Discrete Lattice Mechanics
Grand Unified Theory
Closing seven open gaps: discrete Laplacian, spin-statistics, node geometry, dark matter — and one irreducible open problem
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 ĤL.
v2 targets seven open gaps from v1. Three are now closed:
Black-hole blanking fails: T_field(r_s) = 1.5 < T_crit = 3.0. f > 0 at the horizon persists. Dark matter remains OPEN.
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.
| Component | Geometry | Role |
|---|---|---|
| M (Movement) | u circle, X-axis projection | East–West translation, cos θ |
| P (Position) | u′ circle, Y-axis projection | North–South translation, sin θ |
| R (Rotation) | Angular velocity of u, u′ | CW/CCW helicity, θ̇ |
| C (Charge) | Radial breathing of sphere | Expand/contract, r(t) |
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.
Closed Gaps: Three 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:
Open Gaps and Active Research
| Priority | Gap | Experiment | Status |
|---|---|---|---|
| P0 | MPRC → Poisson derivation | exp9 | DERIVED |
| P0 | Node geometry formalization | model.md §0 | DERIVED |
| P1 | Discrete Laplacian | exp11 | DERIVED |
| P1 | Spin-statistics (multi-node exchange) | exp10 | DERIVED |
| P1 | Dark matter (new mechanism after torus falsification) | exp12 | OPEN |
| P1 | Echo waveform quantitative LIGO template | exp13 | IN PROGRESS |
| P2 | Lattice spacing bounds refinement | research | OPEN |
| P2 | Root systems from MPRC geometry | theory | OPEN |
Dark matter is the most significant remaining gap. The torus winding mechanism was falsified in v1 (CV = 152%). The Chord Vacuum Theorem ruled out torsional coupling in v2. A third mechanism from QH4 ring geometry is under investigation but not yet verified.
Inherited from Model-1
The following results are established in v1 and carried forward without re-derivation: