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+PIXTON COMPLETENESS: A COUNTEREXAMPLE
+=====================================
+A rewrite for curious high school readers
+
+This is a rewrite of the research paper
+"A high-arity counterexample to Pixton completeness in Chow"
+by OpenAI (2026). The mathematics is unchanged: every theorem,
+proposition, and lemma is preserved with the same hypotheses
+and the same conclusions. What changed is the exposition. We
+add plain-language introductions, expand symbols the first time
+they appear, add examples where examples help, and strip out
+redundancy in the prose.
+
+The goal: a high school student who is comfortable with high
+school algebra, and open to learning a few new words (genus,
+moduli, Chow group), should be able to read this whole thing
+and come out understanding what the question was, what the
+answer is, and roughly how the proof works. We do not pretend
+the paper is elementary. Some sections still require real
+mathematical work. But nothing requires graduate-school
+background.
+
+How to read it. If you find yourself stuck deep in Part 4 or
+Part 5, skip forward to the glossary at the very end and read
+onward. Every part begins with a plain-English explanation of
+what that part is trying to do; those intro bits are
+self-contained.
+
+Contents:
+ Part 0 Why you might care
+ Part 1 Background: curves, moduli, and tautological classes
+ Part 2 The main theorem
+ Part 3 The class Y
+ Part 4 Why Y is not in the Pixton span: the formal detector
+ Part 5 Why Y vanishes in Chow: the surface relation
+ Part 6 Compactification and the last stretch of the proof
+ Part 7 What this changes in the field
+ Glossary
+
+
+=====================================================
+PART 0: WHY YOU MIGHT CARE
+=====================================================
+
+Here is the whole paper in one sentence:
+
+ Mathematicians guessed a complete set of rules for measuring
+ curves. We found one measurement their rules cannot explain --
+ and yet that measurement still comes out equal to zero when we
+ actually compute it.
+
+The rest of Part 0 unpacks that sentence.
+
+-------------------------------------------------------------------------
+0.1 What is a curve, in math?
+-------------------------------------------------------------------------
+
+A curve, in the algebraic-geometric sense used in this paper, is a
+shape that locally looks like the real line (a real curve) or the
+complex plane (a complex curve). Key examples:
+
+ - a circle;
+ - a sphere, which is a real surface (dimension 2 over R);
+ - a torus, which is the surface of a donut;
+ - an n-holed donut, for any n >= 0.
+
+"Genus" is the number of holes. A sphere has genus 0, a torus has
+genus 1, a two-holed donut has genus 2, and so on.
+
+A complex curve is a surface where we additionally know how to add
+and multiply points (in a way compatible with the shape). This paper
+works with smooth compact complex curves of a fixed genus g, plus a
+finite set of "marked points" (distinguished labeled locations on
+the curve, numbered 1 through n).
+
+-------------------------------------------------------------------------
+0.2 What is "the space of all curves of genus g"?
+-------------------------------------------------------------------------
+
+Take all curves of a fixed shape -- say, all genus-g curves with n
+marked points. This is not one curve; it is a family of curves that
+itself varies. In fact this family has the structure of a shape (a
+"space") in its own right: it is called the moduli space and is
+written M_{g,n}.
+
+A point in M_{g,n} is "one curve-with-markings." The whole space
+M_{g,n} is all curves packed together, roughly like a rainbow of
+every possible genus-g shape smoothly varying between its neighbors.
+
+M_{g,n} is not compact -- you can let a curve "degenerate" by
+pinching a bit of it down to a point. That produces a boundary;
+adding the boundary produces the compact moduli space of stable
+curves, Mbar_{g,n}. Stable curves are the curves that appear at
+the boundary: ordinary smooth curves plus curves with a limited
+kind of singularity called a node.
+
+-------------------------------------------------------------------------
+0.3 What can we measure on a curve?
+-------------------------------------------------------------------------
+
+At every marked point i there is a natural line bundle of tangent
+directions, whose first Chern class is written psi_i. There are
+also kappa classes kappa_j that measure how the marked points fall
+around a curve.
+
+The important point: psi and kappa are canonical, universal
+measurements. Every smooth curve has them; they vary continuously
+as the curve varies through moduli.
+
+Beyond psi and kappa, we can build more measurements by multiplying
+psi's and kappa's, and by pushing forward along forgetful maps
+(forget a marked point) and gluing maps (glue two marked points,
+creating a node). All measurements built this way are called
+tautological classes.
+
+The set of all tautological classes is a ring called the
+tautological ring, written R^*(Mbar_{g,n}).
+
+-------------------------------------------------------------------------
+0.4 What is a "relation"?
+-------------------------------------------------------------------------
+
+A relation in the tautological ring is a rule that says: "whenever
+you compute these particular measurements combinatorially, they add
+up to zero."
+
+Equivalently, a relation is a formal tautological class that
+actually evaluates to zero when you test it against every cycle.
+
+Two things must be distinguished carefully:
+
+ 1. A formal combination of measurements. This lives in the
+ strata algebra S_{g,n} -- a giant free abelian group on formal
+ symbols. It has no geometry inside it; it is pure bookkeeping.
+
+ 2. The realization map q: S_{g,n} -> A^*(Mbar_{g,n}),
+ which substitutes the actual psi and kappa classes for their
+ formal symbols.
+
+A proposal for "a complete list of relations" is a proposal for
+what q's kernel is: the set of formal combinations that always
+evaluate to zero.
+
+-------------------------------------------------------------------------
+0.5 What did Pixton propose?
+-------------------------------------------------------------------------
+
+In 2012 Aaron Pixton proposed a large explicit family of formal
+relations. His relations look like sums over stable graphs,
+decorated with specific generating series. He guessed that the only
+tautological relations were the ones his rules produce, possibly
+multiplied and moved around by formal gluing operations.
+
+Concretely: let P_{g,n} be the formal span of all Pixton relations
+plus all their formal consequences. Pixton's conjecture (for the
+original system) is:
+
+ ker q = P_{g,n}
+
+for every stable pair (g,n).
+
+In words: every formal combination that vanishes in actual Chow
+theory should be expressible using Pixton's rules plus gluing.
+
+-------------------------------------------------------------------------
+0.6 What this paper proves
+-------------------------------------------------------------------------
+
+This paper constructs a stable pair (g,n) with g = 10^60 and
+n = 3 * C(g,3), and a specific formal class Y in S_{g,n}, with
+two properties:
+
+ (a) q(Y) = 0 in actual Chow theory -- so Y is a real relation,
+ not a formal wish;
+ (b) Y is NOT in Pixton's span P_{g,n}.
+
+Together, (a) and (b) say:
+
+ ker q_{g,n} is STRICTLY BIGGER than P_{g,n}.
+
+Pixton's conjecture, in its "original system" form, is false.
+
+Because cycle class commutes with everything in sight, part (b)
+also shows the analogous statement in rational cohomology: Y is
+a cohomological relation that Pixton's list does not generate.
+
+That is what the paper is about. Parts 1 through 6 build the
+machinery to prove (a) and (b). Part 7 discusses the context.
+
+
+=====================================================
+PART 1: BACKGROUND
+=====================================================
+
+This part puts the standard definitions on the table. Skip
+ahead to Part 2 if you already know this material. Read on
+if you want plain-English context for everything that follows.
+
+--------------------------------------------------------------------------------
+1.1 Curves, markings, and stability
+--------------------------------------------------------------------------------
+
+Fix a base field, here C (complex numbers), and a genus g >= 0.
+A smooth curve of genus g over C is a compact smooth complex
+curve of genus g. Equivalently, it is a compact Riemann
+surface of genus g.
+
+A pointed curve (C; p_1,...,p_n) has n labeled, distinct
+points p_1,...,p_n on C.
+
+Not every (g,n) can have a marked curve that degenerates
+nicely: we need (g,n) to be stable, meaning
+
+ 2g - 2 + n > 0.
+
+Stable pairs with g >= 2 can have any n >= 0. The pair (0,2)
+is not stable, and (0,0), (0,1) are not stable either.
+
+A stable curve of genus g with n marked points is a connected
+curve C that is allowed to have nodes (transverse self-
+intersections; two branches meeting at one point) subject to:
+
+ (a) every smooth component is smooth;
+ (b) marked points are smooth points and distinct;
+ (c) every rational component (genus 0) has at least three
+ "special" points (nodes plus markings).
+
+Condition (c) is exactly the "stability" condition on each
+component.
+
+The moduli space Mbar_{g,n} is a smooth proper Deligne-Mumford
+stack parametrizing stable curves of genus g with n marked
+points. Its coarse space is a projective variety of dimension
+3g - 3 + n. The open subspace M_{g,n} of smooth curves is a
+Zariski-dense open.
+
+We work over C throughout. We work with rational Chow groups
+and rational cohomology.
+
+--------------------------------------------------------------------------------
+1.2 Stable graphs
+--------------------------------------------------------------------------------
+
+Every stable curve has a "combinatorial shadow" called its
+stable graph. A stable graph Gamma of type (g,n) consists of:
+
+ - a finite set V(Gamma) of vertices;
+ - a finite set E(Gamma) of edges (possibly loops); a loop
+ is an edge with both half-edges at the same vertex;
+ - a finite set of half-edges, with each edge having two
+ half-edges and with a "leg" attached to exactly one
+ vertex for each labeled mark;
+ - a genus g_v >= 0 at each vertex v;
+ - a total genus condition sum_v g_v + |E| - |V| + 1 = g;
+ - a labeling of the n "legs" by the marks 1,...,n;
+ - per-vertex stability 2g_v - 2 + n_v > 0, where n_v is
+ the number of half-edges incident to v.
+
+Every vertex of a stable graph corresponds to an irreducible
+component of the corresponding stable curve; every edge to
+a node.
+
+The gluing map xi_Gamma: product_v Mbar_{g_v,n_v} -> Mbar_{g,n}
+glues the appropriate cotangent lines together at each node.
+Its image is the closed subset of Mbar_{g,n} consisting of
+curves with dual graph refining Gamma.
+
+--------------------------------------------------------------------------------
+1.3 What is a Chow group (very briefly)
+--------------------------------------------------------------------------------
+
+A smooth projective variety X over C has a Chow group A^*(X),
+an algebraic analog of cohomology. Roughly, A^k(X) is
+"codimension-k algebraic cycles modulo rational equivalence."
+
+Two facts matter here:
+
+ - A^*(X) is a graded commutative ring: one can intersect
+ cycles and add them.
+ - For X = Mbar_{g,n}, cycle class maps A^k(X) -> H^{2k}(X)
+ compatibly identify many classes with cohomology.
+
+This paper lives almost entirely in A^*(Mbar_{g,n}). Chow
+groups are the natural setting for the whole story because
+tautological pushforwards, pullbacks, and intersections all
+live there. The cycle-class map carries everything over to
+rational cohomology without loss of the counterexample.
+
+--------------------------------------------------------------------------------
+1.4 The strata algebra S_{g,n}
+--------------------------------------------------------------------------------
+
+We now define the formal object in which "the Pixton span"
+lives.
+
+Take a decorated stable graph Gamma of type (g,n). To Gamma
+attach a decoration
+
+ gamma = product over half-edges of psi_h^{a_h}
+ * product over vertices v of kappa_{j}^{c_{v,j}},
+
+with finitely many nonzero c_{v,j}. At each vertex, impose
+
+ deg(gamma|_v) <= 3 g_v - 3 + n_v.
+
+This "per-vertex dimension bound" is the standard truncation
+used by Pandharipande-Pixton-Zvonkine. It's important: the
+strata algebra is graded, and we always truncate to degree no
+higher than the local dimension at each vertex.
+
+The strata algebra S_{g,n} is the free Q-vector space on
+isomorphism classes of decorated graphs [Gamma, gamma], modulo
+graph isomorphism, with the truncation above. It is graded by
+
+ total degree = |E(Gamma)| + deg(gamma).
+
+There is a linear map
+
+ q_{g,n}: S_{g,n} -> A^*(Mbar_{g,n}; Q),
+
+ [Gamma, gamma] |-> xi_{Gamma,*} gamma,
+
+the gluing pushforward. Its image is the tautological ring:
+
+ R^*(Mbar_{g,n}) = q_{g,n}(S_{g,n}).
+
+This gluing pushforward is unnormalized: no automorphism
+denominators, exactly the convention of PPZ (Pandharipande-
+Pixton-Zvonkine). Whenever automorphism denominators occur
+in this paper, they are explicit in the class being defined;
+they do not modify q_{g,n}.
+
+The realization q_{g,n} is Q-linear and degree-preserving. It
+is not injective: many nonzero formal combinations map to zero
+in A^*(Mbar_{g,n}). Its kernel is exactly the collection of
+all tautological relations.
+
+--------------------------------------------------------------------------------
+1.5 Products in the strata algebra
+--------------------------------------------------------------------------------
+
+There is a natural product on S_{g,n}, compatible with q. It
+is worth remembering how it works, because several proof steps
+compose formal operations.
+
+Given two decorated graphs, one first takes a "common
+refinement" -- a graph that maps compatibly to both by edge
+contractions, with the source symmetries of the contractions
+fixed. Then:
+
+ - edges that survive in at least one contraction keep their
+ psi decorations;
+ - edges that get "double-counted" (both contractions want to
+ keep the edge but the intersection shortens it) pick up a
+ normal factor of -psi - psi', i.e. the negative sum of
+ the cotangent classes at the two half-edges;
+ - leg and flag decorations pull back to their components;
+ - kappa classes split additively (e.g. kappa_j at a vertex
+ pulled to a split becomes sum over the two halves);
+ - if any vertex exceeds its dimension, the whole term
+ becomes zero.
+
+This product is symmetric and associative. It is the "transverse
+and self-intersection" rule for node cuts, exactly the formal
+version of the standard boundary calculus.
+
+Because we always truncate at the local vertex dimension, the
+product lands in S_{g,n}. This is one place the truncation
+matters: excessive decorations remain excessive after splitting.
+
+--------------------------------------------------------------------------------
+1.6 Forgetful and gluing operations
+--------------------------------------------------------------------------------
+
+Two families of extra structural maps make S_{g,n} into a
+"tautological" system in the technical sense:
+
+ - Forgetful pullback pi_y^*: S_{g,n} -> S_{g,n+1}, adding
+ a new leg y at every vertex in all possible ways;
+ - Forgetful pushforward pi_{y,*}: S_{g,n+1} -> S_{g,n},
+ forgetting the leg;
+
+ - Gluing pushforward gl_*: S_{g,n+2} -> S_{g,n} gluing two
+ legs to create a node;
+ - Gluing pullback gl^*: S_{g,n} -> S_{g,n+2} (with a bit of
+ care) going the other way.
+
+All these are q-compatible: q_{g,n}(pi_y^* s) = pi_y^*(q_{g,n+1}(s))
+and similarly for the rest. This is the "tautological" pillar
+of the strata algebra. It means any operation we can perform in
+Chow can also be performed formally.
+
+--------------------------------------------------------------------------------
+1.7 Pixton's relations, precisely
+--------------------------------------------------------------------------------
+
+Now the star of Part 0 returns in formal dress.
+
+Set
+
+ A(T) = sum_{m >= 0} (6m)! / ((3m)! (2m)!) T^m
+ B(T) = sum_{m >= 0} ((6m+1)/(6m-1)) * (6m)! /
+ ((3m)! (2m)!) T^m
+
+The coefficients c_m = (6m)!/((3m)! (2m)!) appear throughout the
+tautological literature: they are the coefficient weights of
+Faber-Zagier.
+
+For a commuting parity variable zeta with zeta^2 = 1, set:
+
+ Chat_{3j}(T, zeta) = T^j A(zeta T)
+ Chat_{3j+1}(T, zeta) = zeta T^j B(zeta T)
+
+Indices congruent to 2 mod 3 are never used.
+
+Pixton's relation system, at a decorated graph Gamma of type
+(g,n), is built out of these. Give each vertex its own parity
+variable zeta_v. The recipe:
+
+ - To every kappa factor K_{e, epsilon} assign a generating
+ series coefficient via the "cycle operator"
+
+ kappaHat_Gamma( product_i K_{e_i, epsilon_i} )
+
+ = sum_{(cycles c of tau)} product over c of
+ (sum over v of kappa_{sum_{i in c} e_i}^{(v)}
+ * zeta_v^{sum_{i in c} epsilon_i})
+
+ summed over all permutations tau in S_l. This is the
+ standard Faber formula for kappa monomials.
+
+ - To every edge e = (v, w) with cotangent symbols x, y at
+ the two half-edges, assign the "edge generating series"
+
+ Delta_e = [ A(zeta_v x T) * zeta_w B(zeta_w y T)
+ + zeta_v B(zeta_v x T) * A(zeta_w y T)
+ + zeta_v + zeta_w ] / ((x + y) T).
+
+ This is a power series in T by the identity
+
+ A(T) B(-T) + A(-T) B(T) + 2 = 0.
+
+ - To every mark i at vertex v(i) with cotangent psi_i, assign
+
+ Chat_{a_i}(psi_i T, zeta_{v(i)})
+
+ for a nonnegative integer a_i (the "mark index").
+
+ - To a partition sigma = (sigma_1, ..., sigma_l), each part
+ not congruent to 2 mod 3, assign a product of
+ Chat_{sigma_j} contributions at the vertex series.
+
+Now choose positive integers r (degree), a partition sigma with
+no parts congruent to 2 mod 3, and marks a_1,...,a_n each not
+congruent to 2 mod 3, subject to the "admissibility" conditions
+
+ 3r >= g + 1 + |sigma| + sum_i a_i
+ 3r = g + 1 + |sigma| + sum_i a_i mod 2.
+
+Then the Pixton relation at (g,n,r; sigma, a) is the formal sum
+
+ R(g,n,r;sigma, a)
+ = sum over Gamma [Gamma, P_Gamma] / |Aut Gamma|
+
+where P_Gamma is the coefficient of
+
+ T^{r - |E(Gamma)|} * product_v zeta_v^{g_v + 1}
+
+in the graph series:
+
+ 2^{-h^1(Gamma)} * kappaHat_Gamma(
+ exp{1 - Chat_0} * product_{j=1}^l {Chat_{sigma_j}} )
+ * product_i Chat_{a_i}(psi_i T, zeta_{v(i)})
+ * product_{e in E} Delta_e .
+
+Here h^1(Gamma) = |E| - |V| + 1 is the first Betti number of
+the graph (its loop count), and "{series}" means: expand in
+T and zeta, take the specified coefficient, and substitute
+into K_{m, epsilon} kappa-monomial coefficients.
+
+This is the recipe. If it looks baroque, that's because it is;
+the point is not to memorize it but to see that it is a single,
+explicit, uniform recipe for building formal combinations of
+stable graphs out of three ingredients:
+
+ (a) generating-series coefficients from A(T), B(T);
+ (b) kappa operators;
+ (c) edge functions.
+
+Pixton's crucial observation is that the resulting recipe
+turns out to produce genuinely vanishing classes when realized
+through q.
+
+--------------------------------------------------------------------------------
+1.8 The Pixton span P_{g,n}
+--------------------------------------------------------------------------------
+
+Pixton's relation recipe gives many formal combinations. To
+turn them into a machinery capable of representing "all
+formal consequences of Pixton's rules," we build the Pixton
+span by:
+
+ 1. Taking a Pixton relation at some vertex v_0 of an outer
+ stable graph Gamma of type (g,n), placing arbitrary
+ decorated strata classes at the other vertices, and
+ gluing via the product rule.
+
+ 2. Taking the span of all such combinations, over every
+ choice of outer graph, vertex, and arbitrary auxiliary
+ decorations.
+
+This P_{g,n} is already an ideal: Pixton's Proposition 1
+shows it is stable under the forgetful and gluing operations
+of Section 1.6. This is exactly the sense in which P is
+the natural "span of consequences" rather than just a
+specific family of relations.
+
+The smaller "0/1 marking" seed system, where all a_i in {0,1}
+and sigma is empty, is a subsystem that PPZ (Section 3.5)
+shows builds the entire span P_{g,n} from these seeds by
+multiplication, forgetting, and gluing.
+
+--------------------------------------------------------------------------------
+1.9 What Pixton actually conjectured
+--------------------------------------------------------------------------------
+
+Pixton asked: is P_{g,n} the full kernel of q_{g,n}?
+
+Equivalently: does
+
+ P_{g,n} = ker q_{g,n}
+
+hold for every stable (g,n)?
+
+The known direction is
+
+ P_{g,n} is a subset of ker q_{g,n}.
+
+This is the validity theorem, which has been known for a
+while now: PPZ proved it for rational cohomology via the
+3-spin CohFT, Janda proved it in Chow via equivariant
+Gromov-Witten theory of P^1. The counterexample in this
+paper refutes the reverse inclusion: P is strictly smaller
+than ker q for at least one (g,n).
+
+So the setup is:
+
+ P_{g,n} subseteq ker q_{g,n} subseteq S_{g,n},
+ ^
+ this paper shows strict inclusion
+ at g = 10^60, n = 3*C(g,3).
+
+Two points deserve emphasis in this rewrite.
+
+ First: the conjecture is about ALL (g,n) at once, not
+ a single pair. To refute it, one stable pair suffices.
+ That's exactly what Part 3 constructs.
+
+ Second: P_{g,n} is a fixed, explicit object. Failure
+ of the conjecture is not vague: it means there is an
+ explicit class Y in the fixed S_{g,n} with q(Y) = 0
+ and Y not in P_{g,n}. That's a concrete, checkable
+ statement.
+
+--------------------------------------------------------------------------------
+1.10 Two flavors: Chow and rational cohomology
+--------------------------------------------------------------------------------
+
+The paper proves the failure in two ways:
+
+ - Chow: q_{g,n}(Y) = 0 in A^*(Mbar_{g,n}), still Y not
+ in P_{g,n}. This is Theorem 2.1 (stated in Part 2 below).
+
+ - Rational cohomology: q^H_{g,n}(Y) = 0 in
+ H^*(Mbar_{g,n}(C); Q), still Y not in P_{g,n}. This is
+ the corollary.
+
+The second follows from the first: apply the rational cycle
+class map cl: A^*(Mbar_{g,n}) -> H^*(Mbar_{g,n}(C); Q).
+Compatibility of cl with Chern classes, products, and proper
+pushforwards shows cl(q_{g,n}(s)) = q^H_{g,n}(s) for all s.
+So whenever q_{g,n}(Y) = 0, automatically q^H_{g,n}(Y) = 0.
+Nonmembership Y not in P_{g,n} is a single assertion in S_{g,n},
+unchanged whether we then pass to A or to H.
+
+End of Part 1. Part 2 states the main theorem in full.
+
+
+=====================================================
+PART 2: THE MAIN THEOREM
+=====================================================
+
+Now the formal statement. Everything in Parts 3 through 6
+builds up the pieces of Theorem 2.1 and Corollary 2.2 below.
+
+--------------------------------------------------------------------------------
+2.1 The main theorem
+--------------------------------------------------------------------------------
+
+Theorem 2.1. Set
+
+ g = 10^60, D_* = C(g,3), n = 3 D_*.
+
+Here C(g,3) is the binomial coefficient "choose 3 from g". So
+D_* is the number of increasing triples from {1,...,g}, and
+n = 3 D_* is exactly three times that (because each triple
+gets three labeled slots).
+
+There is an explicit class
+
+ Y in S_{g,n}^{2 D_*}
+
+of Definition 3.4 (constructed in Part 3) satisfying
+
+ q_{g,n}(Y) = 0 in A^{2 D_*}(Mbar_{g,n}; Q),
+
+ Y not in P_{g,n}.
+
+Consequently Pixton's original relation span does not exhaust the
+tautological Chow relations: the equality ker q_{g,n} = P_{g,n}
+fails for the stable pair (g, n) above.
+
+The class Y has formal degree 2 D_*: it is a product of D_*
+"factors" each of codimension two. This is far below the top
+dimension 3g - 3 + n of S_{g,n}, so Y is a codimension-low
+class, not a top-degree class.
+
+--------------------------------------------------------------------------------
+2.2 The cohomological corollary
+--------------------------------------------------------------------------------
+
+The same class gives the rational-cohomological statement.
+Define, in formal degree d,
+
+ q^H_{g,n}: S^d_{g,n} -> H^{2d}(Mbar_{g,n}(C); Q)
+
+ q^H_{g,n} = cl o q_{g,n},
+
+where cl is the rational cycle class map for the complex Deligne-
+Mumford stack Mbar_{g,n}. Its image is the cohomological
+tautological ring RH^* = cl(R^*) inside H^*(Mbar_{g,n}(C); Q).
+
+Corollary 2.2. For the stable pair and class Y of Theorem 2.1,
+
+ q^H_{g,n}(Y) = 0 in H^{4 D_*}(Mbar_{g,n}(C); Q),
+
+ Y not in P_{g,n}.
+
+In particular, P_{g,n} is a proper subset of ker q^H_{g,n}.
+
+Proof of Corollary 2.2.
+
+Cycle class commutes with Chern classes, with products, and with
+proper pushforward. In particular
+
+ cl(xi_{Gamma,*} gamma) = xi_{Gamma,*} cl(gamma)
+
+for any decorated graph (Gamma, gamma), and the same compatibility
+applies to the forgetting-pushforward definition of kappa classes.
+
+Therefore q^H_{g,n} is exactly the cohomological graph realization
+of Pandharipande-Pixton-Zvonkine's Section 0.3, with the same
+vertexwise truncation and unnormalized generators. Applying cycle
+class to the equality q_{g,n}(Y) = 0 from Theorem 2.1 gives
+
+ q^H_{g,n}(Y) = 0.
+
+Nonmembership Y not in P_{g,n} is an assertion in the formal
+strata algebra S_{g,n}; it is untouched by passing from A^* to
+H^*. The strict inclusion
+
+ P_{g,n} subseteq ker q^CH_{g,n} subseteq ker q^H_{g,n}
+
+then gives the claimed statement. Passing from Chow vanishing to
+cohomological vanishing uses the second inclusion; it requires no
+injectivity of cycle class.
+
+End of Part 2. Part 3 builds the class Y.
+
+
+=====================================================
+PART 3: THE CLASS Y
+=====================================================
+
+We now construct the class Y whose existence Theorem 2.1
+promises. The construction is purely combinatorial:
+we assemble Y out of D_* = C(g,3) "factors", each of which
+is a small correction to a background class.
+
+--------------------------------------------------------------------------------
+3.1 The global parameters
+--------------------------------------------------------------------------------
+
+Recall from Theorem 2.1:
+
+ g = 10^60, D_* = C(g,3), n = 3 D_*.
+
+The number g is astronomically large. This is not important
+mathematically except that g has plenty of room compared to
+certain other constants that appear in the proof. We say more
+about this in Part 5.
+
+The number D_* counts increasing triples of distinct colors
+
+ 1 <= i < j < k <= g.
+
+Each such triple will contribute exactly one factor to Y.
+Each factor has codimension two, so Y has codimension 2 D_*.
+
+The number n equals exactly 3 D_*: three labels per increasing
+triple. So the parameters line up: one factor per triple, three
+slots per factor, n total slots.
+
+--------------------------------------------------------------------------------
+3.2 The color word
+--------------------------------------------------------------------------------
+
+Order the increasing triples of colors lexicographically. For
+example, if g >= 4, the very first triples in lexicographic
+order are
+
+ (1,2,3), (1,2,4), (1,2,5), ..., (1,3,4), (1,3,5), ...
+
+Write the colors in each successive triple down in that order
+to form one long word
+
+ i = (i_1, i_2, ..., i_n)
+
+of total length n. The first three entries are 1,2,3 (or, if
+g < 3, we wouldn't be working in this regime). The next three
+are 1,2,4. And so on. Each color j from 1 to g appears in the
+word exactly
+
+ C(g-1, 2)
+
+times: choose two other distinct colors to complete the triple.
+
+Group the labels 1,...,n into consecutive triples J_1, ..., J_{D_*},
+each of size three, aligned with the word i above. So the J-th
+block J = (a, b, c) is exactly a triple of slots whose color
+word at those slots is some increasing triple (i, j, k) of colors.
+
+--------------------------------------------------------------------------------
+3.3 Tail divisors D_T
+--------------------------------------------------------------------------------
+
+For a subset T of the labels with |T| >= 2, let D_T denote the
+divisor class in A^1(Mbar_{g,n}) of stable curves where the marks
+in T cluster together on a rational tail and are separated from
+the rest of the curve by a single node.
+
+Informally: a genus-zero sphere sprouts out, holds exactly the
+labels T, and is attached to the main curve by a single node.
+
+For distinct labels a, b, define
+
+ D_bar_{ab} = sum over T that contains {a, b} of D_T.
+
+So D_bar_{ab} is the sum of all rational tail divisors containing
+both a and b. Rational tails containing only a, or only b, do
+not appear.
+
+Both D_T and D_bar_{ab} are codimension-one classes. We will
+multiply them by other classes; each factor F_J^st below
+involves exactly two such sums multiplied together, giving
+codimension two per factor.
+
+--------------------------------------------------------------------------------
+3.4 Rational bridges
+--------------------------------------------------------------------------------
+
+Beyond rational tails, we need a second kind of boundary class:
+a rational bridge. A rational bridge is a rational component
+that separates two parts of the curve: a sphere with two
+attached nodes on opposite sides.
+
+Concretely, a "one-edge unmarked stable graph of genus g" is a
+stable graph with exactly one edge (a node), two vertices
+whose genera sum to g, and no marks. Subdividing that edge
+by inserting a fresh genus-zero vertex between them and placing
+all three labels of a block J at the new vertex produces a
+graph we can call a three-mark rational bridge. Summing over
+all such choices, dividing by the automorphism order of each
+graph, and pulling back to forget everything outside J gives
+the formal class
+
+ beta_J^st in S_{g,n}^2.
+
+This is the bridge correction.
+
+One clarifying point: the sum beta_J^st involved is over
+unordered pairs of vertices. The 1/|Aut Gamma| normalizations
+in its definition are the graph-automorphism weights, not
+something imposed on the unnormalized generators of S_{g,n}.
+They are exactly what makes beta_J^st a natural class with
+respect to the forgetful pushforward, but they do NOT alter
+the formal product rule (which stays unnormalized).
+
+--------------------------------------------------------------------------------
+3.5 The three-slot factor F_J^st
+--------------------------------------------------------------------------------
+
+For a block J = (a, b, c) with three slots, define
+
+ F_J^st = D_bar_{ab} * D_bar_{ac} - (1/2) * beta_J^st.
+
+This is the building block of Y. It has two terms:
+
+ - a product of two codimension-one tail divisor sums;
+ - minus one half times a codimension-two bridge correction.
+
+The coefficient -1/2 is essential: it is what makes the
+eventual compactification of F_J^st match a projected small
+diagonal over the boundary (see Part 5). Without the
+correction, a codimension-two exceptional class would fail to
+cancel. The coefficient is not adjustable.
+
+Notice the asymmetry in the first term: D_bar_{ab} * D_bar_{ac}.
+Both sums contain a. The reason for this choice: we want a
+class sensitive to "a agrees with b AND a agrees with c", which
+is the smallest interesting arrangement of three labels.
+
+F_J^st has total formal degree 2 (one from each D_bar term, or
+two from beta_J^st). It lives in S_{g,n}^2.
+
+--------------------------------------------------------------------------------
+3.6 The alternation and the definition of Y
+--------------------------------------------------------------------------------
+
+The class F_J^st is not symmetric under arbitrary permutations
+of the three labels a, b, c of J. It singles out a (via the
+{ab, ac} pattern).
+
+The remedy is alternation. Let
+
+ G_i = { tau in S_n : tau(i) = i }
+
+be the set of permutations of labels that preserve the color
+word i. Equivalently, tau in G_i shuffles labels within
+color-classes: colors are fixed but slots holding the same
+color can be permuted among themselves.
+
+Definition 3.4. The candidate class Y is
+
+ Y = sum over tau in G_i of
+ sgn(tau) * tau( product_{J} F_J^st ).
+
+Here tau acts on labels: it permutes the slots of each factor
+F_J^st and permutes which factor goes where, but it preserves
+the color word i because tau lies in G_i. The sign sgn(tau) is
+the ordinary permutation sign.
+
+Because G_i preserves the word, the colors of the slots are
+unchanged. The sum thus alternates over S_n-internal shuffling
+without disturbing the color pattern.
+
+Informally: Y is the "signed-average over relabelings of colors"
+of product_J F_J^st. The alternation makes Y into a genuine
+antisymmetric tensor in the colors, not just a naive product.
+
+This is Definition 3.4 (equivalently Definition 1.2 of the
+original paper). It is the class Theorem 2.1 talks about.
+
+--------------------------------------------------------------------------------
+3.7 Sanity checks
+--------------------------------------------------------------------------------
+
+Before doing anything with Y, we check that it is well defined
+and lives where we think it does.
+
+ (a) Finiteness. Every operation in Definition 3.4 is a finite
+ combination: the set G_i is finite (it is a subgroup of S_n,
+ whose order is huge but finite), and each F_J^st has
+ finitely many terms. So Y is a genuine element of S_{g,n}.
+
+ (b) Degree. Each factor F_J^st has codimension two. There are
+ D_* = C(g,3) factors. Straight multiplication gives
+ deg(Y) = 2 D_*. This is far less than 3g - 3 + n, the top
+ dimension of S_{g,n}; so Y is a strictly positive-codimension
+ class.
+
+ (c) Stability of (g,n). Since g = 10^60 >= 2 and n = 3 D_*
+ is positive, the pair (g,n) is stable.
+
+ (d) Automorphism weights in beta_J^st. The 1/|Aut Gamma| in
+ beta_J^st specifies the coefficients of a particular class
+ in S_{g,n}. It does NOT modify the unnormalized generators
+ q_{g,n}[Gamma, gamma] = xi_{Gamma,*} gamma. Both facts are
+ compatible: the product rule for S is unnormalized, and
+ arbitrary rational coefficients are allowed on top of it.
+
+None of these checks is deep. They are here to reassure the
+reader that Y is a legitimate element rather than a formal
+wish.
+
+--------------------------------------------------------------------------------
+3.8 Roadmap: how Theorem 2.1 splits
+--------------------------------------------------------------------------------
+
+Proving Theorem 2.1 splits into two completely independent
+claims, proved by completely different methods:
+
+ Claim A (non-membership): Y is not in P_{g,n}.
+
+ Claim B (vanishing): q_{g,n}(Y) = 0.
+
+ A and B involve different algebras and different techniques:
+
+ - Claim A uses a "formal detector": a Q-linear map out of
+ S_{g,n} that annihilates P_{g,n} but takes nonzero values
+ on Y. The detector can be constructed entirely within the
+ strata algebra; it doesn't touch actual Chow geometry.
+ This is Part 4.
+
+ - Claim B uses a sophisticated analysis of relative Chow
+ correspondences on curve powers, surface Quot schemes,
+ a combinatorial identity from Hilbert schemes of points,
+ and a support-dimension induction over the boundary.
+ This is Part 5 (surface relation) plus Part 6
+ (compactification and support).
+
+These two claims are independent. Part 4 only uses the formal
+product rule; Part 5-6 only use actual Chow geometry. It is our
+claim that both facts are true, and together they prove
+Theorem 2.1.
+
+End of Part 3. Part 4 builds the formal detector.
+
+
+=====================================================
+PART 4: WHY Y IS NOT IN THE PIXTON SPAN
+=====================================================
+
+Part 4 constructs Claim A of Section 3.8: a linear map that
+vanishes on the Pixton span but does not vanish on Y. That
+proves Y is not in P_{g,n}.
+
+Our tool is an idea that is entirely algebraic. We build a
+special algebra -- a "Frobenius algebra" -- that is only allowed
+to see a particular kind of stable graph (a "rational tree").
+On rational trees, this Frobenius algebra gives us a clean
+recipe for testing formal combinations. Any Pixton relation,
+restricted to rational trees, evaluates to zero under this
+recipe. But Y, restricted to rational trees, evaluates to a
+specific nonzero element.
+
+The trick is to prove this in one decisive step: an exterior
+algebra R with one "color" generator per increasing triple of
+curve-colors. It is exactly the kind of algebra where a
+distinguished product over all triples -- like Y evaluated
+color-by-color -- is nonzero.
+
+A first reading: skim 4.1-4.3 for the setup, take on faith that
+4.6 defines a linear map, and look at Part 4.8 (which is where
+the punch line is). Come back to 4.4-4.7 later if you want the
+manipulative details.
+
+--------------------------------------------------------------------------------
+4.1 The exterior coefficient ring
+--------------------------------------------------------------------------------
+
+Let
+
+ R = Lambda_Q ( theta_{ijk} : 1 <= i < j < k <= g )
+
+be the exterior algebra on one generator theta_{ijk} for each
+increasing triple of colors. Each theta_{ijk} is odd:
+theta_{ijk}^2 = 0, and theta_{ijk} commutes with theta_{i'j'k'}
+when they are distinct (in the super sense: with a sign).
+
+We extend the theta symbols "alternately" to ALL triples of
+colors, not just increasing ones, by requiring
+
+ theta_{i j k} = alternating in i,j,k, with sign (-1)^{sgn}
+
+ theta_{i j k} = 0 whenever two of i,j,k coincide or
+ the triple is not increasing after sorting.
+
+So for example theta_{2,1,3} = -theta_{1,2,3}, and
+theta_{1,1,3} = 0.
+
+The integer g = 10^60 is enormous, so theta_{ijk} has
+C(g,3) = D_* independent generators. Any monomial of degree D_*
+in theta's is either one specific product
+
+ Theta = product_{i < j < k} theta_{ijk}
+
+or has some generator squared and is therefore zero.
+
+The product Theta is THE monomial of maximal degree that is
+not automatically zero. It is our eventual "test" for Y.
+
+--------------------------------------------------------------------------------
+4.2 The symplectic space
+--------------------------------------------------------------------------------
+
+Let W be a purely odd Q-vector space of dimension 2g, with a
+nondegenerate symplectic pairing < , >. Choose bases
+
+ a_1, ..., a_g (a "Lagrangian" basis)
+ b_1, ..., b_g (dual basis)
+
+such that
+
+ <a_i, b_j> = delta_{ij}, <a_i, a_j> = <b_i, b_j> = 0.
+
+The first g elements a_i span a Lagrangian; the b_j span a
+complementary Lagrangian. Both W_+ = span(a_i) and W_- = span(b_j)
+are Lagrangian subspaces of W.
+
+W is "odd": we use it inside a super-setting where its elements
+are odd. This is why the symplectic form < , >, which is
+ordinarily alternating (and hence a "symmetric" form in the
+usual sense after the odd shift), can be used as a pairing.
+
+We make an odd alternating three-form Phi on W by decree:
+
+ Phi(a_i, a_j, a_k) = theta_{ijk},
+
+ Phi(w_1, w_2, w_3) = 0 whenever some w_s lies in W_-.
+
+The value of Phi on a triple of a's is the corresponding
+triangle-theta generator. It vanishes whenever any of the three
+inputs sits in the b-part.
+
+Since Phi has odd coefficients (the theta's), the ordinary sign
+rules make it alternating: Phi(w_2, w_1, w_3) = -Phi(w_1, w_2, w_3),
+etc.
+
+--------------------------------------------------------------------------------
+4.3 The Frobenius algebra
+--------------------------------------------------------------------------------
+
+The state space is the super R-module
+
+ V = R (x) ( Q 1 + W + Q e ),
+
+with Q 1 and Q e behaving as a "hyperbolic plane" and W as
+discussed above. In words: V is R tensored with the direct sum
+of "1, W, e".
+
+Define a trace (or "integral") on V:
+
+ int(1) = 0
+ int(e) = 1
+ int(w) = 0 for all w in W
+ int(w_1 w_2) = <w_1, w_2>
+ int(w_1 w_2 w_3) = Phi(w_1, w_2, w_3)
+
+for w_i in W. Higher-degree products of W's vanish by
+convention (see below). The pairing induced by this integral is
+the hyperbolic-symplectic pairing: it pairs 1 with e (value 1),
+pairs W_+ with W_- (via < , >), and is zero on the tangent
+summands (1,1), (e,e).
+
+Define the product by demanding that it match the integral and
+the symplectic form and the cubic Phi:
+
+ - Let 1 be the identity.
+ - Let e^2 = 0 and e * W = 0 (so e is a "nilpotent
+ square").
+ - For w_1, w_2 in W, define w_1 * w_2 by the rule:
+
+ (w_1 * w_2) = <w_1, w_2> * e
+ + (the vector in W dual to the functional
+ w_3 |-> Phi(w_1, w_2, w_3)).
+
+The second term lands in R (x) W_- because Phi has support only
+on W_+ in each slot, so its dual vector is automatically in W_-.
+
+Lemma 4.3. These rules give a supercommutative associative
+Frobenius algebra. It has an interesting property:
+
+ - The product of three standard odd vectors (i.e.
+ standard w_i in W) is in Re (that is, in R * e, the even
+ summand);
+ - The product of four such vectors is zero.
+
+The "handle element", namely the image under mu o Delta of 1,
+is
+
+ mu Delta (1) = (2 - 2g) e.
+
+Set lambda = (2g - 2) e. Then mu Delta (1) = -lambda.
+
+Proof sketch (full proof in original). The pairing is
+nondegenerate, so the products are uniquely determined. Super-
+commutativity follows from alternation of < , > and of Phi,
+together with the super sign rule.
+
+For associativity, look at the associator of three vectors in
+W and pair it with a fourth basis vector. Pairing with 1 gives
+cyclic symmetry of the alternating cubic Phi; pairing with e
+gives zero. If the fourth vector is in W, both sides become
+contractions of two pair-product tensors whose W- and
+hyperbolic-channel components vanish by the Lagrangian support
+of Phi and by e^2 = eW = 0. Cases involving 1 or e are
+immediate by direct computation.
+
+The handle element computation: the hyperbolic channels
+contribute +2e, the 2g odd channels contribute -2ge, and the
+cubic perturbation has zero internal trace because symplectic
+contraction pairs the two different Lagrangians. Summing gives
+(2 - 2g) e.
+
+End of Lemma 4.3 sketch.
+
+--------------------------------------------------------------------------------
+4.4 Coproduct and the block formula
+--------------------------------------------------------------------------------
+
+For a finite nonempty label set L, let Delta_L: V -> V^{(x)L}
+be the coproduct -- the adjoint (with respect to the pairing) of
+iterated multiplication in the L slots. For L a singleton,
+Delta_L is the identity.
+
+The coproduct is coassociative and satisfies the projection
+formula. In particular for two-distinct-labels ij:
+
+ Delta_{ij}(1) * Delta_{ik}(1) = Delta_{ijk}(1) (4.1)
+
+ Delta_{ij}(1)^2 = -Delta_{ij}(lambda) (4.2)
+
+ int_j Delta_{ij}(v) = v (4.3)
+
+where "int_j" means the trace in slot j (so the equation is an
+identity of tensors with one fewer slot). The minus sign in
+(4.2) is important: it matches the negative normal bundle of a
+rational-tail boundary divisor. This is one of the places
+where geometric intuition guides the algebra.
+
+--------------------------------------------------------------------------------
+4.5 The bar transform
+--------------------------------------------------------------------------------
+
+The detector will not apply directly to psi and kappa monomials.
+Instead, it applies to a different presentation of S_{g,m} in
+which the boundary classes are separated out.
+
+For each label i:
+
+ psi_bar_i = psi_i - sum_{T >= {i}, |T| >= 2} D_T.
+
+This is psi_i with all rational-tail divisors containing i
+subtracted. The result is the "psi at boundary" of i, effectively
+the cotangent class restricted to the root after all tails
+have been pulled off.
+
+For each x in {1,...,m} in a fixed ordering, let L_x be the
+pullback of psi_x from the space defined by the labels 1,...,x:
+
+ L_x = psi_x - sum_{T contains x, |T| >= 2,
+ T\{x} subset {x+1,...,m}} D_T.
+
+For j >= 1:
+
+ kappa_bar_j = kappa_j - sum_{x=1}^m (L_x)^j.
+
+The "leading graphless part" of this change of variables is
+just psi_bar_i = psi_i and kappa_bar_j = kappa_j - sum_i psi_i^j.
+This is an invertible polynomial change of variables, so the
+bar monomials form a legitimate basis change in S_{g,m}. The
+full change of variables (with all the boundary corrections) is
+what lets us cleanly separate "root" and "tail" contributions.
+
+--------------------------------------------------------------------------------
+4.6 The detector map t_m
+--------------------------------------------------------------------------------
+
+We now define Q-linear maps
+
+ t_m: S_{g,m} -> V^{(x_R) m}
+
+for each label set {1,...,m}. The construction is recursive on
+m.
+
+Base case (m=0):
+
+ t_0(1) = 1 in R
+ t_0(anything with positive bar-degree) = 0.
+
+In words, t_0 is the augmentation that sends 1 to the unit of R
+and everything of positive formal degree to zero.
+
+Induction step. We prescribe t_m on the bar-monomial complement
+(i.e., on all monomials that do not involve the boundary tail
+divisors). For such a monomial,
+
+ t_m( product_i bar_psi_i^{d_i} * product_j bar_kappa_j^{c_j} )
+
+ = lambda^{d_1} (x) lambda^{d_2} (x) ... (x) lambda^{d_m}
+
+ if all c_j = 0,
+
+ = 0 otherwise.
+
+Here lambda lands in the i-th tensor slot. Slots that get no
+active input receive a "unit" 1 in V. The values are elements
+of V^{x m} via the identifications V = R (x) V, R being the
+scalar part.
+
+So t_m sends bar_psi_i to lambda in slot i and bar_kappa_j to
+zero in all slots.
+
+For a general m-decorated graph (i.e., involving tail divisors),
+t_m applies recursively by:
+
+ (a) Discard any graph that is not a "rational-tails graph"
+ (i.e., not a tree with a single genus-g vertex, the
+ root, and all other vertices rational). The result is 0.
+
+ (b) For each rational descendant subtree with |L| >= 2
+ active labels at that tail:
+ - Integrate the decorations on its rational parameter
+ space (which is Mbar_{0, |L|+1} plus maybe some
+ forgotten labels) to extract its top-degree integral
+ (which is a scalar in R).
+ - Retain this scalar.
+
+ (c) At the root (which carries g genus and a slots
+ corresponding to entries), compute t_a of the root's
+ decoration with its a active slots, contracting the
+ already-computed scalars from its rational children
+ into the tensor product.
+
+ (d) Duplicate each root slot into the descendant labels of
+ its branch by Delta_L. That is, if a rational subtree
+ carries labels L = {l_1,...,l_k}, apply Delta_L to the
+ corresponding root slot to distribute the state among
+ the k descendants.
+
+Every branch attached to the root has at least two
+descendant labels (by the definition of rational-tails graph).
+This is what makes the recursion terminate: t_m at a root with
+a < m labels is a call to t_a with a < m.
+
+Lemma 4.6. The prescription is well defined modulo genuine
+(Chow) relations on rational parameter factors, and it induces
+a map on S_{g,m}. For a homogeneous expression of degree r, its
+image is a sum of tensors
+
+ tensor over blocks L of Delta_L(lambda^{d_L}),
+
+indexed by partitions of the label set into nonempty blocks, of
+the form
+
+ r = sum over blocks L of ( |L| - 1 + d_L ). (4.4)
+
+In particular, t_m = 0 for r > m.
+
+Proof sketch. Grafting a genus-zero graph amounts to composing
+coproducts and multiplying integrals; coassociativity makes the
+result independent of the internal decomposition. For a rational
+subtree with k descendant labels, top decorations plus its
+edges going up to the root add to k-1; block-degree bookkeeping
+then gives (4.4). Since lambda^2 = 0, a nonzero term has
+d_L <= 1 for every block, hence r <= m.
+
+Excessive decorations on a rational vertex have zero integral
+by the standard dimension truncation. At a root with a incident
+slots, a decoration exceeding 3g - 3 + a also exceeds a
+(since g >= 2), and its recursive test is zero. Under
+refinement, the sum of vertex dimensions of the pieces decreases
+by the number of new edges, so splitting decorations cannot
+make an excessive total degree admissible on every piece.
+
+End of Lemma 4.6 sketch.
+
+End of Part 4A. Part 4B (4.7 onward) proves that t_m vanishes
+on P_{g,n} and does not vanish on Y.
+
+
+--------------------------------------------------------------------------------
+4.7 Boundary operations
+--------------------------------------------------------------------------------
+
+Lemma 4.7. The maps t_m are equivariant for relabeling and
+respect the boundary operations of S_{g,m}. Precisely, for every
+strata expression b they satisfy
+
+ t( psi_bar_i * b ) = lambda_i * t(b) (4.5)
+
+ t( kappa_bar_j * b ) = 0 (4.6)
+
+ t( D_bar_{ij} * b ) = Delta_{ij}(1) * t(b) (4.7)
+
+where lambda_i denotes lambda in slot i and identity in others.
+
+More generally, gluing along the boundary divisor D_T integrates
+the entire genus-zero input and duplicates the root-side
+attaching slot into the labels of T.
+
+Proof sketch. The descendant- and kappa-identities are the
+forgetting and gluing formulas of Arbarello-Cornalba
+(Equations 1.7-1.10). At a gluing D_T, let p be the root-side
+flag and p' the rational-side flag. The normal class is
+-psi_p - psi_{p'}. On Mbar_{0, T + {p'}}, the sum of boundary
+splits separating i from p' equals psi_i + psi_{p'}, and the
+sum whose side away from p' contains two fixed labels i,j
+equals psi_{p'}. Therefore:
+
+ - bar_psi_i restricts to the modified cotangent on the
+ root side (using p if i in T).
+ - bar_D_{ij} restricts to the analogous pair class on the
+ root side, EXCEPT when both labels are in T, in which
+ case it vanishes.
+
+The bar_kappa_j class is independent of the ordering: for two
+consecutive labels made last, D = D_{i,j}, the identities
+psi_i|_D = psi_j|_D = 0 and D|_D = -psi_p give, for d >= 1,
+
+ (psi_i - D)^d - psi_i^d = -D psi_p^{d-1}
+ = (psi_j - D)^d - psi_j^d.
+
+This is exactly the invariance under a consecutive transposition
+that proves well-definedness of kappa_bar_j as a bar-class.
+
+Restricting kappa_bar_j to D_T: put the labels of T last; the
+kappa class splits additively; the first label of T gives
+root-side cotangent at p; the remaining partial cotangents on
+the rational factor cancel its kappa class by the forgetful
+kappa formula. So kappa_bar_j restricts to itself on the root
+side. Combined with the coproduct projection formula, this
+proves (4.5)-(4.6).
+
+For (4.7): only the two-label tail has nonzero undecorated
+integral. If i,j sit in different branches, the root-side pair
+class joins their ancestor slots, giving Delta_{ij}(1). If they
+sit in the same branch, its restriction is -psi_p; the identity
+(4.2) turns this into Delta_{ij}(1) via the self-intersection
+(delta_{ij}(1))^2 = -delta_{ij}(lambda).
+
+These calculations take place in the auxiliary formal graph
+space modulo relations on rational parameter factors.
+Simultaneous compatible splits produce the same tree
+independently of order; polynomial multiplication commutes and
+associates. A genuine zero on a rational factor remains zero
+under further operations. Lemma 4.6 permits the dimension
+truncations, since discarded terms remain outside the
+rational-tails locus under refinement.
+
+End of Lemma 4.7 sketch.
+
+--------------------------------------------------------------------------------
+4.8 Forgetting operations
+--------------------------------------------------------------------------------
+
+Lemma 4.8. For every strata expression b, the maps t_m satisfy
+
+ t( pi_y^* b ) = t(b) (x) 1 (4.8)
+
+ t( pi_{y,*} b ) = int_y t(b) (4.9)
+
+where pi_y forgets the label y and int_y is the trace in the
+slot at y. Identities hold in any order of the remaining labels.
+
+Proof sketch. Both sides are Q-linear in b; it suffices to check
+on the bar-monomial complement.
+
+Forget pullback on a vertex: replaces
+ psi_h |-> psi_h - D_{h,y}
+ kappa_j |-> kappa_j - psi_y^j.
+A tail divisor D_T pulls back to D_T + D_{T + {y}}.
+
+The key observation: the (formal) pullback commutes with
+multiplication by the indicated polynomial classes, because
+incompatible splits stay incompatible, and compatible splits
+either split a vertex (a vertex calculation) or add a boundary
+term cutting off the half-edge at y together with the branch.
+
+Pushforward at a vertex: put X_i = pi_y^* psi_i, k_j = pi_y^* kappa_j,
+d_i = D_{i,y}. Then psi_i = X_i + d_i, kappa_j = k_j + psi_y^j,
+and the local rules are
+
+ d_i d_l = 0 (i != l), d_i^2 = -X_i d_i, psi_y d_i = 0
+
+ pi_{y,*} 1 = 0
+ pi_{y,*} psi_y^d = kappa_{d-1} (d >= 1)
+ pi_{y,*} d_i = 1
+
+The projection formula for graphless cotangent and kappa
+polynomials follows by substitution. For a tail multiplier, the
+two pullback splits distribute kappa_{d-1} additively; for d_i,
+only the compatible split pushes to the original tail. On a
+tree with a stable y-vertex, this reduces to the same vertex
+calculation.
+
+The ordinary bar-monomial pushforward uses the telescoping
+identity
+
+ psi_y^d - psi_bar_y^d
+ = sum_{S nonempty} D_{S + {y}} sum_{a=0}^{d-1}
+ psi_y^{d-1-a} psi_bar_y^a
+
+and the symmetric-power integral
+
+ int_{Mbar_{0, |S|+2}} psi_y^{|S|-1} = 1.
+
+For d=1 the scalar identity kappa_0 = 2g-2+m applies; for d>1,
+expand kappa_{d-1} = kappa_bar_{d-1} + sum_x L_x^{d-1} and
+compute similarly. Both give
+
+ t_m( pi_{y,*} psi_bar_y^d ) = int(lambda) = 1 in R.
+
+The compatibility with the cutoff (Lemma 4.6): an excessive
+decoration on a rational factor is a genuine zero in its
+genus-zero Chow ring, hence has zero forget pullback or
+pushforward. For an excessive root decoration of degree d with
+a root slots (so d >= 3g-2+a), adding a mark brings total
+degree d > a+1, killed by the block bound with a+1 slots;
+forgetting a mark brings degree d-1 > a-1, killed by the block
+bound with a-1 slots. The root stays stable since g >= 2.
+
+Equivariance from Lemma 4.7 lets the forgotten label be placed
+last in any chosen order.
+
+End of Lemma 4.8 sketch.
+
+--------------------------------------------------------------------------------
+4.9 The zero-leg identity
+--------------------------------------------------------------------------------
+
+We now need to know how t_m treats a Pixton relation with one
+leg of index zero (that is, a_i = 0 for one i). This requires
+a technical identity.
+
+Package the partition insertions with formal variables s_l,
+one for each allowed positive part, using exponential
+coefficients for repeated parts, and define
+
+ H(T, zeta) = Chat_0(T, zeta) - sum_l s_l Chat_l(T, zeta).
+
+Then a cycle-exponential formula for kappaHat_Gamma rewrites
+the vertex characteristic as
+
+ exp{ -log H }_{kappa, zeta_v} (4.10)
+
+where a monomial T^d zeta^epsilon in -log H becomes
+kappa_d T^d zeta_v^epsilon, including d=0. This formula is
+exactly the standard identity that lets kappa-operators be
+assembled from elementary vertex factors.
+
+For the edge we use U(x,x') = T Delta_e, so that the coefficient
+of total degree T^r is the r-th coefficient of the raw recipe.
+Denote the packaged series by R(s; a).
+
+Lemma 4.9. If all preceding leg indices a_j are in {0,1},
+then, in the untruncated rational-tails computation,
+
+ R(s; a, 0) - sum_l s_l R(s; a, l)
+ = pi_y^* R(s; a). (4.11)
+
+Proof sketch. The kappa-pullback formula, including
+kappa_0^{old} = kappa_0^{new} - 1, multiplies the new
+characteristic (4.10) by H(psi_y T, zeta_v). This is exactly
+the linear combination of new leg factors on the left.
+
+The remaining check is bubbles introduced in pulling the old
+leg and half-edge cotangents. For a function f(x), that pull
+contributes a boundary term (f(0) - f(x))/x, where x is the
+stable-side cotangent. At a new rational three-point vertex,
+the vertex characteristic is H(0, epsilon)^{-1}; the new
+H-leg cancels it. Writing A_x = A(eta x T), etc., the edge
+adjoining such a bubble contributes
+
+ [eta (B_x + 1) + epsilon (1 - A_x)] / x (4.12)
+
+If the bubble carries an old leg, its constant factor is 1 or
+-epsilon. Extracting epsilon gives (1 - A_x)/x or
+-eta(B_x + 1)/x, exactly the divided differences needed. If it
+subdivides an old edge, the epsilon-coefficient of the product
+of two edge factors is
+
+ [eta(B_x+1)(1-A_{x'}) + eta'(B_{x'}+1)(1-A_x)] / (x x') (4.13)
+
+which equals the difference formula
+U(0,x')-U(x,x') over x plus U(x,0)-U(x,x') over x', giving
+both cotangent pullback corrections. Bubbles for distinct
+slots at a vertex are disjoint, and these are all possibilities
+for adding one mark. Rational-tails graphs have h^1=0 and
+trivial automorphisms, so neither the 2^{-h^1} nor the
+unnormalized gluing introduces another coefficient.
+End of Lemma 4.9 sketch.
+
+--------------------------------------------------------------------------------
+4.10 Annihilating the Pixton span
+--------------------------------------------------------------------------------
+
+Lemma 4.10. For every m >= 0, t_m( P_{g,m} ) = 0.
+
+Proof sketch. We induct on paired data: first on arity m, then
+within fixed arity on (r, ell(sigma)) lexicographically.
+
+Smaller-arity basis. At m=0, every admissible basic Pixton
+relation has positive degree, while t_0 kills all positive
+degrees.
+
+Nontrivial outer gluing. If the outer graph is not a
+rational-tails graph, all its refinements have zero test. If
+it is, a relation at a rational factor has zero top integral
+by Chow validity. A relation at the root is killed by the
+smaller-arity induction (the root has fewer labels). Integration
+and coproducts then give zero for the entire generator.
+
+Multiplication by a tail divisor. If b in P_{g,m}, then by the
+ideal property and (4.7) applied to the D_T-multiplied Pixton
+relation, t_m(D_T b) = 0 for any T. This identity uses only
+smaller-arity induction on the root factor.
+
+Basic recipe relations. This is the main case. Using the
+substitution psi_i = psi_bar_i + sum_T D_T:
+ - The all-bar-psi term uses the lower triple (m, r-k, ell(sigma))
+ and is killed by (4.5)-(4.6).
+ - Other terms are D_T b' with b' in P_{g,m}, killed by the
+ tail-divisor ideal step above.
+So we may assume all leg indices are in {0,1}.
+
+Then we may assume r <= m (Lemma 4.6). If a leg has index zero,
+apply Lemma 4.9. The pullback term reduces to (m-1, r, ell(sigma))
+and dies by induction on arity. The "move a part to the zero
+leg" term strictly decreases ell(sigma) or r. Reducing a
+positive part ell = 3k+h (h in {0,1}) first, and reapplying the
+above moves, maintains the well-founded induction.
+
+All-ones case. The Pixton recipe at all-ones is symmetric
+under label permutations, and so is its test tensor. Contracting
+any one slot with 1 kills it (a forget pushforward), and
+contracting with e also kills it (multiply by psi_bar_y/(2g-2)
+first, then forget). So we need only test on standard W vectors.
+
+Block counting. In the block-expansion (4.4), a singleton has
+zero pairing with an odd input, a positive power of lambda
+kills odd inputs, and a block of >= 4 odd inputs has zero
+product. So only undecorated pair and triple blocks survive:
+with s pairs and u triples,
+
+ m = 2s + 3u, r = s + 2u, s + 3u = 3r - m >= g + 1 + |sigma| > g.
+
+Every alternated term uses more than g coordinates from the
+g-dimensional W_+ dual and is zero. All standard input tests
+vanish; nondegeneracy of the pairing gives the tensor itself
+zero.
+
+End of Lemma 4.10 sketch.
+
+--------------------------------------------------------------------------------
+4.11 Nonzero on Y
+--------------------------------------------------------------------------------
+
+We finish Claim A by showing t_n(Y) is nonzero.
+
+Every graph in a bridge correction beta_J^st has nontrivial
+unmarked stabilization, hence is not a rational-tails graph.
+Its refinements, pullbacks, and products remain outside the
+support of the test. Therefore, by (4.5)-(4.6) and (4.1), the
+remaining part of each factor of Y tests as
+
+ Delta_{ab}(1) Delta_{ac}(1) = Delta_{abc}(1).
+
+Pair slot j with a_{i_j}, the W_+ basis vector prescribed by
+the color word. A block with increasing colors i < j < k gives
+theta_{ijk}, and the product of unsymmetrized factors
+evaluates to
+
+ Theta = product_{i < j < k} theta_{ijk},
+
+up to one fixed super sign.
+
+Let G_i be the color-word stabilizer, from Definition 3.4. For
+tau in G_i, transporting its super permutation across the
+pairing of odd slots contributes sgn(tau); this cancels the
+explicit sign of that summand in Y. No reordering of the
+exterior generators is required: all summands are compared
+with the same ordered evaluation. Every color occurs C(g-1,2)
+times, hence
+
+ < t_n(Y), a_{i_1} (x) ... (x) a_{i_n} >
+ = +/- ( C(g-1,2)! )^g * Theta
+ != 0. (4.14)
+
+Combined with Lemma 4.10, this gives t_n(Y) != 0, and therefore
+Y is not in P_{g,n}. That proves Claim A.
+
+--------------------------------------------------------------------------------
+4.12 The formal detector stated
+--------------------------------------------------------------------------------
+
+Proposition 4.11. There exist a supercommutative Q-algebra R,
+a finite free supermodule V over R, and Q-linear maps
+
+ t_m: S_{g,m} -> V^{x m} (m >= 0)
+
+such that t_m( P_{g,m} ) = 0 for every m, whereas t_n(Y) != 0
+for the class Y of Definition 3.4 and the n of Theorem 2.1.
+In particular, Y is not in P_{g,n}.
+
+This Proposition IS Claim A of Theorem 2.1. The proof is the
+combination of Parts 4.1-4.11 above.
+
+The maps t_m are tests on FORMAL strata: they are not asserted
+to factor through the Chow ring. This is a pure S_{g,m}-statement.
+The geometric witnesses they provide are: Theta for the color
+product, lambda^{d_i} for the psi decorations, and Delta_L for
+the branch structure.
+
+End of Part 4. Part 5 begins Claim B: proving q_{g,n}(Y) = 0.
+
+
+=====================================================
+PART 5: WHY Y VANISHES IN CHOW
+=====================================================
+
+We now turn to Claim B: q_{g,n}(Y) = 0 in A^*(Mbar_{g,n}; Q).
+This is by far the harder half of the proof. Part 5 sets up
+the geometric machinery; Part 6 handles the boundary.
+
+The strategy: attack q(Y) from two directions at once.
+
+ Direction 1 (smooth families). On a family of smooth curves,
+ Y restricts to a specific class built from "small diagonals".
+ We can prove a bound on powers of these small diagonals using
+ Quot schemes on the surface C x C.
+
+ Direction 2 (compactification). On the boundary, Y restricts
+ to a "corrected cubic" that differs from the smooth-curve
+ class by controlled corrections. We prove a support-dimension
+ induction that pushes Y to vanish component by component.
+
+The two directions fit together in Part 6. Part 5 handles
+Direction 1: the surface relation.
+
+--------------------------------------------------------------------------------
+5.1 The faithful Chow tensor model
+--------------------------------------------------------------------------------
+
+Let pi: C -> S be a smooth projective family of connected curves
+of genus b over an irreducible smooth quasi-projective complex
+variety S, with a section. Fix a rational divisor class
+e_C in A^1(C) of relative degree one. The class e_C need not
+be the class of the section.
+
+Set:
+
+ a_C = pi_*(e_C^2) (a base class)
+ e'_C = e_C - (1/2) pi^*(a_C) (a "corrected degree-one")
+ eta = [Delta_{C/S}] - e'_{C,1} - e'_{C,2} (the "projector")
+
+Here Delta_{C/S} is the diagonal, and e'_{C,i} means e'_C pulled
+back via i-th projection of C x_S C.
+
+The point of the correction: e'_C is a degree-one class whose
+self-intersection lands in the base (not in C). Combined with
+the base-section algebra, this gives a decomposition of C into
+a "hyperbolic plane" (unit and correction) plus a leftover
+summand. The projector eta isolates that leftover summand: it
+is a self-transpose (up to sign, in the super sense) idempotent
+whose trace equals -2b.
+
+This is the standard tensor-model decomposition used all over
+the tautological literature (Gross-Schoen, O'Sullivan). Its
+whole reason to exist is that the surface Quot relation of
+Section 5.2 can be stated cleanly in it.
+
+The category we work in is the rational correspondence
+category Ch with:
+
+ - objects: smooth projective S-families (of curves, curve
+ powers, etc.);
+ - morphisms Hom(P,Q) = A^*(P x_S Q);
+ - composition: pull to triple product, intersect, push;
+ - tensor: relative product;
+ - finite direct sums and split idempotents allowed.
+
+We may quotient by classes that vanish on S minus a codimension-a-
+subset; call this "the base-support quotient." It is not
+numerical equivalence, and it is not testing individual fibers.
+
+Proposition 5.1 (Faithful Chow tensor test). In this setup there
+is a supercommutative rational algebra B_b with:
+
+ (i) A faithful tensor model of the curve algebra on
+ B_b (x) (Q 1 + W_b + Q e), where W_b is an odd
+ symplectic space of dimension 2b. The classes
+ 1_C and e'_C correspond to 1 and e; the projector
+ eta projects onto W_b. Integration gives the hyperbolic
+ pairing <1,e>=1 on the even summands and the symplectic
+ pairing on W_b. All actual Chow correspondences between
+ curve powers are faithful in this model, not only
+ pairing diagrams.
+
+ (ii) The projected small diagonal is an alternating three-
+ form phi: (W_b)^3 -> (B_b)_odd. Cup product, tensor,
+ permutation, contraction are computed with Koszul signs.
+
+ (iii) For an ordinary rational color space U: a pure colored
+ tensor in
+
+ ( eta^{(x)k} A^*(C^k) (x) U^{(x)k} (x) sgn_k )_{S_k}
+
+ is zero iff all its evaluations using the same ordinary
+ linear map z: U -> W_b in every slot are zero in B_b.
+ Finitely many rational maps z suffice for each fixed U,k.
+
+ The same holds in the base-support quotient.
+
+The most important clause here is faithfulness: this is not just
+an analog model. It is an actual isomorphism onto the Chow
+classes. In particular, any class the model cannot see has some
+specific evaluation into B_b vanishing, rather than some analogy
+failing.
+
+For a pointed curve over a field, the projected small diagonal
+is the modified diagonal of Gross-Schoen. In a family, the base
+corrections (the (1/2) pi^*(a_C) terms) are part of the same
+construction. This is exactly what F_J^st - (1/2) beta_J^st is
+modeling at the compactified level.
+
+--------------------------------------------------------------------------------
+5.2 The surface Quot relation
+--------------------------------------------------------------------------------
+
+Now suppose we have, in addition, a smooth family of curves of
+genus b over a smooth base S, as in Proposition 5.1. We form the
+surface
+
+ X = C x_S C,
+ pi_X: X -> S,
+ K = K_{X/S}.
+
+Every fiber of X is a surface with K ample, K^2 = s_K = 8 chi,
+Euler characteristic chi = (b-1)^2.
+
+Definition 5.2. Let Quot be the relative Quot scheme of
+rank-zero quotients
+
+ 0 -> I' -> V -> Q -> 0
+
+where V = L_1 + L_2 = K + O_X, subject to:
+
+ D = d K = 3K,
+ det Q = D in Pic(X/S) (a determinant constraint),
+ int_{X_s} ch_2(Q_s) = nu + L_1 D - D^2/2,
+ nu = t_0 - 1.
+
+Here t_0 = 25 chi - 1 (this is the t_0 from Theorem 5.3 below).
+
+The numerical parameters:
+
+ d = 3, s_K = 8 chi, chi = (b-1)^2
+ t_0 = chi + d(d-1)/2 * s_K - 1 = 25 chi - 1
+ m = 3 t_0.
+
+Quot is a projective moduli space over S, by Nitsure's Quot
+theorem plus the Kleiman Picard scheme for the determinant
+constraint.
+
+The key structural facts about Quot that we need, in order:
+
+ (a) Derived obstruction theory. The determinant constraint
+ changes both the obstruction space and the virtual
+ dimension. The map to based Picard (via trace of
+ determinant) is surjective on Ext^2, giving a two-term
+ perfect obstruction theory. This is a standard application
+ of the perfect-complex moduli theory (TV, STV).
+
+ (b) Fixed loci and virtual classes. The torus acts with fixed
+ subsheaves splitting into the two weight components. Each
+ rank-one factor has a unique "divisor + residual point"
+ factorization (Fogarty). Setting up a low/high index
+ choice lets us express the fixed virtual class as a
+ virtual pullback from a low-side Quot scheme, which
+ reduces to a product of a divisor scheme with a Hilbert
+ scheme of points.
+
+ (c) Marked localization. We attach m = 3 t_0 independent
+ surface positions, insert ch_2^{T}(Q) at each, and take
+ the localized pushforward to X^m. Tracking the codimension
+ budget shows only a specific low-index branch survives,
+ and only its bare-triple partitions contribute a nonzero
+ scalar.
+
+ (d) Marked universality (Lemma, EGL relative version).
+ Pushing forward finitely many universal point-ideal
+ Chern characters (with marked surface outputs retained)
+ gives a universal combination of decorated equality
+ diagonals. This lets us identify the surviving scalar
+ without committing to a fiberwise computation first.
+
+The upshot is the localization relation in the following form:
+
+Theorem 5.3 (Surface bound, Chow). Suppose b >= 5 and
+dim S < chi - 1 = (b-1)^2 - 1. Let (w_alpha)_{alpha in I} be a
+finite collection of vectors in the odd symplectic standard
+space W_b with isotropic span. For any matrix of scalars
+gamma = (gamma_{alpha beta}), define
+
+ T_gamma = sum over alpha_1,alpha_2,alpha_3
+ beta_1,beta_2,beta_3
+ ( product_{j=1}^3 gamma_{alpha_j beta_j} )
+ phi(w_{alpha_1},w_{alpha_2},w_{alpha_3})
+ phi(w_{beta_1},w_{beta_2},w_{beta_3}).
+
+Then
+
+ T_gamma^{t_0} = 0 in B_b.
+
+The assertion remains valid after any characteristic-zero scalar
+extension.
+
+Interpretation. The right object to think about is not "small
+diagonals" but "products of two cubic phi values tied by the
+matrix gamma". If we plug in isotropic vectors w_alpha -- that is,
+vectors whose symplectic pairings vanish on the span -- the
+surface relation says the t_0-th power of that product is zero.
+
+This is the "surface Quot relation" of the paper. It is the
+geometric axiom that produces the boundedness we need in
+Part 6.
+
+--------------------------------------------------------------------------------
+5.3 The scalar a_{N,t_0} is nonzero
+--------------------------------------------------------------------------------
+
+The proof of Theorem 5.3 has a scalar factor that could in
+principle vanish. The next result shows it doesn't.
+
+Proposition 5.4 (Lehn-type scalar). With the parameters of
+Theorem 5.3, the scalar a_{N,t_0} produced by the marked
+localization computation is a nonzero rational number.
+
+The scalar a_{N,t_0} is defined by a residue:
+
+ a_{N,t_0}
+ = (1 / (6^{t_0} t_0!))
+ [q^{d s_K - 1}]
+ (1 + w_0)^{d s_K} (1 + 2q)^{t_0} (1 + 4q)^E
+
+where w_0(1 + w_0) = q with w_0(0) = 0, d = 3, s_K = 8 chi,
+t_0 = 25 chi - 1, chi = (b-1)^2, and
+
+ E = d(d+1) s_K / 4 - 3 t_0 / 2.
+
+We won't reproduce the whole derivation here -- it lives in
+Sections 5-6 of the original paper, a difficult Hilbert-scheme
+interpretation. What matters is the conclusion: substitution
+v = 1 + 2 w_0 turns the coefficient into a residue at v = 1,
+
+ Res_{v=1} [ (1 + v^2)^{t_0} v^{2E+1} ] / (v-1)^{d s_K} * dv.
+
+This residue computes (up to sign) the residue at v = 0:
+
+ [v^{27 chi - 5}] (1 + v^2)^{25 chi - 1} (1 - v)^{-24 chi}.
+
+The v = 0 residue is nonzero and positive, because:
+ - the factor (1 + v^2)^{25 chi - 1} has nonnegative coefficients
+ and constant term 1;
+ - the factor (1 - v)^{-24 chi} has strictly positive coefficients
+ in every nonnegative degree;
+ - the exponent 27 chi - 5 >= 0.
+Hence the v = 1 residue is a negative scalar multiple of a
+positive integer and is nonzero.
+
+The exact bookkeeping of prefactors 2^{d s_K - t_0 - 1} and
+1/(6^{t_0} t_0!) is nonzero too. Combining: a_{N,t_0} != 0.
+
+Consequently Theorem 5.3 holds: since a_{N,t_0} is invertible
+in B_b, T_gamma^{t_0} = 0 follows from a_{N,t_0} T_gamma^{t_0} = 0.
+
+--------------------------------------------------------------------------------
+5.4 Arbitrary color products
+--------------------------------------------------------------------------------
+
+Theorem 5.3 gives nilpotence only for products whose inputs are
+isotropic vectors in W_b. That's a harsh restriction: our
+classes Y involve arbitrary color inputs, not isotropic vectors.
+
+To remove the restriction, we prove a purely linear-algebra
+lemma.
+
+Lemma 5.5 (An exterior ideal). Let F subset H be vector spaces
+of dimensions 2r and D, let Omega in Lambda^2 F be nondegenerate
+(a symplectic form on F). If 0 <= t <= r, then the ideal in
+Lambda H generated by Omega^t contains Lambda^q H for every
+q >= D - r + t.
+
+Proof sketch. The linear symplectic Lefschetz decomposition
+gives surjectivity
+
+ Omega^t ^ (-): Lambda^{k - 2t} F -> Lambda^k F
+
+for k >= r + t. Split H = F + F'. A nonzero summand of Lambda^q H
+has F'-degree at most D - 2r, hence F-degree at least
+q - (D - 2r) >= r + t. That summand is in the ideal Omega^t
+(Lambda H). So the whole Lambda^q H is in the ideal.
+
+Interpretation: this says a large positive power of the
+symplectic form kills everything of high enough exterior degree.
+"Large enough" is D - r + t.
+
+Now the application to colors. Take an m-dimensional color space
+E. Fix an even integer e with e >= m/2 - 2 and e <= m/2. Write
+
+ D_E = C(m,3), d_e = C(e,3).
+
+Lemma 5.6 (Arbitrary color products). Let a smooth curve family
+of genus b >= 5, with a section, be over a smooth irreducible
+quasi-projective base S satisfying dim S < (b-1)^2 - 1 = chi - 1.
+Put
+
+ t_b = 25 (b-1)^2 - 1 = 25 chi - 1 = t_0.
+
+If t_b <= d_e/2, then every product of
+
+ q >= D_E - d_e/2 + t_b
+
+pure cubic entries whose colors lie in E vanishes modulo
+support in base codimension at least b-1.
+
+Proof sketch. Fix a color test z: E -> W_b. The pullback of the
+symplectic form to E has a totally isotropic subspace of
+dimension at least m/2: take the radical plus a Lagrangian in the
+symplectic quotient.
+
+Choose an e-dimensional subspace E' of it. Its image under z is
+isotropic. Choose an invertible skew matrix gamma in a basis of
+E'. Grouping the ordered triples of E' into a quadratic expression
+in cubic generators, the coefficients of the resulting quadratic
+form are a nonzero scalar times the 3x3 minors of gamma, hence a
+nondegenerate bivector
+
+ Omega in Lambda^2 ( Lambda^3 E' ).
+
+Theorem 5.3 says (via a specialization) that Omega^{t_b} kills
+the whole cubic coefficient space in the specified support
+quotient.
+
+Lemma 5.5 then says: with F = Lambda^3 E' (dimension 2r where
+2r = d_e, and r = d_e/2), H = Lambda^3 E (dimension D_E), any
+exterior element of degree >= D_E - d_e/2 + t_b is in the ideal
+Omega^{t_b}. This is exactly the threshold in Lemma 5.6.
+
+Apply Proposition 5.1 (faithfulness). All color maps z agree on
+the threshold, so the vanishing is uniform in z. Faithful
+polarization (Prop 5.1(iii)) upgrades this to a vanishing in
+Chow, modulo the base-support bound.
+
+The base-support bound b-1 is required everywhere: it is the
+quotient by which we already lost information when we restricted
+to the surface.
+
+--------------------------------------------------------------------------------
+5.5 What Part 5 has given us
+--------------------------------------------------------------------------------
+
+We now have:
+
+ (i) A faithful model of Chow correspondences on curve powers
+ in a coefficient algebra B_b with an odd symplectic W_b
+ and a projected cubic phi.
+
+ (ii) A surface Quot relation: T_gamma^{t_0} = 0 for
+ isotropic-gamma contractions, t_0 = 25 (b-1)^2 - 1.
+
+ (iii) The scalar a_{N,t_0} in the surface relation is
+ nonzero.
+
+ (iv) An exterior-ideal argument: arbitrary color products
+ of >= D_E - d_e/2 + t_b pure cubic entries vanish
+ modulo base codimension b-1.
+
+What's missing, and what Part 6 supplies:
+
+ (v) The compactification step: the class Y, restricted to
+ the boundary, agrees with a corrected cubic that we can
+ compare to the pure cubic entries from (iv), up to
+ controlled errors over a dense open.
+
+ (vi) Support induction: iteratively reduce the dimension of
+ the base-image of the residual loca, using (v),
+ until the support is empty.
+
+Part 6 does (v) and (vi). That completes Claim B and the proof
+of Theorem 2.1.
+
+End of Part 5. Part 6 handles the compactification.
+
+
+=====================================================
+PART 6: COMPACTIFICATION AND THE LAST STRETCH
+=====================================================
+
+Part 5 proved a bound for pure cubic entries over smooth curve
+families. Part 6 lifts this to the entire compactified moduli
+space Mbar_{g,n}. Two things happen here:
+
+ 1. Boundary analysis: we build a "corrected cubic" class
+ F^0_{h,B} on weighted compactifications whose restriction
+ to the smooth core agrees with the projected small
+ diagonal (from Part 5).
+
+ 2. Support induction: on the boundary, F^0_{h,B} differs from
+ the smooth-curve class by controlled errors. We iteratively
+ lower the dimension of the base image of the residual
+ cycle until it is empty.
+
+The key geometric objects are Hassett's weighted stable curves
+with light and heavy markings. We use these because plain
+Mbar_{g,n} is not fine enough for the comparison: we need to
+allow lights to collide while heavy marks stay separate.
+
+The chapter is long. The overall shape:
+
+ 6.1-6.2 Weighted conventions and the corrected cubic
+ 6.3 The blowup at a triple flag: three-position
+ calculation
+ 6.4 Pure lifts: F^0 controlled by G_h,B
+ 6.5 Support advancement lemma (dimension induction)
+ 6.6 Finite color budget (arithmetic)
+ 6.7 Wrap
+
+--------------------------------------------------------------------------------
+6.1 Weighted configurations and cut charts
+--------------------------------------------------------------------------------
+
+Let (h,B) be "heavy data": B is a finite ordered set (the heavy
+marks) and 2h-2+|B| > 0 (heavy stability). Write
+
+ X_{h,B}(k) = Mbar_{h, (1^B, epsilon^k)} for 0 < k epsilon <= 1,
+
+ X_{h,B}(0) = Mbar_{h,B}.
+
+The first argument is the genus; the entries in (1^B, epsilon^k)
+specify heavy weights (all equal 1) and light weights (all equal
+epsilon, small and positive, total <= 1).
+
+Hassett's theory: for any positive rational weights with total
+at most 1, the moduli space of weighted pointed stable curves
+is a smooth proper Deligne-Mumford stack. Lights can collide
+with one another (but not with heavy marks); one-light space is
+the universal curve; weight reduction gives forgetting push and
+pull maps.
+
+A rational component with one node and only light markings is
+forbidden by stability. A rational component with two nodes and
+positive light mass is allowed and stable.
+
+Now fix a stable graph Gamma of the heavy type. At a vertex v
+with genus h_v and heavy set B_v (original marked points plus
+incident half-edges, all ordered), write
+
+ B_Gamma = product_v Mbar_{h_v, B_v} (heavy base)
+ S_Gamma = product_v M_{h_v, B_v} (smooth core)
+
+The cut chart with k lights:
+
+ X^Gamma(k) = coproduct over assignments a: {1,...,k} -> V(Gamma)
+ product_v X_{h_v,B_v}( a^{-1}(v) ).
+
+So a cut chart records how k lights are distributed among the
+vertices. It has a proper gluing map to X_{h,B}(k). Assignment
+conditions are part of the indexing: a class supported on a
+specific assignment is zero on the other components.
+
+Cutting and light forgetting commute: the heavy flag replacing
+a node branch retains its stability contribution, so stabilizing
+after gluing gives the same curve as forgetting-then-stabilizing.
+
+We work in rational Chow with operational (bivariant) behavior.
+For an ordinary color space U, the operation algebra we use is
+
+ (+)_{k >= 0} ( A^*(X^Gamma(k)) (x) U^{(x)k} (x) sgn_k )_{S_k}.
+
+The product pulls operations to disjoint sets of labels by
+forgetting and then multiplies. Cubic and singleton operations
+are odd, pair operations are even. All expressions below are
+first identities of labeled classes, then identities in this
+algebra.
+
+--------------------------------------------------------------------------------
+6.2 The corrected cubic F^0_{h,B}
+--------------------------------------------------------------------------------
+
+For three light labels 1,2,3, define
+
+ F^0_{h,B} = [1=2=3]
+ - (1/2) beta_{h,B}
+ + (1/2) * sum_{p in B} delta_{p,123} psi_p (*)
+
+in A^2(X_{h,B}(3)). Each term:
+
+ - [1=2=3]: coincidence class, product of [1=2] and [1=3].
+
+ - beta_{h,B}: take each stable one-edge graph of the heavy type,
+ subdivide its edge by a genus-zero vertex carrying exactly
+ the three lights 1,2,3, sum the resulting gluing classes
+ divided by their graph automorphism orders. This is the
+ "rational bridge" correction on the weighted compactification.
+
+ - delta_{p,123} psi_p: for p in B (heavy mark), the divisor
+ where p and 1,2,3 sit on a rational tail; psi_p is the heavy
+ cotangent class on the weighted space.
+
+All terms are symmetric in the three light labels. This is
+F^0 ("raw cubic"). It is NOT the final smooth pure diagonal --
+that gets built below.
+
+Lemma 6.1 (Weighted pullback). Let g >= 2 and heavy set B empty.
+Pull three-light operations to any larger number of light labels
+by forgetting. Under weight reduction from ordinary stable
+pointed curves:
+
+ [i=j] pulls back to D_bar_{ij},
+
+ beta_{g,empty} pulls back to the normalized two-edge bridge sum
+ used to define beta_J^st.
+
+Consequently the coefficient of the prescribed color word in the
+signed product of all F^0_{g,empty} pulls back to q(Y), up to a
+nonzero rational scalar.
+
+Proof sketch. In codimension one, the inverse image of [i=j] is
+exactly the rational tails containing both labels. At the generic
+point of each tail, the two coarse positions differ by a
+node-smoothing parameter, and every multiplicity is one.
+
+For the bridge: both bounding nodes exist already in the source
+(weight reduction does not contract a bridge with positive light
+mass). Therefore the inverse image is supported on the
+corresponding ordinary strata of codimension at least two. At a
+generic two-edge stratum the three labels are distinct on the
+bridge and no component is contracted, so the multiplicity is
+one. Further collisions add edges and have higher codimension.
+
+Forgetting commutes with reduction. Extracting a fixed color
+word from signed averaging retains exactly the permutations
+preserving that word, with signs. Depending on averaging
+convention there is a positive factorial denominator, but no
+zero scalar.
+
+End of Lemma 6.1 sketch.
+
+--------------------------------------------------------------------------------
+6.3 The local three-position calculation
+--------------------------------------------------------------------------------
+
+Over the smooth core S_Gamma, let C_v denote the smooth curve at
+vertex v. For a fixed assignment of three light labels, there is
+a coarse position map
+
+ c: X^Gamma(3)|_{S_Gamma} -> C_{a(1)} x_{S_Gamma}
+ C_{a(2)} x_{S_Gamma}
+ C_{a(3)}
+
+(all products over S_Gamma).
+
+Lemma 6.2 (Three-position blowup). The map c is the iterated
+blowup obtained, at each flag p, by:
+
+ - first blowing up the locus where all three assigned
+ positions equal p (when present);
+ - then blowing up the strict transforms of the loci where two
+ assigned positions equal p.
+
+The latter centers are disjoint after the first blowup.
+
+Proof sketch. Extra components above a smooth pointed core are
+chains attached at its flags. Write a local parameter x_i for
+each label's coarse coordinate near a flag. The zero divisor of
+x_i is the sum, with multiplicity one, of the cuts moving label
+i onto the expanded chain. Cutting and smoothing satisfy
+x*y = t, so at a fixed point the cuts form an initial segment
+of the ordered chain. Each relevant pair/triple ideal is
+principal after pullback, giving the first blowup then the
+successive pair blowups.
+
+Blowup ratios recover the expanded configuration: on a level,
+the leading ratios of coarse-coordinate differences are the
+ratios of light coordinates on that level, modulo common scaling.
+Labels on separate levels give zero/infinity. Forgetting to
+pairs gives all these ratios compatibly, so the ratios determine
+levels and their configurations.
+
+Automorphisms fixing the core and marked lights are trivial. The
+resulting representable map is quasi-finite, proper, birational
+to a smooth target [Stacks 02LS and 0AB1]. Working on an etale
+chart of the core data, it is an isomorphism.
+
+In the first exceptional plane, the strict pair centers meet at
+three distinct coordinate points, proving their disjointness.
+Constructions at distinct flags have disjoint centers.
+
+End of Lemma 6.2 sketch.
+
+Denote the total first exceptional divisor at a triple flag
+locus by E_{p,123}, and pair exceptional divisors (preferably
+pulled back from two-light spaces) by E_{p,ij}.
+
+The blowup formula for Chow groups (Fulton Prop 6.7(e)) gives
+that any codimension-two class on c has, in addition to its main
+pullback, terms:
+
+ E_{p,ij} A_{p,ij} + E_{p,123} D_p + a_p E_{p,123}^2 (**)
+
+ A_{p,ij} a divisor class on the remaining curve over the base;
+ D_p a base divisor class;
+ a_p a scalar on each connected component of the base.
+
+If we work with the STRICT pair-exceptional divisor E_{p,ij}^str
+versus the PULLED BACK pair-exceptional divisor E_{p,ij}^pull,
+then
+
+ E_{p,ij}^pull = E_{p,ij}^str + E_{p,123}.
+
+Multiplying by a curve divisor D_{p,ij}, this changes only the
+D_p term, not the a_p E_{p,123}^2 coefficient. So using pulled
+pairs is harmless.
+
+When a raw class F^0 is pulled from an ancestor chart, its main
+component is the small diagonal if all three labels are assigned
+to one relevant descendant vertex, and zero otherwise. The
+coincidence/bridge/heavy-tail corrections project to loci where
+all three coarse positions are flags, of codimension three; they
+cannot alter a main codimension-two class. Hence the divisor
+coefficients A_{p,ij} of F^0 are supported on flag sections of
+the remaining curve, up to base terms.
+
+Lemma 6.3 (Triple-exceptional cancellation). For the pullback
+of a raw cubic F^0 from an ancestor chart, the coefficient a_p
+in (**) is zero at every triple flag locus.
+
+Proof sketch. Over a field point of the smooth heavy base, the
+full fiber over three positions at a flag is the weighted chain
+space with two heavy endpoints and three lights: the Losev-Manim
+space. By Lemma 6.2 it is the first exceptional P^2 blown up at
+its three pair-collision points.
+
+The total class E_{p,123} restricts to the pullback of
+O_{P^2}(-1); hence
+
+ int E_{p,123}^2 = 1.
+
+Main pullbacks, pair terms times remaining-position divisors, and
+base terms have integral zero on this fiber. So this integral
+detects exactly a_p.
+
+The coincidence term gives 1: its locus is the single transverse
+configuration where all three lights coincide away from endpoints.
+
+If p is a NEW cut flag (from a fuller graph), there is one pair
+of bounding nodes isolating the chain carrying the three lights.
+With these cuts distinguished, the bridge locus is a regular
+two-node condition in a versal deformation chart. Its two
+normal lines are tensor products of branch tangent lines; the
+core factors are constant, so its self-intersection restricts to
+
+ (-psi_0)(-psi_infinity) = psi_0 psi_infinity.
+
+The normalized bridge sum counts this local branch once (this
+also handles self-nodes).
+
+Reduction from Mbar_{0,5} to the chain space preserves both
+heavy cotangent lines. Genus-zero string recurrence gives
+
+ int_{Mbar_{0,5}} psi_0 psi_infinity
+ = int_{Mbar_{0,4}} psi_0 + int_{Mbar_{0,4}} psi_infinity
+ = 2.
+
+Hence the bridge contribution to F^0 is -1/2 * 2 = -1, canceling
+coincidence +1.
+
+If p is an INHERITED heavy flag, the relevant term is the unique
+outer tail containing that heavy mark and all three lights. Its
+normal factor is -psi_0 on the chain; the inserted psi_p is
+psi_infinity. Thus delta_{p,123} psi_p has integral -2, and its
+coefficient +1/2 again cancels the coincidence term.
+
+No other bridge or heavy-tail type meets this fiber: after lights
+are forgotten, only the specified node or specified heavy mark
+survives.
+
+So a_p = 0 on every connected smooth-core base component.
+
+End of Lemma 6.3 sketch.
+
+--------------------------------------------------------------------------------
+6.4 Controlled pure lifts
+--------------------------------------------------------------------------------
+
+For a smooth heavy core pi: C -> S, choose a degree-one rational
+class
+
+ e_C = [p] if a heavy flag p is chosen,
+ = c_1(omega_{C/S}) / (2h-2) if B is empty (h >= 2).
+
+Then put
+
+ e'_C = e_C - (1/2) pi^* pi_*(e_C^2)
+ eta_C = [Delta_C] - e'_{C,1} - e'_{C,2}
+ Delta_C^pure = small diagonal projected by eta_C in each of its three slots.
+
+The expansion
+
+ [Delta_{123}] - Delta_C^pure = (sum of pair-singleton products
+ with singleton factors 1_C or e'_C)
+
+is a factual decomposition of the diagonal correspondence (not
+a numerical equivalence). This is the smooth-core analog of the
+surface corrections from Part 5.
+
+Proposition 6.4 (Controlled pure lifts). For every stable heavy
+type (h,B) one can choose a symmetric class G_{h,B} in
+A^2(X_{h,B}(3)) with the following properties:
+
+ (i) Over the smooth heavy base, G_{h,B} equals the pullback of
+ Delta_C^pure by the coarse-position map.
+
+ (ii) Pull a raw cubic F^0 or one of these chosen pure lifts
+ G_{h,B} from any ancestor chart to a compact descendant
+ cut chart. Its operation is a sum of the chosen pure
+ lifts at relevant descendant vertices (all three labels
+ at that vertex), together with pair-times-singleton
+ operations and a residual cubic operation.
+
+ (iii) The singleton operations in (ii), and all their further
+ pullbacks through cuts and light forgettings, lie
+ componentwise in the span of unit, canonical class, and
+ flag points (over fixed smooth core data). On any smooth
+ test family the same holds modulo classes supported
+ over positive base codimension.
+
+ (iv) On the current smooth-core open, the residual operation
+ in (ii) is a finite sum of classes containing a pulled
+ base divisor factor (from current or ancestor heavy base).
+ Consequently its pullback to every smooth test family is
+ zero operationally after removing a positive-codimension
+ subset of that test base.
+
+All expansions preserve the color multiset. They may always use
+one and the same chosen G and centering at a given current
+vertex, regardless of which ancestor supplied the operation.
+
+Proof sketch. On the smooth heavy base of the class itself, the
+blowup decomposition (**) and Lemma 6.3 express F^0 as the
+pullback of the small diagonal, plus pair-exceptional-times-
+singleton terms and triple-exceptional-times-base-divisor terms.
+Subtract these, and also subtract the [Delta] - Delta^pure
+decomposition.
+
+These subtractions have compact extensions with the required
+structure:
+ - pair exceptional extends as the heavy-flag tail boundary on
+ the two-light space;
+ - triple exceptional extends as its three-light counterpart;
+ - divisor coefficients extend by Chow localization;
+ - singleton factors use units, heavy sections, relative
+ dualizing line on one-light universal curve, and base
+ classes.
+
+Define G by making these structured subtractions from F^0 and
+averaging over the three labels. This gives (i).
+
+For (ii), pull to a deeper cut chart. The raw F^0 has the
+decomposition just proved there. Its main component is a sum of
+small diagonals of relevant descendant vertices. Replace each by
+the corresponding G plus structured corrections.
+
+For (iii): inherited pair-times-singleton subtractions retain
+the form; the singleton spans as claimed by the normalization
+formula for the dualizing line (canonical class plus branch
+points on each component, base terms stay base terms).
+
+For (iv): the only other corrections are exceptional divisors
+times base divisor classes. On pulling an inherited correction,
+the exceptional factor need not have a special form, but its
+base divisor factor remains pulled from an ancestor heavy base.
+So the residual is of the form
+
+ sum_j H_j * rho_j^* D_j,
+
+ H_j a one-codimensional three-label operation,
+ D_j a divisor class on a current or ancestor heavy base.
+
+Represent each pulled rational divisor class by the first Chern
+class of a rational line bundle. After removing the finite union
+of divisors on the test base, every summand is zero as an
+operational class. This remains true when S maps into a special
+locus of an ancestor base: what matters is the pulled line
+bundle on S, not the original support of D_j.
+
+Assignment conditions can be imposed separately on pair and
+singleton factors by making their operations zero on other
+components of the disjoint union X^Gamma(k). All steps occur
+before inserting colors, so preserve their multiset.
+
+End of Proposition 6.4 sketch.
+
+--------------------------------------------------------------------------------
+6.5 Support advancement
+--------------------------------------------------------------------------------
+
+We now formalize the induction that repeatedly lowers the
+dimension of the base image of a residual cycle.
+
+Setup. A "term" in the induction is a set of active operations,
+pulled to disjoint sets of light labels, acting on a cycle on
+the full labeled chart. The cycle need not factor over the
+labels or the vertices. This is the freedom that lets us push
+through the induction even after odd operations have "consumed"
+some labels.
+
+Lemma 6.5 (Support advancement). Let a cycle multiplier on
+X^Gamma(n) have base image contained in an irreducible closed
+Z subset of B_Gamma. After a refinement if necessary, suppose
+its generic base image lies in the smooth core S_Gamma. Let L be
+a factor involving a specified subset of the active entries.
+
+Suppose that the pull of L vanishes operationally over a dense
+open of a smooth test family dominating Z intersect S_Gamma.
+Then its action can be written as a finite sum of cycles with
+strictly smaller base-image dimension. Only the entries in L
+are consumed.
+
+More precisely: over a smooth test base S proper generically-
+finite over Z intersect S_Gamma, if the vanishing holds off
+base codimension >= a, then the terms remaining over S_Gamma
+have base-image dimension at most dim Z - a. Other terms lie
+over Z minus S_Gamma and require a strict graph refinement.
+
+If one replaces S by the total space of one of its curve
+families (to get a section), the corresponding bound is
+dim Z - a + 1.
+
+Proof sketch. A cycle with a finer generic dual graph can be
+lifted to a chart of that graph; heavy gluing is finite, so
+this doesn't increase base-image dimension.
+
+Use the following property of rational Chow groups: if
+T' -> T is proper and surjective, every integral cycle on T has,
+after multiplication by a nonzero rational, a proper lift
+generically finite over it. Close up a closed fiber point, and
+the pushforward is the original cycle times the generic degree.
+Same property holds for finite-type Deligne-Mumford charts
+here [BaeSchmitt B.19].
+
+For the test base, take a finite level cover of the smooth
+heavy-curve stacks and resolve the component over
+Z intersect S_Gamma (Wlodarczyk). A full level of order >= 3
+removes automorphisms in positive genus (Deligne-Mumford,
+Deligne-Rapoport). Lift the cycle multiplier properly.
+
+Let A subset S be codim >= a, with L operationally zero off A.
+Chow localization expresses the action as a pushforward of
+cycles lying over A. Return to the original family by
+projection/pushforward. These cycles have base-image dimension
+< dim Z. Closing up inside the preimage of Z introduces cycles
+only over Z minus S_Gamma, a proper closed subset.
+
+To get a section of a smooth curve family C -> S, pull back
+along C -> S, where it has its diagonal section. New test base
+is smooth of dim Z+1. A support of codim >= a in the new base
+has image dim <= dim Z - a + 1.
+
+All operations concern the chosen factor L. Other operations are
+carried along by pullback and projection formula.
+
+End of Lemma 6.5 sketch.
+
+--------------------------------------------------------------------------------
+6.6 Elementary vanishing tests and repeated color
+--------------------------------------------------------------------------------
+
+We need concrete tests that provide the operational vanishing
+required by Lemma 6.5.
+
+Define, for a graph Gamma,
+
+ R_Gamma = 2|V(Gamma)| + 2|E(Gamma)|.
+
+Lemma 6.6 (Noise and small-genus tests). If R_Gamma + 1 noise
+entries have their singleton at the same color, their product
+advances support while consuming those entries. If a genus-h
+vertex has more than C(2h,3) pure entries, the product of any
+C(2h,3)+1 of them advances support.
+
+Proof sketch. Over smooth generic core data, the one-light space
+at a vertex is its curve. Permitted singleton classes lie in
+the span of unit, canonical class, and flag points, so overall
+dimension is at most
+
+ sum_v (2 + |B_v|) = 2|V| + 2|E| = R_Gamma.
+
+A one-slot operation is odd in the sign-coinvariant algebra, so
+a product of R_Gamma + 1 same-color singletons vanishes by
+alternation. Accompanying pair operations can be grouped
+separately.
+
+For the pure entry bound: by Proposition 5.1, each pure entry
+contributes a value of the same odd cubic phi in
+B_h (x) Lambda^3 W_h^dual. Values span at most C(2h,3) odd
+coefficients; a longer product vanishes by supercommutativity.
+Faithfulness gives the operation vanishing modulo positive base
+support. Combine with Lemma 6.5.
+
+End of Lemma 6.6 sketch.
+
+Lemma 6.7 (Repeated-color advancement). Suppose a vertex of
+genus h has a flag and k > C(h+1,2) pure entries contain one
+common color i. Then their product is a finite sum of products
+L * H with:
+
+ (i) L involves at most h+2 entries and vanishes modulo
+ positive base support;
+ (ii) H is still a product of individual pure entries at that
+ vertex;
+ (iii) each term has the same color multiset as the original.
+
+Consequently each term admits a support advancement consuming at
+most h+2 entries.
+
+Proof sketch. A symmetric cubic class defines an alternating
+three-color operation. Put the common color first in each entry;
+the product then factors through Lambda^k(Lambda^2 U_0). This
+holds already on the compact chart (alternation inside a pair is
+from cubic symmetry; interchange of two cubic entries has sign
+minus because arity is odd).
+
+By Schur-Weyl and the row cutoff, the corresponding projector p
+on the 2k input slots lies in the two-sided ideal of the group
+algebra generated by the alternator on h+2 slots. In a finite
+sum decomposition, each term uses at most h+2 entries during
+alternation. Group those into L; the remaining entries go into H.
+
+For vanishing of L: work modulo positive base support, center
+the curve projector at the flag section. Then int(e) = 1,
+int(e^2) = 0, evaluation at the section is multiplicative and
+kills W_h. For standard odd vectors u,v,
+
+ u v = omega(u,v) e + (u v)_pure
+
+(no unit component, since coefficient of u v in 1 is its
+evaluation at the section).
+
+Test via z: U_0 -> W_h, a = z(i). Each entry becomes
+T_a(u,v) = phi(a,u,v), an alternating 2-form with odd
+coefficients. The critical fact: contracting a variable slot
+between two such entries gives zero (super: a odd, a^2 = 0 in the
+Frobenius contraction int(a u * a v)).
+
+After alternation, L is an ordinary alternating (h+2)-form on
+W_h. Every summand of its double symplectic contraction is
+zero: if a contraction joins distinct entries, do it first and
+get zero; otherwise two internal contractions in distinct
+entries give (odd)^2 = 0.
+
+For h >= 2, double contraction
+Lambda^{h+2} W_h^dual -> Lambda^{h-2} W_h^dual is injective by
+symplectic Lefschetz. For h=0,1, source alternation is already
+zero. All color tests of L vanish. Apply Proposition 5.1 and
+Lemma 6.5.
+
+End of Lemma 6.7 sketch.
+
+--------------------------------------------------------------------------------
+6.7 Finite color budget
+--------------------------------------------------------------------------------
+
+Fix positive INTEGER constants
+
+ K = 10^5,
+ m_0 = 2000 * g^{2/3},
+ C_K = C(2K, 3).
+
+Reserve 4K + 20 disjoint blocks of m_0 colors each. Let S_0 be
+the remaining colors. We use
+
+ (4K + 20) m_0 / g = 8.0004 * 10^-12 < 0.1,
+ |S_0| > 0.9 g.
+
+For a current graph put
+
+ b = max_v h_v, l = g - b.
+
+Under refinement the maximum genus cannot increase (so l cannot
+decrease), and the number of edges cannot decrease either.
+Unmarked stability gives
+
+ |V| <= 2g - 2, |E| <= 3g - 3, R_Gamma <= 10 g.
+
+Advancements fall into two regimes:
+
+ Ordinary advancements: cost entries, from residual, noise,
+ small-genus, or repeated-color tests. Each consumes at most
+ ~11g entries.
+
+ Expensive advancements: from Lemma 5.6 (long products of pure
+ entries), only when a single vertex carries almost all genus.
+ These lower base-image dimension by >= g - K - 2.
+
+We count available entries against these costs. The key
+inequality is
+
+ 10^17 g << 0.01 D_0 for D_0 = C(m_0, 3),
+
+which holds easily at g = 10^60. The reserve blocks suffice for
+the total of at most 3g - 2 advancements and at most
+3K + 5 expensive ones. We do not need the details here; the
+bookkeeping lives in Section 8 of the original.
+
+--------------------------------------------------------------------------------
+6.8 Completion
+--------------------------------------------------------------------------------
+
+Theorem 6.8. The class Y of Definition 3.4 satisfies
+q_{g,n}(Y) = 0 in A^*(Mbar_{g,n}; Q).
+
+Proof sketch. Begin on X_{g,empty}(n) with the product of cubic
+operations F^0_{g,empty}, one for each increasing color triple,
+and the fundamental cycle as multiplier. Apply Proposition 6.4
+and the preceding induction (Lemmas 6.5, 6.6, 6.7).
+
+At every stage, each nonzero branch admits an advancement: use a
+residual or noise test when available, the small-vertex and
+reserved-block tests if l < K, and otherwise the repeated-color
+test. The counts show all required active entries and unused
+blocks are available at the moment they are needed.
+
+Each individual expansion and localization involves finitely many
+cycles. Each branch has length at most 3g - 2 (the advancement
+bound from Section 8 of the original paper). The process therefore
+ends with empty support, proving that the full product is zero in
+the sign-coinvariant Chow group of the weighted space.
+
+The weighted-to-unweighted comparison (Lemma 6.1) identifies the
+pullback of the raw cubic with q(F_J^st). After signed averaging,
+the coefficient of the ordered color word i is a nonzero rational
+scalar times q(Y). Pullback of the zero weighted class proves
+the theorem.
+
+End of Part 6.
+
+
+=====================================================
+PART 7: WHAT THIS CHANGES
+=====================================================
+
+We've now seen the whole proof. Here: what it means in
+context, and what remains open.
+
+--------------------------------------------------------------------------------
+7.1 The main consequence
+--------------------------------------------------------------------------------
+
+Pixton's original 2012 conjecture asked whether his proposed
+system of stable-graph relations was COMPLETE: whether
+ker q_{g,n} = P_{g,n} for every stable (g,n). This paper shows
+the answer is no: there is at least one stable pair for which
+ker q is strictly bigger than P.
+
+The proof is a single counterexample. That's enough. Once
+completeness fails somewhere, the uniform conjecture is false.
+
+Equivalently: the family of Pixton relations, though it produces
+many true relations, is not exhausted by the ones the recipe
+gives. There is a tautological relation of a different kind,
+one that no amount of Pixton-recipes (multiplied, glued,
+forgotten) can produce.
+
+The specific witness Y is amusingly small in a certain sense:
+it is a codimension-2 D_* class, built from pairs of rational
+tail divisors and a rational-bridge correction, alternated
+against color-symmetric shuffles. Its existence was not
+suspected from any particular low-complexity example, which is
+why the conjecture lasted as long as it did.
+
+--------------------------------------------------------------------------------
+7.2 Pixton's own follow-up work: non-Gorenstein at M_25
+--------------------------------------------------------------------------------
+
+Completeness is a different question from the GORENSTEIN
+PROPERTY.
+
+ - The Gorenstein property asks: is multiplication into
+ complementary degree a nondegenerate pairing?
+ - Completeness asks: are all relations among generators
+ produced by Pixton's rules?
+
+These are independent questions. The Gorenstein property can
+fail while completeness holds, and vice versa.
+
+Pixton himself proved that R^*(M_25) is not Gorenstein
+(Pixton 2026, Theorem 1.1, Section 2.2), while establishing
+Faber-Zagier completeness in that genus. That is a case where
+the "nice geometric property" fails but the "list of relations
+is complete" property holds.
+
+So the two failure modes are truly different, and this paper's
+failure is of the "completeness" kind. That's important
+conceptually: you cannot use this paper to deduce anything new
+about Gorenstein-ness at M_25, or vice versa.
+
+--------------------------------------------------------------------------------
+7.3 Related work
+--------------------------------------------------------------------------------
+
+Canning-Larson-Schmitt (2026). Proves completeness together with
+non-Gorenstein behavior on the COMPACT-TYPE spaces
+M^ct_6, M^ct_{5,2}, M^ct_7. These are compactifications that
+include only curves of compact type (no separating nodes). The
+technique relies on the special structure of compact-type. Our
+counterexample lives on the FULL stable compactification; there,
+Pixton's rules are not complete. So there is no contradiction:
+completeness on a smaller space (compact type) does not imply
+completeness on the full Mbar.
+
+Janda's discriminant paper (2018). Compares two methods for
+producing relations from reconstructions of semisimple CohFTs
+with nonempty discriminant. That comparison identifies the
+systems arising from two different cancellation schemes.
+It does NOT identify the entire Chow kernel with Pixton's
+system. Our paper's comparison is with the specific Pixton
+system P_{g,n}; Janda's comparison is orthogonal to that.
+
+PPZ (2015) and Janda (2017). Proved validity of Pixton's
+system: P_{g,n} is a subset of ker q_{g,n}. These proofs use
+3-spin CohFT and equivariant GW theory of P^1 respectively.
+Validity is a strong form of "these are indeed relations".
+Completeness is the reverse direction, and this paper refutes it.
+
+Canning (2025). Surveys the tautological ring and its
+Gorenstein property for full Mbar and various subspaces,
+discussing exactly the "original system" being refuted here.
+That paper states the Chow and rational-cohomological
+completeness of the original Pixton system as a conjecture
+(Conjecture 1 in Canning 2025). This paper provides a
+counterexample to that specific statement.
+
+--------------------------------------------------------------------------------
+7.4 Different flavors of "Pixton conjecture"
+--------------------------------------------------------------------------------
+
+One caution about language. The phrase "Pixton's conjecture" is
+sometimes used to refer to different statements depending on
+the community. Three distinct versions:
+
+ V1. Original Pixton 2012 graph formula gives a COMPLETE system.
+
+ V2. Weaker version restricted to specific colors, specific
+ decorations, or specific subspace of Mbar_{g,n}. Failure
+ of V1 does NOT refute all such weak versions.
+
+ V3. Complete-system conjecture only for the "operational"
+ closure P_{g,n} (Pixton span including all Pixton-ideal
+ consequences).
+
+This paper refutes V1 and V3, but does not directly refute
+narrower V2 variants that, e.g., restrict to compact type
+(Canning-Larson-Schmitt have positive results there) or to
+low genus (Pixton at M_25).
+
+When you see "Pixton's conjecture is false" in the popular
+press: check which version is being discussed. This paper
+specifically refutes the ORIGINAL full-system version, and
+that's what makes it non-trivial.
+
+--------------------------------------------------------------------------------
+7.5 What remains open
+--------------------------------------------------------------------------------
+
+Three directions remain open and interesting:
+
+ (a) Other counterexamples in lower genus. The construction
+ uses g = 10^60 to make the color budget and the
+ support-dimension clear. There may be infinitely many
+ lower-genus counterexamples, and characterizing the
+ exact set of (g,n) where Pixton completeness holds is
+ open. This paper only refutes the conjecture at one
+ explicit (g,n).
+
+ (b) Characterize ker q. This paper's Y is a specific class
+ that's in ker q but not in P. What the FULL kernel is
+ (i.e., how to describe an enlarged relation system that
+ IS complete) remains open. What kind of relations do you
+ need to add to Pixton's system to get ker q exactly?
+
+ (c) Gorenstein-ness beyond M_25. Pixton proved
+ non-Gorenstein at M_25. Canning-Larson-Schmitt
+ completed compact-type cases. The full
+ R^*(Mbar_{g,n}) Gorenstein behavior across all (g,n)
+ remains under investigation. This paper doesn't touch
+ that question except by example: the failure of
+ completeness does NOT imply failure of Gorenstein.
+
+Beyond these three, there is a longer-term question: what is
+the right conceptual framework for the tautological ring that
+would give completeness in a broader sense? Candidates exist
+(e.g., adding cross-boundary "Mumford-type" relations, or
+working in a quotient that naturally produces the correct
+kernel). This paper's Y is meant to be a stepping stone:
+it tells you the original system was too small, and the proof
+techniques (exterior-ring detectors, surface Quot relations,
+support induction) may generalize to a full description.
+
+--------------------------------------------------------------------------------
+7.6 Summary of what just happened
+--------------------------------------------------------------------------------
+
+To recap, in one paragraph: we built an explicit class Y in
+the strata algebra S_{g,n} for g = 10^60 and n = 3*C(g,3). We
+proved two independent facts about Y:
+
+ (i) Y is not in the Pixton span P_{g,n}. (Part 4: formal
+ detector over an odd Frobenius algebra.)
+
+ (ii) Y vanishes in rational Chow. (Parts 5-6: surface
+ Quot relation plus compactification plus support
+ induction.)
+
+Together, (i) and (ii) say ker q_{g,n} is strictly bigger than
+P_{g,n}. By the cycle-class map, the same failure holds in
+rational cohomology. This refutes Pixton's original
+completeness conjecture.
+
+The two proofs live in completely different worlds (formal vs
+geometric) and neither implies the other. This is why the whole
+apparatus -- surface Quot schemes, Hilbert-scheme scalars,
+support induction -- is necessary. If we could prove non-membership
+from geometry alone, or vanishing from formalism alone, we
+wouldn't need either the detector or the surface machinery.
+
+That's the paper. The glossary follows.
+
+
+=====================================================
+GLOSSARY
+=====================================================
+
+The alphabetized terms appear throughout. Entries that are
+fairly technical (marked with *) are only needed if you want
+to read the formal proofs.
+
+A^* (X), Chow group of X
+ Algebraic analog of cohomology. Cycles modulo rational
+ equivalence, graded by codimension. Fact: A^*(X) is a
+ graded commutative ring.
+
+A-curve
+ Algebraic curve.
+ A compact smooth complex curve of genus g. So it is
+ simultaneously a Riemann surface of genus g and an
+ algebraic curve over C.
+
+algebraic curve of genus g
+ A compact smooth complex curve (Riemann surface) with
+ g "holes".
+
+Aut Gamma (automorphism group of a graph Gamma)
+ Permutations of half-edges preserving the graph structure
+ and the marked vertices. Finite. Same for stable graphs.
+
+bar transform (*)
+ The change of variables psi -> psi_bar and kappa -> kappa_bar
+ that separates boundary divisors from the "root" part.
+
+bar_psi, bar_kappa (*)
+ psi and kappa after subtracting all boundary contributions.
+ The formal "root" cotangent and kappa.
+
+B_b (faithful Chow coefficient algebra) (*)
+ A supercommutative rational algebra constructed in Part 5
+ so that the tensor model of curve correspondences is
+ faithful. Contains all Chow correspondences between curve
+ powers, not just pairing diagrams.
+
+b (in the surface Quot section)
+ The genus of the smooth curve family underlying the
+ surface. The "b-1" appears in base codimension bounds
+ because the surface Quot relation holds modulo that.
+
+base-support quotient (*)
+ The quotient of A^* by classes vanishing off a closed
+ subset of codimension at least a. Used to track loss
+ of information on parameter families. Not numerical
+ equivalence.
+
+beta_J^st, beta_{h,B} (bridge class)
+ A codimension-two class summing over one-edge unmarked
+ graphs of the relevant type, subdivided by a
+ genus-zero vertex carrying the three labels.
+
+Chow group (see A^*)
+ Algebraic cycles modulo rational equivalence.
+
+CohFT
+ Cohomological Field Theory. An algebraic structure encoding
+ tautological relations from Gromov-Witten theory. Not
+ needed for our proof, but historically crucial for
+ Pixton validity.
+
+compact type
+ A boundary stratum with only compact-type stable curves
+ (no separating nodes). M^ct_{g,n} is open inside Mbar_{g,n}.
+
+coproduct (*)
+ The adjoint of multiplication with respect to the
+ Frobenius pairing. Delta_L: V -> V^{(x)L} for a label set L.
+
+cycle class map cl
+ A^* (X) -> H^{2 *}(X), the map from Chow groups to
+ singular cohomology (over Q). Used for Corollary 2.2.
+
+D_T (tail divisor)
+ A codimension-one boundary divisor on Mbar_{g,n}
+ consisting of curves where the labels in T sit on a
+ rational tail attached to the rest by a single node.
+
+D_bar_{ab}
+ Sum over all T containing {a, b} of D_T.
+
+Delta_C, Delta_C^pure (*)
+ Small diagonal in C x_S C x_S C, and its projection via
+ eta_C in each of the three curve slots. Delta_C^pure is
+ a pure odd-class in the tensor model.
+
+Delta_e (edge series) (*)
+ Pixton's edge function in T, built from A(T), B(T) with
+ the identity A(T)B(-T)+A(-T)B(T)+2=0.
+
+e'_C (*)
+ e_C - (1/2) pi^*(a_C). A corrected degree-one divisor
+ class. Ensures the "hyperbolic plane" decomposition works.
+
+eta (*)
+ [Delta_C] - e'_{C,1} - e'_{C,2}. A projector onto the
+ odd symplectic summand.
+
+eta_C (*)
+ Same as eta but for a smooth core curve pi: C -> S.
+
+exterior algebra
+ Lambda_Q(...) means: multiply symbols antisymmetrically.
+ Square of any odd generator is zero.
+
+F^0_{h,B} (corrected cubic)
+ The raw boundary cubic on the weighted compactification:
+ [1=2=3] - (1/2) beta_{h,B} + (1/2) sum_p delta_{p,123} psi_p.
+
+F_J^st (three-slot factor)
+ D_bar_{ab} * D_bar_{ac} - (1/2) beta_J^st, the building
+ block of Y.
+
+Faber-Zagier
+ An early system of relations among kappa classes on
+ M_g (smooth curves). Its proving (Pandharipande-Pixton
+ 2021) motivated Pixton's extension to Mbar.
+
+Frobenius algebra (*)
+ An algebra V with a nondegenerate pairing (the integral),
+ such that V is self-dual for that pairing. The odd
+ symplectic examples here carry both a symplectic form
+ and an alternating cubic form.
+
+g, n (stable pair parameters)
+ g = genus, n = number of marked points. Stability is
+ 2g-2+n > 0. The paper's counterexample uses g = 10^60
+ and n = 3*C(g,3).
+
+Gorenstein property
+ The tautological ring R^*(Mbar_{g,n}) is Gorenstein if
+ multiplication R^d x R^{top-d} -> R^{top} is a perfect
+ pairing. This is a "nice geometric property". It is
+ independent from Pixton completeness.
+
+Gross-Schoen cycle (*)
+ A corrected small diagonal in the triple product of a
+ pointed curve. The prototype of the projected small
+ diagonal in Part 5.
+
+Hassett (weighted) moduli space
+ Mbar_{g,(w_1,...,w_n)} for positive rational weights
+ summing to at most 1. Allows light weights to collide.
+ Named after Brendan Hassett.
+
+heavy / light marks
+ In weighted stable curves, "heavy" marks (weight 1) are
+ kept separate; "light" marks (weight epsilon) may collide.
+
+Hilbert scheme of points (X^{[r]})
+ Moduli of finite flat subschemes of length r. Smooth if
+ X is a smooth surface. Used in the surface Quot relation.
+
+kappa class
+ A class measuring how the marked points wrap around a
+ curve. kappa_j = pi_*(psi^{j+1}) where pi forgets an
+ extra mark. In the strata algebra: kappa_j is a formal
+ symbol at each vertex.
+
+ker q_{g,n} (the kernel)
+ The set of formal combinations in S_{g,n} mapping to zero
+ under q_{g,n}. These are the actual relations.
+
+lambda
+ (2g-2) e in the odd Frobenius algebra. Plays the role of
+ the "handle element" at a curve. Properties: lambda^2 = 0.
+
+Losev-Manim space
+ Mbar_{0,(1,1,epsilon,epsilon,epsilon)}. The weighted
+ chain space with two heavy endpoints and three lights.
+ In Lemma 6.3, it is the first blowup P^2 of the
+ triple-exceptional locus.
+
+M_{g,n}, Mbar_{g,n}
+ Moduli of smooth curves / stable curves of genus g with
+ n marked points.
+
+marked point
+ A distinguished labeled location on a curve.
+
+node
+ A transverse self-intersection of a curve.
+
+Pixton relation R(g,n,r;sigma,a)
+ The formal sum over stable graphs of [Gamma, P_Gamma] /
+ |Aut Gamma| built from the recipe of Section 1.7.
+
+Pixton span P_{g,n}
+ The formal span of all Pixton relations, multiplied and
+ glued by arbitrary auxiliary strata. Already an ideal
+ (Pixton Prop 1).
+
+psi class
+ The cotangent class at a marked point (or at a
+ half-edge). psi_i in A^1(Mbar_{g,n}).
+
+q_{g,n} (realization map)
+ S_{g,n} -> A^*(Mbar_{g,n}), the formal-to-geometric map
+ that sends [Gamma, gamma] to xi_{Gamma,*} gamma.
+ Unnormalized (no Aut denominators).
+
+Quot scheme
+ Moduli of quotients of a fixed coherent sheaf. In our
+ surface setting, parametrizes rank-zero quotients of
+ V = K + O_X with fixed determinant and ch_2.
+
+rational tails graph (*)
+ A tree with one genus-g root and all other vertices
+ rational, every rational branch having >= 2 descendant
+ labels. Not the same as "curves with rational tails".
+
+rational tail divisor
+ Same as a tail divisor D_T (see above).
+
+S_{g,n} (strata algebra)
+ The free Q-vector space on decorated stable graphs modulo
+ isomorphism, with per-vertex dimension truncation.
+
+small diagonal
+ The diagonal Delta in C x C x C (three copies). Its
+ "projection" via eta corresponds to a modified diagonal
+ of Gross-Schoen type.
+
+stable graph
+ A graph with genus-labelled vertices whose dual graph
+ is the combinatorial shadow of a stable curve.
+
+Stable pair (g,n)
+ 2g - 2 + n > 0. Necessary and sufficient for the moduli
+ space Mbar_{g,n} to exist as a nonempty stack.
+
+support induction
+ The method of reducing the codimension-a support of a
+ cycle until it vanishes, used in Part 6.
+
+symplectic space (*)
+ A vector space with a nondegenerate antisymmetric form
+ < , >. In the odd setting, the form provides the pairing
+ on V's middle summand.
+
+tautological (class / ring / space)
+ Classes built from psi and kappa and gluing. The full
+ subring R^* = q(S) of A^*(Mbar_{g,n}) is the
+ tautological ring.
+
+Theta = product_{i<j<k} theta_{ijk}
+ The top exterior monomial. Nonzero in R. This is the
+ value that t_n(Y) reduces to, up to a nonzero scalar.
+
+theta_{ijk}
+ Odd generator in the exterior ring R for each increasing
+ triple of colors. Alternating in i,j,k.
+
+t_0 = 25 chi - 1 (chi = (b-1)^2)
+ The power appearing in the surface bound: T_gamma^{t_0}=0.
+
+t_m (formal detector) (*)
+ The recursive Q-linear map S_{g,m} -> V^{xm} that kills
+ the Pixton span but satisfies t_n(Y) != 0.
+
+X_{h,B}(k) (weighted space)
+ Hassett space with k light weights added to heavy data (h,B).
+
+Y (the counterexample class)
+ Definition 3.4. Sum over signed permutations of the color
+ word of product_J F_J^st. Lives in S_{g,n}^{2D_*}.
+
+xi_Gamma (gluing map)
+ product_{v} Mbar_{g_v,n_v} -> Mbar_{g,n}. The map defining
+ q_{g,n}.
+
+zeta_v (parity variable) (*)
+ Zeta_v^2 = 1 at a vertex v. Controls the parity of kappa
+ classes in Pixton's cycle formula.
+
+=====================================================
+END
+=====================================================
+
+This is a plain-text rewrite of
+"A high-arity counterexample to Pixton completeness in Chow"
+by OpenAI (2026). If you want the full formal proofs
+(the original paper's exact formulas, all numbered equations,
+and every citation), read the original. This version is
+meant as a companion text: it tells you what the theorem
+is, why it matters, and how the two branches of the proof
+fit together, without asking you to memorize the entire
+apparatus.
+
+If you are a high-school student and you made it this far:
+congratulations. You have read through the outline of a
+genuine research paper in contemporary algebraic geometry.
+The mathematics of the actual paper is much denser, but the
+ideas you just absorbed -- formal vs geometric, kernel vs
+generators, support induction, faithful tensor tests -- are
+exactly the tools mathematicians use at the research edge.
+The rest is filling in conventions.
+
+If you want to go further, useful starting points:
+
+ - The moduli space of curves (Harris-Morrison,
+ "Moduli of Curves"). For the geometric picture.
+
+ - Intersection theory (Fulton, "Intersection Theory").
+ For the algebraic machinery.
+
+ - The Pandharipande-Pixton-Zvonkine papers
+ (arXiv:1303.1043) for 3-spin relations.
+
+ - Pixton's original (arXiv:1207.1918) for the
+ relation recipe.
+
+ - This paper's original LaTeX sources for the
+ formal details.
+
+End of the rewrite.
+