The Capacity Harmonization Theorem: A Forcing-Harmonizing Duality and Exact Curvature Gap in Intelligence-Bounded Capacity Allocation
Justin Hart, Aristotle (Harmonic) · Zenodo (CERN European Organization for Nuclear Research) · 2026
Standalone thematic-series record of the Viridis Canon (S4 Stewardship/Governance). The core Intelligence-Bound canon spine is unchanged (frozen at v10.0.0, record 20801185); this record links to the spine via isDerivedFrom the concept DOI 10.5281/zenodo.19317982. We prove that allocating a conserved budget R across two capacity-writing subsystems as (x, R − x), each with a strictly concave quadratic yield g_i(r) = c_i·r − (k_i/2)·r², exhibits a sharp forcing/harmonizing duality governed by the Intelligence Bound. The emergent equimarginal (shadow-price, “wu wei”) allocation is x* = (c₁ − c₂ + k₂R)/(k₁ + k₂). Below the Landauer ceiling (concave yields, k_i > 0) the harmonized split x* is the unique global maximizer of aggregate capacity, and the loss of any other allocation is an exact positive-definite Bregman curvature form ICB(x*) − ICB(x) = ((k₁+k₂)/2)·(x − x*)², vanishing iff the marginal profile is flat. The paradox. At the ceiling the yield becomes exactly linear (k₁=k₂=0), its marginal is constant, and the optimum flips to bang-bang forcing (cht_forcing_optimal_at_ceiling): with a strictly richer subsystem (c₂ < c₁), x = R — all budget to the richer subsystem — strictly dominates every interior split. Harmonizing dominates everywhere the Intelligence Bound is slack (cht_equimarginal_is_strict_global_max); forcing dominates only at the unreachable saturation limit. Real hardware runs 1019–1020× below the ceiling, so harmonizing is the physically operative regime. The symmetric case recovers exact equipartition, xStar c c k k R = R/2 (cht_equipartition_symmetric), and the capacity-writing rate itself obeys the Intelligence Bound, rate ≤ P·D/(k_B T ln 2) (cht_capacity_IB_ceiling). All six results are machine-checked in Lean 4 (Aristotle, zero sorry, axioms ⊆ {propext, Classical.choice, Quot.sound}), with an explicit interior non-vacuity witness (cht_nonvacuous: instance (c₁,c₂,k₁,k₂,R)=(2,1,2,2,1) gives x* = 3/4 ∈ (0,1) with strict gap ICB(0) < ICB(x*)). The distinctive content relative to the rest of the shadow-price water-filling family is the Intelligence-Bound forcing/harmonizing duality and the exact curvature/Bregman gap. Scope. The Lean proofs certify the validity of the reasoning, not empirical magnitudes. Working paper; not peer-reviewed.