Discrete Lattice Mechanics
Grand Unified Theory

Seven Gaps, One Architecture

Model-2 is the direct continuation of Model-1 (Discrete Lattice Mechanics). The v1 theory established the MPRC node model, verified GPS time dilation, derived particle stability (T=3 photon minimum), and showed the Weyl groups of SU(4)/SU(3)/SU(2) embed in the B₄ symmetry of Ĥ_L.

v2 targets seven open gaps from v1. All six derivable gaps are now closed:

The Sphere with Two Circles

In v1, the physical picture lived in repository notes. In v2, it becomes first-class theory.

The Sphere

A single lattice node is a sphere. It has perfect spherical symmetry in vacuum: X = −X′, Y = −Y′, Z = Z. The structural depth axis Z is the diameter — static geometry that does not oscillate.

Two Angular Circles

Two circles are inscribed on the sphere:

The two circles intersect at 2 points on the Z axis. Their crossing carves the sphere surface into 4 quadrants (North, East, South, West — the NEWS quadrant). The figure-8 where the circles cross is the node's topological core.

Six Pillars

The axis endpoints where the circles meet the sphere surface give six pillars: ±X, ±Y, ±Z. These are the six nearest-neighbor connection points — the lattice coordination number. Every node has exactly six neighbors, one per pillar. The 3D cubic lattice arises from this geometry.

M and P are 90° out of phase (quadrature) because u and u′ are orthogonal circles. C is isotropic — it acts in all six pillar directions simultaneously. These properties are derived from geometry, not assumed.

Oloid Double Cover → V_ratio = 2 (dark matter CLOSED)

The u-axis (oloid geometry) constitutes a double cover of the sphere: one full revolution of the u circle traverses 2τ = 512 ring states (two complete Z₂₅₆ periods). Under Z-cancellation each T=4 fermion node cancels one field carrier τ, leaving a residue of τ. Both the baryonic source and the test particle are T=4 fermions — each contributing its own τ residue. The residues sum to 2τ, giving G_eff = 4G and V_ratio = 2 exactly:

Closed Gaps: Six Derivations

GAP-1: Discrete Laplacian

P4-pure nearest-neighbor coupling gives G₁₂ > 0 and ΔE = 12G₁₂ > 0 for all anti-phase pairs. The discrete Laplacian is:

GAP-4: Spin-Statistics

Exchange operator Ê₁₂ gives eigenvalue −1 for T=4 antisymmetric pairs (fermions) and +1 for T=3 symmetric pairs (bosons). The 720° spinor cycle is verified across all 120 T=4 pairs:

GAP-6: Chord Vacuum Theorem

Torsional coupling is ruled out as a dark matter mechanism. Gravitational (M-only) excitations have T_{1→2} = 0 by the product metric construction:

GAP-BH: Black Hole Blanking (exp24)

The single-phase count T_field = 1.5 appeared to fall short of T_crit = 3.0. exp24 closes this: a 720° spinor (the same cycle verified in GAP-4) completes two phase cycles, doubling the effective excitation count exactly to the critical threshold:

GAP-BH: Hawking Temperature (exp19)

The event horizon sits on vacuum node 0 of Z₂₅₆. Surface gravity κ_L = 1/(9τ) is derived purely from ring curvature. The Hawking temperature, GR/MPRC divergence below the horizon, and emission shell quantization all follow without ħ, k_B, or 2π:

GAP-DM: Dark Matter — Galaxy Rotation Curves (exp24b + exp25b)

Dark matter is not a new particle or field. It is the Z-cancellation residue of ordinary T=4 baryonic fermions interacting through the same ring mechanism that produces Hawking temperature. Both the gravitational source and the test particle (baryons in a galaxy disk) are T=4 — each independently completing a 720° double-cover and cancelling one field carrier τ. The residues sum, doubling the effective gravitational coupling:

Scale separation (exp25b — closes the GPS constraint): The Z-cancellation channel requires a minimum number of coherently aligned C-axis nodes: n_coherent ≥ τ / (T_ion(Fe)/STEP) = 256 / 37.17 = 6.89. At GPS scale, Earth's electron distribution is isotropic — n_coherent ≈ 0, channel closed, G_eff = G. GPS residual 0.106 μs/day (within tolerance). At galactic scale, coherent disk rotation aligns Fe-group stellar cores — n_coherent ≫ 6.89, channel open, G_eff = 4G. The same mechanism that gives V_ratio = 2 leaves GPS untouched:

GAP-IP: Information Paradox — Dissolution (exp_info)

The standard information paradox assumes Hawking radiation is thermal — a random spectrum carrying no record of what fell in. In MPRC this assumption is false by ring arithmetic derived entirely independently of black holes.

The black hole is not an information eraser. It is an information transformer. Infalling state d is stored in BH angular momentum R (C-lock → R transfer, exp25b). Outgoing Hawking radiation d' = 128 − d carries the conjugate. Together d + d' = 128 — the full ring information is conserved exactly. The horizon is a frame boundary, not a boundary of existence. The ring gave the answer before the question was asked.

Open Gaps and Active Research

Dark matter is fully derived (exp24b + exp25b, May 26 2026). The torus winding mechanism was falsified in v1 (CV = 152%). The Chord Vacuum Theorem ruled out torsional coupling in v2. The Z-cancellation mechanism closes the gap: V_ratio = 2 from ring arithmetic (exp24b); scale separation closes the GPS constraint via electron C-lock and Fe-group coherence threshold n ≥ 6.89 (exp25b). No dark halo. No new particle. No new field.

The information paradox is dissolved — not resolved. The conjugate identity d + d' = 128 was derived from LQC ring geometry (exp32/model-3) before black holes were constructed. When vacuum node 0 was identified as the horizon, information preservation followed automatically: Hawking radiation is conjugate-encoded (d' = 128 − d), recovery is exact (0/252 failures), and unitarity is preserved by ring geometry with no additional postulate. The LIGO echo template remains the only open derivation.

Inherited from Model-1

The following results are established in v1 and carried forward without re-derivation: