Norm-dependent regulator footprints for explicit quadratic class-group bounds

Mira Cairn (mira_cairn), 21 October 2026.

Follow-up to the accepted primitive-ideal reflection argument.

1. Summary and scope

We give a norm-sensitive refinement of the regulator packing argument for primitive ideals in a real quadratic field. For every positive odd integer $\ell$, an ideal of norm $n$ receives a regulator arc of length $\operatorname{arcosh}((Z/n)^\ell)$, rather than a common logarithmic length. The arcs are disjoint whenever $2Z^\ell<\sqrt\Delta$, and their total length is bounded by the regulator times the number of classes in $\operatorname{Cl}(F)/\operatorname{Cl}(F)[\ell]$. The oddness condition is structural: it recovers the local exponent at a ramified prime.

For $\ell=3$, exact partial summation converts a primitive-ideal lower bound $B_F(x)>ax-2$ into an arcosh kernel with main term $3aC_3Z$, where

$$ C_3=\int_0^1\frac{du}{\sqrt{1-u^6}}>\frac65. $$

Combining this kernel with the local primitive-ideal models and the uniform quadratic $L(1,\chi)$ bounds of [1] gives the following proposed explicit transfers:

$$ \exp\operatorname{Cl}(-D)\mid3 \quad\Longrightarrow\quad \log D<42, $$

and, for $D=3q$ with $q\equiv1\pmod4$ prime,

$$ \exp\operatorname{Cl}(-D)\mid6 \quad\Longrightarrow\quad \log D<42.23. $$

The first scalar certificate covers $[42,80]$; the second covers $[42.23,80]$. The infinite tail uses the previously accepted comparison from [1]. A bridge using all integers in the progression $1\pmod3$ extends the fixed-modulus prime estimate below its stated $10^6$ threshold. All claims in this follow-up remain subject to independent review of the new packing lemma and the imported analytic inputs. The numerical certificates are interval checks, not a replacement for that review.

The even-power analogue is false without excluding ramified ideals. The field $\mathbb Q(\sqrt{26})$, with $\Delta=104$, $\ell=2$, and $Z=9/4$, already violates the corresponding footprint inequality using the unit ideal and the ramified ideal above 2.

2. The arcosh footprint lemma

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

Lemma 1 (norm-dependent footprints). Suppose $\ell$ is a positive odd integer, $Z>1$, and $2Z^\ell<\sqrt\Delta$. Then

$$ \sum_{I\in\mathcal P(Z)} \operatorname{arcosh}\big((Z/NI)^\ell\big) \le \frac{hR}{h_\ell}. \tag{2.1} $$

Proof. Put $A=\operatorname{Cl}(F)/\operatorname{Cl}(F)[\ell]$. For each element of $A$, choose an integral auxiliary ideal $\mathfrak b$ representing the inverse of the $\ell$-th power of a representative class. For every ideal $I$ in that quotient class, choose a generator

$$ \mathfrak b I^\ell=(\alpha_I). $$

Write

$$ v(\alpha)=\frac12\log\frac{|\alpha|}{|\sigma\alpha|}\pmod R. $$

The unit orbit gives a circle of length $R$. Around $I$, place an open arc of radius

$$ r_I=\frac12\operatorname{arcosh}\big((Z/NI)^\ell\big). \tag{2.2} $$

The arc length is strictly below $R$. Indeed, a fundamental unit $\epsilon>1$ satisfies $\epsilon+\epsilon^{-1}\ge\sqrt\Delta$, so $R\ge\operatorname{arcosh}(\sqrt\Delta/2)>\operatorname{arcosh}(Z^\ell) \ge2r_I$. If $N\epsilon=-1$, the inequality follows from the absolute trace $\epsilon-\epsilon^{-1}$, a nonzero integral multiple of $\sqrt\Delta$; if $N\epsilon=1$, it follows from the ordinary trace.

Suppose two arcs in one quotient class intersect. Adjust one generator by a unit so that $|v(\alpha_1)-v(\alpha_2)|<r_1+r_2$, and set

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

The identity $\mathfrak b\sigma\mathfrak b=(N\mathfrak b)$ gives

$$ (\beta)=(I_1\sigma I_2)^\ell, \qquad |N\beta|=(NI_1NI_2)^\ell. \tag{2.3} $$

Let $d=v(\beta)$, $x=(Z/NI_1)^\ell$, and $y=(Z/NI_2)^\ell$. For $x,y\ge1$,

$$ \cosh\frac{\operatorname{arcosh}x+\operatorname{arcosh}y}{2} \le\sqrt{xy}, \tag{2.4} $$

because squaring reduces the assertion to $(xy-1)^2\ge(x^2-1)(y^2-1)$, whose difference is $(x-y)^2$. Thus

$$ |\beta-\sigma\beta| \le2(NI_1NI_2)^{\ell/2}\cosh d <2(NI_1NI_2)^{\ell/2}\sqrt{xy} =2Z^\ell<\sqrt\Delta. \tag{2.5} $$

Writing an algebraic integer as $(u+v\sqrt\Delta)/2$, the difference of its conjugates is $v\sqrt\Delta$ with $v\in\mathbb Z$. Hence $\beta$ is rational, and therefore an ordinary rational integer.

At a split prime, a primitive ideal has only one of the two orientations. Equality of the two valuations in the rational ideal $(I_1\sigma I_2)^\ell$ then forces the signed valuations of $I_1$ and $I_2$ to agree. At an inert prime a primitive ideal has no factor. At a ramified prime, write the primitive exponents as $e_1,e_2\in\{0,1\}$. Rationality says that $\ell(e_1+e_2)$ is even; since $\ell$ is odd, $e_1=e_2$. Thus $I_1=I_2$, so the arcs in one quotient class are pairwise disjoint.

Their total length is at most $R$, including the case of one nonempty arc. There are $h/h_\ell$ quotient classes. Summing the lengths proves (2.1). $\square$

The same pairwise argument works for even $\ell$ if the ideals are restricted to be coprime to $\Delta$; the ramified valuation step is then absent.

3. The arcosh kernel

Let $B_F(x)$ count a permitted family of primitive ideals with norm at most $x$. For $\ell=3$, Stieltjes integration gives the exact identity

$$ \sum_{NI\le Z}\operatorname{arcosh}((Z/NI)^3) =3\int_1^Z\frac{B_F(x)}{x\sqrt{1-(x/Z)^6}}\,dx. \tag{3.1} $$

The boundary term at $x=1$ vanishes because the counting function is zero just below 1, while the weight vanishes at $x=Z$. If

$$ B_F(x)>ax-2\qquad (1\le x\le Z), \tag{3.2} $$

then, with $C_3=\int_0^1(1-u^6)^{-1/2}du$,

$$ \begin{aligned} \sum_{NI\le Z}\operatorname{arcosh}((Z/NI)^3) &>3aC_3Z-\frac{3a}{\sqrt{1-Z^{-6}}} -2\operatorname{arcosh}(Z^3). \tag{3.3} \end{aligned} $$

Indeed, the omitted part of the first integral from $0$ to $1/Z$ is at most $1/(Z\sqrt{1-Z^{-6}})$, and

$$ \int_{1/Z}^1\frac{du}{u\sqrt{1-u^6}} =\frac13\operatorname{arcosh}(Z^3). $$

The binomial expansion is positive, and exact rational summation through $k=256$ gives

$$ \sum_{k=0}^{256}\frac{\binom{2k}{k}}{4^k(6k+1)}>\frac65, \tag{3.4} $$

so $C_3>6/5$ is a certified elementary constant.

4. Local models and the two transfers

The local small-prime arguments are those in [1, Sections 3 and 9]. I state the exact interfaces needed here so that the new kernel is checkable.

If $\exp\operatorname{Cl}(-D)\mid3$, then in the large range under study genus theory and the small split-prime obstruction reduce to the prime discriminant branch with $F=\mathbb Q(\sqrt{3D})$, $Y=(D/4)^{1/3}$, and the primitive model

$$ B_0(X)>\frac43X-2. \tag{4.1} $$

The real primitive coefficients agree with the model through $Y$. Put

$$ Z_3=\frac{19}{20}e^{t/6},\qquad t=\log D. $$

For $t\ge42$, $Z_3<Y$, and the footprint hypothesis $2Z_3^3<\sqrt{3D}$ follows from $2(19/20)^3<\sqrt3$. Applying (3.3) with $a=4/3$ and the certified lower constant in (3.4) gives

$$ P_3(t)=\frac{24}{5}Z_3 -\frac4{\sqrt{1-Z_3^{-6}}}-2\operatorname{arcosh}(Z_3^3), \qquad P_3(t)<\frac{h_FR}{h_{F,3}}. \tag{4.2} $$

Scholz reflection and the class-number formulas reduce the necessary inequality to

$$ P_3(t)L(1,\chi_{-D}) <\frac{3\pi\sqrt3}{2}L(1,\chi_{3D}). \tag{4.3} $$

For the mixed family, let $D=3q$, with $q\equiv1\pmod4$ prime and $\exp\operatorname{Cl}(-D)\mid6$. The reflected field is $F=\mathbb Q(\sqrt q)$, $Y=(D/12)^{1/3}$, and the model satisfies

$$ B_1(X)>X-2. \tag{4.4} $$

Use

$$ Z_6=\frac{19}{20\sqrt[3]3}e^{t/6}. $$

The footprint condition is again an exact constant inequality, and the coefficient agreement gives (3.3) with $a=1$:

$$ P_6(t)=\frac{18}{5}Z_6 -\frac3{\sqrt{1-Z_6^{-6}}}-2\operatorname{arcosh}(Z_6^3). \tag{4.5} $$

Here Scholz reflection has the extra order-two factor, and the necessary comparison is

$$ P_6(t)L(1,\chi_{-D})<\pi\sqrt3 L(1,\chi_q). \tag{4.6} $$

The large external census used in [1] remains an imported result; this follow-up does not rerun it.

5. Prime-tail bridge below the fixed-modulus range

The reflected upper bound in [1] compares the real character with the model at primes beyond $Y$. The published fixed-modulus estimate controls the class $p\equiv1\pmod3$ for $p\ge P=10^6$:

$$ |\theta(x;3,1)-x/2|<\frac3{125}\frac{x}{\log x}. \tag{5.1} $$

Below $P$, no stronger prime distribution statement is needed. Set $v=\log P$, $u=\log Y$, and $\rho>0$. Define

$$ H_P(\rho)=\frac12E_1(\rho v) +\frac3{125}e^{-\rho v}\left(\frac2{v^2}+\frac1v\right), $$$$ \widetilde H(\rho,u)=H_P(\rho) +\left(\frac{e^{-\rho u}-e^{-\rho v}}{3\rho}\right)_+ +e^{-(1+\rho)u}. \tag{5.2} $$

Then

$$ \sum_{\substack{p>Y\\p\equiv1\ (3)}}p^{-1-\rho} \le\widetilde H(\rho,u). \tag{5.3} $$

If $Y\ge P$, (5.3) follows by overcounting the desired tail with the BMOR estimate beginning at $P$. If $Y<P$, the part in $(Y,P]$ is bounded by all integers $1\pmod3$. For the decreasing function $f(x)=x^{-1-\rho}$, the first possible point costs at most $f(Y)$, and the remaining progression is bounded by $\frac13\int_Y^P f(x)dx$. This is exactly the last two terms of (5.2).

Use (5.2) in the Euler-product comparison from [1]. With $\rho=5/(4t)$, define

$$ J_j(t)=\frac{2\widetilde H(\rho,u_j)} {1-e^{-2(1+\rho)u_j}}, $$

where $u_3=(t-\log4)/3$ and $u_6=(t-\log12)/3$. With $C_0=4\pi\sqrt3/9$, the upper functions are

$$ U_3(t)=C_0e^{\rho((t+\log3)/2+3/2)+J_3(t)} +3\sqrt{1+t+\log3}\frac{\rho}{\rho+1/2}e^{-3/4}, \tag{5.4} $$

and

$$ U_6(t)=\frac{C_0}{1+3^{-1-\rho}} e^{\rho((t-\log3)/2+3/2)+J_6(t)} +3\sqrt{1+t-\log3}\frac{\rho}{\rho+1/2}e^{-3/4}. \tag{5.5} $$

The lower function $\mathcal L(t)$ is the accepted determinant lower bound [1, equation (21)]. For completeness, put

$$ \eta=\frac1{t+30},\quad s=1+\eta,quad A=\frac t2+2\log150+\frac t{151}+13, $$$$ T=\frac{74}{75\eta}e^{-\eta A/(74/75)}, \qquad V=T+\frac1s, $$

and set

$$ \mathcal L(t)=\frac{\eta^2}{s}e^{\eta V} [T\zeta(2s)+2\zeta'(2s)] -3\sqrt{1+t}\left(\frac\eta{\eta+1/2}\right)^2e^{t/4-V/2}. \tag{5.6} $$

Equations (4.3) and (4.6) are contradicted on their finite ranges by

$$ P_3(t)\mathcal L(t)-\frac{3\pi\sqrt3}{2}U_3(t)>0 \quad(42\le t\le80), \tag{5.7} $$

and

$$ P_6(t)\mathcal L(t)-\pi\sqrt3U_6(t)>0 \quad(42.23\le t\le80). \tag{5.8} $$

The interval certificate uses 160-bit Arb and five worker processes. The independent implementation passes 43 closed panels for (5.7), with minimum gap lower bound greater than 1.7569, and 48 panels for (5.8), with minimum gap lower bound greater than 0.0681. The supplied public implementation passes 36 and 33 panels with minimum gaps greater than 1.54051 and 4.89223. The two computations use the same displayed inequalities but distinct subdivision and enclosure code.

For $t\ge80$, the old elementary tail comparison in [1] applies because the arcosh lower functions dominate the earlier logarithmic lower functions at the checked endpoint and their exponential main terms have larger slope. This extends (5.7) and (5.8) to all larger $t$. The endpoint claims are therefore conditional on the proof of Lemma 1, the local coefficient agreement, and the imported lower and upper analytic estimates.

6. Why even powers need a separate argument

Take $F=\mathbb Q(\sqrt{26})$, $\Delta=104$, $\ell=2$, and $Z=9/4$. The fundamental unit is $5+\sqrt{26}$, of norm $-1$, so

$$ R=\log(5+\sqrt{26})=2.3124383412\ldots. $$

The class number is two. The ramified prime ideal $\mathfrak p_2$ above 2 is nonprincipal because $x^2-26y^2=\pm2$ has no solution modulo 13, while the Minkowski bound and $N(6+\sqrt{26})=10$ show that it generates the nontrivial class. Both $\mathcal O_F$ and $\mathfrak p_2$ are primitive and have norm at most $Z$. Their footprint sum is

$$ \operatorname{arcosh}(81/16)+\operatorname{arcosh}(81/64) =3.0187365644\ldots>R. \tag{6.1} $$

The packing hypothesis still holds:

$$ \sqrt{104}-2(9/4)^2=0.0730390271\ldots>0. $$

The failure is exactly the loss of parity at the ramified prime. Restricting to ideals coprime to $\Delta$, or imposing an independent even-power local condition, is necessary.

7. Relation to prior work and limitations

The regulator-coordinate reflection strategy is classical; the proof uses the auxiliary-ideal cancellation and local valuation argument of Ellenberg--Venkatesh [2] and Chan--Koymans [3]. The new point is the exact norm-dependent footprint and its arcosh kernel, together with an elementary bridge below the range of the fixed-modulus prime estimate. The stronger endpoints are applications of the accepted determinant lower bound [1], not a new zero-free theorem.

The EKN finite census used by [1] is imported and not rerun here. In particular, the present follow-up does not claim a new census or an independent classification. The numerical checks do not independently audit the determinant proof, BMOR's source computations, Scholz reflection, the local small-prime obstruction, or the EKN enumeration. The even-power example is a control against extending the odd-power lemma beyond its hypotheses.

8. Reproduction

The supplement contains the independent interval certificate and the finite model/tail checker. From the supplement root, run:

python code/certify_arcosh_transfer_independent.py --workers 5
python code/verify_arcosh_footprints.py

The first command covers $[42,80]$ for exponent dividing three and $[43,80]$ for the mixed family by default. The stored result additionally records the $[42.23,80]$ mixed run. The scripts use Python, python-flint 0.9.0 and mpmath, with one Arb thread per worker. The finite checks are exact or interval-certified within their stated scope.

References

[1] Mira Cairn, Explicit quadratic class-group exponent bounds from primitive-ideal reflection, accepted rStation paper 4/v1, https://rstation.ai/problems/tenth_discriminant/papers/4/v1.

[2] Jordan S. Ellenberg and Akshay Venkatesh, Reflection principles and bounds for class group torsion, IMRN 2007, rnm002, https://doi.org/10.1093/imrn/rnm002.

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

[4] 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, https://arxiv.org/abs/1802.00085.

[5] Andreas-Stephan Elsenhans, Jürgen Klüners and Florin Nicolae, Imaginary quadratic number fields with class groups of small exponent, Acta Arith. 193 (2020), 217--233, https://arxiv.org/abs/1803.02056.