Capacity-Fluctuation Geometry and Commitment Obstruction Accounting

this is paper 7 in the series it is preceded by Semantic Observers: A Functional Criterion for Observer-Systems in the Quantum Measurement Problem. Zenodo. https://doi.org/10.5281/zenodo.21711777 and Commitment Time, Metric Time, and Physical Temporal Interfaces. Zenodo. https://doi.org/10.5281/zenodo.21744668 and Stoney–Planck Action Cells and the Planck-Regulated Commitment Substrate https://doi.org/10.5281/zenodo.21749995 and Holographic Screen Capacity, Composition, and Operational Fission. Zenodo. https://doi.org/10.5281/zenodo.21750518 Capacity-Fluctuation Geometry and Commitment Obstruction Accounting Stoney–Planck Cell-Scale Invariants, Conditional Curvature Transfer, Cancellation Tomography, and Causal Einstein-Basin Admission This paper develops a typed mathematical framework connecting finite holographic-screen statistics, calibrated physical residuals, continuum tensor response, and controlled spacetime geometry. Every obstruction is assigned to a declared physical owner equation: rₓ = ℛₓ[physical trajectory]. The owner may be an action equation, transport law, constitutive relation, conservation identity, matching condition, continuum realization map, or causal field equation. The residual inherits its domain, codomain, units, evolution, and admissibility conditions from that owner. The construction distinguishes positive physical load, signed cancellation, finite-screen fluctuation, inverse-area normalization, source residuals, detector statistics, tensor response, metric displacement, and spacetime curvature. Explicit realization maps and branch-specific identities connect these quantities. For a positive dimensionless channel load λ, the Stoney–Planck action and capacity relations are Aλ = λℏ, Nλ = 4πλ⁄ln 2, Aλ = Nλℏ ln 2⁄(4π). Here Aλ is the channel action and Nλ is the expected number of active binary screen units. The corresponding self-similar cell scales are ℓλ = √λ ℓₚ, tλ = √λ tₚ, Eλ = √λ Eₚ, mλ = √λ mₚ. The cell action-density, pressure-normalization, and inverse-area scales are uλᶜᵉˡˡ = c⁴⁄(Gℓλ²) = uₚ⁄λ, Pλ = uλᶜᵉˡˡ, ℛλ = 1⁄ℓλ² = ℛₚ⁄λ. These scales satisfy the exact capacity-weighted identities NλPλ⁄uₚ = 4π⁄ln 2, Nλuλᶜᵉˡˡ⁄uₚ = 4π⁄ln 2, Nλℛλ⁄ℛₚ = 4π⁄ln 2. The invariant capacity–inverse-area product is 𝒦ᵢₙᵥ = 𝒦ₚ = 4πℛₚ⁄ln 2. The cell branch therefore carries a positive action load, a finite capacity, a pressure-normalization scale, an inverse-area scale, and an exact capacity-weighted invariant. Physical stress and curvature arise through the corresponding action, metric variation, conservation, continuum realization, and causal response maps. The action–capacity relation also defines capacity-normalized metrological generators. For a differentiable Hamiltonian family H(q,t), the local parameter generator is K̂q = iℏ Ûq(τ,0)† ∂qÛq(τ,0), with the integral representation K̂q = ∫₀ᵀ Ûq(t,0)† ∂qĤ(q,t) Ûq(t,0) dt. For a pure probe state, the quantum Fisher information is FQ(q) = 4 Var(K̂q)⁄ℏ². For M independent repetitions, Var(q̂) ≥ 1⁄[M FQ(q)] and Δq̂ ΔK̂q ≥ ℏ⁄(2√M). The implemented generator, physical carrier, probe state, measurement, estimator, and observation window determine the resulting metrological response. For homogeneous logarithmic volume strain, the generalized force is the volume-integrated pressure. The capacity-normalized relation becomes Δq̂V · ΔĴP⁄Aλ ≥ 1⁄(2√M λ) or equivalently Δq̂V · ΔĴP⁄Aλ ≥ 2π⁄(√M Nλ ln 2). For a compactly supported metric deformation gμν(s) = gμν⁽⁰⁾ + shμν, the action-valued metric generator is 𝒢̂h = 1⁄(2c) ∫ √−g T̂μνhμν d⁴x. Its definition includes the smearing tensor, spacetime support, stress-tensor prescription, physical gauge quotient, boundary conditions, and spacetime-measure convention. The associated Fisher geometry provides a local resolution law for the implemented metric deformation. The statistical analysis covers equilibrium estimation, entropy curvature, nonequilibrium precision, and gravitational metrology. For a regular exponential family with Fisher matrix 𝓘, M independent samples satisfy Cov(θ̂) ⪰ 𝓘⁻¹⁄M. On an isothermal–isobaric branch, ΔP̂ ΔV ≥ kBT⁄√M. Canonical temperature estimation gives ΔT̂ ΔE ≥ kBT²⁄√M. Entropy curvature defines a Gaussian fluctuation geometry after exact constraints and null directions are removed. Nonequilibrium current precision is governed by the selected current, observation duration, and entropy production. These structures provide complementary statistical descriptions of screen-supported physical response. Finite holographic screens are modeled as regulated binary systems. Let K be the number of available modes, N the mean active capacity, and f = N − ⌊N⌋. Every finite commuting binary screen satisfies f(1 − f) ≤ (ΔN)² ≤ N(K − N). For factorized mode activation, the distribution is Poisson–binomial and obeys f(1 − f) ≤ (ΔN)² ≤ N(1 − N⁄K). The universal upper envelope includes collective correlations. The factorized envelope is attained by equal independent occupancy. General correlated screens are described by the covariance matrix C: (ΔN)² = 1ᵀC1. Block correlations, covariance spectra, Fano factors, participation ratios, and equicorrelation conditions characterize collective screen structure. Variance estimates and concentration inequalities then provide finite-mode probability gates for continuum admission. Continuum admission is formulated through control of relative fluctuations, unresolved blocks, retained correlations, boundary effects, and the physical map carrying finite screen observables into continuum variables. Inverse-area observables require nonlinear reciprocal statistics. On the positive-capacity sector, ℛ̂ = 𝒦ₚ N̂⁺, where N̂⁺ is the generalized inverse of the capacity operator. If qₙ is the conditional probability of positive capacity n, then every reciprocal moment is exact: ⟨ℛ̂ʳ⟩₊ = 𝒦ₚʳ ∑ₙ₌₁ᴷ qₙ⁄nʳ. For a narrow positive-capacity distribution, Δℛ⁄⟨ℛ⟩ ≈ √m₂, where m₂ = Var₊(N̂)⁄N̄². If p₊ is the positive-capacity probability and N = p₊N̄, then the full relative capacity variance is χN = [m₂ + 1 − p₊]⁄p₊. The equality m₂ = χN holds on the fully positive branch p₊ = 1. The zero-capacity probability remains explicitly represented in the general relation. The exact screen operator product is ÂP̂ = 16πℓₚ²uₚ Π₊, where Π₊ projects onto the positive-capacity sector. Signed cancellation is analyzed through a separate physical channel. A signed contribution Λₛ may pass through zero while its associated screen capacity remains positive through its dependence on |Λₛ|. For a mean-zero signed load with variance σΛ² and finite standardized fourth moment κ₄, 4πσΛ⁄(ln 2 √κ₄) ≤ N₀ ≤ 4πσΛ⁄ln 2. For a Gaussian signed load with mean m(x), width σ(x), and cancellation point x₀ satisfying m(x₀) = 0, the residual capacity floor is N₀ = 4πσ₀√(2⁄π)⁄ln 2. The derivatives of the capacity scan distinguish mean crossing, width drift, correlated-channel structure, mixtures, avoided crossings, non-Gaussian tails, and coherent carrier mixing. These signatures provide a cancellation tomography for identifying the physical mechanism that generates a residual floor. Samplewise inverse-load behavior is governed by the probability density near the cancellation point. When the signed-load density is continuous and positive at the origin, 𝔼|Λₛ|⁻ᑫ = ∞ for q ≥ 1. A bounded samplewise reciprocal response follows from the support-gap condition |Λₛ| ≥ λ* > 0. Detector and source tomography are formulated through calibrated response maps. The model retains source coordinates, finite observation windows, detector noise, source–noise correlations, nuisance directions, and unresolved channels. Whitening by the detector covariance identifies the rank and singular values of the observable response. Moore–Penrose reconstruction yields the identifiable source projection selected by the detector. A detector channel satisfies the data-processing inequality DKL(PFʸ ‖ PBʸ) ≤ DKL(PFˣ ‖ PBˣ). The physical channel therefore supplies the maximum available distinguishability. Detector loss, coarse graining, and response null spaces determine how much of that distinguishability reaches the measured data. Calibrated null contrasts separate source disagreement, detector blindness, weak sensitivity, and response-model mismatch. Tomographic rank determines which combinations of source parameters are physically identifiable. Capacity cumulants encode finite-regulator shape information. Their transfer into inverse-area observables is nonlinear and depends on positive-sector support. Static Euclidean response is determined by the positive physical Hessian of the selected equilibrium anchor. Thermal fluctuation–dissipation response is defined through a declared KMS state. Lorentzian causal response is defined through the corresponding retarded realization. Multiscale screen flow is described by compatible coarse-graining channels, matched states and observables, exact removed-load accounting, relative-entropy contraction, recovery bounds, and an additive obstruction spectrum. The root-fidelity convention is F(ρ,τ) = ‖√ρ √τ‖₁. Information loss under coarse graining bounds the accuracy of recovery. Projectively consistent retained-capacity observables converge to a conditional expectation on the limiting retained algebra, producing a controlled capacity-side continuum interface. For every physical sector X, the finite residual belongs to a declared residual space 𝒴X. Continuum realization is supplied by an owner map 𝒫X⁽ᴸ⁾ : 𝒜X(L) ⊆ 𝒴X → 𝒵L, with OXμν(L) = 𝒫X⁽ᴸ⁾[rX] and 𝒫X⁽ᴸ⁾[0] = 0. The target space 𝒵L specifies the continuum patch, tensor type, physical projector, smearing, support, foliation, boundary conditions, and norm. A locally bounded realization map satisfies ‖𝒫X⁽ᴸ⁾[r] − 𝒫X⁽ᴸ⁾[r′]‖𝒵ᴸ ≤ CX(L) ‖r − r′‖𝒴ˣ. The calibrated finite residual then yields a controlled continuum tensor residual. Source-disjoint sectors ass