The Five-Dimensional One-Spin phtn of the Aether Fabric: Paired Half-Spin Loxodromes, Quadratic Cardioid Projection, and Species-Scale Transport

This paper derives the minimal five-dimensional geometry of a freely transported one-spin phtn from the four half-spin loxodromic channels of the Aether Unit. It addresses a structural question left open by the preceding Aether-fabric papers: if a materially occupied primitive particle uses only one half-spin Ligamen Circulatus channel, which combination of latent Aether channels can support a neutral one-spin carrier in transit? The four Stage 11D loxodromic channels are indexed by electrostatic-sphere polarity q=±1 and temporal-torque handedness h=±1. Their established assignments are e⁻=(-1,+1), $$p=(+1,+1),$$ p=(-1,-1), and e⁺=(+1,-1). For a two-channel transit state, define the sphere-polarity sum Qₚₐᵢᵣ=q₁+q₂ and the total half-spin torque Jₚₐᵢᵣ=h₁+h₂/2. Requiring a free carrier to be neutral, Qₚₐᵢᵣ=0, and to carry one nonzero spin, Jₚₐᵢᵣ=±1, uniquely selects opposite-sphere channels with the same temporal-torque handedness: γ₊=e⁻⊕ p, and γ₋= p⊕ e⁺. These pairings describe normalized transit participation in two latent loxodromic positions; they do not imply simultaneous material occupation by an electron and a proton or by an antiproton and a positron. The alternative pairings are excluded explicitly. The historical electron–positron assignment is sphere-neutral but has cancelling temporal torques and therefore gives Jₚₐᵢᵣ=0. The electron–antiproton pair remains on one electrostatic sphere and also has cancelling torque. The visible outer and hidden inward traces are not two separate half-spin carriers because they form the ordered 2π+2π=4π history of one loxodromic channel. The paper therefore replaces the earlier electron–positron carrier proposal with a ledger-derived electron–proton pairing for one helicity and the complementary antiproton–positron pairing for the opposite helicity. The intrinsic temporal coordinate is represented by the half-angle lift ζₕ(Φ)=exp(ihΦ/2). At the 2π branch boundary, ζₕ(2π)=-1, while complete intrinsic closure occurs at ζₕ(4π)=1. The observable quotient ζₕ²(Φ)=exp(ihΦ) has ordinary one-spin 2π periodicity. The five-dimensional state therefore retains the branch sign required for the complete 4π chronogeometric history, while its observable quotient presents one-spin periodicity. The paper next derives the projected geometry directly from the authoritative Stage 11D loxodrome. Orthographic projection through the z axis gives the raw radial curve r=asin(φ/2). This curve is cusp-bearing but is not itself an exact Euclidean cardioid. The exact cardioid appears in the quadratic swept-area coordinate R⊥=2r²/a. Substitution gives R⊥=a(1-cosφ). Thus the historical “cardioid of velocity squared” is recovered as an exact quadratic, time-integrated projection of the loxodrome rather than as an independently imposed photon shape. Opposite-sphere channels with the same handedness are indistinguishable when viewed through the z axis, allowing their two aligned half-spin contributions to appear as one projected angm quantity. For the discrete propagation scale aₙ=nλC at quantum moment tₙ=n/Fq, the Aether handoff law remains c=λCFq, so aₙ=ctₙ. The projected cardioid root remains fixed at R⊥(0,n)=0, while its forward extremity is R⊥(π,n)=2aₙ=2ctₙ. Consequently, the projected envelope nose advances at the apparent rate dR⊥,maxdt=2c, even though the underlying local Aether transfer scale remains c. The $2c$ quantity is therefore a quadratic projected-envelope rate, not the velocity of a locally transported object and not a first-arrival signal speed. The channel topology is independent of the transported mass-string scale. For a species label x, the paper defines Sₓ=mₓλC, angmₓ= Sₓc=mₓλC²Fq, and phtnₓ=angmₓc=mₓλC³Fq². This gives a species-indexed phtn family that includes both electron- and proton-scale carriers while preserving the same neutral one-spin channel geometry. In particular, phtnₚ/phtnₑ=mₚ/mₑ=6π⁵. The existence of a proton-scale phtn does not, by itself, determine whether that carrier is captured directly, converted into daughter phtn states, distributed among multiple receiver seats, retained as receiver response, or partially returned to an unoccupied dark-string state. To preserve those later possibilities without presupposing their dynamics, the paper introduces the conservation-complete receiver ledger phtnx=∑yNyphtny+phtnesc+phtnrec+c² Sdark. This ledger permits future derivations of direct capture, multi-seat conversion, daughter radiation, receiver recoil or deformation, and dark-string remainder while prohibiting lost or duplicated phtn content. No nuclear branching operator, neutrino state, dark-string return probability, or material-specific receiver evolution is assumed in the present paper. The channel structure also permits the compatibility relation e⁻⊕ e⁺→γ₊+γ₋ for two-phtn electron–positron annihilation. This is presented only as a channel-completion result. Propagation directions, relative phases, polarization correlations, branching probabilities, recoil, and three-phtn annihilation remain subjects for later dynamical analysis. The derivation package verifies the channel-pair theorem, temporal double cover, axial projection degeneracy, raw projected curve, quadratic cardioid identity, rooted-envelope relation, species scaling, and receiver ledger. Symbolic and numerical validation completed $46$ checks with a maximum residual below 7×10⁻¹⁶. The paper’s principal conclusion is that a free one-spin phtn is a species-scaled, neutral, paired-lodoxodromic transit state whose observable cardioid is the quadratic projection of its five-dimensional geometry: γ₊=e⁻⊕ p, γ₋= p⊕ e⁺, and R⊥=2r²/a=a(1-cosφ) This establishes the carrier geometry required for the next stage of the series: receiver-induced endpoint identification, winding conservation, complementary half-spin accounting, and persistent Ligamen Circulatus closure.