bibkey: dunn2024cubicreciprocity authors: Alexander Dunn; Maksym Radziwiłł year: 2024 title: “Bias in cubic Gauss sums: Patterson’s conjecture” doi: 10.48550/arXiv.2109.07463 url: https://arxiv.org/html/2109.07463v3 claim: “Equations (1.4)-(1.5) give the classical cubic reciprocity and supplementary laws used by GCR; this use is unconditional.” strata_touched: [] license: citation-only triage: anchor

Cubic characters of original golden depth vectors

Primary locator and exact scope

Alexander Dunn and Maksym Radziwill, Bias in cubic Gauss sums: Patterson’s conjecture, arXiv:2109.07463v3, revised 14 May 2024. The arXiv version record identifies the authors and this version: https://arxiv.org/abs/2109.07463v3 .

The source equations (1.4)-(1.5), read in the research underlying GCR, state cubic reciprocity and its supplementary laws in Z[omega], with omega^2+omega+1=0 and lambda=1+2omega. In the normalization a=1 modulo three, coprime primary a,b satisfy (a/b)_3=(b/a)_3. If

$$d\equiv 1+u{\lambda }^2+t{\lambda }^3(\mathrm{mod}9),$$

then (omega/d)_3=omega^u and (lambda/d)_3=omega^(-t). These are classical algebraic laws. The paper’s analytic prime-average, Patterson and large-sieve results involving GRH are not premises of GCR. No statement about those averages on exact Fibonacci-rank supports is attributed to the authors.

Rasmus Frigaard Lemvig, Cubic and quartic reciprocity (2021), revised manuscript, Section 3.2, Theorem 3.9(2), pp. 9-12, https://rasmusfl.github.io/Documents/project_revised.pdf, gives a proof of the ramified supplementary law. Lemvig uses primary denominators a + b omega with a = 3m - 1 and b = 3n, and proves chi_(a+b omega)(1-omega) = omega^(2m) after establishing cubic reciprocity through Gauss and Jacobi sums. For a primary denominator pi = a + b omega in this dossier’s a = 1 (mod 3) convention, apply that formula to -pi. Since negating a denominator does not change its prime-ideal factors, the converted exponent is (a-1)/3 modulo three. For eta_j = -2 + (x_j-1) omega, this gives chi_eta_j(1-omega) = omega^2. The identities lambda = omega (1-omega) and N(eta_j) = 1 (mod 9) then give chi_eta_j(lambda) = omega^2. This source supplies a proof route for the classical input; cubic reciprocity and this conversion are not yet Lean theorems in the pinned repository.

Consumer in the existing WSS dossier

Problems/wall-sun-sun-golden-unit-lift.md, GCR, keeps the original Fibonacci sequence, x_j=L_(3^j), B_j=x_j^2+3 and h_p=v_p(F_(p-(5/p))). PCL7A supplies v_p(B_j)=h_p at each factor. The auxiliary factor eta_j=-2+(x_j-1)omega has norm B_j and is 1+lambda^3 modulo nine. Its factorization uses the unique primary prime above each p selected by eta_j, with its actual exponent h_p.

GCR derives (3/eta_j)_3=omega, a nonzero depth balance modulo three, constant interlevel symbols, and a separate condition for every earlier prime. Under an actual factorization B_j=P^2 Q^3, cubic characters delete the Q^3 factor and constrain P. The old quadratic characters instead constrain Q. This specialization is written out in the owner; the classical reciprocity theorem is not claimed as a new result.

Kummer source and degree convention

J. S. Milne, Fields and Galois Theory, course notes v5.10 (September 2022), Theorem 5.30 and Remark 5.32, printed pages 75-76: https://www.jmilne.org/math/CourseNotes/FT.pdf . The author’s version page identifies the frozen v5.10 notes: https://www.jmilne.org/math/CourseNotes/ft.html .

The Kummer correspondence computes the degree from the subgroup of E^times/E^(times 3) generated by the selected elements when E contains mu_3. GCR proves independence of its selected prime-element classes by individual prime-ideal valuations. The degree 3^t and Frobenius-coordinate identity concern this auxiliary extension. They are not a prime-size bound or a statistical independence theorem for Fibonacci factors.

Reused exponential-equation and quadratic prerequisites

Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation, Compositio Mathematica 142 (2006), 31-62, DOI 10.1112/S0010437X05001739; https://arxiv.org/abs/math/0405220 . Theorem 1 and Section 16, the D=3 classification, are the classical input previously located in the DCE section of #8343. For x>=2 and y>=2 the case e>=3 is excluded by that classification; e=2 follows from (y-x)(y+x)=3. This implies gcd of the actual block depths is one. GCR0 restates this input and proves the required consequence locally. The main cubic identities GCR1-GCR3 do not depend on this classification.

The older GPC quadratic condition is also reproved in GCR0 using ordinary quadratic reciprocity: Bj=3 modulo an earlier prime p, p=1 modulo three, and Bj=3 modulo four. It uses all earlier primes, including those of even depth. The old DCE/GPC research provenance is #8343 at 5b54a51edb8ef08802311cdc4c5df28c2fd5f838; no absent old section is silently used as a local theorem in #9761.

Frontier context and remaining arithmetic

B. Grechuk and A. Ratcliffe, On the shortest open cubic equations, arXiv:2603.29831v1, submitted 31 March 2026: https://arxiv.org/html/2603.29831v1 . Its Section 2 recalls cubic reciprocity; its integer-solvability obstruction gives methodological context only. No reduction of the fixed-golden equation to that paper’s equation has been established.

GCR supplies necessary character constraints and infinitude of witnesses with depth not divisible by three, which still permits depth two. It does not eliminate the full P^2 Q^3 branch, exhibit a WSS prime, decide a new WSS prime family, establish global priority, or carry Lean kernel certification. The existing Scribe reference marks this as ordinary research context and leaves its authored Lean statement unchanged.

Conjugate completion and the inert-prime-two input

The continuation GCR.7-GCR.12 keeps both primary primes above every previous rational prime. The defining cubic residue exponent implies (bar(a)/bar(b))_3=overline((a/b)_3); bar(lambda)=-lambda and the fact that minus one is a cube fix the inverse phase. Multiplying both directions produces the rational-prime cubic balance in GCC1.

For the primary inert prime -2, the residue field is F4 and the defining cubic exponent is one. The original eta_j reduces to omega modulo two. Cubic reciprocity therefore gives (2/eta_j)_3=omega, without assuming a split denominator. Grechuk-Ratcliffe uses primary generators equal to two modulo three, whereas Dunn-Radziwill and this dossier use one. Negating generators converts the normalizations; it does not change the symbols. The source definition, rather than a choice of the other prime above a split rational prime, determines every phase.

The Kummer field with both directions has degree 3^(2t) over E. Adding cube roots of two and three raises this to 3^(2t+2). Valuations at the individual selected primes, at two and at lambda prove independence. Milne’s Theorem 5.30 supplies the general correspondence; GCC3 proves its specific input and the complex-conjugation action explicitly.

Chebotarev source and exact comparison-prime scope

Andrew V. Sutherland, MIT 18.785 Number Theory I, Fall 2021, Lecture 28, Theorem 28.9, printed page 6: https://math.mit.edu/classes/18.785/2021fa/LectureNotes28.pdf .

The theorem assigns density |C|/|G| to an unramified Frobenius set stable under conjugacy in a finite Galois extension. GCC4 proves the specific field intersection with Q(zeta_(80*3^(j+2))) and computes the conjugacy class of size two before applying it. This yields a positive Dirichlet density for unrestricted comparison primes satisfying the specified P-side characters and a fixed admissible congruence class. It is not a density assertion on the finite exact-period support of B_j, and provides neither a WSS example nor an effective least-prime bound.

Cubic Thue source and the exact integer descent

Rafael von Kaenel and Benjamin Matschke, Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture, arXiv:1605.06079: https://arxiv.org/abs/1605.06079 ; https://arxiv.org/pdf/1605.06079 .

Sections 5.1-5.3 describe the cubic Thue framework, classical methods, explicit reconstruction from Mordell equations, and Algorithms 5.2/5.4. The latter needs a Mordell-Weil basis in its input. Classical invariant theory is credited there to earlier work; it is not attributed to this WSS project as a new general theory.

GCC5 derives the particular form directly from eta_j=pi^2gamma^3: with pi=a+bomega, A=a^2-b^2 and D=2ab-b^2, it is f_pi(u,v)=Au^3-3Du^2v+3(D-A)uv^2+Av^3=-2. Its discriminant is 81P^4. Irreducibility is proved using valuation at the actual prime (pi), not assumed from the discriminant. The second coordinate must still equal L_(3^j)-1, and the norm of u+vomega must be a distinct prime. The fixed-form finiteness theorem does not supply a uniform bound as P varies. No complete Thue solver or solution list was executed or asserted in this continuation.

GIR. An actual golden rank tower and its exact initial-depth index

This is an ordinary mathematical continuation for the SAME problem owner Problems/wall-sun-sun-golden-unit-lift.md. The arguments below are not assertions attributed to Dunn-Radziwill. The classical sources used for orders, Kummer extensions, elliptic torsion and heights are specified in GIR.7. No new parallel Problems entry or Lean declaration is introduced.

GIR.0 Fixed integers and the noncube input

Retain x_j=L_(3^j), f_j=F_(3^j), B_j=x_j^2+3 for j>=1, and the ORIGINAL initial depths h_p. PCL7A and GCR0 in the owner give

$$v_p(B_j)=h_p(p\mid B_j),x_j^2+4=5f_j^2,B_j\equiv 1(\mathrm{mod}9),v_2(x_j)=2,v_2(f_j)=1.$$

All prime factors of B_j exceed five and are one modulo 2*3^(j+1). Different B_j have disjoint prime supports. Each B_j is a noncube: GCR1 gives product_(p|B_j)(3/varpi_(j,p))_3^(h_p)=omega, whereas a cube would make every exponent divisible by three and the product one. Thus this noncube input already follows from the proved cubic balance; GIR does not need another perfect-power classification.

Write uniquely

$$B_j=d_j{c}_j^3,d_j=\prod _{p\mid B_j}{p}^{e_p},e_p=h_p\mathrm{mod}3\in \{0,1,2\},$$

$$R_j=\mathrm{rad}(d_j)=\prod _{p\mid B_j,3\nmid h_p}p.$$

Here d_j is cubefree, c_j is positive, and R_j>1. These symbols are local to GIR; d_j is not a modified Fibonacci generator. Let theta_j be the positive real cube root of the ACTUAL integer B_j, and k_j=Q(theta_j). The polynomial T^3-B_j is irreducible over Q. The constructions of theta_j and the order below do not require factoring B_j; the formulas for R_j and the maximal-order index still encode arithmetic information.

GIR.1 A factorization-free order with precisely the WSS index support

Theorem GIR1. Put a_j=(B_j-1)/9 and beta_j=(1+theta_j+theta_j^2)/3. Then

$${\mathcal{A}}_j=\mathbb{Z}\cdot 1+\mathbb{Z}\cdot {\theta }_j+\mathbb{Z}\cdot {\beta }_j$$

is an order in k_j. Its multiplication and discriminant are

$${\theta }_j^2=3{\beta }_j-{\theta }_j-1,{\theta }_j{\beta }_j={\beta }_j+3a_j,{\beta }_j^2={\beta }_j+a_j{\theta }_j+2a_j,$$

$$\boxed{\mathrm{disc}({\mathcal{A}}_j)=-3B_j^2,\Delta (k_j)=-3R_j^2,I_j:=[{\mathcal{O}}_{k_j}:{\mathcal{A}}_j]=B_j/R_j.}\text{\hspace{1em}(GIR1)}$$

In particular

$$\boxed{p\mid B_j:p\text{ is WSS}⟺p\mid I_j,}\boxed{\mathrm{rad}(I_j)=\prod _{p\mid B_j,h_p\ge 2}p.}\text{\hspace{1em}(GIR2)}$$

Proof. Reduce the products by theta_j^3=B_j to obtain the displayed integral multiplication table. Hence A_j is a rank-three ring lattice, so every element is integral, and it is an order. On the ordered basis (1,theta_j,beta_j), the trace Gram matrix is

$$\left(\begin{array}{ccc}3& 0& 1\\ 0& 0& B_j\\ 1& B_j& (2B_j+1)/3\end{array}\right).$$

Its determinant is -3B_j^2. In particular the unavoidable index-three correction to Z[theta_j] has already been made before testing maximality.

We compute the FIELD discriminant locally. The fields Q(cuberoot(B_j)) and Q(cuberoot(d_j)) are equal. At p!=3 with e_p=1 or 2, a root has p-adic valuation e_p/3, so the local cubic is irreducible and totally ramified. It is tame, with discriminant exponent two. At p!=3 outside R_j, delete the cube factor; the resulting unit cubic has unit derivative at its roots and is etale, so it is unramified.

At three the original B_j is a cube in Q_3. Indeed B_j=1+9a_j and

$$(1+3z{)}^3=1+9(z+3z^2+3z^3).$$

The equation z+3z^2+3z^3=a_j has derivative one modulo three and a solution modulo three, so Hensel lifting gives z in Z_3. Consequently k_j tensor Q_3 is Q_3 times Q_3(omega), whose discriminant exponent is one. The cubic field has one real embedding and one complex pair, fixing the negative sign. The local tame-different formula now gives Delta(k_j)=-3R_j^2. The order-index/discriminant relation proves GIR1. These are classical discriminant mechanisms, not a new general theorem about pure cubic fields.

At a prime of B_j the index valuation is exactly

$$v_p(I_j)=\{\begin{array}{ll}{h}_p-1,& 3\nmid h_p,\\ h_p,& 3\mid h_p.\end{array}$$

It is zero precisely when h_p=1. Since p>5, this is precisely the original non-WSS condition. No prime outside B_j divides I_j, proving GIR2. Thus A_j is maximal iff every factor of this actual block is non-WSS. A different integral generator for k_j is not being ruled out.

Specialization. If B_j=P^2 Q^3 with distinct primes, then k_j=Q(cuberoot(P)), Delta(k_j)=-3P^2 and I_j=P Q^3. These formulas concern the SPECIFIED order A_j. They do not assert that such a block exists. Determining its maximal order is still an arithmetic task, so GIR2 is not advertised as a factorization-free fast WSS decision algorithm.

GIR.2 Two simultaneous, actual non-torsion twists at every layer

Theorem GIR2. For EVERY j, without a WSS assumption, the cubefree twists

$$E_j^{-}:Y^2=X^3-3d_j^2,E_j^{+}:Y^2=X^3+125d_j^2$$

have the explicit rational integral points

$$S_j^{-}=(d_j{c}_j,d_j{x}_j),S_j^{+}=(5d_j{c}_j,25d_j{f}_j).\text{\hspace{1em}(GIR3)}$$

Both points have infinite order. The twist classes for distinct j are distinct. The same point formulas with d_j replaced by B_j and c_j by one provide integral points without a factorization of B_j.

Proof. Substitution uses B_j=d_j c_j^3, x_j^2=B_j-3 and 5f_j^2=B_j+1. Both abscissae are odd; the ordinate valuations at two are two and one. The doubling formula on Y^2=X^3+b is

$$X(2S)=\frac{9X(S{)}^4}{4Y(S{)}^2}-2X(S).$$

Its two-valuations here are -6 and -4. The doubles are finite and nonintegral. By the classical Nagell-Lutz integrality theorem they cannot be torsion, and hence neither original point is torsion. The same argument applies to the raw B_j twists. Distinct cubefree d_j have disjoint, nonempty support, so their ratios are not rational cubes. For curves Y^2=X^3+b*d_j^2 this also prevents a rational isomorphism of the twists. This constructs points rather than assuming a positive rank.

Theorem GIR2a (the earlier descent conditions hold on the actual family). Neither S_j^- nor S_j^+ is in the rational image of the degree-three isogeny from E_(-27b) to E_b for its respective b=-3d_j^2 or 125d_j^2.

Proof. The usual map is

$${\phi }_b(s,t)=\left(\frac{s^3-108b}{9s^2},\frac{t(s^3+216b)}{27s^3}\right),$$

and t^2=s^3-27b implies the direct identity

$$Y+\sqrt{b}={\left(\frac{t+9\sqrt{b}}{3s}\right)}^3.\text{\hspace{1em}(GIR4)}$$

A finite rational image has s!=0. On the minus curve the relevant element is d_j(x_j+sqrt(-3)). At a prime p with e_p!=0, the two primes above p in E=Q(sqrt(-3)) have valuations h_p+e_p and e_p. This uses the coprime factorization (x_j+sqrt(-3))(x_j-sqrt(-3))=B_j and p>5. The valuations are 2e_p and e_p modulo three, both nonzero. It is not a cube.

For the plus curve the element is

$$25d_j{f}_j+5d_j\sqrt{5}=5\sqrt{5}{d}_j(\sqrt{5}{f}_j+1).$$

The factors sqrt(5)f_j+1 and sqrt(5)f_j-1 multiply to B_j and differ by two. Each p|B_j splits in the ORIGINAL golden field. Conjugation exchanges these factors up to sign; one prime over p has valuation h_p in the first factor and the other zero. After the displayed prefactor their valuations are again h_p+e_p and e_p. This is also not a cube. Apply GIR4. An e_p!=0 exists by GIR.0. No Selmer-group dimension or Tate-Shafarevich claim is included. In particular positive rank and this nonimage condition alone do not isolate the hypothetical depth pattern (2,3): they occur on every actual layer, including the checked depth-one layers.

GIR.3 The explicit common field, signature and exact discriminants

For J>=1, using the positive real roots, define

$$F_J=\mathbb{Q}({\theta }_1,\dots ,{\theta }_J),N_J=F_J(\omega ),R^{(J)}=\prod _{j=1}^J{R}_j.$$

Theorem GIR3. One has

$$\boxed{[F_J:\mathbb{Q}]=3^J,[N_J:\mathbb{Q}]=2\cdot 3^J,\mathrm{Gal}(N_J/\mathbb{Q})\simeq (\mathbb{Z}/3{)}^J⋊C_2.}\text{\hspace{1em}(GIR5)}$$

Complex conjugation acts by inversion on (Z/3)^J. The field F_J has signature (1,(3^J-1)/2). It is contained in the reals in the chosen embedding, but it is NOT totally real. The exact absolute discriminants are

$$\boxed{|\Delta (F_J)|=3^{(3^J-1)/2}(R^{(J)}{)}^{2\cdot 3^{J-1}},}\text{\hspace{1em}(GIR6)}$$

$$\boxed{|\Delta (N_J)|=3^{3^J}(R^{(J)}{)}^{4\cdot 3^{J-1}},\mathrm{rd}(N_J)=\sqrt{3}(R^{(J)}{)}^{2/3}.}\text{\hspace{1em}(GIR7)}$$

Proof. If product_j B_j^(a_j) is a cube in E, choose for each j a prime p|B_j with 3 not dividing h_p. Valuation at either E-prime over p gives three dividing a_j*h_p, hence three dividing a_j. The classes of B_1,…,B_J in E^times/E^(times 3) are therefore independent. Kummer theory proves degree 3^J over E. Positive cube roots are fixed by conjugation, which inverts omega; the stated semidirect action follows. Since F_J is real in its chosen embedding, F_J intersect E=Q, proving its degree and identifying it with the fixed field of this conjugation. An embedding is real only if every positive cube root maps to its unique real conjugate. Thus exactly one embedding is real, giving the signature.

At a rational p in R^(J), precisely one radicand has valuation nonzero modulo three. Over the maximal unramified local extension, units have cube roots since p!=3. Consequently the inertia group in N_J has exact order three, acting as a nonzero translation in that one coordinate. No other rational prime apart from three ramifies. At three every B_j is already a cube in Q_3 by GIR1, so a completion of N_J is exactly Q_3(omega); its inertia has order two and residue degree one.

All this ramification is tame. For a degree n extension the tame discriminant exponent equals sum f*(e-1), or equivalently n minus the number of inertia orbits on its embeddings. The 3^J embeddings of F_J are labelled by (Z/3)^J. A nonzero coordinate translation has 3^(J-1) orbits; inversion has one fixed point and (3^J-1)/2 two-cycles. These counts give GIR6. On the regular action for N_J, inertia orders three and two give exponents 43^(J-1) and 3^J respectively, proving GIR7. Taking the 23^J-th root gives the root discriminant. The signature fixes the signed discriminants if desired; GIR6-GIR7 report absolute values. No bounded-root-discriminant assertion is made as J increases.

GIR.4 Explicit independent points on TWO fixed elliptic curves

Now fix the curves once and for all:

$${\mathcal{E}}^{-}:y^2=x^3-3,{\mathcal{E}}^{+}:y^2=x^3+125.$$

Theorem GIR4. The ACTUAL points

$$P_j^{-}=({\theta }_j,x_j),P_j^{+}=(5{\theta }_j,25f_j)(1\le j\le J)\text{\hspace{1em}(GIR8)}$$

belong to their respective fixed curves over F_J. For either sign these J points are Z-linearly independent, even modulo the points over Q. Over N_J, with iota(x,y)=(omega*x,y), the 2J points P_j^sign, iota(P_j^sign) are Z-linearly independent, even modulo the points over E. In particular

$$\boxed{\mathrm{rank}{\mathcal{E}}^{\pm }(F_J)\ge \mathrm{rank}{\mathcal{E}}^{\pm }(\mathbb{Q})+J,}\text{\hspace{1em}(GIR9)}$$

$$\boxed{\mathrm{rank}{\mathcal{E}}^{\pm }(N_J)\ge \mathrm{rank}{\mathcal{E}}^{\pm }(E)+2J.}\text{\hspace{1em}(GIR10)}$$

Both fixed curves consequently have infinite rank over the explicit union of the F_J. The cost is the displayed growing degree 3^J, not a fixed number field or a uniformly bounded-degree construction.

Proof. Membership is exactly x_j^2=B_j-3 and 625 f_j^2=125(B_j+1). Over k_j, divide the coordinates of the raw B_j version of GIR3 by theta_j^2 and theta_j^3=B_j. This is an isomorphism from the rational twist to the indicated fixed curve, so GIR2 proves that every P_j is non-torsion.

Let sigma_j in Gal(N_J/E) multiply theta_j by omega and fix all other cube roots. Then sigma_j P_j=iota P_j and sigma_j P_i=P_i for i!=j. The three points P,iota P,iota^2 P lie on a horizontal line, so 1+iota+iota^2=0 as endomorphisms, and

$$(\iota -1)({\iota }^2-1)=[3],(a+b\iota )(a+b{\iota }^2)=[a^2-ab+b^2].\text{\hspace{1em}(GIR11)}$$

Given sum a_i P_i equal to a rational point, apply sigma_j-1 and then iota^2-1. It follows that [3a_j]P_j=O. Non-torsion forces a_j=0 for all j. This proves the first independence, including modulo rational points and torsion after multiplying a putative relation by its order.

For a relation sum_j(a_j+b_j iota)P_j equal to an E-rational point, apply sigma_j-1, then iota^2-1, then a_j+b_j iota^2. This yields [3(a_j^2-a_j*b_j+b_j^2)]P_j=O. Non-torsion and the positive integer norm force a_j=b_j=0. This proves the stronger independence and the rank bounds. Every point and every isolating automorphism is specified; no rank oracle, BSD assumption or independence heuristic is an input.

GIR.5 The constructed height lattice is explicitly orthogonal

Use one fixed absolute normalization of the canonical height and put H_j^sign=hat(h)(P_j^sign)>0. This section concerns only the subgroup just constructed, with its induced height pairing.

Theorem GIR5. Different layers are orthogonal, including all their CM rotations. For either fixed curve the Gram matrix on the 2J points in GIR4 is block diagonal with j-th block

$$H_j\left(\begin{array}{cc}1& -1/2\\ -1/2& 1\end{array}\right),$$

and hence its determinant is

$$\boxed{(3/4{)}^J\prod _{j=1}^J{H}_j^2>0.}\text{\hspace{1em}(GIR12)}$$

The Gram matrix on the J real points is diagonal with entries H_j.

Proof. Canonical height and its bilinear pairing are Galois invariant; iota preserves the height since its x-coordinate multiplier is a root of unity. If i!=j, sigma_j fixes the i-th point and cycles the three j-th points, so all three pairings are equal. Their sum is zero by 1+iota+iota^2=0, proving orthogonality. For one layer, P+iota P=-iota^2 P has the same height H as P. Bilinearity gives H=2H+2<P,iota P>, hence the off-diagonal entry -H/2. Compute each 2-by-2 determinant and multiply. This is the determinant of the generated sublattice, NOT a full Mordell-Weil regulator: neither saturation nor a basis of all rational points has been established. No numerical canonical height computation is claimed.

GIR.6 What has been constructed and what remains arithmetic

GIR supplies positive witnesses, not only a condition on a hypothetical WSS prime: there are J independent points on each of TWO fixed curves over the same specified degree-3^J field, and their CM companions give 2J over its normal closure. The construction retains the actual B_j, with exact discriminants and an order whose index detects precisely the original WSS factors in each block. The raw construction of the fields, points and order uses the recurrence without factoring B_j.

The index/discriminant formulas still contain the true depths. They do not force I_j=1 or I_j>1 at a new layer. The positive-rank and nonimage properties hold for the entire actual family and therefore do not alone exclude the minimal all-WSS pattern. No new integer WSS witness, decided WSS prime family, uniform rational-rank result over Q, or new solved externally stated open problem is claimed. The two fixed curves, the explicit independent points, and the exact degree/discriminant/index statements are the mathematical outputs of this continuation.

GIR.7 Classical sources and prior-art boundary

Maciej Ulas, A note on higher twists of elliptic curves, Glasgow Mathematical Journal 52 (2010), 371-381, DOI 10.1017/S0017089510000066: https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/note-on-higher-twists-of-elliptic-curves/611411025610A3311F8E614F0BF08AED . The publisher abstract already gives simultaneous positive cubic twists for any pair of rational j=0 elliptic curves over Q(u,v). Thus simultaneous twisting is prior art. Only that abstract’s scope was used in this check; no unread numbered theorem or uniform specialization assertion is invoked. GIR proves its prescribed golden integer family, common tower, point independence and exact discriminants directly. Global priority of this specialization has not been established.

Andrew V. Sutherland, MIT 18.785, Fall 2021, Lecture 12, Section 12.4, Theorem 12.27 and Proposition 12.28: https://math.mit.edu/classes/18.785/2021fa/LectureNotes12.pdf . These are the order-index/discriminant relation, tame different exponent e-1 and tower law. Their parsed statements were read. The requested page images failed to load, so no successful visual verification is asserted. The pure-cubic and tower specializations in GIR1/GIR3 are proved locally above, rather than attributed as new general discriminant formulae.

Andrew V. Sutherland, MIT 18.782, Fall 2013, Lecture 24, Theorem 24.21: https://math.mit.edu/classes/18.782/2013fa/LectureNotes24.pdf . The parsed Nagell-Lutz integrality theorem was read; its page-image fetch also failed. Only integrality of rational torsion on the stated integral short Weierstrass model is used, after the actual double is computed.

Milne’s Kummer Theorem 5.30 and Remark 5.32, already located above, are used after proving independence of the actual block cube classes. The page image of printed page 76 was successfully inspected in this pass. The normalized canonical-height pairing used in GIR5 is also recorded in John Cremona’s reviewed LMFDB entry: https://www.lmfdb.org/knowledge/show/ec.canonical_height . The Gram determinant is derived here; no regulator value is taken from a table or from a rank computation. These are ordinary proofs, with no Lean kernel certification, independent-model review or priority claim.

GNT. Golden normalization, point blow-ups and marked topology

This continuation uses the SAME actual integers and order as GIR. Its arithmetic scheme, its real tori, its old Fibonacci mapping torus, and its complex comparison curve are distinct constructions. They are connected below by computed modules and maps. An abstract ring is not itself a geometric circle. No new WSS prime or prime-family decision is asserted.

GNT.0 Objects and the prior result retained

Fix j>=1 and abbreviate B=B_j=L_(3^j)^2+3, theta=cuberoot(B)>0, k=Q(theta), O=O_k and A=Z+Z theta+Z(1+theta+theta^2)/3. GIR proves that A is an order, B is a noncube, and I=[O:A]=B/R, where R=product_(p|B,3 not dividing h_p) p. Every p|B exceeds five, and its exponent h_p=v_p(B) is the ORIGINAL Fibonacci initial depth. In particular h_p>=2 is the original WSS condition, with no index-multiplier square. Put D=O/A. The aim below is to compute more than its already-known order I.

For p|B set h=h_p, u=B/p^h in Z_p^times, S=A tensor Z_p and T=O tensor Z_p. Since three is a unit here,

$$S={\mathbb{Z}}_p[\theta ]\simeq {\mathbb{Z}}_p[t]/(t^3-p^{h}u).\text{\hspace{1em}(GNT0)}$$

The completed algebra can have more than one generic component. T is its normalization in the full finite etale Q_p-algebra, not necessarily a single local field. Write h=3a+r, r in {0,1,2}. Cyclic groups of order p^0, or of order one, mean the zero group throughout this section.

GNT.1 Exact normalization modules and the arithmetic singular locus

Theorem GNT1. The normalization defect is

$$\boxed{D\otimes {\mathbb{Z}}_p\simeq \mathbb{Z}/p^{\lfloor h/3\rfloor }\oplus \mathbb{Z}/p^{\lfloor 2h/3\rfloor }.}\text{\hspace{1em}(GNT1)}$$

Its Z_p-length is

$${\delta }_p=\lfloor h/3\rfloor +\lfloor 2h/3\rfloor =h+(\gcd (3,h)-3)/2.\text{\hspace{1em}(GNT2)}$$

Proof. If r=0, put alpha=theta/p^a. It satisfies alpha^3=u. The algebra Z_p[alpha] is finite etale because three and alpha are units, so it is T. The original S has basis 1,p^a alpha,p^(2a)alpha^2. If r=1, the same alpha satisfies alpha^3=pu, an Eisenstein polynomial. It is a uniformizer, T=Z_p[alpha], and the same diagonal basis comparison applies. Its valuations justify maximality directly: in a sum c_0+c_1 alpha+c_2 alpha^2, the three term valuations are distinct modulo three, so integrality forces every c_i to be in Z_p. If r=2, put pi=theta^2/p^(2a+1). Then pi^3=pu^2 is Eisenstein, T=Z_p[pi], theta=p^a pi^2/u and theta^2=p^(2a+1)pi. In the basis 1,pi,pi^2 the inclusion therefore has diagonal coefficients 1,p^(2a+1),p^a/u. Discarding the unit and sorting gives GNT1 in all three cases. Their sums give GNT2. No h=1 assumption occurs.

Global corollary. Write uniquely B=s t^2 c^3 with s,t squarefree and coprime, and c>0. The integer c need not be coprime to s or t. Then

$$\boxed{D\simeq \mathbb{Z}/c\oplus \mathbb{Z}/(c^{2}t),I=c^{3}t.}\text{\hspace{1em}(GNT3)}$$

Indeed the two exponents in GNT1 are v_p(c) and 2v_p(c)+1_(p|t). Chinese remaindering gives the result. GIR proves that there is no normalization defect at three or outside B, so no other factors occur.

Theorem GNT2. The closed point above p|B on the TOTAL arithmetic scheme Spec(A) is regular iff h_p=1. Thus its singular closed points are exactly the original WSS prime factors of B. Nevertheless, every such p has the SAME nonreduced special fiber

$$S/pS\simeq {\mathbb{F}}_p[t]/(t^3).\text{\hspace{1em}(GNT4)}$$

Proof. S has maximal ideal m=(p,t) and dimension one. In the regular ambient local ring Z_p[t]_(p,t), the defining equation has a nonzero linear p-term when h=1, giving embedding dimension one. When h>=2 the equation lies in (p,t)^2, so dim_Fp(m/m^2)=2 and S is not regular. Regularity is preserved and detected by completion for this finite-type arithmetic local ring. Outside B, GIR gives A localized equal to its normalization; its nonzero prime localizations are DVRs, hence regular. Equation GNT4 follows by reduction for every h>=1. Regularity here is a property of the total one-dimensional arithmetic scheme. It is NOT the assertion that the map to Spec(Z) is smooth at a ramified prime. The standard dimension-one regular/DVR criterion is Stacks, Lemma 10.119.7, tag 00PD; the embedding-dimension computation is explicit above.

GNT.2 Conductor as a square ideal, with exact multiplicities

Let f=(A:O)={x in O:xO subset A}, the conductor ideal. The definition is classical; see Conrad, The conductor ideal of an order, Definition 1.1.

Theorem GNT3. For p|B the local conductor is

$$\boxed{f\otimes {\mathbb{Z}}_p=\{\begin{array}{ll}{p}^{2a}T,& h=3a,\\ {\pi }^{2h-2}T,& 3\nmid h,\end{array}}\text{\hspace{1em}(GNT5)}$$

where pi is any uniformizer of the totally ramified cubic T in the second case. Hence length_Zp(T/fT)=2 delta_p. Globally f is the SQUARE of an integral O-ideal J, and

$$\boxed{f=J^2,N(J)=I,N(f)=I^2.}\text{\hspace{1em}(GNT6)}$$

More explicitly, if P_p is the unique prime above p when 3 does not divide h_p, then

$$J=(\theta )\prod _{p\mid B,3\nmid h_p}{P}_p^{-1}.\text{\hspace{1em}(GNT7)}$$

Proof. For any full Z_p-lattice L in the separable algebra write L^vee={x:Tr(xL) subset Z_p}. Monogenic trace duality gives S^vee=(3theta^2)^(-1)S; Conrad records this formula in the proof of Corollary 4.3, printed page 12. Double duality and the T-module property of T^vee imply

$$xT\subset S⟺x{S}^{\vee }\subset T^{\vee }⟺x/(3{\theta }^2)\in T^{\vee }.$$

For the last equivalence use 1 in S and stability under S subset T. Thus the conductor is 3theta^2 T^vee. If h=3a then T is etale, its different is a unit, and theta is p^a times a unit, giving the first case of GNT5. Otherwise T=Z_p[pi] with pi^3=p times a unit, so its different is generated by 3pi^2. Since v_pi(theta)=h, the second case follows. The residue degree of this ramified cubic is one; the total residue degree in the etale case is three. The conductor lengths are therefore 2h-2 and 6a, respectively, exactly 2 delta_p.

For 3 not dividing h, the ideal in GNT7 has exponent h-1 at P_p. When h=3a it has exponent a at each prime above p. These are nonnegative and are half the conductor exponents. No other prime contributes, by GIR. Their norm exponents are delta_p, proving GNT6-GNT7. The square property concerns the conductor of THIS specified cubic order. It is not claimed for conductors of arbitrary orders or for a new WSS instance.

GNT.3 An explicit intrinsic singular-point blow-up chain

For a fixed p>5 and unit u let S_h=Z_p[t]/(t^3-p^h u). For h>=2 blow up its singular closed point (p,t), and repeat at the singular closed point of the result as long as one remains. This is the intrinsic blowup of the arithmetic curve. It does not require an embedded normal-crossing resolution of an ambient surface.

Theorem GNT4. Each step and its normalization-length contribution are explicit:

$$\boxed{h\ge 3:S_h⟶S_{h-3},t\mapsto py,\mathrm{length}(S_{h-3}/S_h)=3.}\text{\hspace{1em}(GNT8)}$$

For h=2 the blowup is the normal Eisenstein ring

$$\boxed{{\mathbb{Z}}_p[v]/(v^3-p/u),t=u{v}^2,}\mathrm{length}(T/S_2)=1.\text{\hspace{1em}(GNT9)}$$

The prescribed process reaches normalization after exactly

$$\boxed{\ell (h)=\lfloor (h+1)/3\rfloor }\text{\hspace{1em}(GNT10)}$$

steps for h>=1. Its successive index lengths sum to delta_p in GNT2.

Proof. The two standard affine blowup charts are S_h[t/p] and S_h[p/t], as in Stacks, Lemma 31.33.2, tag 01OF. Work inside the total quotient algebra so that extraneous p- or t-torsion from a naive chart presentation is discarded. For h>=3, the first chart is exactly Z_p[y]/(y^3-p^(h-3)u). Its basis 1,y,y^2 contains S_h with diagonal coefficients 1,p,p^2. In the second chart put v=p/t. Cancelling t^3 in t^3=p^h u gives 1=t^(h-3)v^h u, so v is a unit. This entire second chart lies in the first. Thus the first chart covers the blowup and GNT8 holds, including h=3, where S_0 is finite etale.

For h=2 the t-chart gives t=u v^2 and v^3=p/u. This Eisenstein ring is normal. The p-chart instead has p y^3=u; hence y is a unit and that chart lies in the t-chart. The inclusion basis is, up to units, 1,v^2,pv and has index p. This proves GNT9. The terminal cases h=1 and h=0 are respectively a DVR and a finite etale algebra. Subtracting three until 0,1,2 remains and, in the last case, taking one extra step gives GNT10 and the stated sum of index lengths. This is an exact count for this closed-point process, not a lower bound over every possible choice of blowup centers or ideals.

GNT.4 An arithmetic differential module equals the earlier three-manifold torsion

Theorem GNT5. For p|B, the module of relative Kahler differentials is

$$\boxed{{\Omega }_{S/{\mathbb{Z}}_p}\simeq S/(3{\theta }^2)d\theta \simeq ({\mathbb{Z}}_p/(B){)}^2\simeq (\mathbb{Z}/p^h{)}^2.}\text{\hspace{1em}(GNT11)}$$

The two generators in the middle description are dtheta and theta dtheta. In particular p is WSS iff p Omega_(S/Zp) is nonzero. The F_p-dimension of Omega tensor F_p is two for EVERY h>=1, so that coarse tangent-space observation alone cannot detect the first exceptional lift.

Proof. The polynomial conormal sequence gives the presentation by the derivative 3theta^2; see Stacks, Section 10.131, tag 00RM. After killing theta^2, the remaining relation theta^3=B kills B on each of the basis vectors 1,theta. Equivalently, multiplication by theta^2 on the basis 1,theta,theta^2 has Smith factors 1,B,B over Z_p. This proves the exact module statement and both consequences.

For the literal manifold comparison keep the repository’s OLD CG.5 construction: Q=((1,1),(1,0)), C=Q^2, and S_0=2Q-I. For the positive even index n=p-(5/p)=p-1 at these block primes, form the closed oriented three-manifold

$$M_n=({\mathbb{T}}^2\times [0,1])/((z,1)\sim (C^{n}z,0)).$$

The old identity C^n-I=F_n S_0 Q^n and det(S_0)=-5 are recorded in Library/notes/katz2015goldeninterfaces.md, Section 5. This old torus bundle and its old torsion calculation are not counted again as new.

Corollary GNT5a. With the chosen coordinate bases there is an explicit Z_p-module isomorphism

$$\boxed{{\Omega }_{S/{\mathbb{Z}}_p}\simeq \mathrm{Tor}{H}_1(M_n,\mathbb{Z})[p^{\infty }].}\text{\hspace{1em}(GNT12)}$$

Proof. The fundamental group of the mapping torus is Z^2 semidirect_(C^n) Z. Abelianization gives H_1=Z direct-sum coker(C^n-I). At p>5, S_0 Q^n is invertible over Z_p, so its image lattice is F_n Z_p^2. The ORIGINAL valuation equality v_p(F_n)=h=v_p(B) makes the ideals (F_n) and (B) equal in Z_p. Both sides of GNT12 are therefore the explicitly based quotient Z_p^2/(B)Z_p^2. This connects the arithmetic differential and the old manifold using the same depth. It does not bound that depth by topology.

GNT.5 Real torus covers and the full mapping-cone homology

Use the real vector space V=k tensor R, isomorphic to R times C and hence to R^3 as a real vector space. Minkowski embedding makes A and O full lattices. The identity of V induces an actual smooth finite covering

$$q:{\mathbb{T}}_A=V/A⟶{\mathbb{T}}_O=V/O.$$

Theorem GNT6. The covering has degree I and deck group D=O/A. In the notation B=s t^2 c^3 of GNT3 its based fundamental-group map has Smith form diag(1,c,c^2t). Let C(q) be its mapping cone. Then

$$\boxed{{\pi }_1(C(q))\simeq D,{\tilde{H}}_1(C(q),\mathbb{Z})\simeq D,}\text{\hspace{1em}(GNT13)}$$

$$\boxed{{\tilde{H}}_2(C(q),\mathbb{Z})\simeq D\oplus \mathbb{Z}/I,{\tilde{H}}_3(C(q),\mathbb{Z})\simeq \mathbb{Z}/I,}\text{\hspace{1em}(GNT14)}$$

and all other reduced integral homology groups vanish.

Proof. Deck translations are exactly O/A, and the fiber size is its order I. The induced fundamental-group map is the inclusion A into O, whose Smith form follows from GNT3. Van Kampen for the attached cone quotients O by the normal closure of A; since O is abelian, this is D. On homology, H_m(V/L,Z)=exterior^m L for m=1,2,3, and the induced map is the respective exterior power of the inclusion. These maps are all injective. The mapping-cone long exact sequence therefore identifies its reduced homology with their cokernels. Their Smith factors are (1,c,c^2t), (c,c^2t,c^3t), and (c^3t), proving GNT13-GNT14. The map on H_0 is an isomorphism and there is no kernel in top degree, so there are no further groups. Covering theory, van Kampen, the exact sequence, and the exterior algebra of a torus are classical; Hatcher, Algebraic Topology, Sections 1.2-1.3, 2.1 and Example 3.16 provide the inputs. Here the actual lattices and every Smith factor are calculated.

Both T_A and T_O by themselves are diffeomorphic to the standard three- torus. Arithmetic information is retained by the specified COVERING, not by their unmarked manifold types. C(q) is a topological CW-space; no smooth-manifold claim is made for that cone. Its first homology has nontrivial p-primary torsion exactly at WSS factors of the actual B.

GNT.6 Exact binomial comparison with complex singularities and links

To make the connection to classical singularity links precise, introduce the separate equal-characteristic complete ring

$$R_h=\mathbb{C}[[s,z]]/(z^3-s^h),h\ge 1.$$

It is a comparison model for the same exponent h. The original completed order has mixed characteristic. There is no asserted isomorphism from Spec(S) to a complex analytic germ and no topological link in C^2 assigned to Spec(S) by forgetting its residue characteristic.

Theorem GNT7. The normalization quotient of R_h, as a C[[s]]-module, has elementary exponents floor(h/3) and floor(2h/3), exactly as in GNT1. It has delta invariant delta_p from GNT2, b=gcd(3,h) branches, and

$$\boxed{\mu =2(h-1)=2{\delta }_p-b+1.}\text{\hspace{1em}(GNT15)}$$

The link of its convergent binomial plane curve is the torus link T(3,h). Its Milnor fiber has b boundary components and genus

$$g=h-(b+1)/2.\text{\hspace{1em}(GNT16)}$$

Proof. For h=3a+1 or 3a+2, normalization is C[[w]], with s=w^3, z=w^h. Comparison with its C[[s]]-basis 1,w,w^2 gives elementary exponents (a,2a) or (a,2a+1). For h=3a the curve has three branches z=omega^i s^a. Its normalization is the product of three copies of C[[s]]. The invertible constant Vandermonde matrix for 1,omega,omega^2 reduces the inclusion to diagonal 1,s^a,s^(2a). These computations prove the module and branch assertions. Directly, the Jacobian algebra is C[[s,z]]/(z^2,s^(h-1)), of dimension 2(h-1), including dimension zero at h=1. This proves GNT15 from the independently computed delta. The classical binomial link theorem identifies the link as T(3,h); Gorsky-Kivinen-Simental (2023), Section 6.1, Example 6.1, records this with its gcd component count. The Milnor fiber of a reduced complex plane-curve singularity is a connected surface with first Betti number mu and b boundary components; its Euler characteristic gives GNT16. The smooth h=1 case is a disk bounded by the unknot.

For comparison with arithmetic branches, pass only on the arithmetic side to a strictly henselian unramified base. Since three is a unit, u has a cube root there. The tame local normalization then has one branch if 3 does not divide h, and three otherwise. This is a matching branch count and binomial calculation, not a change of characteristic isomorphism.

Information boundary. Depths three and four both give the additive normalization quotient Z/p direct-sum Z/p^2, and the same delta=3. Thus even the full deck group of GNT6 cannot recover h by itself. Their geometric branch counts differ (three versus one), and their differential modules in GNT11 also differ. Once delta and b are both retained,

$$h=\delta +(3-b)/2.$$

Likewise T(3,3) has three components while T(3,4) has one. A nontrivial comparison link at h>=2 does not construct a WSS prime: assigning that h to a new ACTUAL block factor is still the missing arithmetic assertion.

GNT.7 Source roles, verification scope, and the remaining question

The new ordinary proofs compute the local normalization module, total- scheme singularity criterion, conductor ideal, exact point-blowup process, Kahler module, its based connection to the old mapping-torus torsion, and the marked-cover homology. These are specializations of classical commutative algebra and topology to the fixed golden order. Global first-discovery priority is unestablished. They decide no previously undecided WSS prime family and construct no singular actual golden block.

Primary source locators and precise roles:

• Stacks Project, Lemma 10.119.7, tag 00PD: https://stacks.math.columbia.edu/tag/00PD . Dimension-one regular local/DVR/normal criteria. Total-space regularity and smoothness of the structural morphism are kept separate.
• Stacks Project, Section 10.131, tag 00RM: https://stacks.math.columbia.edu/tag/00RM . Kahler differentials and the polynomial conormal presentation.
• Stacks Project, Lemma 31.33.2, tag 01OF: https://stacks.math.columbia.edu/tag/01OF . Affine blowup algebras S[I/a]. The two charts and index jumps above are derived, not inferred by counting terms in an embedded-resolution graph.
• Keith Conrad, The conductor ideal of an order, Definition 1.1 and proof of Corollary 4.3, printed page 12: https://kconrad.math.uconn.edu/blurbs/gradnumthy/conductor.pdf . The conductor definition and monogenic trace dual. The page-12 image and parsed proof were inspected. The square ideal formula above is proved for this order; no unrelated quadratic-order corollary is used.
• Allen Hatcher, Algebraic Topology, Chapters 1-3 (author-hosted): https://pi.math.cornell.edu/~hatcher/AT/ATchapters.html . Van Kampen, covering degree/index (Proposition 1.32), homology exact sequences, and Example 3.16 for the torus exterior algebra. The input passages were read; no generated cone is called a manifold.
• E. Gorsky, O. Kivinen and J. Simental, Algebra and geometry of link homology: Lecture notes from the IHES 2021 Summer School, Bulletin of the London Mathematical Society 55 (2023), 537-591, DOI 10.1112/blms.12761; arXiv:2108.10356. The publisher’s indexed Section 6.1, Example 6.1 supplies the classical binomial-link identification and component count. No link-homology conjecture or modern affine-Springer result is assumed. The direct arXiv HTML and full publisher fetch failed; only the retrieved section and publication locator are claimed inspected.
• H. D. Nguyen, Invariants of plane curve singularities and Plucker formulas in positive characteristic, arXiv:1412.5007: https://arxiv.org/abs/1412.5007 . Its abstract records the classical characteristic-zero identity mu=2delta-b+1 and warns of wild corrections in other settings. GNT15 is checked by direct binomial computation, with no extension to an arbitrary mixed-characteristic Milnor number.

The prior CG.5 construction is located in the existing Katz companion; GIR supplies the original order, initial depths and absence of a defect at three. Current branch GoldenCubicBlockCongruences source descriptions also retain the block recurrences. No new Lean endpoint, kernel check, Scribe compilation or independent-model review is asserted for GNT. Finite algebraic tests of higher h are labelled synthetic throughout. Their cusps, conductors and cover groups do not furnish actual WSS samples.

GMI. A uniform power-basis obstruction and exact generator-index restrictions

GMI.0 The same order and two different indices

Retain the actual golden blocks B_j=L_(3^j)^2+3 and the order A_j of GIR/GNT, for j>=1. Write theta=cuberoot(B_j)>0, a=(B_j-1)/9 and beta=(1+theta+theta^2)/3. The proved integral basis of this ORDER is (1,theta,beta). Its fraction field has degree three since B_j is not a cube. Let O_j be the maximal integer ring and I_j=[O_j:A_j].

For an element alpha in A_j with Q(alpha)=k_j, define instead

$${\mu }_j(\alpha )=[A_j:\mathbb{Z}[\alpha ]].$$

The index I_j is the normalization index that detects WSS in GIR/GNT. The index mu_j(alpha) measures a specified power basis inside A_j. These are distinct inclusions and must not be substituted for each other. Monogenicity of A_j means that some mu_j(alpha) equals one.

Use the original oriented Eisenstein factor eta_j=-2+(L_(3^j)-1)omega, with omega^2+omega+1=0. For an integer n coprime to B_j, set chi_j(n)=(n/eta_j)_3. GCR1, GCC1 and GCC2 already give

$${\chi }_j(3)=\omega ,{\chi }_j(2)=\omega ,{\chi }_j(\ell )=1\left(\ell \mid \prod _{i<j}{B}_i\text{ prime}\right).\text{\hspace{1em}(GMI0)}$$

These are characters of the actual factorizations with the original exponents h_p, not assumptions that h_p=1. The first identity follows from eta_j=1+lambda^3 modulo nine, lambda=1+2omega: the classical supplementary laws give (lambda/eta_j)_3=omega^2 and 3=-lambda^2. The other identities are proved in the existing conjugate-completed continuation. All of them use unconditional classical cubic reciprocity.

GMI.1 Exact index form for every integral element of this order

Theorem GMI1. For integers r,b,c with (b,c)!=(0,0), the element alpha=r+b theta+c beta generates k_j over Q and

$$\boxed{{\mu }_j(\alpha )=|F_j(b,c)|,F_j(b,c)=3b^3+3b^{2}c+b{c}^2-a{c}^3=\frac{(3b+c{)}^3-B_j{c}^3}{9}.}\text{\hspace{1em}(GMI1)}$$

In particular F_j(b,c) is nonzero on every such integer pair.

Proof. Since the field degree is prime and alpha is not rational, Q(alpha)=k_j. Its powers 1,alpha,alpha^2 are a Z-basis of Z[alpha]. The integer r does not affect their signed determinant. For alpha=b theta +c beta, the GIR multiplication table gives the theta and beta coordinates of alpha^2 as

$$-b^2+a{c}^2,3b^2+2bc+c^2.$$

The determinant of the columns 1,alpha,alpha^2 in (1,theta,beta) is b(3b^2+2bc+c^2)-c(-b^2+a c^2), exactly the cubic in GMI1. Its absolute value is the subgroup index. Expanding (3b+c)^3 and using B_j=9a+1 proves the second expression. This constructs the particular index form; it does not import an index form for a different maximal integer ring. The general determinant/index-form convention is Kang-Kim, Section 3.

GMI.2 Cubic phases of the actual generator indices

Theorem GMI2. Suppose mu=mu_j(alpha) is coprime to B_j. Then

$$\boxed{{\chi }_j(\mu )=\omega .}\text{\hspace{1em}(GMI2)}$$

For every individual prime p|B_j with p not dividing mu, one also has

$$\boxed{{\mu }^{(p-1)/3}\equiv 3^{(p-1)/3}(\mathrm{mod}p).}\text{\hspace{1em}(GMI3)}$$

The second statement only requires coprimality to the indicated p.

Proof. Put m=F_j(b,c) and w=3b+c. Equation GMI1 gives w^3=B_jc^3+9m. If m is coprime to B_j, w is a unit at every prime of eta_j. Taking the cubic symbol modulo eta_j gives

$$1={\chi }_j(9m)={\omega }^2{\chi }_j(m).$$

Since minus one is a cube, chi_j(m)=chi_j(|m|). This proves GMI2. At an individual p, the same congruence gives (9m/varpi_(j,p))3=1, so (m/varpi(j,p))3=(3/varpi(j,p))_3, because cubing a symbol is one. The residue field is F_p. Its defining symbol exponent gives GMI3; changing m to |m| is harmless since (p-1)/3 is even.

Consequences. A generator index coprime to B_j cannot be an integer cube. It cannot have all its prime factors among the earlier block supports, because the product of their characters is one. If its support is contained in {2,3} together with those earlier primes, then

$$v_2(\mu )+v_3(\mu )\equiv 1(\mathrm{mod}3).\text{\hspace{1em}(GMI4)}$$

These conditions are necessary. No sufficiency for arbitrary integers satisfying a character condition is asserted.

GMI.3 Uniform nonmonogenicity, with an exact local distinction

Theorem GMI3. Every actual order A_j is nonmonogenic:

$$\boxed{\text{for every }j\ge 1\text{ and every }\alpha \in A_j,A_j\ne \mathbb{Z}[\alpha ].}\text{\hspace{1em}(GMI5)}$$

Equivalently the two integer equations

$$\boxed{w^3-B_j{c}^3=9\text{or}-9}\text{\hspace{1em}(GMI5a)}$$

have no solutions with integers w,c, for any j>=1. Any such solution would have w=c modulo three and hence w=3b+c for an integer b.

Proof. A rational alpha cannot generate the rank-three ring. For any other alpha, index one would contradict GMI2 because chi_j(1)=1 while omega!=1. This is a simultaneous proof for all j; it is not an extrapolation from a finite list of blocks or a bounded Thue search.

Theorem GMI3a. Nevertheless A_j tensor Z_l is generated as a Z_l-algebra by one element for EVERY rational prime l. Furthermore

$$\boxed{\gcd \{{\mu }_j(\alpha ):\alpha \in A_j,Q(\alpha )=k_j\}=1.}\text{\hspace{1em}(GMI6)}$$

Proof. For l!=3, theta generates since beta=(1+theta+theta^2)/3. At three, use theta+beta, whose index by GMI1 is |7-a|. Here a_1=2, and a_j=0 modulo three for j>=2. Indeed B_1=19 and the block recurrence B_(j+1)=B_j^3-3B_j^2+3 give B_2=1 modulo 27 and preserve one modulo 27. Thus 7-a is a three-adic unit. The determinant is a unit over Z_3, so 1,theta+beta,(theta+beta)^2 form a local integral basis. Finally theta has index three and theta+beta has index coprime to three. Their greatest common divisor is one, proving GMI6.

This is local ALGEBRA monogenicity. It is not the stronger assertion that the globally normalized index form represents +/-1 over every Z_l. In fact GCR1a supplies a prime p|B_j at which three is a noncube. Then GMI3 forbids index-form value +/-1 already over F_p and Z_p. Thus these orders DO have a local index-form obstruction. The distinction is explicit in Alpoege-Bhargava-Shnidman, Definition 3 and Section 2.3; their global Hasse-failure theorems do not describe this example.

GMI.4 Sharp minima and a complete prime-power subproblem

Let m_j be the least positive value of mu_j(alpha) over field-generating alpha in A_j. Then

$$\boxed{m_j\in \{2,3\},m_1=2,m_3=3.}\text{\hspace{1em}(GMI7)}$$

Proof. GMI3 excludes one, and theta has index three. For j=1, beta has index a_1=2. For j=3, the ACTUAL factorization is

$$B_3=192900153619=3079\cdot 62650261.$$

Both factors are prime and

$$3^{1026}\equiv 1(\mathrm{mod}3079),2^{1026}\equiv 546(\mathrm{mod}3079).$$

Thus GMI3 excludes mu=2 in that block, proving m_3=3. Trial division up to each square root certifies these two fixed primes; modular exponentiation verifies the displayed residues. The universal exclusion of all alpha with index two follows from GMI3, not enumeration of alpha. The value of m_2 is not claimed determined here.

Theorem GMI4. All generator indices which are powers of three are classified exactly, uniformly in j:

$$\boxed{\{{\mu }_j(\alpha ):{\mu }_j(\alpha )=3^e,e\ge 0\}=\{3^{3k+1}:k\ge 0\}.}\text{\hspace{1em}(GMI8)}$$

Proof. Since three is prime to every B_j, GMI2 requires omega^e=omega, so e=1 modulo three. Conversely alpha=3^k theta has index F_j(3^k,0)=3^(3k+1). This proves both exclusion and realization. The varying k scales a primitive element, not the original Fibonacci recurrence or its initial depths.

GMI.5 Localizing does not remove every obstruction

Theorem GMI5. Let S be any subset of the rational prime factors of the earlier blocks B_1,…,B_(j-1). Then

$$\boxed{A_j\otimes \mathbb{Z}[S^{-1}]\text{ is not monogenic over }\mathbb{Z}[S^{-1}].}\text{\hspace{1em}(GMI9)}$$

In contrast A_j[1/3]=Z[1/3][theta].

Proof. Suppose alpha generates after inverting S. Clear its denominators to get beta_0=N alpha in A_j, where N is supported on S. At every prime l outside S, N is a unit and beta_0 still generates the localized algebra. Therefore the finite integer index mu_j(beta_0) is supported only on S. The supports of the earlier and current blocks are disjoint, so this index is coprime to B_j. Each of its prime factors has character one by GMI0. This contradicts GMI2. The second assertion follows directly from beta=(1+theta+theta^2)/3.

Theorem GMI5a. For the actual third block, even

$$\boxed{A_3[1/2]\text{ is not monogenic over }\mathbb{Z}[1/2].}\text{\hspace{1em}(GMI10)}$$

Proof. Clearing powers of two as above would produce a generator index 2^e for some e>=0. At the actual prime p=62650261 dividing B_3,

$$2^{20883420}\equiv 1(\mathrm{mod}p),3^{20883420}\equiv 31105528\not\equiv 1(\mathrm{mod}p).$$

Every 2^e therefore violates the individual necessary condition GMI3. This excludes all exponents and all possible generators at once. It does not use a bounded search for solutions of F_3(b,c)=+/-2^e. No stronger claim about inverting two together with arbitrary earlier primes is made: their individual characters at this p need not be one.

GMI.6 A proved separation from WSS and the two marked covers

Counterexample to a candidate shortcut. Nonmonogenicity of the specific A_j does NOT imply I_j>1, a singular point of Spec(A_j), or a WSS factor in B_j. The actual first block already refutes each implication:

$$B_1=19,I_1=1,A_1=O_{\mathbb{Q}(\sqrt[3]{19})},m_1=2,F_{18}=2584=19\cdot 136,19\nmid 136.\text{\hspace{1em}(GMI11)}$$

Maximality follows also directly from GIR’s discriminant/index formula for the squarefree block 19. GNT then makes the total arithmetic scheme regular. Yet GMI3 proves no element gives an integral power basis. This refutes the stated candidate inference, not the WSS existence conjecture and not a claim attributed to a cited author.

A genuine sufficient direction within this family is

$$\boxed{O_{k_j}\text{ monogenic }⟹I_j>1⟹B_j\text{ has a WSS prime factor}.}\text{\hspace{1em}(GMI12)}$$

Indeed I_j=1 would identify this monogenic ring with the nonmonogenic A_j. No monogenic maximal integer ring in this actual family is constructed here. The first three checked blocks have I_j=1.

To relate this to GNT’s topology, keep the marked chain of lattices Z[alpha] subset A_j subset O_j in V=k_j tensor R. It gives two covers

$$V/\mathbb{Z}[\alpha ]⟶V/A_j⟶V/O_j$$

of degrees mu_j(alpha) and I_j, respectively. The composite degree is mu_j(alpha) I_j. Nontriviality of the first cover is forced at EVERY layer by GMI3, including the regular block 19. Nontriviality of the second is exactly the original WSS-support condition of GIR/GNT. An unmarked torus type does not distinguish these inclusions.

The other monogenicity criterion in the existing Katz note, concerning the SPECIFIED polynomial X^(2p)-X^p-1, has a different field and order. No contradiction to that Jones criterion follows from GMI11. Likewise the cubic Thue equation f_pi(u,v)=-2 in GCC5 concerns factorization of eta_j and is not the index equation F_j(b,c)=+/-1. The present theorem does not eliminate B_j=P^2Q^3 or decide the full original WSS zero set.

GMI.7 Primary sources and validation boundary

• Minchan Kang and Dohyeong Kim, The proportion of monogenic orders of prime power indices of the pure cubic field, arXiv:2306.13295, Section 3, definition of the order index form and Lemma 3.1: https://arxiv.org/abs/2306.13295 . The determinant/index criterion is used after deriving GMI1. The paper’s main density theorem assumes m^2 is NOT one modulo nine. Our actual B_j are one modulo nine. That density theorem is not applied here.
• Levent Alpoege, Manjul Bhargava and Ari Shnidman, A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so, arXiv:2011.01186v2; Mathematische Annalen 391 (2025), 5535-5551, DOI 10.1007/s00208-024-03054-w: https://arxiv.org/abs/2011.01186 . Definition 3 and Section 2.3 distinguish representing a p-adic unit from representing +/-1 with the fixed global index form. Its example is Q[X]/(X^3-7/5), not a golden block. GMI3a has the former local property and has a genuine obstruction of the latter kind. No new Hasse-principle counterexample or use of their averaging theorem is claimed.
• Dunn-Radziwill, arXiv:2109.07463v3, equations (1.4)-(1.5), has the classical supplementary laws already cited and used in GCR/GCC. No GRH hypothesis from its analytic theorems enters the proof.

The relevant primary PDF text was read, including the index convention, local-obstruction distinction and supplementary formulas. The attempted PDF page screenshots failed, so no successful visual-page check is claimed. The general index-form machinery and reciprocity laws are classical. These fixed-family deductions have no established global first-discovery priority. Repository and bounded literature searches are not proofs of novelty.

The attached verify_gmi.py checks the symbolic determinant, fixed exact matrix determinants, all stated numerical prime/residue certificates, actual block recurrences, the required character values and the pure- three-power realizations. Finite checks supplement the proofs of the unbounded statements. No Lean elaboration, Scribe compilation, complete Thue solver, rank/Selmer computation, independent-model review or new WSS witness is claimed. No previously undecided WSS prime family has been settled, and the full P^2Q^3 branch remains unexcluded.