The First-Occupation Law of the One-Spin phtn: Competing Receiver Seats, Bright-State Reduction, and the Four-Dimensional Material Record

This paper develops the multi-receiver and first-occupation dynamics of the free one-spin phtn. The preceding papers determine the five-dimensional free-carrier geometry and the topological conditions under which one identified receiver seat can become a persistent occupied Ligamen Circulatus. The present work determines how one normalized phtn competes among many admissible receiver seats, how successive incomplete interactions compose into the first completed occupation, what portion of the complete event can be registered by four-dimensional occupied matter, and why receiver closure does not send a physical notification backward to the source. At Aether moment n, the primitive one-phtn event space is Hₙ⁽¹⁾= HF,n⊕ HLC,n⊕ HR,n, where the sectors represent the surviving free carrier, the port-labeled occupied-seat alternatives, and an explicit remainder. Overlapping receiver labels are reduced through the Gram matrix Gyz= LCy LCz, so the number of independent closure alternatives is Meff=rankG. The completed-occupation projector is idempotent, Nocc²= Nocc, and therefore one normalized direct event contains either no completed occupation or one completed occupation. A coherent sum of receiver alternatives does not multiply the transported phtnₓ=mₓλC³Fq² or angmₓ=mₓλC²Fq. The most general primitive linear coupling from the one-dimensional free channel into the orthogonal receiver closure space has one complex one-moment coordinate ϑy,n per admitted seat. Its anti-Hermitian generator is Aₙ=∑y(ϑy,n LCy Fₙ-ϑy,n* Fₙ LCy). This generator reduces exactly to a two-mode rotation between the free phtn and the unique bright receiver mode Bₙ=Θₙ⁻¹∑yϑy,n LCy, where Θₙ²=∑y|ϑy,n|². All receiver combinations orthogonal to Bₙ are dark during that update. Beginning from the free state, the coherent free and closure weights are cos²Θₙ and sin²Θₙ. When completed records are orthogonal, absorbing, and supplied by one unique surviving free history, the one-moment weights compose into the exact first-occupation law Sₙ₊₁=Sₙ(1-qₙ) and Wn,y=Sₙqy,n, where qₙ=sin²Θₙ and qy,n=qₙ|ϑy,n|²/Θₙ². The finite event ledger satisfies SN+1+∑ₙ₌₀N∑yWn,y=1. The cumulative closure depth is DN=-ln SN, and the receiver fractions conditional on closure during moment n are W(y n)=|ϑy,n|²/∑z|ϑz,n|². The intrinsic temporal lift ζₕ(Φ)=exp(ihΦ/2) remains 4π periodic, while occupied half-spin matter registers the quotient QT(ζₕ)=ζₕ². A temporal deck-pair construction shows that durable material observables are invariant under ζₕ↦-ζₕ and omit both deck-odd branch information and coherence between mutually exclusive material records. The resulting four-dimensional restriction preserves the first-occupation measure but cannot, by linearity alone, select one realized extremal record. An additional Aether-internal registration principle remains necessary. Receiver attachment cancels the terminal free boundary without constructing a reverse chain to the source. If ∂ΓT(N)=vN-v₀ and ∂ By=y-vN, then ∂(ΓT(N)+By)=y-v₀. The source remains the historical root of the completed event, while the terminal active free boundary is replaced locally by the occupied receiver boundary. Every receiver-local trace-preserving operation leaves the unconditioned source reduced state unchanged, ρ'S=ρS. Global five-dimensional event completion is therefore compatible with local four-dimensional causality and cannot be used as a backward or faster-than-c material communication channel. The paper derives eleven coefficient-free prediction and falsifier classes, including effective receiver rank, rank-one bright coupling, coherent dark-mode extinction, geometric first-occupation statistics, additive closure depth, strict single-record occupation, local-accessibility nulls, no backward notification, and temporal quotient invariance. The strongest near-term experimental program combines receiver-response tomography, bright- and dark-mode preparation, first-occupation statistics, closure-depth additivity, and single-record anticoincidence. The most general primitive linear coupling from the one-dimensional free channel into the orthogonal receiver closure space has one complex one-moment coupling value ϑy,n for each admitted seat. These values together form the receiver coupling vector (ϑ1,n,ϑ2,n,…,ϑMₙ,n). The remaining constitutive problems are the QMU law that determines the magnitudes and relative phases of the receiver coupling values ϑy,n and the Aether-internal rule that realizes one material record in an individual event. The reproducibility package contains the manuscript PDF, complete Overleaf source, stage derivations, figures, claims ledger, equation inventory, prediction-falsifier ledger, validation scripts, public validation report, manifest, checksums, and license. The integrated validation suite completed 744,970 automated checks with zero failures. The maximum residual among exact identities was below 1.2×10⁻¹³. These checks establish formal and computational consistency of the stated derivations; they do not constitute independent empirical validation.