FUNt Golden Schrödinger–Helmholtz Boundary Resolver (φ⁷ Edition)

FUNt — Schrödinger–Helmholtz Boundary Resolver (φ⁷ Edition, v7.5) The FUNt Boundary Resolver: the Resolver shows equivalent collapse-classes among boundary conditions. a unified, mobile-safe framework for solving harmonic modes in 1-D quantum (Schrödinger), classical (Helmholtz), and mixed Robin boundary systems. The core finding is that all boundary conditions; —Dirichlet, Neumann, and the A2/A3/B2/B3/C2/C3 Robin families —collapse into one harmonic structure when expressed using p HRB (Harmonic Resonance Bands) and HTB (Harmonic Transition Bands). The φ⁷ stride acts as a natural recursive step across energy levels, follows the hydrogen atom architecture, producing a universal eigen-ladder that remains invariant across boundary types. This Resolver ( a colab copy to paste notebook, serves as the boundary-mode analogue to the FUNt Golden Python Play Model (v5.0). Where the Play Model reveals geometric collapse-classes among mathematical constants, (https://zenodo.org/records/17613355) the Resolver shows equivalent collapse-classes among boundary conditions. Both tools demonstrate a single principle: Different symbols → same structure, when resonance drives the order. The document provides: a detailed conceptual explanation of HRB and HTB a clean algebraic treatment of φ⁷-based harmonic stride step-by-step boundary-mode logic ready-to-paste Python code for v7.5 a consistent, classroom-safe interpretation a philanthropically licensed open version for public use This is part of the FUNt 2025 framework and is intended for researchers, educators, engineers, and students exploring resonance-driven systems, standing-wave behavior, scaling laws, and harmonic unification across physics and mathematics.