Unified Limit and the Harmonic Ladder: A Unified Steepness Quantization Linking Hydrogen to the Neutron-Star Collapse Limit (Under review at <i>Advances in Astronomy</i>, submitted July 23, 2025)

<p dir="ltr">This paper presents a unified geometric framework linking quantum and astrophysical structure through a quantized ladder of energy-density steepness. In the <b>Mass–Energy Gravity (MEG)</b> model, all coherent systems, from hydrogen to neutron stars, occupy discrete rungs on a <b>harmonic ladder</b> governed by a universal stride (C ≈ 9.72) and bounded by the <b>Unified Limit (U)</b>, the steepest sustainable energy-density gradient in nature.</p><p dir="ltr">Two independent derivations, (1) a geometric climb from hydrogen’s quantized energy-density gradient and (2) an observational ceiling anchored in neutron-star collapse (PSR J0740+6620), converge on the same terminal value within ±3 %, confirming that atomic and relativistic thresholds follow one quantized steepness law. The model reproduces the observed <b>3–4 M⊙ compact-remnant gap</b>, explains the symmetric placement of hydrogen between the CMB energy-density floor and the neutron-star ceiling, and predicts half-integer “paired-mode” phenomena such as the CMB harmonic peaks.</p><p dir="ltr">By deriving quantization, stability limits, and gravitational correspondence from a single scalar steepness field, the work unifies quantum, stellar, and cosmological structure under one coherence principle, the harmonic ladder of the Unified Limit.</p>