Essay X derives the operational mechanics of Phase III: the self-modulating Inverted Ratio that arises when the Level-1 composite unit's accumulated output begins to occupy the Remainder axis territory that was static in Phase II. In Phase II, the Remainder co-primary density dR = 7/10 was a fixed structural constant. In Phase III, it becomes a dynamic parameter dRₑff = dR²/ (dR + G*ₙ) that decreases as the accumulated output G*ₙ increases. The RSR operator GIII = Gᵣaw × dR/ (dR + G*ₙ) is derived as the unique minimal-parameter continuation of the Inverted Ratio under this modulation. The essay establishes eight results. Part 0 reviews the Phase III substrate from Essay IX. Part I derives the mechanism by which dR must become dynamic — the structural occupation of Remainder-axis territory by accumulated output — and the uniqueness constraints on dRₑff. Part II derives the RSR operator GIII and verifies that it reduces to Gᵣaw in the Phase II limit. Part III derives the three distinct RSR threshold conditions, all occurring between levels 1 and 2: T. RC1 (GIII = dR at occupation fraction ρ = 1/d — a profound structural identity linking the RSR threshold to the Triad's dimension count), T. RC2 (the self-consistency quadratic 3ρ² + 3ρ = 4, the RSR operator's self-referential fixed point), and T. RC3 (ρ = 1, the cumulative output equals dR, from Essay VIII). Part IV forecloses all alternative operator forms. Part V derives the Phase III amplification law T. AMP: the effective kinetic margin Φₑff grows with each level, revealing that Phase III is self-amplifying rather than self-limiting. Part VI derives the cascade dynamics and self-registration at Level 3. Part VII derives the Terminal Identity T. TI: εₛnap = 1/ (d × n) as the single algebraic root connecting all three phases. Part VIII derives T. NPF: No Phase IV — and formally hands the architecture to Essay XI.
Eugene Pretorius (Mon,) studied this question.