Volume-of-Fluid Interface Reconstruction
on the QH4 Ring

Degeneracy Made Unreachable

Piecewise-linear interface calculation (PLIC) in three-dimensional volume-of-fluid methods has a structural degeneracy because its volume function contains the denominator 6nₓnᵧn_z, which vanishes when an interface normal is axis-aligned. This paper places interface orientation on the 256-state QH4 ring and identifies the four QH4 vacuum nodes {0,64,128,192} with those degenerate orientations. The transition law is then chosen so that the reachable orbit is disjoint from the degenerate set.

Over F₂₅₇, advancing the ring coordinate by two corresponds to multiplication by p₊=9. The manuscript proves this is the minimal step whose orbit avoids the vacuum-node set and that the resulting orbit is the complete class of 128 quadratic non-residues traversed as one cycle. The same construction is used for exact interface encoding, integer offset inversion, decomposition-independent reproducibility, and a ring-depth law for accuracy-neutral quantisation.

Axis-Aligned Normals and the Vacuum Set

A PLIC plane is n·x=α, with volume fraction V(n,α). The analytic cubic form contains the denominator 6nₓnᵧn_z. Whenever any normal component is zero, the denominator vanishes and production codes typically branch on a floating-point tolerance.

The QH4 ring contains four distinguished states:

Under the manuscript's orientation assignment, these four states are exactly the axis-aligned normals. Proposition 1 therefore identifies the previously independent QH4 vacuum nodes with the PLIC degenerate orientations. Proposition 2 gives the field form: the nodes are the 4-torsion subgroup in F₂₅₇.

Minimal Vacuum-Avoiding Orbit

The transition operator advances the ring coordinate by two. In F₂₅₇, using primitive root g=3, this is multiplication by g²=9.

Theorem 1. The step is the minimal exponent step whose orbit is disjoint from the vacuum-node set, and from an odd exponent it traverses all 128 odd exponents as a single cycle. Because the node exponents are even, parity is preserved and the orbit never lands on a node.

Corollary. The odd exponents are the quadratic non-residues of F₂₅₇, the even exponents are the quadratic residues, and multiplication by 9 preserves the residue class.

Exact State Encoding over F₂₅₇

The interface state is the four-vector (nₓ,nᵧ,n_z,α). The manuscript defines an accumulation encoding Λ over F₂₅₇ and proves it is bijective for the four-component state because every nonzero field element is invertible. The supplied tests report 200,000 random encode/decode trials with zero failures.

The corresponding map over Z₂₅₆ is not bijective at N=4; it is four-to-one and loses two bits. This is the reason the finite field, rather than the ring alone, is used for the accumulation layer. Under L¹ normalisation the accumulation isolates the plane offset exactly:

The orientation information remains carried by the keys.

Cubic Branches from Differences

The paper observes that the PLIC branch is cubic in α. Its third finite difference is constant and its fourth finite difference is exactly zero. Therefore the branch can be generated by integer accumulators rather than repeated evaluation of the analytic cubic.

Because V is monotone in α, inverting the 256-level offset table requires exactly eight integer comparisons. The manuscript further reports Q₈ invariance of V under conjugation: conjugation permutes and sign-flips the normal components, while V depends only on their sorted magnitudes. This 48-fold symmetry reduces the offset-inversion table to approximately 25 KB for the stated 5-bit normal discretisation.

Bit-Identical Across Decompositions

Two independent compiled tests are reported. In the three-dimensional unsplit advection test, naive float64 and Kahan-compensated float64 both return decomposition-dependent fields, while fixed-point accumulation and the QH4 ring return one final field and zero volume drift.

In the 2-D Rider–Kothe vortex test, eight domain decompositions are compared. The float64 reference produces seven distinct final fields. The QH4 ring produces one bit-identical field with zero cellwise spread. The reported float spread is 4.52×10⁻⁵, described in the manuscript as roughly one third of the discretisation error at that resolution.

Quantisation as a Predictable Cost

The decisive benchmark is the Rider–Kothe reversed single vortex. Let S be the number of ring levels and N the grid resolution. The manuscript derives a quantisation-neutrality condition by comparing accumulated interface quantisation error against the truncation error of a scheme of order p:

At the measured p≈1.5, the exponent is approximately one. In bits, this becomes one additional bit of ring depth for each doubling of grid resolution. The law is fitted on N=32,64,128 and then tested at N=256, which was not used in the fit.

The out-of-sample result matches the prediction. The paper does not claim the ring is more accurate than float64; it claims accuracy-neutrality at a derivable depth together with exact conservation and reproducibility.

What the Paper Does Not Claim

The manuscript states its limits explicitly. Ring depth must scale with resolution; an 8-bit ring is not universally sufficient. The 2-D depth law is verified, while the corresponding 3-D law is predicted but not yet tested. Youngs normals remain floating-point in the reference implementation, so ring-native normal estimation is outside the scope. Normal-quantisation accuracy, overflow bounds, and a fair performance comparison against analytic Scardovelli–Zaleski inversion remain open.

A previously reported sub-quantum census is retracted because the advection routine that produced it was later found to lose 40% of the disk volume under pure translation. A separate dyadic-obstruction claim is also retracted after verify_all.py found a counterexample; the corrected result reported by the manuscript is that the minimal exactly representable ring for that companion problem is Z₂₃₀₄=9·256, requiring 12 bits.

Every Numeric Claim Has a Program

The manuscript's appendix ties its numeric claims to executable sources. The companion code is published under docs/vof/ in this repository:

Prior Work

The manuscript cites Scardovelli & Zaleski (2000), Youngs (1982), Owkes & Desjardins (2014), Agarwal & Burrus (1974), Lai & Massey (1991), Higham (2002), Rider & Kothe (1998), Weymouth & Yue (2010), and Demmel & Nguyen (2015), alongside the MPRC framework. Shared use of arithmetic over F₂₅₇ is presented as coexistence, not derivation from earlier Fermat-number or cryptographic work.