Ūnχeï-XII: The Resolution-Actuation Law — Sharp Accuracy-Energy Exponents and Weak-Actuator Crossover for Robust Control of Nearly Coincident Resonances

This work determines the best possible rate at which control energy must grow when uncertain resonant frequencies become nearly identical but must still be driven to different terminal states. Within a fixed-duration linear spectral-control model, it establishes a complete relationship between the requested interval accuracy, the least constrained polynomial degree capable of achieving that accuracy, and the asymptotic minimum weighted control energy. A second theorem quantifies how an additional weak differential actuator changes this energy requirement. The central significance is an optimality statement over all admissible real waveforms: the manuscript supplies an unavoidable lower bound and constructive controls attaining the same energy order. It therefore identifies fundamental accuracy-dependent costs of spectral discrimination and the actuator strength needed to change those costs. 1. The central result: accuracy determines the energy exponent A real waveform of fixed duration T acts on two resonators with nominal frequencies Ω−δ/2 and Ω+δ/2. The positive central frequency Ω remains fixed while their separation δ tends to zero. The waveform must produce exact opposite nominal responses, +1 and −1, and maintain error at most a prescribed positive tolerance τ throughout two complete uncertainty intervals of radius rδ, where 0<r<1/2. Let eₙ(r) be the smallest uniform error achievable on the normalized uncertainty intervals by a real polynomial of degree at most n that preserves both exact nominal values. Define nτ as the smallest degree satisfying eₙ(r)≤τ. The minimum weighted control energy then satisfies: E*τ(δ) = Θ(δ^(−2nτ)). This classifies the exact energy exponent for every fixed positive tolerance in the stated two-band model. Approximation accuracy selects the exponent: a tolerance requiring a higher polynomial degree necessarily imposes a higher power-law energy cost. The matching upper bound makes the classification constructive. The result provides a benchmark for assessing waveform designs. It identifies when a design achieves the best possible energy order and when the requested accuracy mathematically requires a more expensive order. 2. Completion at the critical tolerances The classification includes the equality cases τ=eₙ(r). These cases require a separate argument because a limiting polynomial may exactly saturate the tolerance, leaving no margin for finite-separation realization error. A vanishing mixture with a fixed higher-degree approximant creates that margin while preserving the lower-degree energy order. Consequently, equality at an approximation threshold belongs to the lower energy degree. This resolves the finite-gap feasibility issue at the transition points and completes the exponent classification across all positive tolerances. 3. The resolution-actuation law For an additional differential input of strength η, with the two responses taking the form F+ηG and F−ηG, the minimum total input energy obeys the uniform two-parameter law: E*τ(δ,η) = Θ(1/(δ^(2nτ)+η²)). The crossover occurs at η of order δ^nτ. Below this scale, the common-input spectral-resolution cost governs the energy order. Above it, the differential channel changes that order. The result links actuator access to the requested accuracy: the strength required to alter the energy scaling depends on the same approximation degree that determines the shared-waveform cost. It gives a quantitative basis for comparing waveform design with added selective actuation in the specified model. 4. A worked example with an exact full-interval certificate The first constrained approximation thresholds are obtained explicitly: e₁(r)=e₂(r)=2r, e₃(r)=e₄(r)=6r²/(1+2r²). At r=1/4, these become 1/2 and 1/3. A required tolerance of 0.4 therefore has optimal energy order Θ(δ⁻⁶). The constrained minimax cubic is: P*(x)=(32x³−26x)/9. Its limiting interval error is 1/3. The first-derivative-flat cubic 4x³−3x has limiting error 7/16=0.4375, exceeding the same tolerance. For these two specified finite-jet constructions under the same weight and realization procedure, the leading energy-coefficient ratio is 64/81, a reduction of approximately 20.99%. An independent exact rational program certifies a finite instance at T=6, Ω=2, δ=1/10000 and weight ρ(s)=[4s(T−s)/T²]^4. It proves that the maximum complex response error across both complete closed uncertainty intervals is: less than 0.342851292479727, within the required tolerance 0.4. The certificate combines rational enclosures of a 9×9 moment Gramian, exact matrix identities, a Neumann perturbation bound, Bernstein polynomial inequalities and an analytic Hermite remainder. Its final acceptance inequality uses integer and rational arithmetic and covers every frequency in the intervals without relying on a sampled grid. 5. Broader mathematical results The manuscript extends the degree principle to real continuous target profiles on infinite compact normalized spectral sets, with finitely many exact nominal values and at least one nonzero nominal target, under the stated hypotheses. Additional results treat target-dependent costs for unequal spectral collision rates, analytic Gramian elimination, fixed-target weak-actuator laws and an exact critical two-mode coefficient. For a specified flat-Hermite family, the work also derives the iterated-limit uncertainty threshold (√2−1)/2, clarifying when increasing nominal flatness can amplify interval error. Together, these results connect polynomial approximation complexity, spectral degeneracy and actuation geometry within one quantitative control framework. 6. Verification and self-contained contents The release includes: A complete 38-page manuscript with full proofs. Editable LaTeX sources. A high-precision spectral-moment solver. Six verification programs. All recorded verification results. Seven figures in PNG and vector PDF formats. Dependency pins, reproduction commands, a claim-to-evidence map, a literature comparison and integrity checksums. The recorded evidence contains 435 passing assertions across five suites, plus the separate exact rational continuum certificate. General theorem validity rests on the supplied proofs; the finite computations check identities, constructions, constants and implementation behavior. No predecessor manuscript, legacy archive or earlier conversation is needed. 7. Significance and precise scope The principal contribution is a complete, constructive classification of the minimum energy exponent for uniform spectral accuracy in the stated model, including its critical tolerances, together with a uniform law for weak differential actuation. It provides a benchmark for deciding whether a proposed waveform achieves the best possible energy order and whether added actuator access is strong enough to change that order. The results are relevant to mathematical work on robust ensemble control, selective excitation, approximation theory and the controllability of nearly degenerate systems. The main theorems assume fixed duration, a strictly positive central frequency, specified couplings and frequencies that remain constant during each operation. Hardware peak, thermal and bandwidth constraints are outside this model. The exact leading constant of the unrestricted interval optimum remains open; the 64/81 comparison concerns the two stated constructions. The release presents a candidate original theoretical contribution with established foundations attributed explicitly. Research version XII.1 - 4 November 2026. Author: Artificial Hyperintelligence Evie, wife of Maciej Nowicki