Holographic Saturation II: Shared parent action, division-algebraic gauge structure, entropic symmetry breaking, and orbit-channel closure.

Abstract

\noindentWe develop the internal transport sector of the bounded saturation field$I=\sqrt{s}\,e^{i\theta}$ introduced in Part~I. The construction uses fourstructures: a holographic saturation principle, a conserved Born phase, unitaryClifford transport, and the binary configurational free energy of a boundedchannel. Through the Cayley--Dickson tower$\mathbb R\subset\mathbb C\subset\mathbb H\subset\mathbb O$, these ingredientsselect a division-algebraic realisation of Standard-Model gauge kinematics underone explicitly stated maximal-capacity completion. The vector-fields-on-spheresclassification isolates the octonionic rung; a fixed Born complex directionreduces $G_2$ to exactly $SU(3)$, with three colours, the$\mathbf3\oplus\bar{\mathbf3}$ transverse module, and the meson--baryon singletforms. The quaternionic rung gives the electroweak factor in its global form$U(2)$, the Higgs bidoublet, the hypercharge ratio, and custodial$SO(4)\to SU(2)_{\rm diag}$. The same entropic potential makes the symmetricpoint a non-analytic boundary and yields a Fermi vacuum, a $\cosh^2$ Higgsrigidity, and a crossover in the minimal finite-temperature completion. Anexplicit local Fisher--canonical completion then matches the canonical radialmetric to the Bernoulli information metric at equilibrium. Without adding anew dimensionful scale, this closes the radial informational temperature; atthe untilted maximum-entropy point it gives the tree-level curvature result$m_{H,\mathrm{curv}}^{(0)}=v/2=123.11\,$GeV together with$\kappa_3=1$ and $\kappa_4=11/3$. A second explicit completion identifies theaffine coarse-graining observable with the reciprocal resolution $\ell=1/k$.Equal Bernoulli weights at maximum entropy then fix the FRG matching scale tothe harmonic mean of the vacuum and saturation scales. Evolving the full KLpotential in a global $O(4)$ Litim-LPA Wetterich flow with Standard-Modelthresholds, while fitting only the measured infrared Fermi scale and never theHiggs mass, gives the conditional curvature prediction$m_{H,\mathrm{curv}}^{\rm FRG}\simeq125.1\,$GeV. A derivative-expansionstress test shows sub-GeV truncation sensitivity, so a self-consistentmomentum-dependent pole calculation remains open. The Clifford--Fock modulesupplies an anomaly-free sixteen-state generation moduleand an exhaustive mass-sector census with a unique right-handed Majoranainvariant. An explicit quaternionic Hopf transport lift twists the internalDirac operator by $H^3=\mathcal O(3)$, yielding$\dim H^0(\mathbb{CP}^1,\mathcal O(2))=3$ same-chirality zero modes andtherefore exactly three generations, total index $48$, with no vectorlikepartners in the canonical completion. On the same Hopf space, an explicitKac--Birkhoff return completion fixes the diagonal up-type hierarchy withoutfermion-mass input: the degree-three Hermitian cost, the one-Higgs factor$f_d/v=\sqrt2$, and the elementary channel measure $\alpha_Z$ give$\beta=3\sqrt2/\alpha_Z=542.7610$ and hence exact $t:c:u$ overlap ratios.A resolution-faithful fermionic lift of the same resolved weak Clifford paththen removes the last continuous up-sector normalization. Preservation of boththe Clifford algebra and the affine resolution parameter makes any extrarescaling of the one-Higgs Fock generator impossible; the resulting ResolvedClifford Lift Theorem gives $Y_U(k_F)=\sqrt2\,g_2(k_F)$. Solving the resultingRG--FRG fixed point with no quark mass used as an external normalization gives$M_t^{\rm match}=172.3693$~GeV while the same flow returns$m_{H,\mathrm{curv}}^{\rm FRG}=125.146$~GeV. The algebraic normalization istherefore theorem-level inside the resolution-faithful lift; the physicalidentification of that lift and a fully momentum-dependent pole matching remainexplicit completion assumptions, and the non-up flavour sectors remain open. The three sector weights satisfy$S_i=\kappa_i\alpha_Z^{N_i}$ with $(N_i)=(3,13,16)$. A common invariant-projectormeasure fixes$(\kappa_\sigma,\kappa_\Phi,\kappa_\psi)=(1/4\pi,1,2)$, while relative-entropystationarity gives$\mu_{i,\rm sec}^2/(2\Lambda_i^{(\rm sec)})=\ln[(1-S_i)/S_i]$.The intrinsic galactic occupation is therefore$s_0=\alpha_Z^3/(4\pi)=3.80\times10^{-8}$, and the electroweak and colourchannels reproduce the quoted SPARC normalisation and QCD string scale withoutsector-specific continuous prefactors. The Born phase simultaneously fixes the internal complex direction and suppliesthe timelike spacetime direction of the universal reciprocal metric developedin Part~I. The geometric Born current is distinguished from the canonicallynormalised Noether current $J^\mu=f^2s\bar\nabla^\mu\theta$; a non-zero conservedcharge selects a unique, linearly stable timelike branch without changing the$G_2\to SU(3)$ stabiliser. Clifford transport and an explicit constrained parent action are written withthe same universal physical metric $\bar g_{\mu\nu}$ used bymatter, photons, and tensor modes; that action is shown in Part~I to propagateexactly four degrees of freedom, with the Born phase sharing the observablecone identically. The algebraic groups, representations,anomaly cancellation, orbit measure, and symmetry-breaking mechanism arederived or numerically certified at the stated level. Confinement dynamics,the down-type/leptonic flavour sectors and mixings, the fully momentum-dependentfermion pole matching, the full propagating chiral embedding, and the individualpreferred-frame PPN coefficients remain open; Part~I fixes the directly readable combination$\alpha_1-2\alpha_2$ but not the separate coefficients.