Parametric Symmetry in the Cubic Prouhet–Tarry–Escott Problem (v6): Non-Ideal Polynomial Templates, Dual Reconstruction Schemes, and Infinite Arithmetic Families

This release presents version 6 of the manuscript on cubic-only Prouhet–Tarry–Escott (PTE-type) identities for r = 3, viewed as a computationally generated infinite family of balanced N×N constructions. The frequently shown 4×4 case is treated as a minimal canonical demonstrator, not as the scope limit.

At the polynomial level, the core invariant relation is

∑ S₁³ = ∑ S₂³,

while lower moments (r = 1, 2) and r = 4 generally do not match (non-ideal in the classical PTE sense).

What is new in v6:

• Dual reconstruction framework is now formalized:
• Scheme 1 (scale-based): interpolation from three decimal-scale probes (the extended GUI route), including all-nines and uniform-middle masks (middle digit block only 0 or 9).
• Scheme 2 (mask-family over b₀): fixed-edge decimal families of the form
  b₀ = n₁ d...d n₂, with n₁ ∈ {1,...,9}, d ∈ {1,...,8}, n₂ ∈ {0,...,9}.
• For Scheme 2, three neighboring probes in middle-run length are used to recover quadratic slots directly in b₀, then verify the invariant
  P(b₀) = ∑S₁³ = ∑S₂³.
• This extends practical applicability for exploratory scans: broader controlled search domains, more stable slot matching, and invariant-driven validation in structured decimal families beyond the original all-nines trajectory.

Contents of the manuscript: explicit polynomial templates, symbolic verification in Q[x], non-ideality diagnostics, numerical examples, and practical guidance for computational research workflows built around telescopic decomposition experiments.

In short: v6 upgrades the work from a single practical reconstruction track to a two-scheme parametric methodology, improving both theoretical clarity and experimental usability.

Windows build of the extended GUI: https://nvvorobtsov.github.io/sources/decompose_gui_EXT_windows.zip

Prior Zenodo revision: 10.5281/zenodo.19533934 (version 5). This v5 record is superseded for citation of v6-specific material after publication of the present v6 record.