A Quaternion--BCH Framework for the Local Accuracy of SIDER Interpolation

Abstract: Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation method for data on the unit sphere $\mathbb{S}<sup>2$, built from repeated spherical linear interpolation (SLERP). This paper develops a quaternion--Lie algebra framework for proving the local consistency of SIDER for smooth spherical curves sampled at equally spaced parameter values. Points on $\mathbb{S}$ are represented as pure unit quaternions, and interpolation errors are measured in fixed-base quaternion logarithmic coordinates. In this setting, each SLERP operation admits an exact Baker--Campbell--Hausdorff (BCH) representation, which converts the geometric interpolation problem into an algebraic problem involving filtered Lie-polynomial expansions. The BCH expansion shows that SLERP is affine to leading order, has no quadratic correction, and has a first nonlinear correction that is cubic and commutator-valued. Using this structure, we prove that SIDER2 has a third-order divided-error form with the same leading nodal factor as ordinary quadratic interpolation. We then show that the recursive SIDER step raises the order by one: the affine part gives the Neville-type finite-difference cancellation, while the nonlinear BCH remainder preserves the sharp filtered degree structure after the nodal factor is removed. Consequently, for every fixed $n\geq2$, $$d_{\mathbb{S}^{2}\bigl(γ(θh),P_i^{{[n]}(θ;h)\bigr)}} = O(h^{n+1})$$under the stated smoothness and small-stencil assumptions. The proof also identifies the shift-invariance of the leading divided-error coefficient as the algebraic compatibility condition underlying the SIDER recurrence.

• The paper proves that fixed-order SIDER interpolation achieves local spherical error O(h^{n+1}), extending SIDER2’s third-order accuracy to SIDER3 and all n≥2 under smoothness and small-stencil assumptions.
• Its exact quaternionic BCH formulation shows that SLERP has no quadratic correction and that the first nonlinear effect is a cubic commutator term, clarifying how curvature enters interpolation errors.
• A filtered divided-error argument preserves Neville-style cancellation through the recursion, while the result remains local and does not establish stability, monotonicity, non-oscillatory behavior, or accuracy for nonuniform stencils.

Overview and motivation

This paper establishes a rigorous local accuracy theory for SIDER (Spherical Interpolation of orDER $n$), a recursive family of high-order interpolants on the unit sphere introduced previously by Fong and Leung as the reconstruction engine of the SENO method (2606.12833). SIDER replaces every affine blending step of Euclidean Neville interpolation with SLERP, thereby preserving the spherical constraint at each stage, but this substitution introduces curvature- and commutator-dependent corrections that made direct proofs in normal coordinates increasingly unwieldy at higher orders. The contribution here is an alternative, fully algebraic proof: points on $S^2$ are embedded as pure unit quaternions, errors are measured in fixed-base left-trivialized logarithmic coordinates, and each SLERP operation admits an exact Baker–Campbell–Hausdorff (BCH) representation. The main result is that for every fixed $n \geq 2$,

$$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$

under smoothness and small-stencil assumptions, together with a sharper filtered divided-error representation of the logarithmic error.

Quaternion formulation of SIDER

The paper identifies $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$, where the Lie bracket satisfies $[X,Y] = 2X\times Y$, so nested commutators admit a geometric reading as cross-product corrections. A point $p \in S^2$ is embedded as the pure unit quaternion $(0,p)$; although pure unit quaternions do not form a subgroup, SLERP between two such points remains in their real span and hence stays pure, so the quaternion formalism remains faithful to the original sphere problem.

The SIDER2 construction uses two geodesically extrapolated control points ($\lambda = 2$ SLERPs beyond the middle node) followed by three SLERPs with parameter $\xi/2$, where $S^2$0. Higher levels apply the recursion

$S^2$1

the exact spherical analogue of the equally spaced Neville recursion. All analysis is conducted under a small-stencil assumption: for sufficiently small $S^2$2, all data, extrapolated control points, and intermediate values lie in a common normal neighborhood of the exact value, guaranteeing single-valued logarithms, a consistent local branch, and uniform BCH validity.

Exact BCH structure of SLERP

The central identity is exact rather than asymptotic:

$S^2$3

Expanding through degree three yields a structurally decisive fact: the quadratic term vanishes identically, and the first nonlinear correction is cubic and commutator-valued,

$S^2$4

Two consequences follow immediately. First, since commutators vanish for collinear arguments, SLERP is exactly affine along any one-parameter subgroup — the non-affine part measures precisely the failure of endpoint errors to share a common infinitesimal geodesic direction. Second, if both inputs are $S^2$5, the nonlinear remainder is only $S^2$6, an "affine stability" property used repeatedly in the induction. At all higher degrees the map is a Lie polynomial whose coefficients are polynomials in $S^2$7 of degree at most the homogeneous order.

Filtered divided-error calculus

The proof machinery is a filtration on formal series in $S^2$8: a series is degree-filtered if the coefficient of $S^2$9 is a polynomial in the shifted stencil variable $n \geq 2$0 of degree at most $n \geq 2$1. Two elementary lemmas drive everything: shifted finite differences preserve the filtration (because differencing lowers polynomial degree by one), and an error vanishing at nodes $n \geq 2$2 divides coefficient-wise by the nodal product $n \geq 2$3, with quotient coefficients of degree at most $n \geq 2$4. This is the algebraic reason interpolation at $n \geq 2$5 nodes forces consistency order $n \geq 2$6.

Base case: SIDER2

The base estimate shows SIDER2 has exactly the leading error of ordinary quadratic interpolation in logarithmic coordinates:

$n \geq 2$7

where $n \geq 2$8 is the third left-logarithmic Taylor coefficient of the curve. The key verification is that the cubic BCH correction contributes nothing at order $n \geq 2$9: at that order it sees only the first-order data $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$0, which are all scalar multiples of the same tangent element and hence have vanishing commutators. Curvature effects enter only at higher orders, where they are absorbed into the filtered coefficient. This explains why SIDER2 matches Euclidean quadratic accuracy despite the noncommutative geometry.

Recursive order raising

The induction step rests on two facts. The affine part of the final SLERP produces the Neville cancellation: the shifted nodal products combine as

$$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$1

so the affine contribution equals $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$2. Because the leading coefficient of a degree-filtered series is independent of $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$3, this difference carries one extra factor of $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$4 — the shift-invariance of the leading divided-error coefficient is identified as the algebraic compatibility condition underlying the entire recurrence.

The nonlinear BCH remainder is controlled via the diagonal variable $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$5: every term of the nonlinear map contains at least two factors of $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$6, each contributing one degree saving in $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$7, which exactly compensates the powers of $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$8 contributed by the interpolation parameter. Combined with vanishing at all $$d_{S^2}\bigl(\gamma(\theta h), P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}),$$9 nodes, the remainder takes the form $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$0 with $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$1 degree-filtered. Notably, the remainder is $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$2, so for $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$3 it is automatically of order at least $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$4.

Main theorem

Induction from the SIDER2 base through the order-raising lemma yields the filtered divided-error representation

$\operatorname{Im}(\mathbb H) \cong \mathbb R^3$5

with $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$6 degree-filtered and bounded for bounded $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$7, hence the $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$8 distance bound for every fixed $\operatorname{Im}(\mathbb H) \cong \mathbb R^3$9 (SIDER2: $[X,Y] = 2X\times Y$0, SIDER3: $[X,Y] = 2X\times Y$1, etc.). Constants may depend on $[X,Y] = 2X\times Y$2, finitely many curve derivatives, and the normal neighborhood size, but not on $[X,Y] = 2X\times Y$3.

Limitations and open questions

The paper is explicit about scope. The theorem is local and asymptotic: it requires fixed $[X,Y] = 2X\times Y$4 and $[X,Y] = 2X\times Y$5 small enough that all logarithms and SLERP operations share a common branch, and it does not address antipodal ambiguity, large angular separation, or geodesic non-uniqueness. The argument covers only equally spaced parameter values; extension to nonuniform stencils would require reformulating the divided-error calculus with nonuniform Neville weights, though the BCH representation itself would be unchanged. Finally, the analysis concerns the smooth SIDER operators only — it provides the consistency foundation for SENO's candidate reconstructions but says nothing about stability, monotonicity, or non-oscillatory behavior of the stencil-selection mechanism.

A disclosure section states that the derivations were produced with substantial AI assistance (GPT-5.5 Thinking) and that the author acknowledges the mathematical depth exceeds current manual verification capability, presenting the results as AI-assisted hypotheses pending formal peer review. Readers should weigh this caveat when relying on the detailed coefficient-level computations, particularly the cubic BCH expansion and the diagonal-variable degree bookkeeping.

Conclusion

The paper recasts the local accuracy of spherical recursive interpolation as a problem in filtered Lie-polynomial algebra. The exact BCH form of SLERP reveals that its quadratic correction vanishes and its first nonlinear term is commutator-valued, which suffices to show SIDER2 inherits the classical third-order divided-error structure. A diagonal-variable argument then shows the nonlinear remainders preserve the sharp degree filtration through the Neville-type recursion, proving $[X,Y] = 2X\times Y$6 local consistency for all fixed $[X,Y] = 2X\times Y$7 and isolating shift-invariance of the leading divided-error coefficient as the compatibility condition that makes the Euclidean cancellation survive on the sphere.