The Capacity Harmonization Theorem: A Forcing-Harmonizing Duality and Exact Curvature Gap in Intelligence-Bounded Capacity Allocation

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 gᵢ (r) = cᵢ·r − (kᵢ/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, kᵢ > 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 (chtforcingₒptimalₐtceiling): 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 (chtₑquimarginalᵢsₛtrictglobalₘax) ; 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 (chtₑquipartitionₛymmetric), and the capacity-writing rate itself obeys the Intelligence Bound, rate ≤ P·D/ (kB T ln 2) (chtcapacityIBceiling). 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 (chtₙonvacuous: 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 Working paper; not peer-reviewed.