Holographic Screen Capacity, Composition, and Operational Fission

Abstract

This is the 6th 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 and Stoney–Planck Action Cells and the Planck-Regulated Commitment Substrate. Zenodo. https://doi.org/10.5281/zenodo.21749995 Holographic Screen Capacity, Composition, and Operational Fission Positive Stoney–Planck Load Algebra, Local Buffering, Cross-Control Loss, and Daughter Autonomy This work develops a finite physical theory of holographic screen capacity, positive interaction-load composition, gravitational headroom, network separation, and operational fission. A holographic screen is treated as a finite, substrate-supported physical interface with a maximum reliable capacity. Interaction loads occupy that capacity through realized channels. Different physical forms of composition—orthogonal coexistence, coherent interference, and constrained elimination—obey different exact laws. Local compactness determines how much of the available screen capacity remains usable, while network transport and open-system dynamics determine whether a screen-supported domain remains unified or separates into autonomous daughters. The central result is a finite-window fission theorem. A system does not divide into operationally autonomous daughters merely because a graph disconnects, amplitudes become orthogonal, or interference becomes small. Operational fission requires the simultaneous loss of cross-binding capacity, removal of every relevant cross-component route, suppression of the integrated residual interaction generator, and continued viability of every daughter region. When these conditions hold, the product evolution of the daughters approximates the full physical evolution with an explicit diamond-norm bound. Screen-capacity normalization For a realized interaction channel 𝒞 with positive dimensionless load λ𝒞, the channel action A𝒞 and screen capacity N𝒞 satisfy A𝒞 = λ𝒞ℏ = N𝒞ℏ ln 2⁄(4π). The corresponding screen capacity is N𝒞 = 4πλ𝒞⁄ln 2. The same interaction load therefore has three linked descriptions: dimensionless channel load λ𝒞,quantum action A𝒞,and finite binary screen capacity N𝒞. A channel-resolved screen cell carries the physical package 𝒮𝒞 = (λ𝒞, N𝒞, ΔN𝒞, βλ𝒞, Kᵃⁿᶜʰₒᵣ𝒞, μᵐₒdₑ𝒞, O𝒞), where ΔN𝒞 records capacity fluctuations, βλ𝒞 = dλ𝒞⁄d ln μ records scale flow, Kᵃⁿᶜʰₒᵣ𝒞 records anchor response, μᵐₒdₑ𝒞 records mode density, and O𝒞 records the relevant obstruction data. These quantities form a typed physical interface. Interaction load, finite capacity, anchor response, mode structure, and obstruction remain distinct even when they participate in one realized screen process. Three exact capacity-composition laws The paper identifies three different physical composition laws. Orthogonal composition When two retained sectors remain physically orthogonal, their capacities add: N₁ ⊕ N₂ = N₁ + N₂. The corresponding area loads also add: A₁⊕₂ = A₁ + A₂. Coherent composition When two contributions feed the same retained physical outcome, their amplitudes combine before the capacity is evaluated: N₁ ⊞φ N₂= |√N₁ + eⁱφ√N₂|²= N₁ + N₂ + 2√(N₁N₂) cos φ. Constructive and destructive branches are therefore N₊ = (√N₁ + √N₂)², N₋ = (√N₁ − √N₂)². For m coherent contributions, N꜀ₒₕ = |∑ⱼ₌₁ᵐ √Nⱼ eⁱφʲ|². Coherent composition depends on phase and cannot be replaced by ordinary addition. Projected composition When an unresolved decomposition is eliminated under a common linear constraint, the effective positive capacity is N₁ ⋆ N₂ = (N₁⁻¹ + N₂⁻¹)⁻¹= N₁N₂⁄(N₁ + N₂). This operation is associative: (N₁ ⋆ N₂) ⋆ N₃= N₁ ⋆ (N₂ ⋆ N₃)= (N₁⁻¹ + N₂⁻¹ + N₃⁻¹)⁻¹. Projected composition arises from constrained elimination rather than direct coexistence or coherent addition. The three operations ⊕, ⊞φ, and ⋆ are therefore exact in different physical domains. Their use requires identification of the source sectors, carrier maps, retained outcome, phase data, and projection constraints. Positive Gram geometry The scalar composition laws arise from one positive load geometry. Let the typed source space be ℋ꜀ₒₘₚ = ℋᴳ ⊕ ℋʸ ⊕ ℋΣ ⊕ ℋˢ, representing gauge, Yukawa, reciprocal-seam, and screen carrier sectors. For a typed load vector ℓ and a positive semidefinite capacity form K ≽ 0, N[ℓ] = ⟨ℓ, Kℓ⟩. Because K is positive, N[ℓ] = ∥K¹ᐟ²ℓ∥². If Nₐ is the capacity carried by sector a, the normalized coherence coefficients satisfy γₐᵦ = ⟨Rₐℓₐ, Rᵦℓᵦ⟩⁄√(NₐNᵦ), |γₐᵦ| ≤ 1. The total capacity is N[ℓ] = ∑ₐ Nₐ + 2∑ₐ<ᵦ √(NₐNᵦ) Re γₐᵦ. The coherence matrix Γ = (γₐᵦ) must satisfy Γ ≽ 0. For three carrier sectors, positivity imposes the global condition 1 + 2 Re(γ₁₂γ₂₃γ₃₁)≥ |γ₁₂|² + |γ₂₃|² + |γ₃₁|². Pairwise coherence bounds are therefore insufficient. The full set of mixed loads must belong to one globally positive Gram geometry. The minimum number of independent carrier directions obeys Kᵐⁱⁿ꜀ₐᵣᵣᵢₑᵣ ≥ rank Γ. This gives a physical rank bound on the carrier structure required to realize a declared family of mixed interaction loads. Finite substrate-supported screens A screen is constructed on a finite substrate cut with boundary vertices V∂(D). For load λ > 0, Nλ = 4πλ⁄ln 2, Kλ = ⌈Nλ⌉. If the local boundary multiplicity is ν∂ and Kλ ≤ ν∂|V∂(D)|, the Kλ binary modes can be assigned to finite physical boundary support. The screen Hilbert space is ℋˢᶜʳλ = (ℂ²)⊗ᴷλ. Its number operator is N̂λ = ∑ₐ₌₁ᴷλ n̂ₐ, and its microscopic area operator is Âλ = 4ℓₚ² ln 2 · N̂λ. The exact finite-screen state satisfies Tr(ρλN̂λ) = Nλ, Tr(ρλÂλ) = 16πλℓₚ². The integer Kλ counts the physical binary carriers required to realize the expectation-valued capacity Nλ. Maximum spacetime-capacity density The screen realizes one nat of maximum reliable capacity per area 4ℓₚ², equivalently one binary unit per area 4ℓₚ² ln 2. The capacity density is Cᵣₑg⁄A = 1⁄(4ℓₚ²), Nλ⁄A = 1⁄(4ℓₚ² ln 2). If Nλ = nλ + δλ with 0 ≤ δλ < 1, the first nλ carriers are fully available and the final carrier is a flagged fractional-availability channel. The maximum reliable capacity per use is Cᵣₑg = Nλ ln 2 = 4πλ. The unused portion of the kinematic Hilbert space does not create additional physical capacity at fixed λ. When the finite screen admits a continuum geometric realization with area A(B), Cₕₒₗₒ(B) = A(B)⁄(4ℓₚ²), Nₕₒₗₒ(B) = A(B)⁄(4ℓₚ² ln 2). On a realized black-hole horizon, Sᴮᴴ = kᴮCᵣₑg = kᴮAᴴ⁄(4ℓₚ²). The horizon supplies causal and thermal saturation of the same screen-capacity density. Capacity, ambiguity, and thermodynamic export The paper separates four physical quantities: Cᵣₑg = maximum reliable screen-channel capacity, Sₐ꜀ₜ⁄kᴮ = activation-state uncertainty, H꜀ₒₘₘᵢₜ = unresolved conditional ambiguity, ΔSᵗʰᵉʳᵐₒₑₓₚₒᵣₜ⁄kᴮ = entropy exported by a realized protocol. For a source variable Z, screen input X, channel output Y, and retained ledger L, Z → X → Y → L. Define Hₛᵣ꜀ = H(Z), Iᵣₑₜ = I(Z;L), H꜀ₒₘₘᵢₜ = H(Z|L). Then Hₛᵣ꜀ = Iᵣₑₜ + H꜀ₒₘₘᵢₜ, 0 ≤ Iᵣₑₜ ≤ Cᵣₑg. Retained information occupies the screen channel. Conditional ambiguity is the unresolved portion of the source identity. It is not itself identical to the retained physical load. When provenance, correction, durability, and maintenance occupy disjoint subchannels, the necessary capacity condition is Iᵣₑₜ + Cᵣₑₛₑᵣᵥₑ ≤ Cᵣₑg. If every source alternative must remain independently addressable, Hₛᵣ꜀ + Cᵣₑₛₑᵣᵥₑ ≤ Cᵣₑg. Resolution, retention, discard, and reset remain separate operations. A Landauer cost enters only when a physical memory is irreversibly reset. For discarded memory Mdisc with retained side information Lkeep, Wᵣₑₛₑₜ ≥ kᴮT H(Mdisc|Lkeep), ΔSᵣₑₛₑₜₑₙᵥ ≥ kᴮH(Mdisc|Lkeep). Compactness and gravitational headroom For a spherical continuum realization with radius R, area A = 4πR², and total gravitating energy E, Cᵣₑg = A⁄(4ℓₚ²) = πR²⁄ℓₚ². Define compactness by 𝒞g = 2GE⁄(c⁴R) = rₛ⁄R, where rₛ = 2GE⁄c⁴. The Bekenstein capacity is Cᴮᵉᵏ = 2πER⁄(ℏc). The exact filling relation is Cᴮᵉᵏ⁄Cᵣₑg = 𝒞g, Cᴮᵉᵏ = 𝒞gCᵣₑg. On the nontrapped branch R > rₛ, the remaining gravitational headroom is Cʰᵉᵃᵈg= Cᵣₑg − Cᴮᵉᵏ= (1 − 𝒞g)Cᵣₑg= πR(R − rₛ)⁄ℓₚ². At horizon saturation, 𝒞g = 1, Cʰᵉᵃᵈg = 0, Cᴮᴴ = Cᵣₑg. This distinguishes an ordinary screen with positive headroom from a saturated horizon with no remaining one-region gravitational capacity. Local buffer dynamics The paper defines three capacities: Cᵣₑg = maximum screen capacity, Cₐdₘ = gravitationally admissible capacity, Cᵢₙₜ = interior physical entropy load. The entropy headroom is Δₛ = Cₐdₘ − Cᵢₙₜ. The gravitational headroom is Δg = Cᵣₑg − Cₐdₘ. The total unsaturated capacity is Δₜₒₜ = Cᵣₑg − Cᵢₙₜ = Δg + Δₛ. Changes in energy, area, entropy inflow, entropy production, and entropy export determine whether the one-region branch remains viable. When the buffer reaches saturation, the physically available responses are: entropy export,energy redistribution,areal expansion,fission into viable daughters,or transition to a trapped branch. Fission is one possible capacity response. Physical network capacity A screen-supported region can be represented by a weighted physical transport network. Edge weights encode realized cross-patch channel capacities rather than abstract adjacency. For a partition A|B, the cross-binding capacity measures the total implemented transport that binds the two sides. A graph cut becomes physically significant only when the relevant routes cease to support actionable exchange. Thresholding a diagnostic edge weight is therefore not enough. Every physically implemented cross-route must be removed or suppressed within the declared operating window. The algebraic-connectivity condition λ₂(Lκ) = 0 identifies