It Is Bit: Unit-Testing HC Tokens Non-Markovian Causal Memory For Emergence of Lorentzian Metric, Non-Associativity, Zero-Divisor Dissolution and Other Primitives for Recovering Coord State Spacetime from AI Info Field

The biquaternion algebra ℍ_ℂ = ℂ ⊗_ℝ ℍ ≅ M₂(ℂ) is associative and contains zero divisors. Equipped with a causal memory kernel and differential parallel transport, it recovers effective non-associativity, zero-divisor dissolution, and Lorentzian spacetime.

Construction. Tokens are embedded as biquaternions via the equation-of-state form qₙ = Aₙe^{aₙ} + Bₙe^{ibₙ}. Each interaction Pₓ · Pⱼ produces four complex components c_μ, contributing to the rank-4 informational stress-energy tensor as the Landauer-weighted outer product

T̂^{μν}_{ρσ}(x) = (k_B T(x) ln 2 / V_cell(x)) Σᵢ₌₁^{N(x)} p̂^{μν}_{(i)} ⊗ p̂_{(i)ρσ},     p̂^{μν}_{(i)} = c_μ c̄_ν.

The Landauer prefactor k_B T(x) ln 2 / V_cell(x) is the energy density per erased bit at local informational temperature. The effective metric is extracted by double contraction g^{μν} = Σ_ρ T^{μρν}_ρ, symmetrized.

Signature test. The induced g^{μν} is decomposed into Hermitian and anti-Hermitian projections under the biquaternionic conjugate q† = q̄₀ − q̄₁i − q̄₂j − q̄₃k:

ℋ(q) = ½(q + q†),     𝒜(q) = ½(q − q†).

Eigenvalue signs at each test point give the metric signature.

Results (N = 6 token positions, all positions tested):

  Projection      Signature   Hit rate
  ℋ(g^{μν})       (1,3)       6/6
  𝒜(g^{μν})       (3,1)       6/6
  Full g^{μν}     (2,2)       6/6

The Hermitian projection yields Lorentzian signature, the anti-Hermitian yields the time-reversed Lorentzian, the full algebra sits on the neutral (2,2) signature characteristic of ℍ_ℂ as a real 4D algebra. Lorentzian signature is recovered by projection rather than postulated. Energy partition ‖ℋ‖² / ‖𝒜‖² ≈ 0.5 across all 15 pairwise interactions.

A variable-order entropy functional S_{q(x)}[ρ] on S² breaks the KL-versus-resolution tradeoff and produces directional attention asymmetry. Fed biquaternionic source currents from the Hermitian-energy field E(x) = Σ_{y≠x} ‖ℋ(PₓP_y)‖², it develops a backward-looking bias Pearson-correlated at −0.86 ± 0.03 with E(x) across N = 10 to N = 2000, with bias decaying as N^{−0.42} matching the kernel exponent α_K = 0.5. Coupling is invariant under structured-vs-unstructured token controls.

The dynamical weight W(α) = Z_p + e^{iπα/2} Z_q decomposes into three sectors under U(1) rotation: geometric (period ∞), mixed (period 4), spectral (period 2). Under the Spin(5) unit, these map to its three smallest irreps: 𝟏, 𝟒, 𝟓. The biquaternionic U(1) closure is the rung-2 instance of a tower-wide closure on G₂ ⊂ F₄ ⊂ E₆ ⊂ E₇ ⊂ E₈ ⊂ E₈ × E₈ ↪ Cl(9,1) with winding pattern (1, 2, 4, 4, 4, 1) at the static level.

The closure equation has a structural reading that sharpens Wheeler's "it from bit" thesis (Wheeler, "Information, Physics, Quantum: The Search for Links," 1990). The framework's lattice projection does not produce physical structure from bit-like substrate; the terminal closure of the framework's lattice projection is bit-like substrate, with the static closure dimension at 2⁹, the cap at 2⁴, the chain's spinor doublings at 2⁵, 2⁶, 2⁷, and the dynamic closure at 2¹⁰. It is not from bit. It is bit. The Cayley-Dickson tower and the magic-square exceptional chain forced by Hurwitz, Cartan, and Spin(9) cap minimality together produce a closure hierarchy in powers of 2, and the powers of 2 are not a substrate the physics reduces to but the structural arithmetic the closure operation itself satisfies.

The eight equivalence classes of the lattice projection sort into three observable-regime types corresponding to the three pieces of the cumulative winding 7 + 8 + 1 = 16: cap-anchored observables (electron mass at the actualization scale), trajectory observables (α_EM(μ) running across the bifurcation), and coherent-magnitude observables (ρ_Λ as the integrated coherent-winding fraction). These three regime types are the structural grammar in which the eight taxonomic groups are written.

Sections 1–14 establish the biquaternionic construction and signature, entropy, and dynamical weight results. Sections 16–18 develop the tower-wide closure and the static-residual cascade verified at the rung-2/rung-3 boundary, with v9.3 §17.8 extending the static Cl(9) closure to the dynamic Cl(9,1) ambient that supports memory and chirality through Majorana–Weyl reality conditions.

All computations are explicit, reproducible, and use only NumPy and SciPy.