Coherence Cells on the Karmonic Mesh: Spectral Foundations for Hyperfluid Distributed Consensus

Erratum (2026-07-06) — discrete-time mixing. Theorem 10.3 computes tmix = ln(1/ε)/λ₁, which is correct for the continuous-time dynamics ẋ = −Lx but was interpreted as a count of synchronous protocol rounds. A one-hop round xt+1 = (I − αL)xt is stability-constrained by λmax = 4d (α < 2/λmax); the optimal per-round contraction is (λmax−λ₁)/(λmax+λ₁). For T³₈ (λ₁ = 2−√2 ≈ 0.586, λmax = 12, ε = 0.01) this gives ≈47 rounds and ≈144,000 messages (≈1.8× fewer than PBFT, ≈15× bandwidth) — not the stated 7.9 rounds / 24,576 messages / 10× / 85×. With Chebyshev acceleration (still one message exchange per round; each node retains two iterates) the figures improve to ≈12 rounds, ≈37,000 messages (≈7× messages, ≈60× bandwidth vs PBFT). Asymptotically, in the fixed-N growing-d regime the round complexity remains Θ(log M), but total messages are O(M log² M) unaccelerated and O(M log3/2 M) with Chebyshev acceleration, rather than the stated O(M log M); the statement “mixing time is O(log(1/ε)) and independent of M” does not hold for one-hop protocols. All corrected bounds remain asymptotically better than PBFT’s O(M²). Unaffected: the eigenvalue computations, the two-regime spectral-gap analysis, the logical-vs-physical-topology distinction, and the Coherence Cell definitions. A v1.2 PDF incorporating this erratum is in preparation. We also note the abstract’s comparison to “Byzantine consensus protocols” should be read per Limitation 4: the mechanism handles incoherent, not actively adversarial, nodes. We present a unified framework combining the Paraxiom Coherence Cell, a triadic epistemic fusion operator, with the Karmonic Mesh, a discrete Hamiltonian system on toroidal topology. The key contribution is a spectral analysis proving that the toroidal mesh structure yields O(log N) consensus time with O(N log N) total messages—an asymptotic improvement over classical Byzantine consensus protocols. Version 1.1 (2026-04-23) — revisions: Theorem 9.2 (Toroidal Spectral Gap) and Corollary 9.3 rewritten to explicitly distinguish two scaling regimes: Fixed-N, growing-d: λ1 = Θ(1), system scales by adding dimensions. Fixed-d, growing-N: λ1 = Θ(1/N2), reflecting that tori are not expanders under side-length scaling. v1.0 claimed λ1 = Θ(1) unconditionally. The paper's complexity results assume the fixed-N regime, matching QuantumHarmony's N=8 shard architecture. New subsection: "Logical Topology vs Physical Peer Graph." The toroidal mesh is the logical state-space structure for Coherence Cells, distinct from the physical peer overlay (libp2p gossipsub, expander-like). Spectral claims transfer to the physical layer regardless of torus scaling. Theorem 11.1 (Message Complexity) rewritten with explicit M = Nd notation. Regime-specific bounds: O(M log M) for fixed-N, O(M1+2/d log M) for fixed-d. Both beat PBFT's O(M2) asymptotically. Comparison table: added HotStuff row; relabeled Karmonic row as "(fixed-N)". Proposition 11.2 renamed to "Constant-Factor Advantage at Fixed Size." Corrected the 57× claim: ~10× messages, ~85× bandwidth vs PBFT at T38. Summary of Key Results box updated. Key Results (intro) aligned with body. No changes to: Coherence Cell definitions, Kuramoto connection, Hybrid Architectures, Scope and Limitations, or Applications.