Can a compact S³ admit an intrinsic geometric mechanism that suppresses extreme curvature growth without invoking topological surgery? This work addresses that question from a purely mathematical perspective. Unlike the operational Planck-boundary framework—where ℓP represents the limit of physical distinguishability rather than a dynamical ingredient of Ricci flow—this study investigates modified Ricci-type flows as autonomous geometric systems. Its goal is not to revise the Planck-boundary interpretation, but to identify mathematically natural curvature regulators within the flow itself. The paper demonstrates that several intuitive candidates—including bounded traceless forcing, fibre-restoring forcing, and a constant cosmological term—cannot control the dominant cubic curvature reaction. Instead, it introduces a variational curvature barrier, generated by the negative L²-gradient of the curvature functional FΦg = ∫S³ Φ(K) dμ, where K = |Rm|². A singular Planck-wall functional is then constructed as Bₚ,ᵩg = ∫S³ (K + δ)ᵠ / (KP − K)ᵖ dμ, with KP = ℓP⁻⁴, providing a mathematically consistent local high-curvature regulator. A central result is the exact three-dimensional identity K = 4|Ric₀|² + R²/3, showing that K simultaneously detects isotropic and anisotropic curvature. The corresponding conformal variation yields the decisive criterion 4KΦ′(K) − 3Φ(K). For the power-law family Φ(K) = Kᵠ, this produces the sharp threshold q > 3/4, establishing the minimal scaling required for a supercritical curvature barrier in dimension three. On the Berger family, the analysis further identifies an invariant cone governing the anisotropy ratio and derives explicit evolution equations for curvature, scalar invariants, and the Planck-wall Euler tensor. Equally important is the paper's negative result. It proves that these curvature barriers cannot be transformed into a noncollapse theorem through simple additive modifications of Perelman's entropy. Both polynomial curvature potentials and singular Planck-wall corrections fail analytically on Berger geometries. Consequently, the work separates two distinct mechanisms: local suppression of supercritical curvature and global noncollapse monotonicity. The former is achieved by the variational barrier; the latter remains an open mathematical problem requiring a genuinely non-additive entropy or propagation principle. Rather than extending the physical compact S³ Planck-boundary programme, this article establishes an independent mathematical research branch devoted to variational curvature barriers for Ricci-type flows. It clarifies which geometric regulators are mathematically viable, which approaches provably fail, and where future work must focus to develop a complete intrinsic alternative to surgery.
Batenin et al. (Sun,) studied this question.