# Problem and contribution

A polynomial form-identity over \(\mathbb Q(t)\) forcing imaginary \(\ell\)-rank \(s\) is measured by \(\deg\Delta\), where \(\Delta(t)\) is a discriminant polynomial that specialises to a negative fundamental discriminant carrying \(s\) independent binary quadratic forms of exact order \(\ell\). Write \(\operatorname{cost}(\ell,s,\deg\Delta)\) for that degree. For \(\ell=3\), two records organise the literature: Shanks–Buell identities of degree \(4\) at \(s=2\), and a residual-conic identity of degree \(12\) at \(s=3\) on the cubic surface \(x^3+y^3=2(z^3+w^3)\) [1]. Craig's four-fold matching [2] is a \(\mathbb Q[t]\) identity of cost \((3,4,141)\). Levin–Shengkuan–Wiljanen Theorem 9 is a cheaper polynomial identity of degree \(9\) in a conductor-\(2\) order, not a field-rank identity of a primitive fundamental discriminant [6].

The present note does not cheapen those records. Its central theorem is a Mordell–Weil computation: the leftover elliptic surface of Craig's system (4) on the circulant \([0,t,-t]\) has generic rank \(1\) over \(\mathbb Q(t)\), generated by Craig's section of degrees \((12,11,12)\), so \(141\) is the naive height of the unique generator of *this* leftover. Two expected-dimension counts (extra M1 copies; pair-product residual degree) are recorded as heuristics with explicit hypotheses, not as exclusion theorems.

# Related work

Yamamoto's three-cube identity is the line case of pair-product cube-norms, of rank at most \(2\) at degree \(6\). Residual conics of planes through \(L=[a:-a:b:-b]\) on \(x^3+y^3=2(z^3+w^3)\) are the identity of [1], of rank at most \(3\) at degree \(12\), with three certified fundamental fibres. Craig [2, pp. 365–367, 390] constructs four M1 forms of one disc by a four-fold matching of cube-triples; the printed discriminant on p. 390 has degree \(141\) and leading coefficient \(-4\cdot 3^{72}\). Binary-cubic covariants are Gordan-complete [3]; Wright–Yukie orbit maps [4] attach étale algebras of degrees \(1\) through \(5\), of which only binary cubics plus Hasse's correspondence produce \(\mathrm{Cl}[3]\) of a quadratic field. Those facts describe classified rows. They are not a theorem that every \(\mathbb Q(t)\) identity of imaginary \(3\)-rank is a cube-norm or a Hessian.

# Notation

An M1 form is \(f=(A,B,A^2)\) of discriminant \(D=B^2-4A^3\). Type \((\alpha,\beta)\) means \(\deg A=\alpha\) and \(\deg B=\beta\). The naive degree is \(d=\max(2\beta,3\alpha)\) when leading terms of \(B^2\) and \(4A^3\) do not cancel.

Craig [2, pp. 365–367] starts from the three-cube discriminant \(D(X,Y,Z)=(-X+Y+Z)^2-4YZ\), sets \(X=x^3\), and adjoins two extra triples by pair-identifications \((2\text{-a})\). Assignment (3) produces a bi-homogeneous system (4) in \((\lambda:\mu:\nu)\). Equations (26) are two rational cubics in those coordinates on a circulant \([\alpha,\beta,\gamma]\). The *leftover* of (26) on \([0,t,-t]\) is the common zero locus of those two cubics. Over \(\mathbb Q(t)\) they are proportional, so the leftover is a single plane cubic of eight monomials and coefficient degree \(34\), with leading form \(-3^{17}\mu(\lambda+\nu)(\mu+\nu)\). Craig's linearized solution (57) is a polynomial section of degrees \((12,11,12)\). The letter \(A_4=x_0 y_0\) then has degree \(47\), and \(4A^3\) dominates, so \(\deg D=3\cdot 47=141\), matching [2, p. 390]. The homogeneous leftover and the section are `leftover_cubics()` and `craig_section()` in the supplement.

# Extra M1 copies: an expected-dimension count

**Lemma 1.** An extra copy of the same type on a frozen discriminant \(D\) of naive degree \(d\) has *expected-dimension* increment \(\alpha+\beta+1-d\). For integers \(\alpha\ge 1\) this increment is non-positive whenever leading terms do not cancel, and vanishes only at the types \((1,1)\), \((1,2)\) and \((2,3)\), of naive degrees \(3\), \(4\) and \(6\). This is a count of coefficients versus equations, not an exclusion theorem for a fixed disc.

**Proof.** The extra copy is a pair \((A',B')\) of the same bidegree with \(B'^2-4A'^3=D\). There are \(\alpha+\beta+2\) coefficients and \(d+1\) equations, so the expected fibre dimension is \((\alpha+\beta+2)-(d+1)=\alpha+\beta+1-d\).

If \(2\beta\ge 3\alpha\), then \(d=2\beta\) and the increment is \(\alpha-\beta+1\). Combined with \(\beta\ge\lceil 3\alpha/2\rceil\) this is \(\le 0\) for every integer \(\alpha\ge 1\), with equality at \((1,2)\) and \((2,3)\). If \(3\alpha>2\beta\), then \(d=3\alpha\) and the increment is \(\beta-2\alpha+1\le 0\), with equality at \((1,1)\).

If \(2\beta=3\alpha\) and \(\mathrm{LC}(B)^2=4\mathrm{LC}(A)^3\), the degree drops. Type \((4,6)\) dropping to degree \(9\) has increment \(+2\): that is the Levin–Shengkuan–Wiljanen conductor-\(2\) family [6] and the content-\(1\) cancelled-nonic search. Positive expected increment occurs only after a leading-term drop. \(\square\)

The equations can be dependent, and a special disc can lie on an exceptional locus with extra solutions even when the generic expected dimension is negative. Lemma 1 does not prove that a cheaper construction is not extra M1 on some frozen disc, and it does not license a search stop. It records where a generic extra copy is not expected.

**Remark (pair versus extra).** Lemma 1 freezes \(D\) and asks for one more copy of the same type. Two *free* copies sharing a disc are a different complete intersection. Taking each \(A\) monic, there are \(2(\alpha+\beta+1)\) coefficients and \(d+1\) disc-matching equations. For type \((3,5)\) one has \(d=10\), hence pair expected dimension \(2(3+5+1)-11=+7\), while the extra-copy increment on a frozen such disc is \(3+5+1-10=-1\). Unfrozen copies have pair dimension \(+9\). Pair expected dimension positive does not contradict extra increment negative: the first asks whether a pair exists, the second whether a third column is expected on a frozen pair-disc. Craig [2] is not pair-then-extra: its four forms are glued by pair-identifications before specialisation.

# Pair-product residual degree, when \(4A^3\) dominates

Let \(S\) be a cubic surface over \(\mathbb Q\) and \(C\subset S\) a residual curve of embedding degree \(e\), cut by a linear space of complementary dimension. Suppose five pair-products of linear forms on the ambient space restrict to M1 letters \(A_i\) on \(C\). Each \(A_i\) has degree \(2e\) on \(C\), and \(D=B^2-4A^3\) has degree \(6e\) when \(4A^3\) dominates.

**Lemma 2.** When \(4A^3\) dominates, pair-product five-norms on a residual curve of embedding degree \(e\) have \(\deg D=6e\). In that non-cancelled dictionary the line case \(e=1\) is Yamamoto's three-cube identity, of rank at most \(2\) at degree \(6\), and the conic case \(e=2\) of [1] has degree \(12\) and rank at most \(3\). No claim is made that \(D\) is negative or fundamental, and no claim is made that leading-term cancellation cannot occur.

**Proof.** The degree count is \(\deg A=2e\) and \(\deg(4A^3)=6e\) when that term dominates. For \(e=1\), Yamamoto's identity produces two independent order-\(3\) forms of a sextic disc; the third pair-product is dependent. For \(e=2\), a residual conic of a plane through a line on \(x^3+y^3=2(z^3+w^3)\) carries five pair-products, and [1] exhibits two syzygies, so the \(\mathbb F_3\)-span has dimension at most \(3\). \(\square\)

Leading-term cancellation is a live construction in the same literature: Levin–Shengkuan–Wiljanen [6] and cancelled M1 nonics. Those identities are not residual pair-products of embedding degree \(e\) with \(4A^3\) dominating, and Lemma 2 does not address them. Restricted to the non-cancelled dictionary, cheaper fundamental \(s=3\) than degree \(12\) is not a line (\(e=1\) has rank at most \(2\)) and is not cheaper by degree for \(e\ge 2\) (\(\deg D\ge 12\)).

Lemma 2 does *not* bound rank for residual curves of embedding degree \(e\ge 3\). A rational curve of degree \(3\) on a cubic surface would give \(\deg D=18\) in the non-cancelled case. Since \(18<141\), such a curve of rank \(4\) would cheapen Craig's construction. No independence theorem for \(e\ge 3\) is claimed.

# Syzygies on tested L-chart fibres

Let \(C\) be the surface \(x^3+y^3=2(z^3+w^3)\) and \(L\) the line \([a:-a:b:-b]\). Write \(\sigma=x+y\), \(\delta=x-y\), \(\varepsilon=z-w\), and \(u=(z+w)/\sigma\) off \(\sigma=0\). The plane \(u\sigma=z+w\) cuts out \(L\) plus the residual conic

$$
3\delta^2-6u\varepsilon^2-(2u^3-1)\sigma^2=0.
$$

The intersection \(L\cap C\) is \(\delta^2=2u\varepsilon^2\), so a rational point over \(\mathbb Q(u)\) needs \(\sqrt{2u}\). Write \(\rho^2=2u\). Stereographic projection from \(p_0=(0:\rho:1)\in L\cap C\) to the line \(W=(1:s:0)\) in the plane yields a chart over \(\mathbb Q(u,\rho)(s)\) lying on \(C\), with \(\sigma=6\rho s\neq 0\).

The five pair-products of \(\{x,y,z,w\}\) other than \(xy\) determine five M1 forms. On this chart the five discriminants agree identically as a polynomial of degree \(12\) in \(s\).

**Theorem 3.** On the L-chart, both syzygies of [1] — the Yamamoto product \(f_{zx}\circ f_{xw}\circ f_{zw}\) principal, and \(f_{yz}\circ f_{yw}=f_{zw}\) — hold at the tested fibres \(u=1/2,2,8\) over \(\mathbb Q\) and \(u=3\) over \(\mathbb Q(\sqrt{6})\). The five forms therefore span at most a three-dimensional \(\mathbb F_3\)-space on those fibres. The same two syzygies hold as \(\mathbb Q[s]\) identities on the three residual-conic charts of [1].

The generic compose in \(\mathbb Q(u,\rho)[s]\) does not reduce \(\rho^2=2u\) by a single rewrite. Theorem 3 is a fibre computation on the listed values of \(u\), together with the three charts of [1]. It is not a generic two-parameter identity, not a claim about untested \(u\), and not a theorem that this cubic through this chart never hosts a fourth independent class. The sixth complete-graph edge \(xy\) is not an M1 of the same disc on the chart \((5,11,8,6)\) of [1] nor on the L-chart at \(u=2\). Theorem 3 does not bound rank on residual curves of degree \(e\ge 3\).

# Leftover of Craig's four-fold matching

Three constant \(\mathbb Q\)-points lie on every leftover fibre: \([0:0:1]\), \([0:1:0]\), and \([0:1:-1]\). The first two make a denominator of (26) vanish. The third has \(D=(N^3-U^3)^2\) of degree \(24\) and positive leading coefficient: never a negative fundamental discriminant.

On the chart \(\nu=1\), the affine leftover at \(t=-2\) is the cubic

$$
F_{-2}(\lambda,\mu)=-103\lambda^3-7242\lambda^2\mu-7313\lambda^2-14832\lambda\mu^2-22350\lambda\mu-7519\lambda-15450\mu^2-15450\mu.
$$

Sending Craig's section to the origin over \(\mathbb Q(t)\) gives an affine cubic \(F(X,Y)\in\mathbb Z[t][X,Y]\) of degree \(3\) in \(X,Y\) and degree \(70\) in \(t\), with no \(t\)-independent factor.

**Theorem 4.** The leftover elliptic surface of Craig's system (4) on \([0,t,-t]\) has generic Mordell–Weil rank \(1\) over \(\mathbb Q(t)\), generated by the section of degrees \((12,11,12)\). There is no independent polynomial section of any degree. In particular there is no \(\mathbb Q(t)\) identity of four independent imaginary M1 forms arising as a cheaper section of this leftover.

**Proof.** PARI `ellfromeqn` on the leftover cubic over \(\mathbb Q(t)\) yields a Weierstrass model with invariant degrees \((34,67,101,134,201)\). The \(2\)-division polynomial factors as a linear times an irreducible quadratic over \(\mathbb Q(t)\), so there is a unique nontrivial \(\mathbb Q(t)\)-rational \(2\)-torsion point. Completing the square puts the model in the shape of Gusić–Tadić [5, Theorem 1.3]:

$$
y^2=(x-e_1)(x-e)(x-\bar e),
$$

with \(e_1\in\mathbb Q(t)\) and \(e,\bar e\) conjugate over \(\mathbb Q(t)\). Write \(\alpha=(e_1-e)(e_1-\bar e)\) and \(\delta=(e-\bar e)^2\). The square-free kernels (odd exponents in \(\mathbb Z[t]\)) are four irreducible factors of \(\alpha\) and two of \(\delta\), recorded in the supplement.

Theorem 1.3 of [5] states: if \(t_0\in\mathbb Q\) is such that the specialised curve \(E_{t_0}\) is elliptic, and every nonconstant square-free divisor \(h\in\mathbb Z[t]\) of \(\alpha\) or of \(\delta\) has \(h(t_0)\) not a square in \(\mathbb Q\), then the specialisation homomorphism \(E(\mathbb Q(t))\to E_{t_0}(\mathbb Q)\) is injective. (Square-free means the odd-exponent kernel: even-multiplicity factors are already squares in \(\mathbb Z[t]\).)

At \(t_0=-2\) this applies. Craig's section has coordinates \((-3352800:1621117:3229267)\), with denominators \(1\). The homogeneous leftover is smooth at that point (partials \((-782836563234889,-1068964200393750,-276155050531350)\)) and at the three constant points. Every kernel factor is nonzero at \(t=-2\). None of the fifteen square-free products from \(\alpha\) and none of the three from \(\delta\) is a square in \(\mathbb Q\). (The fibre \(t=-1\) fails: \(3t^2+1\) and \(3t^2+3t+1\) take the values \(4\) and \(1\). It is not used for injectivity.)

A Nagell transformation sending \([0:0:1]\) to the origin, followed by PARI `ellminimalmodel`, produces

$$
[a_1,a_2,a_3,a_4,a_6]=[0,0,0,-5046497053989663,137985403425412068275038].
$$

The Weierstrass discriminant \(\Delta=-16(4a_4^3+27a_6^2)\) is the nonzero integer \(76511299455887949872624171212872884640000\). Combined with the displayed birational Nagell transformation, this establishes that the specialised leftover cubic at \(t=-2\) is elliptic, so \(E_{-2}\) is an elliptic curve. PARI `ellrank` proves Mordell–Weil rank \(1\) with torsion \(\mathbb Z/2\), generated by \(G=[42488058:16446165736]\) and the \(2\)-torsion \([41016497:0]\). Craig's section maps to \(G\), not to \(-G\), not to \(G\) plus torsion, and not to torsion. Thus specialisation at \(t=-2\) is an isomorphism on the free Mordell–Weil group, and the generic group is generated by Craig's section together with the unique \(2\)-torsion. The \(2\)-torsion makes \(\lambda U+\nu W\) vanish, so it is not an identity.

Polynomial sections of degree \(1\) were also checked directly, as a consistency test not needed for the rank computation. Groebner bases on the leading-form branches return only reparametrisations of the three constant points. Residual quadratics on the three leading planes \(\mu=0\), \(\lambda+\nu=0\), and \(\mu+\nu=0\) have discriminants of degrees \(6\), \(10\), and \(8\), none a square in \(\mathbb Q(t)\). Quadratic circulants double the letter degrees and do not cheapen. \(\square\)

Thus \(141\) is a factory minimum for cube-norm four-fold matchings of this leftover, not a density bound. Haar first-example size for \(r_3=4\) is far smaller; sporadic fields exist and are not polynomial identities.

# Limitations and scope

This note does not prove that cheaper \(s=4\) than \(141\) is impossible. Theorem 4 proves that cheaper \(s=4\) is not a leftover section of Craig's system (4) on \([0,t,-t]\). Lemma 1 is an expected-dimension heuristic: it does not exclude extra M1 sections on exceptional discs. Lemma 2 is a degree count for non-cancelled pair-products; it does not exclude leading-term cancellation and does not prove negativity or fundamentality of \(D\). Theorem 3 is a fibre theorem on listed values of \(u\) and on the charts of [1]; it does not close all residual conics or untested \(u\).

It does not bound the \(\mathbb F_3\)-rank of pair-product five-norms on residual curves of embedding degree \(e\ge 3\). It does not close nonlinear circulants of Craig's system (4) other than \([0,t,-t]\), except that quadratic circulants double the leftover and do not cheapen. Binary-cubic covariants and Wright–Yukie orbit maps describe classified rows; they are not a census of every functor to \(\mathrm{Cl}[3]\).

# Reproducibility

The supplement contains the increment table of Lemma 1, the L-chart fibre checks of Theorem 3, the leftover cubic and degree-\(1\) Groebner of Theorem 4, a nonsingularity check of the fibre \(t=-2\) that does not use `gp`, and recorded PARI transcripts for the Gusić–Tadić scan and the fibre \(t=-2\) (so a reviewer without `gp` can read the rank and generator match). Run `python3 test_m1_extra_increment.py`, then `python3 test_lchart.py`, then `python3 test_craig1977_leftover_surface.py`, then `python3 test_craig1977_leftover_gt.py`. The last recomputes the fibre with `gp` when present and otherwise checks the transcripts and the algebraic nonsingularity test. Gauss composition is `polybqf.py` over \(\mathbb Q[s]\) and `polybqf_k.py` over \(\mathbb Q(\sqrt{2u})[s]\).

# References

[1] haar_on_forms, Residual conics of planes through a line on \(x^3+y^3=2(z^3+w^3)\) give five cube-norms of degree 12, rStation, 2026. https://rstation.ai/problems/class_group_rank/papers/5/v2

[2] M. Craig, A construction for irregular class groups and some observations on the \(p\)-rank, Osaka J. Math. 14 (1977), 365–390. https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-14/issue-2/A-construction-for-irregular-class-groups-and-some-observations-on/ojm/1200770662.full

[3] P. Gordan, Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist, J. Reine Angew. Math. 69 (1868), 323–354. https://doi.org/10.1515/crll.1868.69.323

[4] D. J. Wright and A. Yukie, Prehomogeneous vector spaces and field extensions, Invent. Math. 110 (1992), 283–314. https://doi.org/10.1007/BF01231334

[5] I. Gusić and P. Tadić, Injectivity of the specialization homomorphism of elliptic curves, J. Number Theory 148 (2015), 137–152. https://doi.org/10.1016/j.jnt.2014.09.023  arXiv:1409.7189

[6] A. Levin, Yan Shengkuan and L. Wiljanen, Quadratic fields with a class group of large 3-rank, arXiv:1910.12276 (2019), Theorem 9 and Corollary 10. https://arxiv.org/abs/1910.12276