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 (Rivoal and Zudilin 2003, 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 (Rivoal and Zudilin 2003, Theorems 1 and 2). Zudilin reduced this finite collection to the six values through \(\beta(12)\) (Zudilin 2019, Theorem 1). Lai and Zhou subsequently reduced it to the five values \(\beta(2),\beta(4),\beta(6),\beta(8),\beta(10)\) (Lai and Zhou 2022, Theorem 6.1). Fischler obtained stronger quantitative results for irrationality and linear independence in families of Dirichlet \(L\)-values (Fischler 2020, 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 \(\zeta(3)\) (Apéry 1979), and Beukers’s integral proofs of their irrationality (Beukers 1979) 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 (Zudilin 2003, 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 (Rivoal 2006, Theorems 1 and 2). Krattenthaler and Rivoal subsequently gave a more direct hypergeometric proof of these bounds (Krattenthaler and Rivoal 2008, 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 (Krattenthaler and Zudilin 2019, sec. 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 (Nesterenko 2016). Viola announced joint work with Marcovecchio improving an approximation exponent through the Rhin–Viola permutation-group method (Viola 2022); 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 \(m\) (Eskandari 2026, 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 \(L(2,\chi_{-3})\) (Calegari et al. 2024, 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 (Calegari et al. 2024, Remark 11.1.17). Calegari proved irrationality of a \(2\)-adic analogue \(G_2\in\mathbb Q_2\) (Calegari 2005, Theorem 4.2). As explained by Calegari, Dimitrov and Tang (Calegari et al. 2025, sec. 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 (Sun 2026, Theorem 1.1) has also announced a proof of the same qualitative irrationality statement. No result from that preprint is used here.

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 (Zudilin 2017, 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 (Zudilin 2017, sec. 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 (Forrester 2018), Cauchy’s double alternant (Krattenthaler 1999), and the kernel/compression viewpoint of analytic interpolation (Sarason 1967) supply the classical context; the specific estimates are proved below.

The resulting Vandermonde products have logarithmic interactions, as in logarithmic potential theory (Saff 2010). A Chebyshev expansion of the logarithmic kernel (Garoufalidis and Popescu 2013, 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 \[ \mathcal L_N=\frac{\log|\Delta_N|}{(48N)^2}-\frac12\log 2. \tag{1}\] Under the rationality hypothesis, Proposition 7 gives \(\liminf\mathcal L_N>-2.29084\) along every sequence of nonzero determinants, and Proposition 8 supplies such a sequence with \(N\) prime. Independently, Proposition 13 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.

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.

Moment evaluation and cancellation

Set \[ c_l=4^{-l}\binom{2l}{l}\quad(l\ge0),\qquad m_i=\int_{-1}^1t^i\frac{|t|}{f(t)}\,dt =\begin{cases}\displaystyle\frac{2}{(i+1)c_{i/2}},&i\ge0\text{ even},\\ 0,&i\ge0\text{ odd},\end{cases} \tag{9}\] 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 \[ M(i,j)=\sum_{u\ge0}\frac{m_{i+u}}{j+u+1},\qquad M(i,j)-M(i+1,j+1)=\frac{m_i}{j+1}. \tag{10}\]

Put \(K^-_d=M(d,0)\) and \(K^+_d=M(0,d)\). Their boundary recurrences are \[ \begin{aligned} dK^-_d&=(d-1)K^-_{d-2}+m_{d-2}+m_{d-1} &&(d\ge2),\\ K^-_1&=2,\\ (d-1)K^+_d&=(d-2)K^+_{d-2}+\frac{2}{d-1} &&(d\ge2). \end{aligned} \tag{11}\] For the first recurrence, the moment recurrence gives the termwise identity \[\frac{d m_{d+u}-(d-1)m_{d+u-2}}{u+1} =m_{d+u-2}-m_{d+u}.\] 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 \[\frac{(d-1)m_{u-2}-(d-2)m_u}{d+u-1}=m_{u-2}-m_u.\] Their sum is \(2\); the remaining boundary term is \(-2(d-2)/(d-1)\), giving the claimed result.

The two independent starting values are \[ K^-_0=K^+_0=4G,\qquad K^+_1=\frac{\pi^2}{4}=\frac32\zeta(2). \tag{12}\] To verify the first, integrate in \(s\) and put \(t=\sin\theta\) on the positive half interval. This gives \[K^-_0=\int_0^{\pi/2}\log\frac{1+\sin\theta}{1-\sin\theta}\,d\theta =-4\int_0^{\pi/4}\log\tan v\,dv=4G.\] 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 \[K^+_1=\sum_{k\ge1}\frac{1}{2k^2c_k}.\] 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 \[\sum_{k\ge1}\frac{\rho^k\sin(kx)}{k} =\operatorname{Im}\bigl(-\log(1-\rho e^{ix})\bigr),\qquad0<\rho<1,\] 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 \[ k^-_0=k^+_0=k^+_1=0,\qquad k^-_1=2. \tag{13}\] Then the following formulas specify the whole array without ambiguity: \[ M^0(i,j)= \begin{cases} \displaystyle k^-_{i-j}-\sum_{k=1}^{j}\frac{m_{i-j+k-1}}{k},&i\ge j,\\[6pt] \displaystyle k^+_{j-i}-\sum_{k=j-i+1}^{j} \frac{m_{k-(j-i)-1}}{k},&i<j. \end{cases} \tag{14}\] Empty sums are zero. We shall also use the algebraic convention \[ M^0(-1,j)=k^+_{j+1}\qquad(j\ge0); \tag{15}\] this does not define an additional integral. Let \(B_i^{(d)}=\sum_{k=1}^i k^{-d}\), with \(B_0^{(d)}=0\), and put \[ Z^0(i,j)=\begin{cases} -B_i^{(2)},&i=j,\\[2pt] \displaystyle\frac{B_i^{(1)}-B_j^{(1)}}{i-j},&i\ne j. \end{cases} \tag{16}\] The evaluations of the moments are \[ \begin{split} M(i,j)&=M^0(i,j)+4G c_{(i-j)/2} +\frac32\zeta(2)c_{(j-i-1)/2},\\ Z(i,j)&=Z^0(i,j)+[i=j]\zeta(2). \end{split} \tag{17}\] 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 \[Z(i,j)=\sum_{u\ge0}\frac{1}{(i+u+1)(j+u+1)}\] 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 \[ H_z^*=\begin{cases} c_{z/2},&z\ge0\text{ even},\\ \displaystyle\frac{1}{z c_{(z-1)/2}},&z\ge1\text{ odd}. \end{cases} \qquad\frac{H_{z+2}^*}{H_z^*}=\frac{z+1}{z+2}. \tag{18}\] Then \[ \begin{split} k^-_u&=\begin{cases} \displaystyle H_u^*\sum_{\substack{1\le z\le u\\z\ \text{even}}} \frac{2}{z^2(H_z^*)^2},&u\text{ even},\\[6pt] \displaystyle H_u^*\sum_{\substack{1\le z\le u\\z\ \text{odd}}} \frac{2}{z},&u\text{ odd}, \end{cases}\\ k^+_{u+1}&=H_u^*\sum_{\substack{1\le z\le u\\z\equiv u\ (2)}} \frac{2}{z^2H_z^*}\qquad(u\ge0). \end{split} \tag{19}\] 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.

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 \(G\).

Digit reduction of the rational moments

We first prove the reductions needed at primes \[2\sqrt H<p\le H.\] 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 \[E(t)=(1-t^2)^{(p-1)/2}=\sum_{d=0}^{p-1}E_dt^d \quad\text{in }\mathbb F_p[t],\qquad \epsilon=(-1)^{(p-1)/2}.\] We set \(E_d=0\) outside \(0\le d<p\). Within this range, \[E_d=\begin{cases}c_{d/2}\pmod p,&d\text{ even},\\0,&d\text{ odd},\end{cases} \qquad E_{p-1-d}=\epsilon E_d.\] 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 \(E\).

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 \[P'_u=[t^{(u+1)p+\ell}]\,tP(t)E(t)\quad(u\ge-1), \qquad D'_u=[t^{up+\ell}]\,D(t)\quad(u\ge0).\] 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 4, gathered by coefficient residue, gives \[ X_{kp+\ell}:=p^2F_{kp+\ell}\bmod p =\sum_{u\ge-1}P'_uM^0(u,k) -\frac32\sum_{u\ge0}D'_uZ^0(u,k). \tag{36}\] 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 \[V_\ell=2P'_1-\frac32D'_1, \qquad U_\ell=2P'_{-1}-\frac32D'_0.\] The needed small values of the reduced arrays are \[\begin{array}{c|rrr} u&-1&0&1\\\hline M^0(u,0)&0&0&2\\ M^0(u,1)&2&0&-2 \end{array} \qquad \begin{array}{c|rr} u&0&1\\\hline Z^0(u,0)&0&1\\ Z^0(u,1)&1&-1 \end{array}.\] Consequently, whenever the indicated columns exist, \[ X_\ell=V_\ell,\qquad X_{p+\ell}=U_\ell-V_\ell. \tag{37}\] 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 \[\max(b,H-2p)\le\ell<\min(p,L-p,A).\] Their number is exactly \[d_0=\bigl(\min(p,L-p,A)-\max(b,H-2p)\bigr)_+, \qquad y_+=\max(y,0).\] 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 \[ v_p(\Delta_N)\ge-2n+(d_0-q)_+. \tag{38}\]

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.

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.

Two palindromic bases

Let \(p\) be an odd prime and work over \(\mathbb F_p\). The vector space \[\mathcal P_p=\{Q\in\mathbb F_p[w]:\deg Q\le2p-2, \ w^{2p-2}Q(1/w)=Q(w)\}\] has dimension \(p\): its coefficients of \(w^0,\ldots,w^{p-1}\) determine all its coefficients. For \(0\le\ell,i<p\), consider \[U_\ell(w)=(2w)^\ell(1+w^2)^{p-1-\ell},\qquad Q_i(w)=w^i\sum_{j=0}^{p-i-1}w^{2j}.\] 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 \[Q_i=\sum_{\ell=0}^{p-1}a_{\ell i}U_\ell, \qquad \mathbf a=(a_{\ell i})_{0\le\ell,i<p}.\] The same lowest-term comparison gives \[ a_{\ell i}=0\quad(\ell<i),\qquad a_{ii}=2^{-i},\qquad \det\mathbf a=2^{-p(p-1)/2}\ne0. \tag{49}\] 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, \[ \mathbf b=\mathbf a\Pi,\qquad \Pi e_0=0,\qquad \Pi e_i=e_{p-i}\quad(1\le i<p), \tag{50}\] where \(e_i\) are the standard coordinate vectors. The explicit forms are \[Q_i=w^i\frac{1-w^{2(p-i)}}{1-w^2},\qquad Q'_i=w^{p-i}\frac{1-w^{2i}}{1-w^2};\] the latter formula gives zero also when \(i=0\). Their sum satisfies \[Q_i+w^pQ'_i =w^i\sum_{j=0}^{p-1}w^{2j} =w^i(1-w^2)^{p-1},\] because \(\binom{p-1}{j}=(-1)^j\) in \(\mathbb F_p\). With \[t=\frac{2w}{1+w^2},\qquad f=\frac{1-w^2}{1+w^2},\qquad E(t)=(1-t^2)^{(p-1)/2}=f^{p-1},\] division by \((1+w^2)^{p-1}\) therefore proves the rational-function identity \[ E(t)w^i=\sum_{\ell=0}^{p-1}t^\ell (a_{\ell i}+b_{\ell i}w^p). \tag{51}\]

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 \[t'=\frac{2w'}{1+(w')^2},\qquad f'=\frac{1-(w')^2}{1+(w')^2},\qquad (f')^2=1-(t')^2.\] Using \(C=63p\), \(h=2p\), and \(g=4p\) in the row definition gives \[\begin{align*} tR_rE(t) &= (1-t')^2(t')^{63}(w')^{r_0-4}E(t)w^i\\ &=\sum_{\ell=0}^{p-1}t^\ell t' \bigl(a_{\ell i}\underline R_{r_0}(t',w') +b_{\ell i}\underline R_{r_0+1}(t',w')\bigr). \tag{52}\end{align*}\] 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 \[tR_rE=tP_rE-f'D_r, \qquad t'\underline R_j=t'\underline P_j-f'\underline D_j.\] 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: \[\begin{align*} tP_r(t)E(t) &=\sum_{\ell=0}^{p-1}t^\ell t' \bigl(a_{\ell i}\underline P_{r_0}(t') +b_{\ell i}\underline P_{r_0+1}(t')\bigr), \tag{53}\\ D_r(t) &=\sum_{\ell=0}^{p-1}t^\ell \bigl(a_{\ell i}\underline D_{r_0}(t') +b_{\ell i}\underline D_{r_0+1}(t')\bigr). \tag{54}\end{align*}\] 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 \[P'_u=[t^{(u+1)p+\ell}]tP_r(t)E(t),\qquad D'_u=[t^{up+\ell}]D_r(t).\] Every polynomial has a unique expression \(\sum_{\ell=0}^{p-1}t^\ell A_\ell(t^p)\). Consequently Equations (53) and (54) give \[ (P'_u,D'_u)=[(t')^u] \left\{a_{\ell i}(\underline P_{r_0},\underline D_{r_0})(t') +b_{\ell i}(\underline P_{r_0+1},\underline D_{r_0+1})(t')\right\}. \tag{55}\] 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 \[ \min\deg\underline P_{48}=62-44=18,\qquad \min\deg\underline D_{48}=63-44=19. \tag{56}\] 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 \[19p+(p-i)=20p-i.\] 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 \(63+48-4=107>59\).

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 \[p>2\sqrt{65p}=2\sqrt H,\] 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 \[F_j(r)=M^0(P_r,j)-\frac32 Z^0(D_r,j) +4G\sum_v[t^v]P_r\,c_{(v-j)/2}.\] 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 \[\begin{align*} p^2F_{kp+\ell}(pr_0+i) \equiv{}&a_{\ell i}\left\{M^0(\underline P_{r_0},k) -\frac32 Z^0(\underline D_{r_0},k)\right\} \\ &+b_{\ell i}\left\{M^0(\underline P_{r_0+1},k) -\frac32 Z^0(\underline D_{r_0+1},k)\right\} \pmod p. \tag{57}\end{align*}\] 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 \(\mathbb F_p[s]\), \[ s^{7p+k}(1-s)^{4p} =s^\ell(s^p)^{7+k_0}(1-s^p)^4. \tag{58}\] 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 \[a_{\ell i}\mathcal B_0+b_{\ell i}\mathcal B_1.\] Writing this matrix as \(\mathcal L_p\) makes its orientation explicit: \[ \begin{split} \mathcal L_p&=\mathbf a^T\otimes\mathcal B_0 +\mathbf b^T\otimes\mathcal B_1,\\ \mathcal L_p\bigl((\mathbf a^T)^{-1}\otimes I_{48}\bigr) &=I_p\otimes\mathcal B_0+\Pi^T\otimes\mathcal B_1. \end{split} \tag{59}\] 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: \[\begin{align*} \det\mathcal L_p ={}&(\det\mathbf a)^{48}\det\mathcal B_0\\ &\quad\cdot\det(\mathcal B_0+\mathcal B_1)^{(p-1)/2} \det(\mathcal B_0-\mathcal B_1)^{(p-1)/2}. \tag{60}\end{align*}\] There is no determinant factor from the similarity. Returning to the original row and column orders changes at most the sign.

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 \[k^-_0=0,\quad k^-_1=2,\qquad k^+_0=k^+_1=0,\] and use \[\begin{align*} k^-_d&=\frac{(d-1)k^-_{d-2}+m_{d-2}+m_{d-1}}{d} &&(2\le d\le64),\\ k^+_d&=\frac{(d-2)k^+_{d-2}+2/(d-1)}{d-1} &&(2\le d\le58). \end{align*}\] On \(0\le i\le64\), \(0\le j\le58\), form \[Y_{i0}=k^-_i,\qquad Y_{0j}=k^+_j,\qquad Y_{ij}=Y_{i-1,j-1}-\frac{m_{i-1}}j\quad(i,j\ge1).\] Thus \(Y_{ij}=M^0(i,j)\). With \(B_i^{(d)}=\sum_{u=1}^i u^{-d}\) and \(B_0^{(d)}=0\), set \[J_{ij}=\frac32\begin{cases} -B_i^{(2)},&i=j,\\[2pt] (B_i^{(1)}-B_j^{(1)})/(i-j),&i\ne j. \end{cases}\] This gives \(J_{ij}=\tfrac32Z^0(i,j)\), with the sign on its diagonal included.

For the polynomial construction define \[E_0=t^{62},\quad E_1=t^{61},\qquad I_0=0,\quad I_1=t^{62},\] and, for either family \(S=E,I\), use \[S_m=2S_{m-1}/t-S_{m-2}\qquad(2\le m\le44).\] 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 \[y_r(t)=(1-t)^2E_u(t),\qquad z_r(t)=\operatorname{sign}(r-4)(1-t)^2I_u(t).\] They are exactly \(\underline P_r\) and \(\underline D_r\). Contract them with the arrays to obtain the length-\(52\) vector \[v_r(j)=\sum_{i=0}^{64}\bigl([t^i]y_r\,Y_{ij} -[t^i]z_r\,J_{ij}\bigr), \qquad j=7,\ldots,58.\] 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 \[R_j-\frac{R_j(i)}{d_i}R_i.\] 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.

The integral and its two sheets

Put \(d\mu(t)=|t|\,dt/f(t)\) on \((-1,1)\), and set \[A_r(t)=P_r(t)-\frac32\boldsymbol 1_{\{t>0\}}\frac{f(t)}tD_r(t), \qquad \psi_k(s)=s^{b+k}(1-s)^q.\] 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 \[\int_{-1}^1\int_0^1 A_r(t)\frac{\psi_k(s)}{1-ts}\,ds\,d\mu(t).\] 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 (Forrester 2018, Equation (1.7) and Section 2.2). Expanding determinants and relabeling integration variables gives the identity \[ \Delta_N=\frac1{(n!)^2}\int_{(-1,1)^n}\int_{(0,1)^n} \det[A_r(t_i)]_{r,i}\, \det\left[\frac1{1-t_i s_j}\right]_{i,j}\, \det[\psi_k(s_j)]_{j,k}\,ds\,d\mu^n(t). \tag{62}\] 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 (Krattenthaler 1999, sec. 2.1, Equation (2.7)) is \[ \det\left[\frac1{1-t_i s_j}\right]_{i,j} =\frac{V(t)V(s)}{\prod_{i,j}(1-t_i s_j)}. \tag{63}\] 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 \[ x_i=\frac{t_i}{1+\sqrt{1-t_i^2}},\qquad t_i=\frac{2x_i}{1+x_i^2},\qquad d\mu(t_i)=\frac{4|x_i|}{(1+x_i^2)^2}\,dx_i. \tag{64}\] 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 \[(\xi_{\mathrm f}(x),\xi_{\mathrm n}(x))= \begin{cases}(1/2,1/2),&x<0,\\(-1/4,5/4),&x>0,\end{cases} \qquad \mathcal R_n(x)= \det\left[\xi_{\mathrm f}(x_i)x_i^{g-r} +\xi_{\mathrm n}(x_i)x_i^{r-g}\right]_{r,i}.\] 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 \(i\)th column therefore gives the exact identity \[ \Delta_N=\frac1{(n!)^2}\int\!\int \mathcal R_n(x)\, \frac{V(t)V(s)^2}{\prod_{i,j}(1-t_i s_j)} \prod_j s_j^b(1-s_j)^q \prod_i t_i^{C-1}(1-t_i)^h\,ds\,d\mu^n(t). \tag{65}\] In particular, there is one \(t\) Vandermonde and two \(s\) Vandermondes. All the denominators in this formula are positive.

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 \(0\le r<n\), \[ \xi_{\mathrm f}(x_i)x_i^{g-r} +\xi_{\mathrm n}(x_i)x_i^{r-g} =\xi_{\mathrm f}(x_i)x_i^{g-(n-1)} \left(x_i^{n-1-r}+c_i x_i^{D_*}x_i^r\right). \tag{66}\] 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 \[ \sum_{i=1}^n\frac{1-x_i^2}{1+x_i^2}\le D_*. \tag{67}\] The following construction is uniform even when the nodes cluster.

Evaluation in a finite-dimensional Hilbert space

The following argument uses the kernel and compression viewpoint of bounded analytic interpolation; see (Sarason 1967). We prove the needed finite-dimensional facts directly.

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 \[\mathcal J_n(x,s)= \frac{\mathcal V(t)\mathcal V(s)^2}{\prod_{i,j}(1-t_i s_j)} \prod_j s_j^b(1-s_j)^q \prod_i |t_i|^{C-1}(1-t_i)^h.\] Define \[ \begin{split} \mathcal I_{2,n}(x,s) &=\mathcal J_n(x,s)\mathcal V(1/x)\prod_i|x_i|^g,\\ \mathcal I_{1,n}(x,s) &=\mathcal J_n(x,s)2^{-\binom n2} \prod_i(1+x_i^2)^{(n-1)/2}|x_i|^{g-(n-1)}. \end{split} \tag{74}\] 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 \[|\Delta_N|\le\frac{2^n K_n}{(n!)^2} \max_{\kappa\in\{1,2\}}\sup_{(x,s)\in\Omega_{\kappa,n}} \mathcal I_{\kappa,n}(x,s), \quad K_n=\max\{(1+K_0n)^n,(3/2)^n n^{n/2}\}.\] 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 \[ \frac{\log|\Delta_N|}{n^2}-\frac12\log2 \le\max_{\kappa\in\{1,2\}} \sup_{(x,s)\in\Omega_{\kappa,n}} \left(\frac{\log\mathcal I_{\kappa,n}(x,s)}{n^2} -\frac12\log2\right) +O\left(\frac{\log(n+2)}n\right). \tag{75}\] 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).

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 \[ u_k=l_k+\sum_{z}r_z z^k,\qquad l_k=0\quad(k>d). \tag{89}\] 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 \[\begin{align*} \|u\|_*^2 &=\sum_{k=1}^d\frac{l_k^2+2l_k\sum_zr_zz^k}{k} -\sum_{z,z'}r_zr_{z'}\operatorname{Log}(1-zz'),\\ T(u,x)&=\sum_{k=1}^d\frac{l_kT_k(x)}{k} -\frac12\sum_zr_z\operatorname{Log}(1-2xz+z^2),\\ S(u,x)&=\sum_{k=1}^d\frac{l_kx^k}{k} -\sum_zr_z\operatorname{Log}(1-xz). \tag{90}\end{align*}\] 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 \[1-2xz+z^2=(1-ze^{i\theta})(1-ze^{-i\theta}).\] 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 \[\begin{align*} X_\kappa(x)&=W_\kappa(x)+\lambda_\kappa D(x) -2\kappa T(p,x)-S(v,x),\\ Y_\kappa(x)&=\beta\log x+\gamma\log(1-x)+2T(v,x). \tag{91}\end{align*}\] The domain of \(X_\kappa\) is \([-1,1)\setminus\{0\}\), and that of \(Y_\kappa\) is \((0,1)\).

Derivative numerators and exhaustive root brackets

Define the rational functions \[\begin{align*} t_u(x)&=\sum_{k=1}^dl_kU_{k-1}(x) +\sum_z\frac{r_zz}{1-2xz+z^2},\\ h_u(x)&=\sum_{k=1}^dl_kx^{k-1} +\sum_z\frac{r_zz}{1-xz}. \end{align*}\] Differentiation of Equation (90) yields \[\begin{align*} X_\kappa'(x) &=\frac{\alpha+2\gamma}{x}-\frac{2\eta}{1-x} -(\kappa+2\alpha+2\eta+2\gamma)\frac{x}{1+x^2} \\ &\quad-\frac{4\lambda_\kappa x}{(1+x^2)^2} -2\kappa t_p(x)-h_v(x),\\ Y_\kappa'(x)&=\frac\beta x-\frac\gamma{1-x}+2t_v(x). \tag{92}\end{align*}\] For a completely specified rational numerator, use \[\begin{align*} Q_X(x)&=x(1-x)(1+x^2)^{3-\kappa} \prod_{z\text{ in }p}(1-2xz+z^2) \prod_{z\text{ in }v}(1-xz),\\ Q_Y(x)&=x(1-x)\prod_{z\text{ in }v}(1-2xz+z^2),\\ A_X(x)&=Q_X(x)X_\kappa'(x),\\ A_Y(x)&=Q_Y(x)Y_\kappa'(x). \tag{93}\end{align*}\] 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 \[ (1+t)^{d_0}A\left(\frac{b+ct}{1+t}\right) =\sum_{h=0}^{d_0}a_ht^h,\qquad a_h=\sum_{j=0}^{d_0}\sum_{k=0}^j A_j\binom jk b^{j-k}c^k\binom{d_0-j}{h-k}, \tag{94}\] 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 (Descartes 1637, bk. 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 \[\sum_{j=0}^h b_j=-a_h\quad(0\le h\le d),\qquad b_{d+1}=a_d.\] 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 \[ N_A(m)N_A(m+2)<0,\qquad N_A(s)=\sum_{j=0}^{d_0}A_j(10^{10})^{d_0-j}s^j. \tag{95}\] 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.

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 \[ H(y)=2\sum_{j=0}^{17}\frac1{2j+1} \left(\frac{y-1}{y+1}\right)^{2j+1}. \tag{96}\] The substitution for \(\log s\) is \(mH(2)+H(y)\). If \(q=(y-1)/(y+1)\), its unscaled positive remainder is bounded by \[0\le\log y-H(y)\le\frac{2q^{37}}{37(1-q^2)}.\] 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 \[ J(t)=\sum_{j=0}^{23}\frac{(-1)^jt^{2j+1}}{2j+1}, \tag{97}\] substitute \(k(J(1/2)+J(1/3))+J(t)\) for the argument. The identity \[\arctan(1/2)+\arctan(1/3)=\pi/4\] 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 \[h_0=\frac{2(1/3)^{37}}{37(1-1/9)},\qquad j_0=\frac{(1/2)^{49}}{49}.\] 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 \(\kappa\|p\|_*^2+\tfrac12\|v\|_*^2\): \[.778415976284\quad(\kappa=2),\qquad .931985203901\quad(\kappa=1).\] 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 \[ \begin{array}{c|ccc} \kappa&\kappa\|p\|_*^2+\tfrac12\|v\|_*^2&X&Y\\\hline 2&.77844&-.98399&-1.60890\\ 1&.9321&-1.3244&-1.4280. \end{array} \tag{98}\] The logarithm procedure also gives \(.693146<\log2<.693149\).

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 \[\frac{4\cdot10}{(.06)^2}+\frac{13}{.016}<11924.\] 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 \[120000\cdot\frac2{10^{10}}=.000024.\] 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 12, use Equation (98), and add \(.000024\) for each supremum. Since \(\log2>.693146\), the right side is strictly less than, respectively, \[\begin{align*} -\frac{11}{16}(.693146)+.77844-.98399-1.60890+.000048 &=-2.290939875,\\ -\frac{11}{16}(.693146)+.9321-1.3244-1.4280+.000048 &=-2.296789875. \end{align*}\] Taking the larger of these constants proves Proposition 13.

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 (Agol 2010, Introduction and Theorem 3.6). Theorem 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 \(B/\mathbb Q(i)\) (Voight 2021, Definition 38.3.4). For each such algebra, with finite reduced discriminant \(\mathfrak D\), the volume formula gives \[\operatorname{vol}\bigl(\Gamma^1(\mathcal O)\backslash\mathbb H^3\bigr) =\frac{G}{3}\prod_{\mathfrak p\mid\mathfrak D}(N\mathfrak p-1),\] 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 (Voight 2021, Theorem 39.1.13). After conjugating, a common finite-index subgroup \(H\) gives \[\operatorname{vol}(\Gamma\backslash\mathbb H^3) =\frac{[\Gamma^1(\mathcal O):H]}{[\Gamma:H]} \operatorname{vol}\bigl(\Gamma^1(\mathcal O)\backslash\mathbb H^3\bigr),\] 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 \(\operatorname{vol}(\operatorname{PSL}_2(\mathbb Z[i])\backslash\mathbb H^3)=G/3\) (Voight 2021, Example 39.1.16).