The Helical-Space Framework: A Four-Dimensional Geometric Theory Without an External Time Coordinate — Axioms, Derivational Boundaries, and Falsifiable Predictions

We propose and systematically stress-test a theoretical framework in which spacetime consists of three spatial dimensions plus one compact "matter dimension" (topologically a circle), with no external time coordinate. All elementary motion is right-handed helical motion subject to the quota constraint (c\,dφ)^2 = dm^2 + d^2, and time emerges as the count of helical phase. Within this axiom system we obtain: (i) a parameter-free derivation of the Lorentz factor; (ii) a gravitational field equation exactly equivalent to the spherically symmetric sector of general relativity (the full suite of Schwarzschild tests passes automatically), together with the frame-dragging coefficient and the transverse tensor wave modes derived conditional on an observer-universality completion of the field equation — a selected principle (Section 2.5) — (concordance with GP-B, LAGEOS/LARES, and GW170817); (iii) a two-winding-number topological encoding of charge quantization, a geometric slot for the three lepton generations, and the CPT structure, with an even-winding rule derived from the spinor double cover; (iv) a reduction of all fundamental "constants" to power laws of a single medium-strain variable χ, predicting a rank-one locking of constant variations not shared by existing theories, whose operational signature is the exact constancy of μ=m_e/m_p (the α-response is branch-dependent, Section 9.6); (v) under the holographic pinning condition A′ (introduced and partially derived in Section 9), a mass quantum k=√ m_P H_0/c^2=4.2±0.2 meV that organizes the neutrino mass spectrum (the pure-integer prediction m_3/m_2=6; Σ m_ν ≈ 59 meV; subject to a current ~2σ tension between its two mass anchors) and the dark-energy density (compressing the cosmological-constant discrepancy from 122 orders of magnitude to one, the direction now derived from gravitational stability), and yields the master identity Gk^4=^3H_0^2/c^3, welding the Hubble tension and the neutrino mass spectrum into two measurements of one quantum. We equally delimit what the framework cannot derive: the electromagnetic–gravitational hierarchy number N≈ 2×10²¹ is shown to be underivable within the scale-free axiom set by four independent arguments; the fine-structure constant is reclassified, once A′ enters, from forbidden-in-principle to reprieved pending the microscopic layer, its geometric identity argued to be the twist-vortex log-impedance 1/α=ln(RIR/rUV); and quantum many-body entanglement constitutes a hard ceiling. Eight falsifiable fingerprints are stated; two of them (the dark-energy equation of state w; a forced bet against MOND) together with the two A′ verdicts of Section 9.3 (the JUNO integer spectrum; Σ m_ν) face scheduled experiments within the decade.