Stoney–Planck Action Cells and the Planck-Regulated Commitment Substrate

Abstract

This is the 5th paper in the series, which is preceded by the three part 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 Stoney–Planck Action Cells and the Planck-Regulated Commitment Substrate Exact Charge–Gravity Normalization, Finite Causal Structure, and Microscopic Screen Realization This work constructs a finite physical substrate connecting dimensionless interaction strength, quantum action, causal support, holographic screen capacity, and the emergence of continuum geometry. The construction begins with the exact Stoney normalization obtained by equating the Newtonian and Coulomb force magnitudes for two equal masses carrying elementary charge magnitude e: Gm² = kₑe², where G is Newton’s gravitational constant and kₑ = (4πε₀)⁻¹. The corresponding Stoney mass is mₛ = √(kₑe²⁄G). Associated length, time, energy, momentum, and action scales are ℓₛ = Gmₛ⁄c², tₛ = ℓₛ⁄c, Eₛ = mₛc², pₛ = mₛc, Aₛ = Eₛtₛ = mₛcℓₛ. Because the fine-structure constant satisfies α = kₑe²⁄(ℏc), the Stoney action becomes Aₛ = αℏ. This identifies α as the elementary electromagnetic channel load measured in quantum-action units. The Stoney and Planck scales are connected by the exact relations mₛ = √α mₚ, ℓₛ = √α ℓₚ, tₛ = √α tₚ, Eₛ = √α Eₚ, pₛ = √α pₚ. The force and power scales remain unchanged: Fₛ = c⁴⁄G = Fₚ, Pₛ = c⁵⁄G = Pₚ. These identities show that the Stoney system is an exact charge–gravity normalization within the Planck scale rather than a separate dimensional system. Channel action For a physical interaction channel 𝒞 with signed dimensionless coefficient Λ𝒞, the positive interaction load is λ𝒞 = |Λ𝒞|. The sign remains available for phase, chirality, attraction, repulsion, cancellation, and interference. The magnitude determines the action and screen load: A𝒞 = λ𝒞ℏ. This relation assigns one quantum-action scale to every declared interaction load. The elementary electromagnetic channel is recovered through λₑₘ = α, Aₑₘ = αℏ. The construction therefore extends the Stoney normalization from elementary electromagnetism to any interaction channel carrying a positive dimensionless load λ. Geometric benchmarks A channel load λ defines the neutral gravitational benchmark quantities mλᵇᵉⁿᶜʰ = √λ mₚ, rλᵇᵉⁿᶜʰ = 2√λ ℓₚ, Aλᵇᵉⁿᶜʰ = 16πλℓₚ². The corresponding Bekenstein–Hawking binary capacity is Nλᴮᴴ = Aλᵇᵉⁿᶜʰ⁄(4ℓₚ² ln 2) and therefore Nλᴮᴴ = 4πλ⁄ln 2. The same quantity is adopted as the native finite-screen expectation: Nλ = 4πλ⁄ln 2. Action, capacity, and area are then linked by A𝒞 = λℏ = Nλℏ ln 2⁄(4π), and Aλ = 4ℓₚ² ln 2 · Nλ = 16πλℓₚ². The factor ℏ ln 2⁄(4π) is the action normalization per unit expected binary screen capacity. Primitive causal-counting substrate The microscopic foundation is a locally finite causal-counting structure ℛₚₗ = (Eₚₗ, ≺ₚₗ, μₚₗ). Here Eₚₗ is a nonempty set of primitive substrate elements, ≺ₚₗ is a strict causal order, and μₚₗ is a finite counting valuation. For comparable elements x ≺ₚₗ z, the open interval is I(x,z) = {y ∈ Eₚₗ : x ≺ₚₗ y ≺ₚₗ z}. Local finiteness requires |I(x,z)| < ∞. For every finite support A ⊂ Eₚₗ, μₚₗ(A) = vₚₗ|A|, where vₚₗ is the primitive volume scale on a continuum-calibrated branch. The substrate therefore contains intrinsic causal order and finite content without presupposing a smooth spacetime manifold. Finite correlation cells are supported on nonempty finite subsets D꜀ ⊂ Eₚₗ. Each cell carries declared correlation data χ꜀ᶜᵒʳʳ and can support registration, memory, control, comparison, or transport roles. A finite physical event is not an abstract point. It is a finite package of substrate support, physical state, implemented operation, and causal route. Intrinsic dynamics The causal order does not force a unique dynamics. The paper develops two controlled dynamical branches. The first is a finite-configuration jump process with probability law pₜ(X) and transition rates q(X,Y): ṗₜ(X) = ∑Y≠X [pₜ(Y)q(Y,X) − pₜ(X)q(X,Y)] + rₜ(X). The exchange term rₜ(X) accounts for declared open-system transfer while preserving total normalization. The second is a covariant sequential-growth branch in which new maximal elements are added to finite causal structures. Natural birth labels are gauge data: physical predictions depend on the resulting unlabeled causal structure rather than the arbitrary order in which labels were assigned. Both branches preserve the distinction between causal structure, stochastic evolution, physical clock time, and semantic interaction time. Physical operations and retarded channels Every realized event has finite substrate support Sₑ and may perform one or more physical roles: registration,record formation,report generation,control,comparison,memory,or action. A local operation has the form 𝒪ₑ : 𝒮ₑⁱⁿ → 𝒮ₑᵒᵘᵗ. Quantum registration is represented by a normalized instrument {ℐₑᵐ}, with outcome probability pₑ(m | ρ) = Tr[ℐₑᵐ(ρ)]. Physical commitment between events requires an implemented retarded route. Abstract causal compatibility is not enough. A later event must be connected to the earlier event by actual channels carrying state, energy, momentum, records, control signals, or comparison data. This produces a derived physical graph whose edges represent implemented interactions rather than merely possible causal relations. Passive distinguishability and comparison access Physical channels contract distinguishability. For two states ρ and σ transported by a completely positive trace-preserving map Φ, D(Φ(ρ), Φ(σ)) ≤ D(ρ,σ), where D is trace distance. The substrate therefore separates passive transport from active comparison. A channel may carry records without comparing them. Comparison requires the records to be jointly recruited into a physical operation with the required provenance, identifiers, decoder, and output register. Physical marks and currents Implemented channels can carry physical marks such as energy, momentum, charge, registration labels, report labels, provenance, or control data. For a mark m transported along a channel α, the corresponding current is Jαᵐ(t). On a finite event region C, the energy balance is d⟨E꜀⟩ₜ⁄dt ∑incoming Jαᴱ(t)−∑outgoing Jαᴱ(t)+∑ₑ∈꜀ Sₑᴱ(t). This is the finite substrate energy balance. Work, heat, entropy export, horizon flux, stress–energy, and curvature response remain distinct physical quantities. They arise only after the relevant reservoirs, controls, continuum maps, averaging procedures, and tensor reconstruction have been supplied. Substrate-supported screens A finite substrate cut D has a boundary support V∂(D). A screen realization assigns binary screen modes to nonempty physical support on that boundary. If no boundary vertex carries more than ν∂ modes, then a screen with Kλ modes must satisfy Kλ ≤ ν∂|V∂(D)|. Since the required screen contains at least ⌈Nλ⌉ modes, the boundary must obey |V∂(D)| ≥ ⌈Nλ⌉⁄ν∂. This is a physical support bound. It states how much finite substrate support is required to carry the selected interaction load. Minimal binary-screen theorem For every λ > 0, define Nλ = 4πλ⁄ln 2, nλ = ⌊Nλ⌋, δλ = Nλ − nλ, Kλ = ⌈Nλ⌉. A finite binary screen exists on ℋλˢᶜʳ = (ℂ²)⊗ᴷλ. Let N̂ˢᶜʳ be the total occupation operator. The minimum-variance state is ρλᵐⁱⁿ (1 − δλ)|nλ;Kλ⟩⟨nλ;Kλ|+δλ|nλ + 1;Kλ⟩⟨nλ + 1;Kλ|. It satisfies Tr(ρλᵐⁱⁿN̂ˢᶜʳ) = Nλ, A𝒞 = λℏ = Nλℏ ln 2⁄(4π), Tr(ρλᵐⁱⁿÂ) = 16πλℓₚ². Its number variance is (ΔN)²ₘᵢₙ = δλ(1 − δλ). This is the smallest variance possible for any positive normalized state with integer occupation spectrum and mean Nλ. The mode count is also minimal: Kλ = ⌈Nλ⌉. Every positive interaction load therefore possesses an explicit finite binary realization with the required mean capacity, action, area load, minimum fluctuation, and minimum mode number. Covariance and continuum geometry The finite substrate is invariant under structure-preserving relabelings. Physical observables depend on causal order, finite support, implemented channels, states, and relations—not on arbitrary element names. Continuum geometry appears through a separate relation between substrate isomorphism classes and Lorentzian geometries modulo diffeomorphism: 𝒢([Xₛᵤᵦ]) = {[(M,g)]ᴰⁱᶠᶠ}. The continuum branch specifies manifoldlikeness, density, scale, regularity, boundary data, approximation error, and the map from order and counting structure into causal and volume geometry. Coordinates enter only after a continuum representative has been selected. This preserves the distinction between the primitive finite substrate and its smooth geometric realization. Principal result The paper establishes a complete finite chain: dimensionless interaction load→ quantum action→ finite binary capacity→ minimum mode count→ minimum capacity fluctuation→ finite substrate support→ conditional geometric area. For every physical channel with load λ > 0, A𝒞 = λℏ, Nλ = 4πλ⁄ln 2, Kλ = ⌈4πλ⁄ln 2⌉, (ΔN)²ₘᵢₙ = δλ(1 − δλ), Aλ = 16πλℓₚ². These relations define the microscopic action–capacity structure of the Planck-regulated commitment substrate and provide the finite physical support from which screens, records, interaction channels, thermodynamic balances, and continuum geometry are constructed.