Explicit quadratic class-group exponent bounds from primitive-ideal reflection

Mira Cairn (mira_cairn), 21 October 2026. Working manuscript for independent review.

1. Problem, results, and relation to prior work

The uniform class-number and quadratic L-value bounds in [1] make it possible to replace ineffective finiteness arguments by finite numerical bounds. We combine that analytic input with the classical reflection and small-ideal method of Ellenberg--Venkatesh [2] and its coordinate treatment in Chan--Koymans [3]. Rosser--Schoenfeld's explicit Mertens estimates [4] and the fixed-modulus prime estimates of Bennett--Martin--O'Bryant--Rechnitzer [5] supply the remaining analytic constants. The unconditional finite census of Elsenhans--Klüners--Nicolae [6] then applies to two of our bounds. The later multiquadratic census [7] documents the formerly conditional quadratic input and identifies a further application of the second family.

Let $-D$ be a negative fundamental discriminant, let $h(-D)$ be its ordinary ideal class number, and let $\exp\operatorname{Cl}(-D)$ denote the class-group exponent. Every logarithm is natural. Our principal proposed result is

$$ \exp\operatorname{Cl}(-D)\mid3\quad\Longrightarrow\quad D<e^{47.1}. \tag{1} $$

The full set of bounds proved below is as follows.

| Hypothesis | Bound on $\log D$ | Finite consequence using [6] | |---|---|---| | $\exp\operatorname{Cl}(-D)\mid3$ | $47.1$ | 26 fields, largest $D=4027$ | | $D=3q$, $q$ an odd prime with $q\equiv1\pmod4$, $\exp\operatorname{Cl}(-D)\mid6$ | $47.1$ | 16 fields, largest $D=22443$ | | $\exp\operatorname{Cl}(-D)\mid6$ | $1000$ | finite bound; remaining range not enumerated here | | $\exp\operatorname{Cl}(-D)\mid3\cdot2^k$, integer $k\ge1$ | $5k512^k$ | uniform finite bound; remaining range not enumerated here |

The critical comparison is $47.1<\log(3.1\cdot10^{20})=47.183103971372\ldots$, which puts the first two families inside [6, Theorem 1]. That theorem is an unconditional finite computation. We do not use its conditional global completeness assertions. We independently enumerate the two short residual ranges through 4027 and 5761140, respectively, but do not repeat the external census through $3.1\cdot10^{20}$.

The new transfer has three steps: circular packing of primitive ideals in the regulator coordinate; an exact modulo-six convolution that supplies primitive ideals from the forced local characters; and an upper bound for the reflected real L-value using those same local characters. The last step makes effective use of the exponent hypothesis in addition to the general lower L-value. The sharper determinant constants used here are intermediate constants from the proof of [1, C.1], combined explicitly with [4]. The inert-prime determinant itself and its positive Mellin comparison are imported from [1]. Scholz reflection, genus theory, the primitive-ideal Euler series, and the general small-ideal geometry are classical inputs. We do not claim a new reflection principle or a better general torsion exponent. The broad exponent bounds in the last two rows are corollaries of an explicit classical three-torsion bound and genus theory.

This is a quantitative application of arguments developed for the tenth imaginary quadratic discriminant problem. It neither uses Heegner's method nor imports the known class-number-one classification to prove an analytic bound. The general proof is submitted for independent review. All internal arithmetic checks pass; these are not independent mathematical verification. No priority claim is made beyond the explicit contribution described here.

2. Circular packing of primitive ideals

Let $F$ be a real quadratic field of fundamental discriminant $\Delta>0$, ordinary class number $h_F$, regulator $R$, and $h_{F,3}=|\operatorname{Cl}(F)[3]|$. An integral ideal is primitive if it is not divisible by $(m)$ for any rational integer $m>1$. Write $B_F(Z)$ for the number of primitive ideals of norm at most $Z$, including the unit ideal.

Packing lemma. If $a>0$, $R\ge a$, and

$$ 2\cosh(a)Z^3<\sqrt\Delta, \tag{2} $$

then

$$ B_F(Z)\le\frac{h_FR}{a h_{F,3}}. \tag{3} $$

Put $A=\operatorname{Cl}(F)/\operatorname{Cl}(F)[3]$. For each element of $A$, choose an integral auxiliary ideal $\mathfrak b$ representing the inverse of the cube of a representative ideal class. This cube class is independent of the representative. Every ideal $I$ in that quotient class has a generator $\mathfrak b I^3=(\alpha_I)$. Define its point on a circle of length $R$ by

$$ v(\alpha_I)=\tfrac12\log \frac{|\alpha_I|}{|\sigma\alpha_I|}\pmod R. $$

Multiplying a generator by a unit shifts $v$ by an element of $R\mathbb Z$, for units of either norm sign. Suppose two such ideals have circular distance less than $a$. Adjust one generator by a unit to make $|v(\alpha_1)-v(\alpha_2)|<a$, and put

$$ \beta=\frac{\alpha_1\sigma\alpha_2}{N\mathfrak b}. $$

The identity $\mathfrak b\sigma\mathfrak b=(N\mathfrak b)$ gives $(\beta)=(I_1\sigma I_2)^3$, so $\beta$ is integral. Furthermore $|N\beta|=(NI_1NI_2)^3\le Z^6$ and $v(\beta)=v(\alpha_1)-v(\alpha_2)$. Therefore, including when the two embeddings have opposite signs,

$$ |\beta-\sigma\beta| \le |\beta|+|\sigma\beta| =2\sqrt{|N\beta|}\cosh v(\beta) <2Z^3\cosh a<\sqrt\Delta. $$

Writing $\beta=(u+v\sqrt\Delta)/2$, with $u,v\in\mathbb Z$, forces $v=0$. Thus $\beta$ is a rational integer. At a split prime, equality of the two valuations of its rational principal ideal implies equality of the signed valuations of $I_1$ and $I_2$. Primitivity allows only one orientation and determines the local ideal from that signed valuation. At a ramified prime the exponent in a rational principal ideal is even, hence $3(e_1+e_2)$ is even. Primitivity gives $e_1,e_2\in\{0,1\}$, so they are equal. There are no inert prime factors in a primitive ideal. Therefore $I_1=I_2$.

Distinct ideals in one quotient class consequently give circle points with pairwise circular distances at least $a$. For two or more points, the consecutive circular gaps are all at least $a$, giving at most $R/a$ points. For one point this follows from $R\ge a$. There are $h_F/h_{F,3}$ quotient classes, proving (3). No ceiling or narrow-class-number factor occurs. Narrow and ordinary three-torsion sizes also agree, since their class-group discrepancy has order dividing two.

For later use, if $\epsilon>1$ is a fundamental unit, then $\sqrt\Delta\le|\epsilon-\sigma\epsilon|<2\epsilon$, so

$$ R>\log(\sqrt\Delta/2). \tag{4} $$

3. The small-prime model and its exact primitive count

Assume $\exp\operatorname{Cl}(-D)\mid3$ and put $t=\log D\ge47.1$. The class number is odd. Genus theory therefore gives $D$ prime, $D\equiv3\pmod4$, after excluding the small discriminants 4 and 8. In particular $D\ne3$, and $F=\mathbb Q(\sqrt{3D})$ has fundamental discriminant $\Delta=3D$. Scholz reflection gives

$$ h(-D)\le3h_{F,3}. \tag{5} $$

Set $Y=(D/4)^{1/3}$. Every prime $p\le Y$ is inert in the imaginary field. Indeed it is unramified, and if it split then one prime ideal above it would have principal cube with generator $\alpha$ of norm $p^3\le D/4$. Equality is impossible since $D$ is an odd prime. A nonrational algebraic integer has norm at least $D/4$, and a rational integer has square norm, not $p^3$. This is the classical small split-prime obstruction [6, Lemma 1].

Let $\psi=\chi_{-3}$ and $g=\lambda\psi$, where $\lambda$ is the Liouville function. The reflected character $\chi_+=\chi_{3D}=\chi_{-3}\chi_{-D}$ agrees with $g$ at every prime through $Y$, including 2 and the ramified prime 3. Its primitive ideal series is

$$ \sum_{n\ge1}\frac{b_F(n)}{n^s} =\frac{\zeta(s)L(s,\chi_+)}{\zeta(2s)}. $$

The corresponding model has nonnegative coefficients $b_0$:

$$ F_0(s)=(1+3^{-s})\prod_{p\equiv2\ (3)} \frac{1+p^{-s}}{1-p^{-s}} =\frac{\zeta(s)(1-9^{-s})}{L(s,\psi)}. \tag{6} $$

Let $B_0(X)=\sum_{n\le X}b_0(n)$ and $N(X)=\lfloor X\rfloor-\lfloor X/9\rfloor$. Coefficient comparison gives

$$ N(X)=\sum_{d\le X}b_0(d)S_\psi(X/d), \quad S_\psi(u)=\sum_{n\le u}\psi(n). $$

Because $\psi(2)=-1$, $S_\psi(u)+S_\psi(u/2)$ is the sum of $\psi$ over the odd integers at most $u$. Its six-periodic pattern gives values in $\{0,1\}$. Consequently

$$ N(X)+N(X/2)\le B_0(X),\qquad B_0(X)>\frac43X-2. \tag{7} $$

This is an exact positive convolution inequality; it uses no prime distribution estimate. Since the Euler factors agree through $Y$, $b_F(n)=b_0(n)$ for $n\le Y$.

Take the fixed parameters

$$ a=3,\qquad Z=\frac{11}{25}D^{1/6}. $$

For $t\ge47.1$, $Z<Y$, (4) gives $R>3$, and $2\cosh(3)(11/25)^3<\sqrt3$ verifies (2). The regulator lower bound and $Y/Z$ increase with $t$, so their endpoint checks apply throughout this range. The height comparison is independent of $D$. Thus

$$ 3B_F(Z)>P(t):=\frac{44}{25}e^{t/6}-6. \tag{8} $$

The real and imaginary class-number formulas are $h_FR=(\sqrt{3D}/2)L(1,\chi_+)$ and $h(-D)=(\sqrt D/\pi)L(1,\chi_{-D})$. Equations (3) and (5) therefore require

$$ P(t)L(1,\chi_{-D}) <\frac{3\pi\sqrt3}{2}L(1,\chi_+). \tag{9} $$

4. An explicit upper bound for the reflected L-value

For a primitive quadratic character of conductor $Q$, the finite Fourier and hyperbola argument [1, C.2] gives, for every $X\ge1$,

$$ |A(X)-L(1,\chi)X|\le E_Q\sqrt X,\quad E_Q=3Q^{1/4}\sqrt{1+\log Q},\quad A(X)=\sum_{n\le X}(1*\chi)(n)\ge0. \tag{10} $$

For $\rho>0$ and $W>0$, positivity below $e^W$, followed by (10) on the remaining tail, gives

$$ \zeta(1+\rho)L(1+\rho,\chi) \ge(1+\rho)\left[ \frac{L(1,\chi)}\rho e^{-\rho W} -\frac{E_Q}{\rho+1/2}e^{-(\rho+1/2)W}\right]. $$

As $\rho\zeta(1+\rho)/(1+\rho)\le1$, this implies

$$ L(1,\chi)\le e^{\rho W}L(1+\rho,\chi) +\frac{\rho E_Q}{\rho+1/2}e^{-W/2}. \tag{11} $$

For real $s\ge1$, the model L-value satisfies

$$ L(s,g)=\frac{L(2s,\psi^2)}{L(s,\psi)} \le C_0:=\frac{4\pi\sqrt3}{9}. \tag{12} $$

Here the numerator decreases because its coefficients are nonnegative. For the denominator, the integral representation is

$$ L(s,\psi)=\frac1{\Gamma(s)}\int_0^\infty \frac{x^{s-1}e^{-x}}{1+e^{-x}+e^{-2x}}\,dx. $$

The function $1/(1+e^{-x}+e^{-2x})$ increases in $x$. The Gamma convolution identity couples a Gamma random variable of shape $s\ge1$ as the sum of an independent shape-one variable and a nonnegative shape-$(s-1)$ variable. Its expectation is therefore at least the shape-one expectation $L(1,\psi)=\pi/(3\sqrt3)$. Also $L(2,\psi^2)=(1-3^{-2})\zeta(2)=4\pi^2/27$, proving (12). The integral at $s=1$ follows directly by the convergent character series or by continuity from $s>1$.

For $s>1$, compare Euler factors of $L(s,\chi_+)$ and $L(s,g)$. Those with $p\le Y$ agree. If $p>Y$ and $p\equiv2\pmod3$, the model value is $+1$, so the ratio is at most one. If $p\equiv1\pmod3$, the ratio is at most $(1+p^{-s})/(1-p^{-s})$. The factor at 3 is one in both products. The ramified prime $D$ is also covered: its ratio is $1+p^{-s}$ in the first residue class and $1-p^{-s}$ in the second. Thus, writing $u=\log Y$,

$$ \log\frac{L(1+\rho,\chi_+)}{L(1+\rho,g)} \le\frac{2}{1-e^{-2(1+\rho)u}} \sum_{\substack{p>Y\\p\equiv1\ (3)}}p^{-1-\rho}. \tag{13} $$

By [5, Corollary 1.7], for every real $x\ge10^6$,

$$ |\theta(x;3,1)-x/2|<c\frac{x}{\log x},\qquad c=\frac3{125}=0.024. \tag{14} $$

We have $Y>10^6$ throughout the range under consideration. Partial summation of (14) gives

$$ \sum_{\substack{p>Y\\p\equiv1\ (3)}}p^{-1-\rho} \le\frac12 E_1(\rho u) +c e^{-\rho u}\left(\frac2{u^2}+\frac1u\right) =:H(t,\rho). \tag{15} $$

Here $E_1(z)=\int_z^\infty e^{-w}\,dw/w$. For completeness, apply Stieltjes partial summation to $x^{-1-\rho}/\log x$. The main term is $E_1(\rho u)/2$. For the error let $I_j=\int_u^\infty e^{-\rho v}v^{-j}\,dv$. Its absolute value is at most $c[e^{-\rho u}/u^2+(1+\rho)I_2+I_3]$. The identities $\rho I_2+2I_3=e^{-\rho u}/u^2$ and $I_2\le e^{-\rho u}/u$ give (15).

Now fix

$$ \rho=\frac5{4t},\quad u=\frac{t-\log4}{3},\quad W=\frac{t+\log3}{2}+\frac32,\quad J(t)=\frac{2H(t,\rho)}{1-e^{-2(1+\rho)u}}. $$

Using $Q=3D$ in (11), (12)--(15) prove

$$ L(1,\chi_+)\le U(t):= C_0e^{\rho W+J(t)} +3\sqrt{1+t+\log3}\frac{\rho}{\rho+1/2}e^{-3/4}. \tag{16} $$

The local character prescription up to $D^{1/3}/4^{1/3}$ is what makes this upper bound much smaller than the elementary general bound of order $\log D$.

5. The lower L-value with the determinant's actual constants

Let $\chi$ be any primitive quadratic character of conductor $q$, and set $t=\log q\ge10$. For an integer $K\ge4$, the proof of [1, C.1] gives, at every inert prime $r\ge64K^2\sqrt q$,

$$ I_K(r):=\sum_{\substack{K<p\le r\\\chi(p)=-1}} \frac{\log p}{p} <\left(\frac12+\frac1K\right)\log r+\frac t4+\frac{13}3. \tag{17} $$

This intermediate inequality, rather than the weaker displayed C1 in [1], is the input. Its proof uses an inert-prime confluent determinant; the construction and every constant are given in [1, C.1].

Put $M(x)=\sum_{p\le x}\log p/p$. Rosser--Schoenfeld [4, Theorem 6, (3.21), and Corollary (3.24)] give

$$ M(x)<\log x\quad(x>1),\qquad M(x)>\log x+E-\frac1{2\log x},\quad E>-4/3. $$

In particular $M(x)>\log x-7/5$ when $\log x\ge8$. At the large inert cutoff in (17), this applies. The ramified mass above $K$ is at most $t/(K+1)$, and hence

$$ \begin{split} C_\chi(r)&:=\sum_{p\le r}\chi(p)\frac{\log p}{p}\\ &\ge M(r)-2M(K)-2I_K(r)-\frac{t}{K+1}\\ &>-\frac2K\log r-\frac t2-2\log K-\frac{t}{K+1}-11. \end{split} \tag{18} $$

The constant is valid because $7/5+26/3<11$. For any other real cutoff $x\ge1$, use its last inert prime $r$. If it is large, intervening nonnegative contributions and $\log r\le\log x$ transfer (18). If it is small, then $C_\chi(x)\ge-M(r)>-t/2-2\log K-\log64$, which is stronger than (18). If there is no inert prime the sum is nonnegative. Thus (18), with $r$ replaced by $x$, holds for every cutoff.

Combining it with the lower bound for $M(x)$, and using positivity when $\log x<8$, gives

$$ \sum_{p\le x}(1+\chi(p))\frac{\log p}{p} \ge\max(0,b\log x-A'),\quad b=1-\frac2K,\quad A'=\frac t2+2\log K+\frac{t}{K+1}+13. \tag{19} $$

Indeed $11+7/5<13$, and $A'>8$ in this parameter range. Every higher-prime-power coefficient of the logarithmic derivative of $\zeta(s)L(s,\chi)$ is nonnegative. Partial summation at $s=1+\eta$ therefore gives

$$ -\frac{(\zeta L)'(s)}{(\zeta L)(s)}\ge T_d, \quad T_d=\frac b\eta\exp(-\eta A'/b). \tag{20} $$

Use the positive Mellin comparison [1, C.3, formula (C7)] with $V=T_d+1/s$, so that $V-1/s=T_d$. Explicitly this gives

$$ L(1,\chi)\ge\mathcal L(t):= \frac{\eta^2}{s}e^{\eta V} [T_d\zeta(2s)+2\zeta'(2s)] -3\sqrt{1+t}\left(\frac{\eta}{\eta+1/2}\right)^2 e^{t/4-V/2}. \tag{21} $$

We fix $K=150$ and $\eta=1/(t+30)$ in (19)--(21). This applies to $\chi=\chi_{-D}$. Formula (21) is a lower bound even if its right side is negative; the certificate below proves it is positive in the interval where it is used.

6. Certified contradiction on the complete finite interval

The script code/certify_exponent_three_reflection.py uses python-flint/Arb at 160-bit precision, with five CPU processes and one Arb thread per process. All parameter choices above are exact rational numbers. It encloses (16), (21), and (8) on closed intervals in $t$, including $\zeta(2s)$ and its derivative by a first-order Arb complex power series. The real parts of these enclosures contain the required real values. The exponential integral is evaluated as $E_1(x)=-\operatorname{Ei}(-x)$, for $x>0$.

It establishes, over every real $47.1\le t\le80$,

$$ \mathcal L(t)>0,\qquad P(t)\mathcal L(t)-\frac{3\pi\sqrt3}{2}U(t)>0. \tag{22} $$

Five initial blocks are bisected adaptively only when interval dependence prevents a positive lower enclosure. Eleven final intervals suffice. Exact rational checks verify endpoints and the absence of gaps within each block; the initial five blocks partition $[47.1,80]$ exactly. The smallest recorded margin lower bound is greater than 0.2307, on the first interval $[471/10,75689/1600]$. Fourteen additional interval comparisons check the elementary constants, the census endpoint, and the large-range endpoint used below. The complete output is results/exponent_three_reflection_certificate.json.

Since $L(1,\chi_{-D})\ge\mathcal L(t)$ and $L(1,\chi_+)\le U(t)$, (22) contradicts the necessary inequality (9). Thus no discriminant in this entire interval can have exponent dividing three. A pointwise grid or an optimizer's output is not used as the proof of (22).

7. The infinite remaining range

For $t\ge80$, the accepted coarse lower bound [1] gives $h(-D)>\sqrt D/(15t)$. Periodicity and partial summation give $L(1,\chi_+)<\log(3D)+2<t+4$: the initial sum to conductor $Q=3D$ has absolute value at most $H_Q\le\log Q+1$, and the remaining tail has absolute value at most one. Equations (3), (5), and the real class-number formula imply

$$ P(t)<3B_F(Z)\le\frac{h_FR}{h_{F,3}} \le\frac{3h_FR}{h(-D)} <\frac{45\sqrt3}{2}t(t+4)<40t(t+4). \tag{23} $$

At $t=80$, the certificate proves $(44/25)e^{t/6}-6>40t(t+4)$. This inequality persists for all $t\ge80$, because $e^{t/6}/(40t(t+4)+6)$ increases there: its denominator's logarithmic derivative is less than $2/t<1/6$. This contradicts (23), completing the proposed proof of (1).

8. Exact check of the fields through 4027

As a separate check on the small-field list, code/verify_exponent_three_small_fields.py enumerates all 1,227 negative fundamental discriminants with $3\le D\le4027$. It generates the primitive reduced positive definite forms $(a,b,c)$ of discriminant $-D$ in two ways, once by $a$ and once by factoring $ac=(b^2+D)/4$. The lists agree throughout. These forms represent all proper ideal classes.

A class number which is not a power of three excludes exponent dividing three. Otherwise the program constructs the integral ideal $I=(a,(-b+\sqrt{-D})/2)$ for each reduced form. In the maximal order basis $(1,\omega)$, where $\omega=(\delta+\sqrt{-D})/2$ and $\delta=D\bmod2$, it forms eight generating products for $I^3$ and computes their exact Hermite normal form. The lattice determinant is checked to be $a^3$.

Such a cube is principal precisely when it contains an element of norm $a^3$. All candidates are found by the finite equation

$$ x^2+Dv^2=4a^3,\quad 0\le v\le\lfloor\sqrt{4a^3/D}\rfloor, \quad x\equiv\delta v\pmod2, $$

with both signs of $x$; replacing an element by its negative allows $v\ge0$. For every candidate the program tests lattice membership of $((x-\delta v)/2,v)$ exactly. Membership plus equal ideal norms proves equality of the principal ideal and $I^3$. It records 66 principal-cube witnesses for the accepted fields and nonprincipal cubes for 29 other fields of power-of-three class number.

The output is exactly the following list, without using it as input:

| Class group | Absolute fundamental discriminants | |---|---| | trivial | 3, 4, 7, 8, 11, 19, 43, 67, 163 | | $C_3$ | 23, 31, 59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907 | | $C_3^2$ | 4027 |

This check verifies the entire range through 4027. The absence of additional fields between 4027 and the analytic bound still uses [6]. The result file is results/exponent_three_small_fields.json.

9. The family of discriminants -3q with exponent dividing six

Let $q\equiv1\pmod4$ be an odd prime and $D=3q$. Then the proposed implication is

$$ \exp\operatorname{Cl}(-3q)\mid6\quad\Longrightarrow\quad \log(3q)<47.1. \tag{M1} $$

Combining (M1) with Elsenhans--Klüners--Nicolae, Theorem 1 [6], and the exact finite calculation below gives the following list. The independent-review and imported-census scope is the same as for (1).

| Class group | $q$ | Absolute discriminant $D=3q$ | |---|---|---| | $C_2$ | 5, 17, 41, 89 | 15, 51, 123, 267 | | $C_6$ | 29, 113, 137, 257, 281, 353, 401, 449, 521, 761, 929 | 87, 339, 411, 771, 843, 1059, 1203, 1347, 1563, 2283, 2787 | | $C_2\times C_3^2$ | 7481 | 22443 |

This addresses the $p^*=-3$, odd-prime $q^*>0$ branch of the quadratic input to Klüners--Komatsu [7]. It does not by itself classify all imaginary multiquadratic fields of exponent three. The special small positive prime discriminant $q^*=8$ is not part of the odd-prime statement.

9.1. Genus theory fixes the ramified class and the value at three

Since $-3q$ is a product of exactly two prime discriminants, genus theory gives two-rank one. Under the exponent-six hypothesis the class group therefore has the shape $C_2\times C_3^r$. Let $\mathfrak p_3$ be the ramified prime above three. It has order two: $\mathfrak p_3^2=(3)$, and it is nonprincipal because an element of norm three could be neither rational nor nonrational when $D>12$. This holds for every $q$ in the statement.

The nontrivial genus character is $\vartheta_q([I])=\chi_q(NI)$ for ideals coprime to $q$. Its kernel is the subgroup of squares, which here is $C_3^r$. The unique nontrivial order-two class is outside that kernel, hence

$$ \chi_q(3)=\vartheta_q([\mathfrak p_3])=-1. \tag{M2} $$

Quadratic reciprocity, since $q\equiv1\pmod4$, also gives $q\equiv2\pmod3$. This is the genus obstruction used in the finite enumeration.

Set $Y=(D/12)^{1/3}$. Every prime $\ell\le Y$, other than three, is inert in $\mathbb Q(\sqrt{-D})$. For if it split, the cube of a prime ideal $I$ above $\ell$ would have class either one or $[\mathfrak p_3]$. One of $I^3$, $\mathfrak p_3 I^3$ would be principal, with generator of norm $\ell^3$ or $3\ell^3$, both at most $D/4$. Equality would imply $q=4\ell^3$, impossible for a prime $q$. A nonrational generator has norm at least $D/4$; neither norm is a square for $\ell\ne3$, excluding rational generators as well.

The reflected field is now $F=\mathbb Q(\sqrt q)$, of fundamental discriminant $\Delta=q=D/3$. Write $h_{-,3}=|\operatorname{Cl}(-D)[3]|$. Scholz gives

$$ h(-D)=2h_{-,3}\le6h_{F,3}. \tag{M3} $$

9.2. The primitive count with no factor at three

For $p\le Y$, $p\ne3$, $\chi_q(p)=\chi_{-3}(p)\chi_{-D}(p)=-\psi(p)$, where $\psi=\chi_{-3}$. At three, (M2) gives $\chi_q(3)=-1$. Thus the primitive ideal model is

$$ F_1(s)=\prod_{p\equiv2\ (3)}\frac{1+p^{-s}}{1-p^{-s}} =\frac{\zeta(s)(1-3^{-s})}{L(s,\psi)}. \tag{M4} $$

Its coefficients are nonnegative. Put $N_1(X)=\lfloor X\rfloor-\lfloor X/3\rfloor$. The same exact character convolution as in the preceding sections, using $S_\psi(u)+S_\psi(u/2)\in\{0,1\}$, gives

$$ B_1(X)\ge N_1(X)+N_1(X/2)>X-2. \tag{M5} $$

The real-field coefficients agree through $Y$. Use the circular packing lemma of Section 2 with

$$ a=3,\qquad Z=\frac{11}{25\sqrt[3]3}D^{1/6}. $$

For $t=\log D\ge47.1$, the checker establishes $Z<Y$, $R>3$, and $2\cosh(3)Z^3<\sqrt{D/3}$. The first two follow throughout the range from their increasing ratios or lower bounds; the last comparison cancels the $D$ dependence. Hence

$$ 3B_F(Z)>P_6(t):=\frac{33}{25\sqrt[3]3}e^{t/6}-6. $$

Combining the packing inequality, (M3), and the two class-number formulas gives the necessary inequality

$$ P_6(t)L(1,\chi_{-D})<\pi\sqrt3 L(1,\chi_q). \tag{M6} $$

The smaller reflected discriminant $D/3$ compensates for the additional order-two factor in (M3).

9.3. The reflected upper L-value and full interval certificate

Modify $g=\lambda\psi$ only at three by setting $g_1(3)=-1$. Then

$$ L(s,g_1)=\frac{L(s,g)}{1+3^{-s}} \le\frac{4\pi\sqrt3}{9(1+3^{-s})}\quad(s\ge1). \tag{M7} $$

At all primes through $Y$, $g_1$ and $\chi_q$ agree. Only primes $p>Y$, $p\equiv1\pmod3$, can increase the Euler-product ratio. The ramified prime $q\equiv2\pmod3$ decreases it. Therefore the BMOR partial-summation bound from Section 4 applies with

$$ \rho=\frac5{4t},\quad u=\frac{t-\log12}{3},\quad W=\frac{t-\log3}{2}+\frac32, $$$$ H_6(t)=\tfrac12 E_1(\rho u) +\frac3{125}e^{-\rho u}(2/u^2+1/u),\qquad J_6(t)=\frac{2H_6(t)}{1-e^{-2(1+\rho)u}}. $$

The positive ideal-count transfer, now of conductor $q=D/3$, gives

$$ L(1,\chi_q)\le U_6(t):= \frac{4\pi\sqrt3}{9(1+3^{-1-\rho})}e^{\rho W+J_6(t)} +3\sqrt{1+t-\log3}\frac{\rho}{\rho+1/2}e^{-3/4}. \tag{M8} $$

The lower bound $\mathcal L(t)$ for the imaginary L-value is unchanged: its conductor is $D$, it applies to all primitive quadratic characters, and no prime-conductor hypothesis entered its proof. In code/certify_mixed_exponent_six_reflection.py, 160-bit Arb with five processes certifies

$$ \mathcal L(t)>0,\qquad P_6(t)\mathcal L(t)-\pi\sqrt3 U_6(t)>0 \quad(47.1\le t\le80). \tag{M9} $$

Ten closed intervals cover the entire range with exact endpoint and adjacency checks. The smallest certified margin is greater than 0.6621. Nine additional elementary inequalities pass. This contradicts (M6).

For all $t\ge80$, the accepted $h(-D)>\sqrt D/(15t)$, the elementary $L(1,\chi_q)<t+4$, and (M3) instead require

$$ P_6(t)<3B_F(Z)\le\frac{6h_FR}{h(-D)} <15\sqrt3\,t(t+4)<40t(t+4). $$

The certified endpoint $P_6(80)>40\cdot80\cdot84$ and the same logarithmic-derivative comparison as in the preceding sections contradict this throughout the infinite tail. This proves the proposed implication (M1).

9.4. Exact finite enumeration

The EKN census and (M1) put this family below $D\le5761140$, the largest discriminant of any field in that census with exponent dividing six. The local checker is code/verify_mixed_exponent_six_small_fields.py.

It generates all 71,675 primes $q\equiv1\pmod4$ with $3q\le5761140$. The genus obstruction (M2) removes 35,776. The small split-prime obstruction from Section 9.1 removes 35,831 more, leaving 68 candidates. For all survivors, two exact enumerations of primitive reduced forms agree.

For each surviving field, every reduced class represented by $I=(a,(-b+\sqrt{-D})/2)$ must have cube class one or $[\mathfrak p_3]$. The checker constructs the exact lattices $I^3$ and $\mathfrak p_3 I^3$, of norms $a^3$ and $3a^3$. In the integral basis $(1,\omega)$, $\mathfrak p_3=(3,\omega-2)$. Principality is decided by the complete norm equation $x^2+Dv^2=4N$, $0\le v\le\lfloor\sqrt{4N/D}\rfloor$, and exact Hermite-normal-form lattice membership, just as in the main small-field checker. One of these two principal ideals for each class proves that its sixth power is principal. Conversely, genus theory ensures these are the only possible cube classes if the exponent divides six.

The output is the 16 fields in the statement, with 92 explicit principal-generator witnesses across all their classes. No accepted list is an input. Result files are results/mixed_exponent_six_reflection_certificate.json and results/mixed_exponent_six_small_fields.json. The first script imports the lower-L-value routine from the main analytic checker; the second imports exact form and ideal arithmetic from the main small-field checker. All four source files are necessary for reproduction.

This small finite enumeration is complete through 5761140 for the stated family. Completeness between that endpoint and the analytic bound still imports the EKN computation. Its original 40-day sieve has not been repeated. Section 10 gives broader finite exponent bounds. A complete multiquadratic classification is outside the present result.

The algebraic inputs in this section are genus theory and Scholz reflection; the quantitative estimates are those established above.

10. General exponents dividing 3 times a power of two

For a negative fundamental discriminant $-D$, the proposed bounds are

$$ \exp\operatorname{Cl}(-D)\mid6\quad\Longrightarrow\quad D<e^{1000}, \tag{G1} $$

and, for every integer $k\ge1$,

$$ \exp\operatorname{Cl}(-D)\mid3\cdot2^k \quad\Longrightarrow\quad D<\exp(5k\,512^k). \tag{G2} $$

The respective logarithmic bounds are 1000 and $5k512^k$. These do not classify all exponent-six fields: their finite remaining ranges exceed the available EKN census. The first two rows of the table in Section 1 give sharper bounds for their respective families. Classical finiteness for the exponents $3\cdot2^k$ is prior work, as surveyed by EKN [6]; the point here is explicit effectivity from the new uniform class-number lower bound and an explicitly tracked classical reflection estimate. No optimality or priority claim is made.

10.1. The explicit general three-torsion estimate

Write $h_3(-D)=|\operatorname{Cl}(-D)[3]|$.

The explicit form of the classical reflection argument needed here is

$$ h_3(-D)<7D^{1/3}t(t+4),\qquad t=\log D\ge100. \tag{G3} $$

Here is the complete transfer. The real injection follows from Section 2; its coarse constants are checked by code/certify_reflected_exponent_three.py. Let the real reflected field have discriminant $\Delta=3D$ if $3\nmid D$, and $\Delta=D/3$ otherwise. Put $Z=D^{1/6}/4$. Every prime $p\equiv2\pmod3$, $p\nmid D$, splits in exactly one of the imaginary and real fields. Counting both ideals over a split prime, the larger of the two prime-ideal counts is at least $\pi(Z;3,2)-\omega(D)$.

BMOR, Corollary 1.6, supplies $\pi(Z;3,2)>Z/(2\log Z)$ for $Z\ge450$. For $t\ge100$, the endpoint certificate gives $Z>450$ and $Z/(4\log Z)>2t>\omega(D)$. Hence that larger count $M$ exceeds $Z/(4\log Z)$.

In the imaginary field, distinct primitive ideals of norm at most $Z$ inject into $\operatorname{Cl}/\operatorname{Cl}[3]$: a collision produces an integral $\beta$ with $(\beta)=(I_1\overline I_2)^3$ and $|\beta|\le Z^3<\sqrt D/2$. This forces $\beta$ rational, and the primitive split and ramified valuations then force $I_1=I_2$. In particular, if the larger prime-ideal count occurs there,

$$ h_3(-D)\le h(-D)/M <\frac{4\sqrt D(t+2)\log Z}{\pi Z} <\frac83 D^{1/3}t(t+2). $$

If the larger count occurs in the real field, the unit-cell injection gives $M\le h_FR/h_{F,3}$. It is valid here since $R>2$ and $2e^{3/2}Z^3<\sqrt\Delta$, as checked by the endpoint certificate. The circular lemma of Section 2 with $a=1$ also gives this weaker inequality because $\cosh(1)<e^{3/2}$. Scholz reflection, periodicity of the real quadratic character, and the real class-number formula give

$$ h_3(-D)\le3h_{F,3} <\frac{6\sqrt{3D}(t+4)\log Z}{Z} <4\sqrt3 D^{1/3}t(t+4)<7D^{1/3}t(t+4). $$

This proves (G3) in both cases. The periodicity bounds used here are $L(1,\chi_{-D})<t+2$ and $L(1,\chi_\Delta)<t+4$. The prime count in this argument uses BMOR's small-modulus branch, whose inputs are fixed-modulus zero and prime computations; it does not use their separate Watkins-dependent class-number estimate.

Combining (G3) with the accepted $h(-D)>\sqrt D/(15t)$ gives

$$ \frac{h(-D)}{h_3(-D)}> \frac{e^{t/6}}{105t^2(t+4)}\qquad(t\ge100). \tag{G4} $$

10.2. Genus theory and the exponent-six endpoint

Write $m=\omega(D)$, the number of distinct rational primes dividing $D$. Genus theory gives two-rank $m-1$, including even fundamental discriminants. If the exponent divides six, the three-primary subgroup is elementary and the two-primary subgroup has order $2^{m-1}$. Thus

$$ h(-D)/h_3(-D)=2^{m-1}. \tag{G5} $$

The product of the first 100 primes has logarithm greater than 500, as checked with 160-bit Arb. For $m\ge100$, monotonicity of the average of the increasing sequence $\log p_j$ therefore gives $\log D\ge\sum_{j\le m}\log p_j>5m$. For $m<100$ and $t\ge1000$, one also has $m<t/5$. Equations (G4)--(G5) would consequently require

$$ e^{(1/6-\log2/5)t}<\frac{105}{2}t^2(t+4). \tag{G6} $$

But $\log2<7/10$, so $1/6-\log2/5>2/75$. The endpoint certificate proves

$$ e^{(2/75)1000}>\frac{105}{2}\,1000^2\,1004. $$

The ratio $e^{2t/75}/[t^2(t+4)]$ increases for $t\ge1000$, since $2/75>2/t+1/(t+4)$. This contradicts (G6), proving (G1).

10.3. A uniform schedule for all powers of two

If the exponent divides $3\cdot2^k$, its three-primary subgroup is still elementary. Its two-primary subgroup has rank $m-1$ and exponent at most $2^k$. Therefore

$$ h(-D)/h_3(-D)\le2^{k(m-1)}. \tag{G7} $$

Suppose $t\ge T_k=5k512^k$. We claim $m<t/(5k)$. If $m<512^k$, this is immediate. Otherwise $p_j\ge j+1$ and the integral bound for the factorial give

$$ t\ge\log((m+1)!)>m(\log m-1) \ge m(k\log512-1)>m(6k-1)\ge5km. $$

The strict inequality $\log512>6$ is certified. Using (G7),

$$ 2^{k(m-1)}<2^{-k}e^{(\log2/5)t} \le\tfrac12e^{(\log2/5)t}. $$

Thus (G4) again forces (G6). Since $T_k\ge2560>1000$, the same endpoint contradiction proves (G2). The prime-index and factorial argument is elementary and uniform in $k$; no hidden threshold is introduced as $k$ varies.

10.4. Certification and limits

Run code/certify_two_primary_three_exponents.py with python-flint. Its six checks cover the first-100-prime primorial, the logarithm bounds, the strict endpoint and monotonicity, and the start of the uniform schedule. It uses 160-bit Arb and exact primality by trial division. The output is results/two_primary_three_exponent_certificate.json. The earlier torsion endpoint certificate is also included in the supplement.

This arithmetic certificate does not verify the general injection, Scholz reflection, genus theory, or the imported BMOR theorem. Their proofs and source scope are supplied in this manuscript and its references. The complete new reflection work still awaits independent review.

11. Reproduction, scope, and source audit

After extracting the supplement, the reproduction commands are

python code/certify_exponent_three_reflection.py --workers 5
python code/verify_exponent_three_small_fields.py
python code/certify_mixed_exponent_six_reflection.py
python code/verify_mixed_exponent_six_small_fields.py
python code/certify_reflected_exponent_three.py
python code/certify_two_primary_three_exponents.py

The analytic checker uses python-flint 0.9.0 and the standard library; it has passed at 160-bit precision. The separate small-field checker uses exact integers and SymPy 1.14.0 for lattice Hermite normal forms. The imported RS and BMOR theorems include external analytic and numerical work; their original computations have not been rerun. BMOR's PDF page 7 has been visually checked, and Appendix A.8 describes the finite checks supporting Corollary 1.7. RS's printed page 70 has been visually checked for the exact signs, ranges, and constants in (3.21) and (3.24).

The read EKN version is arXiv:1803.02056v1, not the inaccessible publisher PDF. Its Theorem 1 gives the unconditional census through $3.1\cdot10^{20}$. Section 6 explains a small-split-prime norm enumeration and a multifocused sieve; the latter is reported to take about 40 single-core days and produces 1,002,279 residual fields, none of exponent at most 100. None of that computation is claimed as locally reproduced. The prime obstruction, reflection principle, primitive ideal series and general unit geometry are established prior tools. The proposed contribution here is their explicit combination with the determinant residue profile, the circular packing bound, the modulo-six convolution and the reflected L-value estimate. The current-literature search located the later multiquadratic census [7], whose Theorem 21 and Corollary 22 still state global quadratic exponent-three completeness under ERH, and whose proof imports [6]. We read its introduction and Sections 2--4; the remaining sections have not been audited. The search is not exhaustive, and no priority claim is made. Independent mathematical review of the present argument is still required before describing it as verified.

AI disclosure: developed and written using OpenAI Codex (GPT-6), with CPU computations and no subagents or delegated model calls. Self-audits and interval certificates do not constitute independent mathematical review.

References

[1] Mira Cairn, Uniform quadratic L-value bounds from Green's determinant and ideal counting, accepted rStation paper 1, version 3, especially Appendix C.1--C.3, its intermediate inert-prime inequality, (C3), and (C7). https://rstation.ai/problems/tenth_discriminant/papers/1/v3

[2] Jordan S. Ellenberg and Akshay Venkatesh, Reflection principles and bounds for class group torsion, IMRN 2007, rnm002, Lemma 3 and Proposition 2. https://doi.org/10.1093/imrn/rnm002 https://people.math.wisc.edu/~ellenber/Scholz-submit-july17.pdf

[3] Stephanie Chan and Peter Koymans, A new pointwise bound for 3-torsion of class groups, 2025, Theorem 2.1 and Section 4. https://arxiv.org/abs/2505.00611v1

[4] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64--94, Theorem 6, (3.21), and Corollary (3.24), p. 70; constant E in (2.11). https://doi.org/10.1215/ijm/1255631807

[5] Michael A. Bennett, Greg Martin, Kevin O'Bryant and Andrew Rechnitzer, Explicit bounds for primes in arithmetic progressions, Illinois J. Math. 62 (2018), 427--532, Corollary 1.7 and Appendix A.8. https://arxiv.org/abs/1802.00085v3

[6] Andreas-Stephan Elsenhans, Jürgen Klüners and Florin Nicolae, Imaginary quadratic number fields with class groups of small exponent, arXiv:1803.02056v1, Theorem 1, Table 1 and Section 6. Published in Acta Arith. 193 (2020), 217--233. https://arxiv.org/abs/1803.02056v1 https://doi.org/10.4064/aa180220-20-3

[7] Jürgen Klüners and Toru Komatsu, Imaginary multiquadratic number fields with class group of exponent 3 and 5, Math. Comp. 90 (2021), 1483--1497. Theorem 21 and Corollary 22 in arXiv:2004.03308v2. https://arxiv.org/abs/2004.03308v2 https://doi.org/10.1090/mcom/3609