z[\zt_{33}]$ are read in $\Qbar_\ell$ through $\iota$; the asserted identities $(u')^3=-1$, $(u')^6=1$, $u'\neq1$ hold in $\Z[\zt_{33}]$, hence under every embedding, so no choice matters. The $\ell$-adic class of a cycle over $K$ is $G_K$-fixed. Take $\mathfrak p\mid67$ in $\Q(\zt_{33})$ ($67\equiv1\bmod33$ splits, good reduction). $\mu_{33}\subset\F_{67}$ makes the $G$-action $\F_{67}$-rational, so Frobenius preserves each $\V(ta_0)$ and acts on $\V(ta_0)(2)$ by $u(ta_0,67)$ as fixed in the Normalization above. An exact integer dynamic program in $\Z[x]/\Phi_{33}$ gives $u(a_0,67)=1+\zt_3=\zt_6$, and the $20$ unit-indexed values $u(ta_0,67)$, $t\in(\Z/33)^\times$ --- the Frobenius scalars on the \emph{twenty} conjugate eigenlines of $M(a_0)$: the ordered stabilizer is trivial ($7t\equiv7$ forces $t=1$), the multiset stabilizer $\langle4\rangle$ has order $5$, so the twenty eigenlines fall into four coordinate-permutation classes of five; the scalar --- a symmetric function of the entries --- is constant on each class, and the four class-values coincide in pairs --- take exactly the two values $\zt_6^{\pm1}$, each on ten eigenlines. The certificate asserts \emph{exact order six} on every conjugate --- $(u')^3=-1$, $(u')^6=1$, $u'\neq1$ --- and the geometric scalar is independent of the Jacobi-sum sign convention, so each eigenvalue is a primitive sixth root of unity, hence $\neq1$. A Frobenius-fixed vector scaled by $\neq1$ vanishes componentwise. Finally, the choice of $\mathfrak p\mid67$ is immaterial: the primes above $67$ in $\Q(\zt_{33})$ form one $\Gal(\Q(\zt_{33})/\Q)$-orbit, and replacing $\mathfrak p$ conjugates the chosen character $\chi$, hence permutes the twenty computed values $u(ta_0,67)$ among themselves; since \emph{each} of them has exact order $6$, the conclusion is the same at every prime above $67$. \end{proof} $W$ is defined over $\Q$ and its translates are $\Q(\zt_{33})$-rational; by the census the $m=33$ class is the unique non-quasi non-standard class at that degree, so its projection vanishes and Question~1 is answered negatively. The YES-branch would have proved the Hodge conjecture at these classes; the NO says only that this candidate cannot certify it, and imposes a sharp local residue-degree constraint on any number field of definition of a certifying cycle. \section{The closed classes' dossiers}\label{sec:dossier} \emph{The $m=33$ witness (closed by Theorem~\ref{thm:App})}: indicated eigenvalue field $\Q(\sqrt{-3})$ --- throughout, \emph{eigenvalue field} means the field generated by the computed values of $\chi_a$ at the sampled split primes, a numerical label and not the fixed field of the multiset stabilizer --- all entries $\equiv1\bmod3$; its Tate-normalized Jacobi-sum Hecke character $\chi_a$ is of finite order (see the Notation; cf.\ \cite{Weil52}), and its computed values at split primes lie in $\mu_6$: kernel primes occur ($p=859$: every conjugate value exactly $1$), and order $6$ is attained ($p=67$, the prime Theorem~\ref{thm:C} uses; only the certified per-prime obstruction is used, and no global order is claimed). The dossier's remaining role is the acceptance test for explicit cycles (Theorem~\ref{thm:App}'s pass by construction). A Weil-class alternative would need a precise carrier: Weil classes on abelian \emph{fourfolds} of Weil type are now unconditionally algebraic (Markman \cite{Markman1}, with Schoen's degeneration argument \cite{Schoen}; for discriminant one there are now further proofs, \cite{FloccariFu} and \cite{vGR}), but the Shioda--Katsura realization carries $\V(w)$ on CM abelian varieties of dimension $\ge10$ (forced by the faithful $\Q(\zt_{33})$-action), and no algebraic compression onto a $\Q(\sqrt{-3})$-Weil fourfold is known --- the route stays conditional on that open compression, and is now unnecessary. \emph{The closed classes $m=45,105$}: their indicated eigenvalue fields $\Q(\zt_9),\Q(\zt_7),\Q(\zt_{21})$ (degrees $6,6,12$) exceed the imaginary-quadratic multiplication field required by the published algebraicity theorems for Weil-type classes known to us (Schoen \cite{Schoen}; Markman \cite{Markman1}); the CM-field secant-sheaf program \cite{Markman2}, which would reach larger fields, is explicitly conditional on a semiregularity ``not addressed yet'' (loc.\ cit.); on the canonical CM carriers (dimensions $12$ and $24$) restriction of scalars to an imaginary quadratic subfield shifts the cohomological degree away from $4$, so no descent to the covered case exists. For the $\Q(\zt_{21})$ class at $105$, Theorem~\ref{thm:Aprime}'s Fermat-internal route is, among the mechanisms compared here, at present the only proof of algebraicity known to the author (a gap class: $\nu\notin S_{105}$, $2\nu\in S_{105}$); the $m=45$ and $\Q(\zt_7)$ $m=105$ classes are also reachable by the standard calculus --- $\nu\in S_{45}$, resp.\ $\nu\in S_{105}$, by explicit four-generator $\pm1$ certificates (re-derivable via the ancillary lattice test), hence algebraic by Lemma~4.1(iii) of \cite{Aoki00} as well, Theorem~\ref{thm:Aprime} supplying the explicit two-pair presentation. The computed per-prime values (orders $9/42/7$ at sampled split primes; computational observations) indicate analogous local residue-degree constraints at the closed levels; explicit equations remain open. \section{Methods, code availability, and provenance}\label{sec:methods} Exact objects throughout (Gauss/Jacobi sums as algebraic integers; the decisive certificate in $\Z[x]/\Phi_{33}$ with no floating point; the exact-twist certificate framework follows the surface companion \cite{P1}); the only floating-point computation anywhere in the paper is the $30$-digit Gauss-sum cross-check of \S\ref{sec:C}, which corroborates a normalization and certifies nothing. Every proof-critical computational claim of Theorem~\ref{thm:B} carries either a stored per-orbit witness certificate or an algorithmically independent reimplementation; the runner re-executes the anchor levels live and re-verifies every stored witness and every supplied certificate against the definitions. The census engines, the closure oracles, the lattice membership test, the exact Frobenius certificate, and a standalone verifier of every closure identity in this paper are provided as ancillary files (\texttt{anc/}; \texttt{run\_all.sh} reproduces the smoke tier --- identities, census anchors, both lattice verdicts, the Theorem~\ref{thm:C} certificate, the engine calibrations, the checksum manifest, the re-verification of all stored per-orbit witnesses (every stored class re-derived to be a grade-$3$ Hodge character, the per-level counts and tallies asserted, the survivor negatives re-screened), the independent modular-kernel lattice witnesses, the independent census at the anchor levels and at the even key levels, the level-summary cross-check on a fresh sub-range, the direct brute-force generator on every census level through $m=45$, the even-sector tier and reader tables, and a second independent implementation of the Theorem~\ref{thm:C} certificate --- in under a minute on the reference machine whose measured runtimes the ancillary README reports; \texttt{run\_full\_census.sh} is the archival tier, the complete from-scratch independent regeneration of all $89$ levels against the pinned summary hashes; the README records the reference configuration and the measured runtimes quoted in \S\ref{sec:B}), together with the deep-closure ledger and the stored degree-$210$ certificate as data artifacts; an algorithmically independent census reimplementation is bundled --- its complete $89$-level run ships as a checksummed receipt, and the runner re-executes it live at the anchor levels --- and the original implementation used during the verification campaign is methodologically independent of both --- a historical instrument, \emph{not} part of the shipped reproducible chain; no claim in this paper rests on it. \emph{Data and code availability.} The complete package --- engines, verifiers, certificates, receipts, checksum manifest, pinned dependencies, and a single-command runner --- is submitted as ancillary files with this manuscript and is licensed for reuse (see \texttt{LICENSE} there). The same package is deposited as an immutable archival release under the concept DOI \texttt{10.5281/zenodo.21670577}, which always resolves to the latest version; that record carries both manuscripts of the two-paper package together with the shared code, certificates and checksum manifest. This manuscript was prepared with AI assistance; all computational claims are backed by the ancillary code and receipts, and the author takes full responsibility for all proofs, code, and bibliographic claims. \appendix \section{The cited statements of da Silva and Aoki}\label{app:aoki} For self-containedness we reproduce the exact statements used, in the numbering and notation of the primary texts. \emph{From \cite{daSilva}} (his $U^r_m$ is the set of characters of $X^r_m$, $C^n_m$ the subset whose eigenspaces are spanned by algebraic cycle classes, $B^n_m$ the Hodge characters; $\langle\;\rangle$ and the semigroup $M_m$ are as in our Notation): \begin{itemize} \item[--] \textbf{Theorem 2.2.} \emph{Let $n=r+s$ with $r,s\ge1$. Then there is an isomorphism} \[ f\colon\bigl[H^{r}_{\mathrm{prim}}(X^r_m,\C)\otimes H^{s}_{\mathrm{prim}}(X^s_m,\C) \bigr]^{\mu_m}\ \oplus\ H^{r-1}_{\mathrm{prim}}(X^{r-1}_m,\C)\otimes H^{s-1}_{\mathrm{prim}}(X^{s-1}_m,\C)\ \xrightarrow{\ \sim\ }\ H^n_{\mathrm{prim}}(X^n_m,\C) \] \emph{with the following properties: {\rm(a)} $f$ is $G^n_m$-equivariant; {\rm(b)} $f$ is a morphism of Hodge structures of type $(0,0)$ on the first summand and of type $(1,1)$ on the second; {\rm(c)} if $n=2p$ then $f$ preserves algebraic cycles, moreover if $Z_1\otimes Z_2\in H^{r-1}_{\mathrm{prim}}(X^{r-1}_m,\C)\otimes H^{s-1}_{\mathrm{prim}}(X^{s-1}_m,\C)$ then $f(Z_1\otimes Z_2)=m\,Z_1\wedge Z_2$, where $Z_1\wedge Z_2$ is the algebraic cycle obtained by joining $Z_1$ and $Z_2$ by lines on $X^n_m$.} --- Lemma~\ref{lem:transport} uses the \emph{second} summand at $r=s=2$, $n=4$: $H^1_{\mathrm{prim}}(X^1_m)^{\otimes2}\to H^4_{\mathrm{prim}}(X^4_m)$, of type $(1,1)$ by (b), carrying algebraic classes to algebraic classes by (c). No $\mu_m$-invariance is imposed on that summand, and no hypothesis on the characters enters. \item[--] \textbf{Notation before Cor.~2.3.} \emph{$U^{r,s}_m=\{(\beta,\gamma)\in U^r_m\times U^s_m\mid\beta=(b_0,\dots,b_{r+1}),\ \gamma=(c_0,\dots,c_{s+1}),\ b_{r+1}+c_{s+1}=0\}$, and for $(\beta,\gamma)\in U^{r,s}_m$, $\beta\#\gamma=(b_0,\dots,b_r,c_0,\dots,c_s)$.} \item[--] \textbf{Corollary 2.3.} \emph{Suppose $n=2p=r+s$, where $r,s\ge1$. {\rm(a)} If $r,s$ are odd and $(\beta',\gamma')\in C^{r-1}_m\times C^{s-1}_m$, then $\beta'*\gamma'\in C^n_m$. {\rm(b)} If $r,s$ are even and $(\beta,\gamma)\in(C^r_m\times C^s_m)\cap U^{r,s}_m$, then $\beta\#\gamma\in C^n_m$.} --- the full hypothesis list of (b): $r,s$ even, both factors of known claim, and the matching condition $b_{r+1}+c_{s+1}=0$. Pairwise distinctness of entries is \emph{not} among them, which is what Remark~\ref{rem:selfpair} uses. \item[--] \textbf{Definition 2.4.} \emph{An element $a\in M_m$ is called decomposable if $a=c+d$ for some $c,d\in M_m$, otherwise indecomposable. An element is called quasi-decomposable if $a+b=c+d$ for some $b\in M_m(1)$ and $c,d\in M_m$ with $c,d\neq a$.} ($M_m$ is his semigroup of \emph{non-negative} solutions, as recalled in the Notation; the clause $c,d\neq a$ is the non-triviality used in Lemma~\ref{lem:complete}(ii).) \end{itemize} \emph{Notation translation:} Aoki's $\mathfrak B^n_m$ is our $B^n_m$ (Hodge characters of $X^n_m$); his $\mathfrak D^n_m$ is our $\Ds^n_m$; his \textsc{claim} is our $\mathrm{claim}$; his $*$ is our juxtaposition; his multiplicity vector $u(\alpha)$ is our $\nu(\alpha)$ (we reserve $u$ for the local Frobenius scalar); his $S_m$ and gap group are the objects of Proposition~\ref{prop:ident}. Indices: he writes $n$ for the cohomological degree, so his $X^n_m$ with $n=4$ is our fourfold and $n=2$ its surface. \emph{From \cite{Aoki87}} (notation as there: $\mathfrak B^n_m$ the Hodge characters of $X^n_m$, $\mathfrak D^n_m$ the classes that are coordinate permutations of $(a_0,-a_0,\dots,a_r,-a_r)$, $r=n/2$; $*$ juxtaposition; \textsc{claim}$(\alpha)$: $\V(\alpha)$ is generated by classes of algebraic cycles on $X^n_m$): \begin{itemize} \item[--] \textbf{Theorem 1-1} (Shioda). \emph{The linear space $L$ represents $\delta\in\mathfrak D^n_m$; more precisely $\omega_\delta(L)\cdot\overline{\omega_\delta(L)}=(-1)^r m^{n+1}$.} \item[--] \textbf{Theorem 1-4} (Shioda \cite{Shioda79}, Ran \cite{Ran}). \emph{Let $r$ and $s$ be non-negative even integers such that $n=r+s+2$, and let $\alpha\in\mathfrak B^r_m$, $\beta\in\mathfrak B^s_m$. Then: (i) if \textup{\textsc{claim}}$(\alpha)$ and \textup{\textsc{claim}}$(\beta)$ are true, then \textup{\textsc{claim}}$(\alpha*\beta)$ is also true; (ii) if there exists $\delta\in\mathfrak D^s_m$ such that \textup{\textsc{claim}}$(\alpha*\delta)$ is true, then \textup{\textsc{claim}}$(\alpha)$ is also true.} \item[--] \textbf{Theorem 2-1.} \emph{For an odd prime $p\mid m$, $d=m/p$, $\gcd(a,d)=1$, the variety $Y$ defined by (2.1) there is a subvariety of $X^{p-1}_m$ of codimension $r=(p-1)/2$ representing $\alpha=\sigma_{p,a}$; more precisely $\omega_\alpha(Y)\cdot\overline{\omega_\alpha(Y)}=(-1)^r p^{p-2}m^p$.} \item[--] \textbf{\S1 definition.} \emph{For a prime $p\mid m$, $d=m/p$, and each $i$ with $d/(i,d)>2$: $\sigma_{p,i}=(i,\,i+d,\dots,i+(p-1)d,\;m-pi)$ for $p\ge3$ --- the ($p$-)standard elements.} \end{itemize} \emph{From \cite{Aoki00}} (verified verbatim against the open-access copy in the Rikkyo University repository, doi:10.14992/00009819). Two errata of the printed text, both verified against the primary and neither affecting the statements as used here: the companion CMUSP \textbf{51} (2002) paper's attribution line after its Theorem~1.2 assigns part (ii) (= Theorem~0.1(i) below) to the Catalan-curves reference; and the $m=28$ gap-generator entry $\xi_{28}$ on p.~185 is a misprint (not a zero-sum character) --- a machine-verified \emph{possible replacement} of exactly the join form the text requires ($\xi_m\in B^2\cup(B^4\cap(\mathfrak A^1*\mathfrak A^1))$) is $\xi_{28}=(1,9,18)*(10,21,25)$, with $\nu\notin S_{28}$, $2\nu\in S_{28}$ (whether it is the intended generator modulo $S_{28}$ is not established; the $168$ closure of \S\ref{sec:even} does not depend on it): \begin{itemize} \item[--] \textbf{Theorem 0.1.} \emph{Suppose the prime factorization of $m$ is one of the following forms: (i) $m=2^a3^b5^c7^d$, where $a,b,c,d$ are non-negative integers such that either $c=0$ or $d=0$; (ii) $m=p^e$ or $2p^e$, $p$ an odd prime. Then the Hodge conjecture is true for all abelian varieties of Fermat type of degree $m$.} Its proof (\S4 there) opens: ``In view of Corollary~3.2, it suffices to show that the Hodge conjecture for $X^n_m$ is true for all $n$'' --- the bridge to the Fermat varieties themselves used in \S\ref{sec:even}; the verification then reduces, via Lemma~4.1 and Theorem~2.1 there, to the gap generators $\xi_m$ at $m=12,15,20,21,28$ (the $m=28$ entry being the misprint recorded above). \item[--] \textbf{Lemma 4.1} (verbatim). \emph{Let $\alpha,\beta\in B_m$. Then: (i) if both $\V(\alpha)$ and $\V(\beta)$ are algebraic, then so is $\V(\alpha*\beta)$; (ii) if $\V(\alpha*\delta)$ is algebraic for some $\delta\in D_m$, then so is $\V(\alpha)$; (iii) if $\alpha\in S_m$, then $\V(\alpha)$ is algebraic.} (Here $B_m=\bigcup_nB^n_m$ and $\V(\alpha)\subset H^n_{\mathrm{prim}}(X^n_m)$ for the $n$ with $\#\alpha=n+2$; we use it at $n=4$.) No hypothesis on $m$ is imposed; the groups $S_m$ and the ``gap group'' are Aoki's own objects (\S2 there), and Proposition~\ref{prop:ident} identifies the implemented lattice with $S_m$ generator-by-generator (his \S2 also proves, via Theorem~2.1, that $2\alpha\in S_m$ for every $\alpha\in B_m$ --- the doubling our $2u$-certificates instantiate). \end{itemize} \section{\texorpdfstring{The implemented lattice is Aoki's $S_m$}{The implemented lattice is Aoki's S\_m}}\label{app:ident} Proved here because the odd-degree novelty labels of \S\ref{sec:intro} rest on it (\S\ref{sec:even}). \begin{proposition}[the implemented lattice is Aoki's $S_m$]\label{prop:ident} Aoki \cite[\S2]{Aoki00} defines $A_m$ as the zero-sum sublattice of the free abelian group on $\Z/m\smallsetminus\{0\}$; $u$ --- our $\nu$ --- as the multiplicity-vector map, additive under juxtaposition, $\nu(\alpha*\beta)=\nu(\alpha)+\nu(\beta)$; the standard elements, verbatim, as $\sigma_{p,a}=(a,\,a+\frac mp,\dots,a+\frac{(p-1)m}p,\,m-pa)$ for $p\ge3$ and $\sigma_{2,a}=(a,\,a+\frac m2,\,m-2a,\,\frac m2)$ for $p=2$, over primes $p\mid m$ ($m>p$) and $0