Catalan’s constant is irrational

1 Introduction

Catalan’s constant is the real number

It is the simplest even value of the Dirichlet beta function, or equivalently L(2,\chi_{-4}) for the odd quadratic character of conductor 4. Its arithmetic illustrates a basic difficulty in the study of special values: rapidly converging rational approximations need not be sufficiently accurate after their denominators are cleared. We prove the following statement about this individual value.

1.1 Earlier results and the approximation problem

For odd positive integers, the beta values are rational multiples of the corresponding powers of \pi. The even values present a different arithmetic problem [19, Introduction]. Rivoal and Zudilin proved that infinitely many even beta values are irrational, and that at least one of \beta(2),\beta(4),\ldots,\beta(14) is irrational [19, Theorems 1 and 2]. Zudilin reduced this finite collection to the six values through \beta(12) [28, Theorem 1]. Lai and Zhou subsequently reduced it to the five values \beta(2),\beta(4),\beta(6),\beta(8),\beta(10) [15, Theorem 6.1]. Fischler obtained stronger quantitative results for irrationality and linear independence in families of Dirichlet L-values [9, Theorems 1 and 2]. Such family results guarantee irrational members without identifying the particular value \beta(2).

For an individual value, the basic approximation criterion requires nonzero integer linear forms A_mG-B_m that tend to zero. Convergence of B_m/A_m to G alone does not suffice: multiplication by A_m may remove the decay. Apéry’s recurrence constructions for \zeta(2) and \zeta(3) [2], and Beukers’s integral proofs of their irrationality [3] succeed because the analytic decay survives the necessary denominator clearing. For Catalan’s constant, Zudilin constructed Apéry-like recurrences, a continued fraction and double-integral representations [26, Theorems 1–3]. His discussion following Theorem 1 makes the obstruction explicit: the displayed integer linear forms do not tend to zero, despite the rapid convergence of their rational quotients.

The denominators themselves became an important part of the problem. Rivoal connected Padé approximation with the hypergeometric construction and proved its conjectured denominator bounds [18, Theorems 1 and 2]. Krattenthaler and Rivoal subsequently gave a more direct hypergeometric proof of these bounds [13, Theorems 1 and 2]. These results establish arithmetic cancellation that is invisible in a crude estimate of the individual summands. They still leave a denominator cost too large for the small-linear-form criterion. Krattenthaler and Zudilin later identified two apparently different hypergeometric constructions of the same approximants [14, Section 2 and Theorem 2]. Such identities matter arithmetically because different expressions can make different denominator factors visible.

Nesterenko developed effective approximations using half-integer hypergeometric series and double Euler integrals [17]. Viola announced joint work with Marcovecchio improving an approximation exponent through the Rhin–Viola permutation-group method [23]; their report gives the exponent 0.6293\ldots for its explicit approximants. A recent preprint of Eskandari constructs further explicit rational approximations satisfying 0<|G-p_m/q_m|\le q_m^{-0.62} for all sufficiently large m [8, Theorem 1.1]. These are bounds for particular constructions, not irrationality measures. An error bound with exponent below 1 does not by itself make the integer forms q_mG-p_m tend to zero.

Results at other characters and other places give useful comparisons. Calegari, Dimitrov and Tang proved the \mathbb Q-linear independence of 1, \pi^2 and L(2,\chi_{-3}) [5, Theorem A]. Their character has conductor 3, rather than the conductor 4 of G. Their discussion of two Catalan approximation families again identifies the denominator cost as an obstruction for those constructions [5, Remark 11.1.17]. Calegari proved irrationality of a 2-adic analogue G_2\in\mathbb Q_2 [4, Theorem 4.2]. As explained by Calegari, Dimitrov and Tang [6, Section 5.2, footnote 2], the related real identity contains an additional \pi^2 term that is absent in the 2-adic identity. This illustrates why the 2-adic result does not settle the real problem. In our construction, cancellation of an additional \zeta(2) term is likewise essential, although the moment identities and the cancellation argument are proved directly below.

Sun [22, Theorem 1.1] has also announced a proof of the same qualitative irrationality statement. No result from that preprint is used here.

1.2 The determinant strategy

We construct determinants \Delta_N of size n=48N whose entries combine moments of two elementary kernels. Each moment is a rational linear combination of 1, G and \zeta(2). The polynomial rows are chosen to have high Taylor contact: two prescribed expressions in the rows have identical initial Taylor coefficients. This agreement cancels the \zeta(2) term in every entry. Consequently, the single hypothesis G\in\mathbb Q makes all the determinants rational.

For a nonzero rational number, its ordinary absolute value is determined by its prime valuations. Bounds on the denominators of \Delta_N therefore give a lower bound on \log|\Delta_N|. An integral formula for the same determinant gives an upper bound. The contradiction comes from making these bounds incompatible as the size grows. Zudilin’s determinantal criterion gives a useful precedent for this comparison [27, Proposition 2 and Section 2]. There, a positive moment representation produces Hankel determinants with squared-Vandermonde integrals. Positivity supplies nonvanishing, and the denominator and integral estimates are compared on the scale of the square of the matrix size. His discussion of Catalan’s constant explains why the denominator growth of the approximation family considered there still prevents application of the criterion [27, Section 6]. Our determinant is mixed and signed; its nonvanishing and its real-place estimate require separate arguments.

Three features of the construction make this comparison possible. First, integral Chebyshev rows provide both the Taylor contact and useful divisibility. At the large odd primes relevant to the leading bound, denominators of order p^2 and p must both be controlled. The arithmetic estimate follows these two layers separately; cancellation at the first layer alone would not give the required denominator bound.

Second, nonvanishing is arithmetic. Under the hypothesis G\in\mathbb Q, when N=p is itself a sufficiently large prime, Frobenius and a pair of polynomial bases reduce the growing matrix to three fixed rational matrices. An exact finite certificate proves their nonvanishing. The reduction then gives nonzero determinants along an unbounded prime sequence, without asserting positivity of the mixed determinant.

Third, the real estimate respects both branches of the rational parametrization used to construct the rows. A bounded holomorphic interpolant controls their mixed evaluations in one range of configurations; a Hadamard bound treats the complementary range. The interpolation estimate is uniform even when nodes approach one another. Its proof uses a finite-dimensional Hardy-space operator, so the evaluation determinant cancels algebraically rather than introducing an inverse-Vandermonde estimate. Andréief’s integration identity [10], Cauchy’s double alternant [12], and the kernel/compression viewpoint of analytic interpolation [21] supply the classical context; the specific estimates are proved below.

The resulting Vandermonde products have logarithmic interactions, as in logarithmic potential theory [20]. A Chebyshev expansion of the logarithmic kernel [11, Lemma 3.1] reduces the estimate to quadratic sums and two one-variable functions in each case. We regularize their interactions together, control the omitted diagonals and endpoint terms, and only then take a supremum and pass to the limit. Explicit rational trial coefficients finish the argument. Their certificate exhausts all stationary points and bounds entire root brackets, rather than relying on a numerical search for the maxima.

For orientation, the two estimates are stated for the same normalized logarithm

Under the rationality hypothesis, Proposition 3.4 gives \liminf\mathcal L_N>-2.29084 along every sequence of nonzero determinants, and Proposition 4.1 supplies such a sequence with N prime. Independently, Proposition 7.1 gives \limsup\mathcal L_N\le-2.290939875<-2.2909. These inequalities are incompatible. The argument proves qualitative irrationality; a bound for an irrationality measure would require additional control of rational approximations. Section 8 also deduces irrationality of the minimum volume of an orientable complete finite-volume hyperbolic three-manifold with exactly two cusps, and of certain arithmetic hyperbolic volumes.

1.3 Organization and conventions

Section 2 constructs the rows and moments, proves cancellation and rationality, and gives the estimate at the prime 2. Section 3 treats odd primes and derives the finite-place lower bound. Its only prime-distribution input is the ordinary prime number theorem; the weighted consequence needed here is proved locally. Section 4 establishes nonvanishing at prime scales. Sections 5 and 6 give the uniform real-place and energy estimates. Section 7 supplies the rational data certifying the real-place bound, and Section 8 assembles the contradiction.

All logarithms are natural. We use v_p(p)=1 and v_p(0)=+\infty; for real estimates we put \log 0=-\infty. Constants in asymptotic estimates may depend on the fixed rational value hypothetically assigned to G, but never on N or on the integration points. The finite certificates are specified by rational data and arithmetic instructions in the text; supplementary programs reproduce them.

2 Polynomial rows and mixed moments

We construct the determinant and prove that its entries are rational under G\in\mathbb Q. The row design serves two purposes: Taylor contact cancels \zeta(2), while integral Chebyshev coefficients permit the finite-prime estimates. After the moment calculations, we establish the estimate at 2; odd primes are treated in the next Section.

For each positive integer N, set

Write f(t)=\sqrt{1-t^2}, with the positive branch on (-1,1), and w=t/(1+f). Then

The involution denoted by a star sends w to w^{-1}, fixes t, and sends f to -f. For 0\le r<n, define

Here and below identities involving f near zero use its Taylor branch with f(0)=1.

Let T_d,U_d be the Chebyshev polynomials, normalized by T_0=1,T_1=x,U_{-1}=0,U_0=1, and the recurrence S_{d+1}=2xS_d-S_{d-1}. The associated Laurent expressions are the standard formulas [16, Equations (18.5.1)–(18.5.2)]. Since (w+w^{-1})/2=1/t and (w^{-1}-w)/2=f/t, Equation (3) becomes, with d=|r-g|,

In particular these are integer polynomials; the second is zero when r=g. The inequality d\le n-1-g gives

Moreover w=t/2+O(t^3), so that

For nonnegative integers i,j, define

Extend these expressions bilinearly to polynomial arguments. The integrals converge absolutely. Indeed, for 0<u<1,

and (-\log(1-u))/\sqrt{1-u^2} is integrable on (0,1). This dominates the absolute M integrand after integration in s; the corresponding assertion for Z is weaker. Bounded polynomial factors preserve this domination. It also justifies termwise integration of the geometric series used below.

For each raw column index b\le j<L, let

The determinant used throughout the proof is

Thus, if \mathcal F is the n by n+q matrix with columns F_j, the matrix in Equation (7) is \mathcal F\mathcal T, where the integer matrix \mathcal T has entries

A binomial coefficient outside its usual range is zero. In particular, all raw columns required by the filter lie in the contact range j<L.

2.1 Moment evaluation and cancellation

Set

and use the convention m_{-1}=0. The moment formula follows from symmetry, m_0=2, and integration by parts, which gives (i+1)m_i=i m_{i-2} for i\ge1. Throughout, c_z means zero unless z is a nonnegative integer. Termwise integration gives

Put K^-_d=M(d,0) and K^+_d=M(0,d). Their boundary recurrences are

For the first recurrence, the moment recurrence gives the termwise identity

The right side telescopes, since m_v\to0 as v\to\infty. The same identity for d=1, with the term multiplied by d-1 omitted, gives K^-_1=m_{-1}+m_0=2. For the last recurrence, shifting the first series by two leaves, for u\ge2, the terms

Their sum is 2; the remaining boundary term is -2(d-2)/(d-1), giving the claimed result.

The two independent starting values are

To verify the first, integrate in s and put t=\sin\theta on the positive half interval. This gives

The last equality follows on setting x=\tan v and integrating the geometric series for (1+x^2)^{-1} against -\log x. For the other start, Equation (10) and Equation (9) yield

The differential equation (1-x^2)y''-xy'=2, with y(0)=y'(0)=0, determines the Taylor series of y=(\arcsin x)^2 as \sum_{k\ge1}x^{2k}/(2k^2c_k). Its nonnegative coefficients allow passage to x=1 by monotone convergence, giving \pi^2/4. For completeness, integrating

over 0<x<\pi and letting \rho\uparrow1 gives 2\sum_{k\text{ odd}}k^{-2}=\pi^2/4. Here the imaginary part is uniformly bounded and tends to (\pi-x)/2, so dominated convergence applies. The odd sum is 3\zeta(2)/4, establishing the final equality in Equation (12).

Define M^0(i,j) by the rational recurrences (10)–(11), replacing only the starts K^-_0=K^+_0 and K^+_1 by zero. More explicitly, let k^-_d,k^+_d satisfy Equation (11) with

Then the following formulas specify the whole array without ambiguity:

Empty sums are zero. We shall also use the algebraic convention

this does not define an additional integral. Let B_i^{(d)}=\sum_{k=1}^i k^{-d}, with B_0^{(d)}=0, and put

The evaluations of the moments are

Indeed, the homogeneous parts of the minus and plus boundary recurrences propagate the starts by precisely these two c-kernels. Diagonal reduction preserves both index differences. The formula for Z follows from

by partial fractions when i\ne j, and directly when i=j.

For later valuation calculations, it is useful to solve the rational boundary recurrences explicitly. Define

Then

To see this, divide each recurrence by the appropriate H^* and use the ratio in Equation (18). For the minus recurrence the inhomogeneous increment after division is 2/(z^2(H_z^*)^2) when z is even and 2/z when z is odd. The odd case starts with k^-_1/H_1^*=2. For the plus recurrence the increment is 2/(z^2H_z^*), with initial values k^+_1=0 and k^+_2=2. This proves all the formulas, including their empty-sum cases.

2.2 A uniform estimate at the prime two

We normalize v_p(p)=1 and set v_p(0)=+\infty. In the rest of this section assume G\in\mathbb Q, viewed also in \mathbb Q_2. Constants allowed to depend on this fixed rational number will carry a subscript G.

3 Odd primes and the finite-place lower bound

Throughout this section assume that G\in\mathbb Q, so that the columns F_j of Section 2 are rational. We regard each F_j as a column in \mathbb Q^n. For an odd prime p, reduction modulo p always means reduction of a member of \mathbb Z_p; in particular, every congruence below includes the assertion that its two sides are locally integral.

The raw matrix has n+q columns, whereas the filtered determinant has size n. At the large primes treated first, each raw column has at most two powers of p in its denominator. We shall make invertible column changes over \mathbb Z_p and bound how many columns can still have denominator p^2 or p. Such bounds control every full minor of the raw matrix, and Cauchy–Binet then transfers them to the integer column filter. The first reduction identifies residues after multiplication by p^2; for p>H/2 a second reduction identifies the remaining residues after multiplication by p.

3.1 Digit reduction of the rational moments

We first prove the reductions needed at primes

We may suppose that p does not divide the denominator of G: as N tends to infinity, every fixed denominator prime eventually lies below 2\sqrt H. Put

We set E_d=0 outside 0\le d<p. Within this range,

The first identity follows from (-1)^k\binom{(p-1)/2}{k}\equiv4^{-k}\binom{2k}{k}; the second follows by reversing the coefficients of E.

3.2 The leading layer and paired raw columns

We now gather the moment reductions into column vectors. This will identify pairs of raw columns whose sum has at most one power of p in its denominator. For a fixed residue 0\le\ell<p, define column vectors over \mathbb F_p by

Here P,D denote the vectors of all row polynomials; the dependence of these extracted vectors on \ell is understood. Their supports are finite. Lemma 3.1, gathered by coefficient residue, gives

For a term [t^i]P, its unique residue d satisfies i+1+d=(i'+1)p+\ell, so that it contributes exactly to P'_{i'}, including i'=-1. The Z^0 reduction survives only when i=up+\ell, which is precisely the extraction defining D'_u. The coefficient of G in F_j is a sum of integral multiples of c_{(i-j)/2}. These coefficients are integral at every odd prime, so this term disappears after multiplication by p^2. Every raw column therefore lies in p^{-2}\mathbb Z_p^n.

If \ell\ge H-2p, the degree bounds \deg(tPE)\le H+p-1 and \deg D<H show that P'_u,D'_u vanish for u>1. Put

The needed small values of the reduced arrays are

Consequently, whenever the indicated columns exist,

Since tP and D have no terms below degree A+1, U_\ell=0 for \ell\le A.

We record explicitly how changes in the full column pool will be used. Let T be the n by (n+q) matrix of all raw columns. If Q\in\operatorname{GL}_{n+q}(\mathbb Z_p) and every full minor of TQ has valuation at least -D, the same holds for every full minor of T=(TQ)Q^{-1} by Cauchy–Binet. The coefficients in this expansion are minors of the integral matrix Q^{-1}. Applying Cauchy–Binet once more to the integer matrix defining the prescribed filter gives v_p(\Delta_N)\ge-D. This transfer applies even when the columns or their leading residues are linearly dependent.

For 2\sqrt H<p\le H/2, use each pair of columns with

Their number is exactly

Replacing its high column by the sum of the pair is an integral elementary column operation with integral inverse. The pair sum lies in p^{-1}\mathbb Z_p^n by Equation (37); all other columns still lie in p^{-2}\mathbb Z_p^n. These pairs are disjoint, since \ell<p. Any selection of n columns from the pool of n+q contains at least (d_0-q)_+ of the improved columns. The transfer just explained gives

3.3 The central layer and integral column elimination

For p>H/2, the next lemma identifies the residues of central columns after multiplication by p. It expresses them in terms of the same vectors V_\ell that occur in the leading residues of noncentral columns. The elimination proof will retain noncentral columns and use p times those columns to cancel the corresponding V_\ell terms.

We now carry out these cancellations in the full column pool. In the resulting pool, R will count the retained columns with possible denominator p^2, and S will bound the number of other columns with possible denominator p; every remaining column will be integral. The proof first uses the V_\ell supplied by retained noncentral columns to modify the central columns. We then use the rank of their remaining scaled residues to bound the number of central columns that can still have denominator p after an invertible column change.

The omission of the optional pair at \ell=A is harmless: its physical column is included in R. The preceding proof also covers K\le0 and J\le b. In particular, if L\le p\le H, there are no high or low noncentral columns, R=0, and S=H-p; all raw columns are central. At the formal endpoint p=H, all of them are integral.

3.4 Summing the prime losses

For a nonzero rational determinant, the valuations weighted by \log p sum to its ordinary logarithmic absolute value. We now combine the odd-prime estimates with the bound at 2 to obtain the lower bound used in the final comparison.

4 Nonvanishing along the prime sequence

The next proposition supplies the nonzero determinants needed in Proposition 3.4.

We prove the proposition by reducing a matrix of size 48p to three fixed rational matrices of size 48. Throughout this section, an underlined row polynomial is formed with N=1, so that

In addition to the usual rows 0,\ldots,47, define the auxiliary row 48 by exactly the formulas in Equation (3). For 0\le r\le48 and 0\le k<48, put

Here M^0 and Z^0 act linearly on the first polynomial argument. Thus \mathcal B is a fixed rational 49 by 48 matrix. Define

The arithmetic certificate below proves

First we show why these three fixed assertions imply the proposition.

4.1 Two palindromic bases

Let p be an odd prime and work over \mathbb F_p. The vector space

has dimension p: its coefficients of w^0,\ldots,w^{p-1} determine all its coefficients. For 0\le\ell,i<p, consider

Both families belong to \mathcal P_p. The lowest terms of U_\ell and Q_i are respectively 2^\ell w^\ell and w^i. Triangularity of their coefficients in degrees 0,\ldots,p-1 proves that each family is a basis. Define a_{\ell i}\in\mathbb F_p by

The same lowest-term comparison gives

In particular, invertibility of this varying-size matrix introduces no exceptional odd primes.

Define Q'_0=0 and Q'_i=Q_{p-i} for 1\le i<p, and write Q'_i=\sum_\ell b_{\ell i}U_\ell. Equivalently,

where e_i are the standard coordinate vectors. The explicit forms are

the latter formula gives zero also when i=0. Their sum satisfies

because \binom{p-1}{j}=(-1)^j in \mathbb F_p. With

division by (1+w^2)^{p-1} therefore proves the rational-function identity

4.2 Frobenius and the two extraction shifts

Set N=p, and write every row index uniquely as r=pr_0+i, with 0\le r_0<48 and 0\le i<p. Put t'=t^p, w'=w^p, and f'=f^p. Frobenius gives

Using C=63p, h=2p, and g=4p in the row definition gives

Here the single factor w' in the second term creates exactly one successor row; its index is at most 48.

To separate the polynomial parts, star fixes t,t' and negates f,f'. Moreover fE=f', so the row identities give

Taking the star-invariant part of Equation (52) and then its anti-invariant part, and cancelling the nonzero rational function f' in the latter, yields two separate polynomial identities:

These are polynomial identities since the row polynomials on both sides are polynomials, and substitution t=2w/(1+w^2) is injective on \mathbb F_p[t].

In the notation of Equation (36), the extractions for a fixed residue \ell are

Every polynomial has a unique expression \sum_{\ell=0}^{p-1}t^\ell A_\ell(t^p). Consequently Equations (53) and (54) give

Thus the t' in the first identity is exactly consumed by the u+1 in the P extraction. The D extraction has no such shift. Negative coefficient indices are zero; in particular P'_{-1}=0 here.

There is a degree issue at the last successor row which is useful to make explicit. The auxiliary polynomials have

These coefficients must be retained, even though the first 48 base rows have minimum degrees at least 19 and 20. For the last block r=47p+i, i>0, Equation (50) gives b_{\ell i}=a_{\ell,p-i}=0 for \ell<p-i, and b_{p-i,i}=2^{-(p-i)}\ne0. The successor contribution in either tP_rE or D_r therefore starts at

The first-row contribution starts no earlier than 20p+i. Directly from the original rows, |r-g|=43p+i, so tP_r and D_r also start at C-|r-g|=20p-i; since E(0)=1, this agrees with the extraction. The leading Chebyshev coefficients are powers of 2, hence are nonzero in odd characteristic. When i=0, the successor term is zero and the first term starts at 20p. The lower degree of the auxiliary row therefore introduces no forbidden coefficient into the growing matrix. Its contact identity is also valid: t\underline R_{48}/f=O(t^{107}), since 63+48-4=107>59.

4.3 The residue blocks and their determinant

For this paragraph take p>260 and exclude primes dividing the denominator of the hypothesized rational number G. Then

so Equations (31) and (36) apply at the same prime p used as the scale parameter. To spell out the role of G, the rational raw entry is

The last sum is p-integral: its polynomial coefficients are integers, and every c_d=4^{-d}\binom{2d}{d} is integral at an odd prime. The G term therefore reduces to zero after multiplication by p^2. All entries of the scaled matrix are p-integral by the cited layer formulas. Substitution of Equation (55) into Equation (36) gives the reduced raw entry

This applies to every raw column in the specified range. The base entries here have denominators with prime factors at most 65, as the rational recipes below also show, so their reductions exist for p>260.

The integer filter also respects residues. Write the column index uniquely as k=\ell+pk_0, where 0\le\ell<p and 0\le k_0<48. In \mathbb F_p[s],

Multiplying each raw entry by p^2 before reduction makes this polynomial congruence applicable to the column operation: coefficient differences divisible by p multiply integral scaled entries. The fourth finite difference on the right uses exactly the five base raw indices 7+k_0,\ldots,11+k_0, all in 7,\ldots,58. Each of the p residue groups thus contains exactly 48 base filtered columns, and no operation changes \ell.

Order rows by (i,r_0) and columns by (\ell,k_0). The reduction of the scaled 48p by 48p matrix has (i,\ell) block

Writing this matrix as \mathcal L_p makes its orientation explicit:

Indeed \mathbf b^T=\Pi^T\mathbf a^T, which explains both the transpose and the side of multiplication.

The nonzero indices 1,\ldots,p-1 fall into (p-1)/2 disjoint pairs \{i,p-i\}. On each pair the vectors e_i+e_{p-i} and e_i-e_{p-i} are eigenvectors of \Pi with eigenvalues 1 and -1. Together with e_0, of eigenvalue zero, they form a basis since 2 is invertible. Thus the eigenvalue multiplicities of \Pi^T are 1,(p-1)/2,(p-1)/2 for 0,1,-1, respectively. A similarity on the p-dimensional factor in Equation (59) now proves the exact identity in this block ordering:

There is no determinant factor from the similarity. Returning to the original row and column orders changes at most the sign.

4.4 An exact certificate for the fixed matrices

All entries in Equation (47) can be constructed by the following rational arithmetic. This also specifies their reduction modulo 101 without any numerical approximation. First form m_0=2, m_i=0 for odd i, and m_i=i m_{i-2}/(i+1) for even 2\le i\le64. Set

and use

On 0\le i\le64, 0\le j\le58, form

Thus Y_{ij}=M^0(i,j). With B_i^{(d)}=\sum_{u=1}^i u^{-d} and B_0^{(d)}=0, set

This gives J_{ij}=\tfrac32Z^0(i,j), with the sign on its diagonal included.

For the polynomial construction define

and, for either family S=E,I, use

These are ordinary integer polynomials: they are E_m=t^{62}T_m(1/t) and I_m=t^{62}U_{m-1}(1/t). For 0\le r\le48 let u=|r-4| and form the coefficient vectors, in degrees 0,\ldots,64, of

They are exactly \underline P_r and \underline D_r. Contract them with the arrays to obtain the length-52 vector

Replace a vector v four times successively by its vector of consecutive differences (v(j)-v(j+1))_j. The resulting length-48 vector is row r of \mathcal B. In particular, the construction includes all coefficients of the auxiliary row, including its degree-18 term.

Every scalar denominator in these recipes is a product of nonzero integers of absolute value at most 65. Every rational array entry therefore has denominator prime factors at most 65, so each denominator is a unit modulo 101. Polynomial division by t above shifts exponents of polynomials divisible by t and introduces no scalar denominator. Consequently the entire construction can be performed over \mathbb F_{101} and agrees with reduction of the rational matrices.

For completeness, the elimination rule producing the certificate is as follows. For \sigma\in\{0,1,-1\}, start with row vectors R_i=\mathcal B(i,{\cdot})+\sigma\mathcal B(i+1,{\cdot}), 0\le i<48, in increasing order. At step i, if R_i(i)=0, swap row i with the first later row having a nonzero entry in column i. Record d_i=R_i(i) and replace every row j>i by

There are no swaps for \sigma=0,-1. For \sigma=1 the only swaps, using indices starting at zero, are (30,31) and (45,46) at their respective steps. The pivots are given in Table 1; its three lines for each \sigma list the pivots consecutively, with 16 entries per line.

Every listed pivot is nonzero. Since the swaps and row additions are invertible, all three matrices are invertible modulo 101 and hence have nonzero determinants over \mathbb Q. This proves Equation (48). The modulus 101 is used only for this fixed finite certificate; the growing determinant uses arbitrary sufficiently large primes as follows.

5 The real-place determinant estimates

We estimate the determinant defined in Equation (7) without a rationality assumption. Write

In particular, 0<D_*<n and n\ge48. For a real list y of length n, let

All constants in the estimates of this Section are independent of the locations and separations of the integration variables.

5.1 The integral and its two sheets

Put d\mu(t)=|t|\,dt/f(t) on (-1,1), and set

The value at t=0 is immaterial; the displayed expression has a finite limit there because D_r(t)/t is a polynomial. The determinant entries are exactly

The measure has mass \mu((-1,1))=2. The scalar majorant \int\!\int(1-|t|s)^{-1}\,ds\,d\mu(t) is finite: integration in s gives at worst a logarithmic singularity at |t|=1, which is integrable against (1-t^2)^{-1/2}dt. The functions A_r and \psi_k are bounded for each fixed n. Thus all permutation expansions below are absolutely integrable.

We apply Andréief’s determinant integration identity twice [10, Equation (1.7) and Section 2.2]. Expanding determinants and relabeling integration variables gives the identity

Here 0\le r,k<n and 1\le i,j\le n. To see the factor 1/n! at each application directly, expand the two determinants sharing an integration list: every one of the n! permutations of that list gives the same determinant of single integrals. Fubini’s theorem applies by the preceding majorant, also after each permutation expansion.

Cauchy’s double alternant in multiplicative variables [12, Section 2.1, Equation (2.7)] is

For completeness, multiplying by the denominator gives a polynomial alternating separately in t and s, of degree at most n-1 in each individual variable. Dividing by V(t)V(s) therefore leaves a constant. Expanding (1-t_i s_j)^{-1} at s=0, the first nonzero homogeneous part of the determinant is V(t)V(s), so that constant is one. Also \det[\psi_k(s_j)]_{j,k}=V(s)\prod_j s_j^b(1-s_j)^q.

Use the real coordinate

The coordinate maps (-1,1) bijectively to (-1,1). Except on a set of measure zero we may assume that all x_i are distinct and nonzero. Define

The subscripts \mathrm n and \mathrm f refer to the near and far evaluations at x_i and 1/x_i, respectively, inside and outside the unit circle. Indeed, fD_r/t=(R_r^*-R_r)/2, so the negative half-interval has the row (R_r+R_r^*)/2, whereas the positive half-interval has (5R_r-R_r^*)/4. Factoring (1-t_i)^ht_i^{C-1} out of the ith column therefore gives the exact identity

In particular, there is one t Vandermonde and two s Vandermondes. All the denominators in this formula are positive.

5.2 A uniform interpolating function

The interpolation problem comes from factoring out the far-sheet weight. Put c_i=\xi_{\mathrm n}(x_i)/\xi_{\mathrm f}(x_i); the denominator is nonzero on both half-intervals. For 0\le r<n,

Thus a holomorphic function with values h_*(x_i)=c_i x_i^{D_*} turns the expression in parentheses into x_i^{n-1-r}+h_*(x_i)x_i^r. We shall bound the resulting evaluation determinant by controlling the norm of this function. The first estimate applies when

The following construction is uniform even when the nodes cluster.

5.3 Evaluation in a finite-dimensional Hilbert space

The following argument uses the kernel and compression viewpoint of bounded analytic interpolation; see [21]. We prove the needed finite-dimensional facts directly.

5.4 The two principal integrands

To state precisely the quantities needed for the real-place energy estimate, put t_i=2x_i/(1+x_i^2) and

Define

Let \Omega_{2,n} be the set of interior configurations with distinct nonzero x_i satisfying Equation (67), and let \Omega_{1,n} be its complementary case among such configurations. We use the interpolation estimate on \Omega_{2,n} and the Hadamard estimate on \Omega_{1,n}; the latter estimate is valid in both cases. After substituting t=2x/(1+x^2), \kappa is the exponent of the absolute x-Vandermonde in the corresponding majorant: \kappa=2 for the interpolation estimate and \kappa=1 for the Hadamard estimate. The s_j always range over (0,1).

The preceding propositions and Equation (65) give

Indeed d\mu^n(t)\,ds has total mass 2^n, and the two case sets partition its domain up to a null set. In logarithmic form, with \log0=-\infty, this yields

The error is independent of both integration lists. Factorials, measure masses, and sheet sums have all been accounted for explicitly.

Repeated nonzero x_i give a zero original row determinant and are also handled by continuity in the first estimate; repeated s_j give a zero s Vandermonde. The sets with a zero node or a boundary node have measure zero for the absolutely continuous measures above. At such points the original expression in Equation (62), rather than its factored Laurent expression, supplies the integrable definition. Approaches to these exceptional sets require no separation condition in any bound proved here. In particular all configurations arbitrarily close to them remain included in the suprema in Equation (75).

6 A uniform energy bound

The purpose of this Section is to replace the two many-variable majorants by one-variable suprema and explicit quadratic terms. The bound must be uniform before taking those suprema. Matched damping and retained endpoint corrections provide this uniformity.

We retain the constants \alpha,\beta,\gamma,\eta, the two cases \kappa=2,1, and the nonnegative majorants \mathcal I_{\kappa,n} of Equation (74). Put

Here D(x) is a scalar function, distinct from the row polynomials D_r(t). For a real sequence u=(u_k)_{k\ge1} with |u_k|\le K r^k for some K<\infty and 0<r<1, define

These series converge uniformly for -1\le x\le1.

The sequences p,v below are freely chosen trial coefficients. The inequality -a^2\le b^2-2ab lets them replace two negative quadratic sums in the logarithmic interactions by affine upper bounds, leaving the two one-variable suprema. Each admissible choice gives a valid bound; the certificate will supply one successful choice for each case.

7 An exact certificate for the two barriers

All finite decimals in this Section denote exact rational numbers. We specify trial sequences for Equation (78), isolate every stationary point of its two one-variable functions, and bound their values using rational arithmetic.

7.1 Exact trial sequences

For \kappa=2 take d=10 and \lambda_2=0. For \kappa=1 take d=8 and \lambda_1=2.47405979. Write each sequence as

The following tables give all coefficients. Every integer coefficient in them is to be multiplied by 10^{-8}. The finite parts are listed in increasing order of k:

Case \kappa=1 has no exponential terms. The exponential terms for \kappa=2 are as follows. For a real base z the displayed a is the coefficient of z^k. For a nonreal base the two entries a,b are the coefficients of \Re(z^k),\Im(z^k), in that order. In Equation (89) such a pair is represented by r_z=(a-ib)/2 and r_{\bar z}=\overline{r_z}, with the factor 10^{-8} applied to a,b. Each real base appears once and each nonreal base together with its conjugate.

In particular these are real exponentially decaying sequences. There are ten p tail terms and thirteen v tail terms when conjugates are counted, and their base moduli are at most .94 and .984, respectively. All the |l_k| and |r_z| are less than 1.

The convergent power series for the logarithm give the finite formulas

Here \operatorname{Log} is the principal complex logarithm; all sums are real by conjugation. There is no branch ambiguity in these formulas. For the norm and power terms the log arguments have positive real part. For the cosine term, writing x=\cos\theta gives

Both factors have positive real part, and their principal arguments sum strictly inside (-\pi,\pi). The sum of their logarithms is therefore exactly the principal logarithm of the product, including when the product has negative real part.

For these choices put

The domain of X_\kappa is [-1,1)\setminus\{0\}, and that of Y_\kappa is (0,1).

7.2 Derivative numerators and exhaustive root brackets

Define the rational functions

Differentiation of Equation (90) yields

For a completely specified rational numerator, use

Empty products equal 1. These equations, the coefficient tables, and T_0=1,T_1=x, U_0=1,U_1=2x, with recurrence V_{k+1}=2xV_k-V_{k-1}, specify the four polynomials over \mathbb Q without any numerical root calculation. The denominators do not vanish on the indicated open domains: |1-xz|\ge1-|z|>0, and |1-2xz+z^2|\ge(1-|z|)^2>0 for real |x|\le1. Conjugate factors in Q_X,Q_Y have positive products, and real-base factors are positive. Hence Q_X has the sign of x on its domain and Q_Y>0 on (0,1).

Here is an exact root-count certificate. If A(x)=\sum_{j=0}^{d_0}A_jx^j is the specified numerator, on a subinterval (b,c) form

where out-of-range binomial coefficients are zero. Delete zero coefficients and count consecutive sign changes. The degrees and the resulting counts, in the order of the consecutive intervals, are

The sign-variation bound is Descartes’ rule of signs [7, Book III, p. 373]. For completeness, the required bound follows by factoring out positive real roots: multiplication of a real polynomial by t-r, r>0, increases the number of variations by at least one. To see this, positive rescaling of the variable reduces to r=1, and zero initial coefficients can be removed. If a_0,\ldots,a_d are the old coefficients and b_0,\ldots,b_{d+1} the new ones, then

Each sign change between nonzero partial sums forces an intervening b_j with the sign of the later partial sum. Starting with b_0=-a_0, these choices give an ordered subsequence with all the variations of the a_h; the final coefficient b_{d+1} has the opposite sign to the last partial sum and supplies one further variation. Factoring successively therefore proves that the variation count bounds the number of positive roots with multiplicity. The substitution in Equation (94) sends t>0 bijectively to (b,c) and does not change multiplicities.

The following integers m specify disjoint open brackets (m,m+2)/10^{10} for roots of the corresponding derivative:

For each integer in this table the exact sign check is

Equations (93), (94), and (95) give rational addition and multiplication recipes for every entry of the root certificate. The numbers of brackets in each consecutive interval equal the displayed variation counts. The intermediate value theorem and the variation bound therefore give exactly one simple root per bracket and no other roots in those open intervals. Possible roots at the division points are harmless because all finite division points are evaluated separately below.

7.3 Rational logarithms and all candidate values

Here is the rational procedure used for the value bounds. For a positive rational s, write s=2^m y with 1\le y\le2, and put

The substitution for \log s is mH(2)+H(y). If q=(y-1)/(y+1), its unscaled positive remainder is bounded by

For a nonzero rational complex number a+ib, its real log part is half the real log of a^2+b^2. For its principal argument choose an octant: rotate by -k\pi/4, -4\le k\le4, so the rotated real part is positive and the imaginary-to-real ratio t satisfies |t|\le1/2. Select the rotation for which k\pi/4+\arctan t is the principal argument; at the negative real axis use the value \pi. The ratio t is rational, since the rotation can be performed, up to a positive scale, by repeated maps (a,b)\mapsto(a+b,b-a) or (a-b,a+b). With

substitute k(J(1/2)+J(1/3))+J(t) for the argument. The identity

follows from the tangent addition formula and the fact that both summands are positive and their sum is less than \pi/2. The alternating-series estimate gives |\arctan t-J(t)|\le |t|^{49}/49.

All positive inputs used here, including squared moduli, lie between 2^{-100} and 2^{100}. Indeed, the real arguments at the evaluation points are between .0007 and 2, and the squared moduli of all complex arguments lie between (.016)^4 and (1+.984)^4, by the factor bounds following Equation (93). Thus at most 100 scaling steps are needed. To make the error allowance explicit, set

A real log substitution costs at most 101h_0, and an argument costs at most 9j_0. In particular 2\cdot101h_0+9j_0<7\cdot10^{-16}, so 10^{-15} is an upper bound for the absolute error of a complex log. This also proves the coarser allowance 10^{-11} per log.

All rational polynomial terms in Equation (90) are evaluated exactly. The total absolute log coefficient mass in \kappa\|p\|_*^2+\|v\|_*^2/2 is at most 2\cdot10^2+13^2/2=284.5, in X_\kappa it is less than 2\cdot10+13+2=35, and in Y_\kappa it is less than 13+1=14. Their resulting substitution errors are respectively less than 2.85\cdot10^{-13}, 3.5\cdot10^{-14}, and 1.4\cdot10^{-14}, all less than 10^{-8}. One may enclose each rational series term between consecutive multiples of 10^{-40}, add the displayed remainder bounds with outward signs, and then propagate intervals linearly. This additional rounding changes the preceding allowances by less than 10^{-32}. For a complex weight c, use \Re(c\operatorname{Log}z)=\Re(c)\Re(\operatorname{Log}z) -\Im(c)\Im(\operatorname{Log}z), retaining the sign of each weight in interval multiplication. This fully specifies a rational interval calculation; no numerical logarithms are needed.

For transparency, Table 2 gives all fifty point-value upper bounds. Its argument column is 10^{10}x. A row marked B is the left endpoint of the corresponding root bracket above; a row marked P is a finite division point. Each displayed bound is a rational upper bound rounded upwards to twelve decimal places after the series remainder has been included. The largest unrounded interval width in this table is less than 2.16146856249281\cdot10^{-16}.

The same calculation gives the following upper bounds for \kappa\|p\|_*^2+\tfrac12\|v\|_*^2:

In particular the rational substituted expressions, before their error allowances are added, satisfy the following convenient stronger cutoffs:

Each entry means a strict upper bound. With the 10^{-8} allowance, the weaker cutoffs we shall actually use are

The logarithm procedure also gives .693146<\log2<.693149.

7.4 From bracket values to global suprema

A single derivative bound suffices on every whole closed bracket in the root table. All its points have distance greater than .0007 from both 0 and 1. In particular the bracket nearest s=1 has right endpoint 9992037198/10^{10}, whose distance from 1 is exactly .0007962802. For |x|\le1, the identity U_{k-1}(\cos\theta)=\sin(k\theta)/\sin\theta gives |U_{k-1}(x)|\le k, including its endpoint limits. Using the coefficient and base bounds above, the polynomial part of X_\kappa' is at most 4\sum_{k=1}^{10}k+10=230 in absolute value; its endpoint terms are bounded by (19/48+1/12)/.0007<685; its x/(1+x^2) term is less than 3; and its multiplier term is less than 4\lambda_1<10. Its two log-tail derivatives are bounded by

Consequently |X_\kappa'|<12852<120000 throughout every listed bracket. For Y_\kappa' the polynomial part is at most 110, the endpoint terms are bounded by (7/48+1/12)/.0007<328, and the log tails are bounded by 26/(.016)^2=101562.5. Hence |Y_\kappa'|<102001<120000 on every whole bracket as well. These estimates hold for both cases, with empty tails in case 1.

It follows from the mean value theorem that moving from a left bracket endpoint to its stationary point increases either value by less than

Every interior stationary point is in one of these brackets or is a division point. All finite division points have been included in Table 2: -1 for X_2; 1/4,1/2,3/4 for Y_2; and -1,-1/2,1/2 for X_1. The remaining endpoints are singular and have limit -\infty: x\to0^\pm and x\to1^- for X_\kappa, and x\to0^+ and x\to1^- for Y_\kappa. The bounded tangent and multiplier terms do not alter these limits. Thus the root exhaustion, the finite endpoint values, and the derivative estimate together certify the global suprema, not merely the sampled values.

Finally -1+\alpha+\gamma=-11/16. Apply Proposition 6.1, use Equation (98), and add .000024 for each supremum. Since \log2>.693146, the right side is strictly less than, respectively,

Taking the larger of these constants proves Proposition 7.1.

8 Conclusion

Hyperbolic volumes. Normalize sectional curvature to -1. Agol’s theorem identifies 4G as the minimum volume of an orientable complete finite-volume hyperbolic three-manifold with exactly two cusps, attained by the Whitehead-link and (-2,3,8)-pretzel-link complements [1, Introduction and Theorem 3.6]. Theorem 1.1 therefore makes this minimum and both link volumes irrational. More generally, every orientable arithmetic hyperbolic three-orbifold defined over \mathbb Q(i) has irrational volume. Here its lattice \Gamma\le\operatorname{PSL}_2(\mathbb C) is, up to conjugacy, commensurable with the projective norm-one group \Gamma^1(\mathcal O)=\mathcal O^1/\{\pm1\} of a maximal order \mathcal O in some quaternion algebra B/\mathbb Q(i) [24, Definition 38.3.4]. For each such algebra, with finite reduced discriminant \mathfrak D, the volume formula gives

where N\mathfrak p=|\mathbb Z[i]/\mathfrak p|; this follows by inserting \zeta_{\mathbb Q(i)}(2)=\zeta(2)L(2,\chi_{-4})=(\pi^2/6)G into [24, Theorem 39.1.13]. After conjugating, a common finite-index subgroup H gives

so this volume is a positive rational multiple of G and is irrational. In the split case B=M_2(\mathbb Q(i)), \mathcal O=M_2(\mathbb Z[i]), the empty product yields \operatorname{vol}(\operatorname{PSL}_2(\mathbb Z[i])\backslash\mathbb H^3)=G/3 [24, Example 39.1.16].