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The finite-horizon five-expert prediction problem

Jeff Calder, Nadejda Drenska

arXiv:2609.38035v1math.APcs.GTcs.LGmath.OC

TL;DR

The paper addresses the five-expert finite-horizon prediction PDE and the characterization of its maximizing strategy. It constructs and verifies a classical solution, showing that the rank direction maximizes everywhere while COMB does so only on a specified equality set.

  • Problem

    The paper addresses how to solve the five-expert finite-horizon prediction PDE and identify its maximizing strategy.

    Source: §1 · p. 3

  • Method

    The authors construct an explicit solution and verify it using Laplace-transform methods, including an inverse-transform argument for regularity across the collision set.

    Source: §1 · p. 3, §4 · p. 20, §4.2 · p. 24

  • Results

    The rank direction attains the maximum everywhere, while COMB attains it exactly when x1 = x2 and x3 = x4.

    Source: §1 · p. 3, §2 · p. 8

  • Takeaways & Limitations

    The five-expert finite-horizon problem has a classical solution with a pointwise characterization of the maximizing strategy.

    Source: §1 · p. 3, §4 · p. 20

    The result establishes two continuous spatial derivatives, but whether the spatial Hessian is Lipschitz across collision sets remains open.

    Source: §4.2 · p. 24

Abstract

from arXiv · show

We give an explicit solution to the five expert prediction with expert advice partial differential equation (PDE) in the finite-time horizon setting. The solution formula establishes that the adversary's rank strategy $(1,0,1,0,0)$ is globally optimal, and the COMB strategy $(1,0,1,0,1)$ is optimal exactly on the set where $x_1=x_2$ and $x_3=x_4$. The formula is derived from the solution of the geometric-stopping problem given in our companion paper through the transform principle of Bayraktar, Ekren and Zhang, which links the two problems by a Laplace transform. Inverting the transform term by term expresses the solution through a series of Gaussian and complementary error function kernels. The optimality of $(1,0,1,0,0)$ is reduced to the signs of $41$ one-variable Gaussian series, which are certified with computer assistance by Poisson summation, first-mode domination and interval arithmetic on $1616$ rational cells. The proofs of our main theorems, certificates included, are also formalized in the Lean proof assistant.

1 Introduction

The paper solves the five-expert finite-horizon prediction PDE, extending exact solution results beyond the previously addressed cases of at most four experts. Its solution identifies an adversary strategy optimal everywhere and shows that COMB is optimal only on a specific tie set.

1 Introduction

  • The finite-horizon value converges from the discrete game through the viscosity solution, while the cited M-asymptotics are not part of the formalization.The M-asymptotics are attributed to Drenska and Kohn.Source: §1 · pp. 2–3
  • The explicit five-expert solution is a classical solution of the PDE, with v* attaining the maximum everywhere.The theorem extends explicit finite-horizon solutions beyond the previously addressed cases of at most four experts.Source: §1 · pp. 2–3
  • COMB attains the maximum exactly where x1=x2 and x3=x4, so it is not globally optimal for five experts.This is the same tie set identified in the geometric-stopping problem.Source: §1 · p. 3
  • The construction uses a transform principle linking the geometric-horizon solution to the Laplace-transformed finite-horizon problem with the adversary fixed at v*.The inverse transform supplies a candidate finite-horizon solution.Source: §1 · p. 3
  • Term-by-term inversion expresses the solution as a series of rational-coefficient kernels built from Gaussians and complementary error functions.The method expands the stationary formulas into exponentials of linear functions of x with polynomial prefactors.Source: §1 · p. 3
  • Optimality verification reduces to signs of 41 one-variable Gaussian series, certified across the positive axis using Poisson summation, first-mode domination, and interval arithmetic on 1616 rational cells.The proofs of Theorems 2.1 and 2.2, including the certificates, are also formalized in Lean.Source: §1 · p. 3

2 Main results

The paper constructs an explicit finite-horizon solution for five experts and proves that rank direction (1, 0, 1, 0, 0) attains the Hamiltonian maximum throughout the ordered sector. COMB is optimal exactly where x1 = x2 and x3 = x4.

Formula construction

  • The finite-horizon formula replaces each stationary mode P(x)e^−X(x) by P(x)Hj(X(x), τ), with j the degree of P.The kernels are elementary combinations of a Gaussian and the complementary error function; the construction uses a four-expert background in Regions I and II and a correction.Source: §2 · pp. 5–6

Regularity and structure

  • The series converge locally uniformly with derivatives, and U is C2 even on the collision set where the mode series degenerate.The formulas agree across the stated region interfaces, and U has parabolic scaling.Source: §2 · pp. 4, 7, 8

Theorem 2.1

  • U is a classical solution of (2.1) on (0, ∞)×R5, with U(τ, ·) ∈ C2(R5) for every τ > 0; its derivatives Uτ, ∇U, and ∇2U are jointly continuous, and the equation holds at every point.It also satisfies |U(τ, x) − φ(x)| ≤ C√τ uniformly in x.Source: §2 · p. 8

Theorem 2.2

  • COMB attains the Hamiltonian maximum exactly when x1 = x2 and x3 = x4; at every other point its curvature gap is positive.Other controls tie with the optimal direction in specified regions, matching the ties of the stationary problem.Source: §2 · p. 8

Verification

  • The Hamiltonian inequality is reduced to finite computations, with the remaining signs certified using exact rational arithmetic and scripts.The paper also reports independent numerical checks.Source: §2 · p. 9

3 Derivation of the finite-horizon formula

This section derives the finite-horizon formula by transforming the stationary solution and inverting its exponential modes, giving candidate formulas across Regions I–III and a separate quadrature on the top-four collision set. It also describes their behavior at region interfaces and records independent checks.

3.1 The transform principle

  • The transform principle defines the finite-horizon candidate by inverting the Laplace transform of the stationary solution with the adversary frozen at v∗.The authors invert mode by mode and prove directly in Section 4 that the candidate is a classical solution, without establishing uniqueness for the frozen linear problem.Source: §3.1 · pp. 9–10
  • At the origin, the candidate satisfies U(τ, 0) = 2√π u(0)√τ.For five experts, the paper identifies this with the value in (2.22).Source: §3.1 · p. 10

3.2 The inversion kernels

  • The kernels’ high-order growth is controlled by a bound that decays faster than any geometric sequence in j, locally uniformly in Z and τ > 0.This control supports expansions that use kernels of unbounded order.Source: §3.2 · p. 11
  • Each admissible exponential mode in the stationary expansion inverts to an exact solution of the linear problem, using kernels Hj and polynomials invariant along v∗.Termwise inversion requires summability, and deriving properties of the resulting function requires the convergence and regularity results of Section 4.Source: §3 · p. 9, §3.2 · p. 12
  • Nonnegative mode exponents support the terminal condition, but the Region III series diverges on the diagonal, so Section 4 proves a uniform |U(τ, x) − φ(x)| ≤ C√τ bound instead.At X = 0, only H0 among the nonzero kernels tends to zero as τ ↓ 0.Source: §3.2 · p. 13

3.3 Four experts

  • The four-expert finite-horizon solution of Bayraktar, Ekren and Zhang [3] is recovered as an absolutely convergent series with explicit rational coefficients and nonnegative exponents.Unlike their branch-cut inversion, the mode-by-mode series consists of bounded Gaussian and complementary error function terms, so singular terms need not cancel.Source: §3.3 · p. 14

3.4 Region III

  • In Region III, the trace series converges for L > 0 but diverges at L = 0, where e(0) = 45π²/512.The value at L = 0 is supplied separately by the closed quadrature evaluation.Source: §3.4 · pp. 15–16
  • Independent checks compare (3.18) with quadrature (2.8) to 40 digits at five L values and verify (3.23) against (2.9) as exact rationals for every m ≤ 200.Source: §3.4 · p. 16
  • Termwise differentiation of (3.18) updates (Am, Bm) to (Bm − 2mAm, −2mBm), and repeated differentiation generates the coefficients used in the finite-horizon formula.Source: §3.4 · p. 16
  • On Region III with a4 > 0, candidate (3.4) is given by (2.19), with nonzero modes having Xm,ϵ ≥ 0 and ∂v∗Xm,ϵ = −ϵ3k.The exponent bounds follow by considering m ≥ 2, m = 1 (where the all-positive sign vector has zero coefficient), and m = 0.Source: §3.4 · p. 17

3.5 Regions I and II

  • In Regions I and II with z1 > 0, candidate (3.4) is given by (2.20) for τ > 0.The correction depends on z3 through |z3|, so the region-wise expressions agree at z3 = 0.Source: §3.5 · p. 19
  • At the Region II–III interface z4 = 0, the formulas agree with their first and second derivatives outside C; at z1 = 0, use quadrature (2.21) because the series requires z1 > 0.Source: §3.5 · p. 19
  • On the top-four collision set C, where x1 = x2 = x3 = x4 ≥ x5, candidate (3.4) is given by the single quadrature (2.21).Source: §3.5 · p. 19
  • At x = 0, the quadrature agrees with the Corollary 3.2 identity, with both sides numerically equal to 0.08204753591704610024 to 20 digits.Source: §3.5 · p. 19

4 Verification

This section verifies the inverse-transform solution and global PDE and strategy conclusions through convergence and transform identities, global regularity across collision points, and Hamiltonian inequalities supported by computer-assisted Gaussian-series certificates. It also identifies the finite-horizon COMB equality set and formalizes Theorems 2.1 and 2.2 in Lean.

4 Verification

  • The Hamiltonian inequalities reduce to exact rational identity checks and positivity of 41 one-variable Gaussian series, certified using Poisson summation, first-mode domination, and interval arithmetic on 1616 cells.The proofs of Theorems 2.1 and 2.2, including the interval certificates, are also formalized in Lean.Source: §4 · pp. 20–21

4.1 Convergence and the transform identity

  • The Region III series (2.19) converges absolutely and uniformly where a4 ≥ δ > 0, |x| is bounded, and τ lies in a compact subset of (0, ∞); the same holds for termwise x- and τ-derivative series.The majorants of the twice-differentiated series are O(τ^-1/2) as τ ↓ 0, uniformly in x on the stated sets.Source: §4.1 · p. 21

4.2 Regularity at the collision set

  • U(τ, ·) is C2 on all R5 with jointly continuous spatial derivatives through order two, and |U−φ|≤C√τ; Hessian Lipschitz continuity across collision sets remains open.The regularity proof continues stationary formulas to complex scaling parameters and inverts the transform on a Hankel contour.Source: §4.2 · p. 24
  • The inverse-transform solution is globally C2, including at collision points, with jointly continuous spatial derivatives through order two.The Hessian is continuous across the collision set, and time derivatives of every order are justified on compact subsets of positive time.Source: §4.2 · p. 28

4.3 Transfer of curvature identities and the COMB equality set

  • The curvature gap vanishes when x1=x2 and x3=x4, and its scaling reduces strict positivity off that set for every τ to the case τ=1.The gap satisfies Δ(τ,x)=τ^-1/2 Δ(1,x/√τ).Source: §4.3 · p. 30

4.4 The Hamiltonian inequalities

  • The rank strategy (1,0,1,0,0) attains the Hamiltonian maximum across the stated regions; COMB is strict except when x1=x2 and x3=x4.The results cover Region III with a4>0 and Regions I and II with z1>0; the COMB comparison is an equality on the specified set.Source: §4.4 · p. 31
  • The Hamiltonian inequalities are reduced by interpolation and positive evolutions to 41 one-variable Gaussian series with rational coefficients.The transformed gaps are certified through completely monotone representations; the reductions use distinguished corner states and, in Region I, a positive Dirichlet kernel.Source: §4.4 · pp. 32–33
  • Positivity is certified over ranges by Poisson summation for 0 < ξ ≤ 1/100, first-mode domination for ξ ≥ 1, and interval arithmetic on [1/100, 1]; 40 of 41 profiles receive interval certificates.The interval proof uses 1616 rational cells, with an explicit bound for the infinite tail; the remaining profile is certified by an exact polynomial argument.Source: §4.4 · p. 33
  • Theorems 2.1 and 2.2 are formalized in Lean 4 against Mathlib using only Lean’s standard axioms, including proofs of all 1616 cell certificates.The formalization covers the explicit formula and the supporting contour representation; some auxiliary results remain proved only in the paper.Source: §4.4 · pp. 33–34, Table 1 · p. 34

4.5 Proofs of the main theorems

  • Continuity extends inequality (4.23) to the closed ordered sector, and permutation invariance yields a classical solution of (2.1) on (0, ∞) × R^5.The proof first approaches boundary states from strictly ordered states, then uses coordinate permutations to extend the equation across the domain.Source: §4.5 · pp. 34–35
  • Theorem 2.2 proves Δ≥0 throughout the ordered sector, with equality exactly when x_1=x_2 and x_3=x_4 and strict positivity elsewhere.The regional proofs identify equality on the specified set and establish strict positivity off it.Source: §4.5 · p. 35

A The Region III certificate

The certificate uses λ as the square root of the Laplace variable and CM for completely monotone functions on λ > 0. The target inequality is reduced to showing that each transformed gap belongs to CM.

A The Region III certificate

A.1 Normalization and positive heat-exit interpolation

The certificate uses normalized coordinates and a cube interpolation argument to extend positivity from corner values to the physical simplex. The interpolation weights are completely monotone and have a Brownian exit-time interpretation.

A.1 Normalization and positive heat-exit interpolation

  • The Region III coordinates satisfy 0 ≤ a1 ≤ a2 ≤ a3 ≤ a4 = L, and the variables yi are scaled adjacent-coordinate differences.For a control v, the text relates Dv a to k qv; it also notes that u = x1 + F/k with x1 linear, so derivatives commute in the stated calculation.Source: §A.1 · p. 35
  • The weights lie in CM, and positivity at the eight cube corners implies positivity throughout the cube and the physical simplex.The other corners are permutations of the stated corners and give the same gaps after the corresponding control permutation.Source: §A.1 · p. 36
  • For Brownian motion with generator ∂bb, Wi is the Laplace-weighted probability of exiting (0, r) at endpoint ir, with an atom at time zero at the endpoints.Source: §A.1 · p. 36
  • If every corner value gλ(rϵ) is completely monotone, then gλ is completely monotone throughout the cube.The stated consequence applies in particular throughout the physical simplex 0 ≤ a1 ≤ a2 ≤ a3 ≤ L.Source: §A.1 · p. 36
  • The interpolation weights W0 and W1 solve the one-dimensional boundary problem g′′ = λg and take boundary values (1, 0) and (0, 1).Applying the boundary representation successively in each variable gives the cube interpolation; complete monotonicity is preserved under products and nonnegative combinations.Source: §A.1 · p. 36

A.2 The corner gaps as Gaussian series

The corner-gap verification reduces each control and corner to a one-variable time-inverse problem. Rational mode coefficients then express the gaps as lattice Gaussian series, with exact-arithmetic checks for exceptional modes and partial fractions.

A.2 The corner gaps as Gaussian series

  • At a fixed corner and control, the gap is a function of L, and the quantity to sign is its time inverse.Source: §A.2 · p. 36
  • For each of ten listed generators, the corner-gap coefficients are rational functions a(k) and b(k), with specified parity and bounded integer poles.The partial fractions have polynomial parts of degree at most two for a and at most three for 2kb.Source: §A.2 · p. 37
  • At a corner, the modes collect by k into coefficients for the Gaussian-series representation of the gap.The finitely many modes with m ≤ 0 are handled separately; the zero mode has half weight, and the script audit checks the partial fractions and 64 Hessian contractions.Source: §A.2 · p. 37
  • The symmetric sum over k is an even extension, equal to twice the positive-k sum plus the half-weighted zero mode.This extension is explicitly distinguished from the geometric exponential series.Source: §A.2 · p. 37
  • For the control 00000, where qv = 0, the text gives E0 = e and specifies the coefficient convention including a(0) = 1.Source: §A.2 · p. 37

A.3 Small scales: a dual representation

The small-scale analysis converts the Gaussian-series expression using partial fractions and cancellation, leaving principal expressions M with explicit positive lower bounds. These bounds and related rational computations support the corner positivity checks.

A.3 Small scales: a dual representation

  • The partial-fraction substitution expresses Γ through Fj and its derivatives, with the zero image included.Source: §A.3 · p. 38
  • Cancellation of the pole contributions removes the pure √ξ and ξ−1/2−h terms, leaving the principal expression M.The text identifies the surviving expression through the simple and double residues of a(k).Source: §A.3 · p. 38
  • M exceeds √ξ/5 for C1, D1, A2, B2, and P3, and exceeds 1/50 for E0, V0, T1, T2, and T3.For E0, the text gives M = 45π²e−ξ/512.Source: §A.3 · p. 38
  • The polynomial contribution is bounded using coefficient norms below 30, while the stated remainders are smaller still.The accompanying estimate uses 9 < π² < 10 and an endpoint bound for ξ ≤ 1/100.Source: §A.3 · p. 38
  • For small ξ, the positivity argument uses rational lower bounds exceeding 1/5 or 1/50, computed exactly from stored partial fractions.The bounds distinguish the cases where the coefficients da sum to zero and where the resulting lower bound is constant.Source: §A.3 · p. 39

A.4 The compact range and the large-scale tail

The certificate bounds the large-scale tail by first-mode domination and a Gaussian integral estimate, while certifying positivity on the compact range with rational-cell Taylor bounds. Its interval checks use exact rational data and outward-rounded exponential bounds.

A.4 The compact range and the large-scale tail

  • For k ≥ 10, the coefficient bounds are |a(k)| ≤ 100k^8 and |2kb(k)| ≤ 100k^8.The bounds use root and numerator estimates without estimated constants.Source: §A.4 · p. 39
  • Taylor bounds on rational cells certify compact-range positivity by requiring each interval lower bound to be strictly positive.Cells are bisected as needed, with widths below 10^-10 refused; exponential values are enclosed by rational intervals with outward rounding.Source: §A.4 · p. 39, Table 2 · p. 40
  • For ξ ≥ 1, first-mode domination applies because the first nonzero mode has α0 > 0 and ck0 = 0.The certificate also bounds the tail beyond N = 128 using an integral estimate.Source: §A.4 · p. 39

A.5 Conclusion in Region III

In Region III, corner decompositions and positivity-preserving arguments establish the curvature-gap inequality for every control when L > 0. The COMB representative has a vanishing gap at one ordered corner and strictly positive generator gaps at the other three.

A.5 Conclusion in Region III

  • All 64 corner gaps decompose into nonnegative combinations of ten generators, vanishing gaps, and two quantities with positive representations.The resulting generator positivity yields complete monotonicity for every control throughout the closed region with L > 0, and hence the inequality.Source: §A.5 · p. 40
  • The COMB representative 01010 has a vanishing gap at ω0 and strictly positive D1, B2, and P3 generator gaps at ω1, ω2, and ω3.At the corner identity, equality holds when a1 = a2 = a3 = 0.Source: §A.5 · pp. 40–41
  • When some ai > 0, a weighted corner contribution with a strictly positive inverse supports strict positivity of the transformed COMB gap.The other corner summands remain completely monotone; the positive contribution is bκ bg.Source: §A.5 · p. 41

B The Regions I and II certificate

The Regions I and II certificate uses coordinates ordered by 0 ≤ c ≤ b ≤ L, with Z ≥ 0. It expresses the transformed gap through Gv and takes the interpolation weights on (0, L).

B The Regions I and II certificate

B.1 Reduction to four corner families

The reduction uses four physical corner families and interpolation in b and c to extend their nonnegative inverse transforms across Regions I and II. The nonphysical corner is handled by an identity whose extra term appears only in Region I.

B.1 Reduction to four corner families

  • In Region I, swapping the first two control entries relates the nonphysical-corner gap to a physical-corner gap plus hv sinh L; Region II has no extra term.Here hv = 1 − (v1 − v2)^2, and the Region I term is included with its interpolation weights.Source: §B.1 · p. 41
  • Four physical corner families are specified by their scaled gaps and trace powers, with three meeting Region III at Z = 0.Family I does not meet Region III; the other three match its corner gaps at the interface.Source: §B.1 · p. 42
  • Nonnegative inverse transforms of the four physical corner families imply the inequality at every state in Regions I and II with L > 0.Source: §B.1 · p. 42
  • Two-point interpolation in b and c, together with the nonphysical-corner identity and Green term, reduces the regional inequality to the physical corner families.The interpolation weights and Green term lie in CM.Source: §B.1 · pp. 41–42

B.2 Positivity-preserving evolutions and the pure traces

The appendix propagates complete monotonicity through evolutions in the transverse variable, turning positive initial traces and forcing terms into positive curvature gaps. A new identity supplies three additional positive traces without extra certification cost.

B.2 Positivity-preserving evolutions and the pure traces

  • The evolution lemma preserves complete monotonicity for solutions and for the solution after subtracting its exponentially weighted initial data, when the initial data and forcing are completely monotone.The argument assumes bounded, continuous mild solutions and bounded, continuous forcing on the stated domain.Source: §B.2 · pp. 42–43
  • The appendix’s Lean formalization checks the boundedness, continuity, and Duhamel hypotheses for every flow used.The application data, forcing, and solutions are explicit bounded continuous functions on each stated truncated domain.Source: §B.2 · p. 43
  • The scaled gap families satisfy closed or forced evolution equations, allowing certified initial gaps and nonnegative forcing to propagate complete monotonicity.For the forced equations, the forcing terms have nonnegative time inverses, and the initial gaps are certified Region III comparisons.Source: §B.2 · p. 44
  • The new identity yields three further positive traces at no extra certification cost.The auxiliary Jm have Region III generator initial values, so complete monotonicity extends to Jm and Hm = Jm + mDm.Source: §B.2 · pp. 44–45

B.3 The derivative cascades and the scalar certificates

Derivative cascades reduce certain corner gaps to finite families of one-variable profiles. Their positivity is certified by analytic bounds and interval checks across separate parameter ranges.

B.3 The derivative cascades and the scalar certificates

  • Ten critical and twenty additional one-variable profiles remain after the derivative-cascade reductions for the nine critical and six further gaps.The cascades propagate positivity upward from the terminal derivative profiles when the initial profiles are completely monotone.Source: §B.3 · pp. 45–46
  • For the profile ranges, Poisson summation and first-mode domination handle the small- and large-parameter regimes, while rational-cell interval bounds certify the compact range.The cited cell counts are 251 for nine critical profiles, 513 for twenty further profiles, and 45 for the separate boundary datum.Source: §B.3 · pp. 46–47
  • The endpoint datum is positive for 0 < ξ ≤ 1 because each nonzero Poisson image reduces to a polynomial with strictly positive coefficients.Its zero image cancels, leaving only the nonzero images in the certificate.Source: §B.3 · p. 47
  • The remaining Region I boundary datum is certified using a compact-range check over 45 cells, with its zero mode dominating the large-ξ remainder.The small-scale principal lower bound is 3347091/2867200 √ξ; for ξ ≥ 1 the remainder is below 7.7167 × 10^-8.Source: §B.3 · p. 47

B.4 All sixty-four corner controls

The appendix expresses the corner gaps as nonnegative combinations of certified objects, completing the reduction to all physical corner controls. It then uses the corner transforms to establish the COMB equality condition.

B.4 All sixty-four corner controls

  • An audit checks 106 exact identities covering all 64 physical corner entries and verifies that each decomposition has nonnegative coefficients.The checks separately cover the four-expert background, continuous spectral kernel, and moving-endpoint atom.Source: §B.4 · p. 48
  • The corner decompositions, evolutions, and certified initial profiles put every corner gap in the completely monotone class, yielding the inequality for U.The proof reduces to four corner families and uses the stated corner-gap decompositions.Source: §B.4 · pp. 47–48
  • The COMB gap vanishes exactly when y1 = y3 = 0, equivalent to x1 = x2 and x3 = x4.The proof uses strictly positive transformed Region III COMB gaps away from that equality set.Source: §B.4 · p. 48

C Independent numerical checks

Independent numerical checks compare the formula with direct evaluations and a finite-difference Bellman solve. The scans agree with the predicted maximizing strategy and COMB ties, while the series become more costly as relevant length scales shrink.

C Independent numerical checks

  • The series and transform identities were checked against closed forms, quadrature, and the companion stationary solution at selected configurations.Reported agreements include 30 digits for the Region III series, 55 digits for Regions I/II, and 12 significant digits for the transform identity checks.Source: C Independent numerical checks · pp. 48–49
  • A monotone finite-difference Bellman solve agreed with the formula to a largest discrepancy of 6 × 10^-6 after one Richardson step.At the symmetric point, the solve gave 0.6921242 against the exact value 45π^3/2/(256) = 0.6921215; fixing v* and taking the full Bellman maximum agreed to 10^-13.Source: C Independent numerical checks · p. 49
  • The strategy v* attained the maximum at all 1466 retained scan points, with max |Uτ − 1/2 D2_v*U| = 5.0 × 10^-14.The scan covered Regions I, II, and III, including strictly ordered points, forced collisions, and points on S.Source: C Independent numerical checks · p. 49
  • On S, although Proposition 4.7 makes the COMB gap vanish identically, computed |∆| reaches 5.4 × 10−10, so the relative tie tolerance is set above that value at 10−9.The last column records the count at 10−12 for comparison.Source: C Independent numerical checks · p. 49
  • The series converge for every a4 > 0 or z1 > 0, but the required mode count grows as the smallest relevant length scale shrinks.Near the face z1 = 0, double precision loses about ϑ/ln 10 digits; extended precision or direct evaluation of the inverse-transform integral is advised.Source: C Independent numerical checks · p. 50
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