One Code, Three Cosmologies: Parameter-Free Derivations of η, ns, N, and r from the [8,4,4] Extended Hamming Code on Z3 ⊗Q3

We derive four parameter-free cosmological observables — the baryon-to-photon ratio η, the scalar spectral index ns, the number of inflationary e-folds N, and the tensor- to-scalar ratio r — from the discrete combinatorics of the [8,4,4] extended-Hamming code on the local matter cell of the Holographic Circlette framework’s Z3 ⊗Q3 substrate. A single combinatorial value, the spectral gap ∆1 = 1/(2·14) = 1/28 emerging from the tensor product of the 2 transverse photonic modes on the C4 gauge bridge and the 14 weight-4 logical operators of the [8,4,4] code, simultaneously determines: (i) baryogenesis η= (3/14)α4 + (1/3)α5 ≈6.145 ×10−10 via colour-singlet projection on the logical-operator phase space (leading O(α4)) plus the bipartite gauge-bridge one-loop correction (subleading O(α5)), matching Planck 2018 (6.12 ±0.04 ×10−10) at∼0.6σ within experimental error; (ii) scalar spectral index ns = 1−∆1 = 27/28 ≈0.9643 via the Markov-chain spectral gap of the boundary printing operation (Planck 2018 match: 0.14σ, well inside experimental error); (iii) number of e-folds N = 2/∆1 = 56 via exact recovery of the slow-roll relation 1−ns = 2/N, where the structural “2” is the bipartite period of the P2 Markov coarse- graining and the Nishimori-line threshold pth = H−1 2 (1/2) ≈0.110 provides the structural reheating criterion (central in the canonical 50 < N < 60 window); (iv) tensor-to-scalar ratio r≈0 at leading substrate order, with subleading correction r∼α4 ≈2.8 ×10−9 from the quadrupole-shear constraint, structurally distinguishable from generic slow-roll inflation predictionsr∼10−2 andfalsifiablebyCMB-S4/LiteBIRD.Theframeworkrecoversthestan- dard slow-roll relation 1−ns = 2/N as the continuum limit of its discrete Markov-relaxation dynamics, without invoking a continuous scalar field, an inflaton potential, or a continuous metric. The three Sakharov 1967 baryogenesis conditions — baryon-number violation, C and CP violation, departure from thermal equilibrium — are satisfied as structural consequences of canonical substrate commitments (Wilson string topology change, bipartite chirality γ5 , Lindbladian non-unitarity) rather than as posited inputs. As a companion structural result we establish the graph-theoretic impossibility of a pure-photon universe on the substrate: photons exist exclusively as boundary modes on P4 gauge bridges anchored on Q3 matter cells. The combined result joins a growing family of parameter-free predictions on the same substrate (heavy-quarkonium widths, Hawking temperature, cosmological constant), all de- rived by the same combinatorial methodology: substrate rate times small integer from [8,4,4] combinatorics.