The Schwarzschild–AOL Lock Index (SALI): Quantized Horizons on the Allen Orbital Lattice

1. Introduction: From Continuum Horizons to Lattice Locks

General Relativity describes the Schwarzschild radius $r_s$ as a continuous function of mass. In practice, numerical relativity and lattice-based methods discretize spacetime only as a computational device. Pattern Field Theory (PFT) takes a different stance: the substrate is fundamentally discrete, described by a π-quantized hexagonal closure lattice—the Allen Orbital Lattice (AOL)—recursively nested within the MDLN (Multi-Definition Lattice Network). On this substrate, radii, orbits, and horizons naturally find locking points, where geometry snaps to lattice shells and nodes.

We formalize this lock as the Schwarzschild–AOL Lock Index (SALI). SALI assigns integer lattice coordinates to a horizon by mapping $r_s$ to the nearest AOL node or shell. That mapping yields discrete invariants (shell counts, prime-anchored closures) and testable predictions: mass bands and radius preferences in black-hole populations, resonance clustering in planetary and lunar systems, and nested-shell analogies down to quantum scales. This article presents a clear explanation first, then gives formal definitions, algorithms, and equations.

2. Allen Orbital Lattice (AOL) and MDLN: The Cosmic Template

The AOL is the hexagonal/triangular Bravais substrate of PFT, with spacing $s$ typically taken in π-native units (e.g., $s=\pi\,\ell_P$ at the Planck scale, or $s=\pi$ in abstract unit analyses). The lattice is defined by basis vectors $\mathbf{a}_1=[s,0]$ and $\mathbf{a}_2=[\tfrac{s}{2},\tfrac{\sqrt{3}}{2}s]$. The Pi-Matrix enumerates allowed node positions; the Allen Fractal Closure Law (AFCL) governs when shells close and stabilize. The Resolutions of Scale™ framework explains how, at coarse (rough) views, only exact hex-class overlays appear, whereas at finer (smooth) views, all lattices and shapes resolve via key locking points and controlled smoothing.

3. Defining SALI: From $r_s$ to Lattice Coordinates

The Schwarzschild–AOL Lock Index (SALI) maps a physical radius to the nearest AOL node or shell. Let $r_s=\tfrac{2GM}{c^2}$ be the Schwarzschild radius for mass $M$. Let $\mathbf{A}=[\mathbf{a}_1\,\,\mathbf{a}_2]$ be the AOL basis matrix. Choose a direction (w.l.o.g. $+x$) and consider the point $\mathbf{p}=[r_s,0]^\top$. Compute fractional axial coordinates by least-squares:

Snap to the nearest node by rounding $\hat{\boldsymbol{\xi}}=\operatorname{round}(\boldsymbol{\xi})$, giving integer coordinates $(i,j)=(\hat{\xi}_1,\hat{\xi}_2)$. The locking node is $\hat{\mathbf{p}}=\mathbf{A}\,\hat{\boldsymbol{\xi}}$ with node radius $r_{\text{node}}=\|\hat{\mathbf{p}}\|_2$ and residual $\Delta r=|r_{\text{node}}-r_s|$.

The corresponding shell index (hex distance) is

The SALI is the tuple $\mathrm{SALI}(M;s)=(i,j,n,\Delta r)$ for a chosen lattice spacing $s$ and resolution regime. In the rough regime ($\sigma\to 0$), only exact hex-class overlays are allowed; in smooth regimes, a tolerance $\varepsilon(\sigma)=k\,\sigma$ is admitted (see §5), and locking points become dense, allowing universal overlays.

4. Methods & Algorithms

4.1 Alignment and Node Snapping

Given a target radius $r$ (horizon, orbit, ring), choose a reference direction and set $\mathbf{p}=[r,0]^\top$. Compute fractional lattice coordinates $\boldsymbol{\xi}$ via least-squares, snap to nearest integer lattice coordinates, and measure $\Delta r$. For annular features (rings), evaluate over angles $\theta\in[0,2\pi)$ and summarize by median/mean residual to characterize a lock band.

4.2 Key Locking Points

Some overlays align with minimal error at specific nodes—key locking points. These often occur where lattice distances approximate required ratios or angles (e.g., square or rectangular motifs). We search shell neighborhoods for triplets/pairs that minimize a motif’s loss function, record the lock periodicity, and relate it to AFCL closure and prime anchoring.

4.3 Resolution Sweep

Define tolerance $\varepsilon(\sigma)=k\,\sigma$ with scale parameter $\sigma=\alpha\,s$. Sweep $\sigma$ and compute coverage: the fraction of points within $\varepsilon$. The critical $\sigma^*$ where coverage passes a threshold (e.g., $95\%$) is the Resolution Equivalence between a given geometry and the AOL.

4.4 Planck vs Astronomical Units

Set $s=\pi\,\ell_P$ for Planck-native analysis, or $s=\pi$ in abstract unit studies; results scale linearly by $s$. SALI values report both $(i,j,n)$ and the physical $r_{\text{node}}$.

// Pseudocode summary
A = [[s,0],[s/2, sqrt(3)/2*s]]
rs = 2*G*M/(c*c)
p  = [rs, 0]
xi = (A^T A)^-1 A^T p
ij = round(xi)
p_node = A * ij
r_node = norm(p_node)
delta_r = abs(r_node - rs)
SALI = {i:ij[0], j:ij[1], n: sqrt(i*i + i*j + j*j), r_node, delta_r}
      

5. Planetary & Lunar Orbits: Resonances on the Template

Celestial mechanics exhibits abundant resonances—mean-motion ratios (e.g., 3:2 Pluto–Neptune), Laplace chains (4:2:1 for Io–Europa–Ganymede), and ring density waves. Within the AOL template, many orbital radii and resonant gaps fall near specific shell indices or key locking points. The claim is not that all orbits must sit exactly on hex shells; rather, that preferred bands occur, and deviations smooth out via Resolutions of Scale™.

5.1 Planetary Systems

• Resonant commensurabilities: map semi-major axes $a$ to $n\approx a/s$ and look for clustering near integers or prime-anchored shells.
• Gap structures: asteroid belts, ring gaps—test alignment with shell boundaries and motif junctions.

5.2 Lunar Systems

• Moons in resonance chains map to successive shell indices; libration centers coincide with motif junctions.
• Nodal precession cycles may reflect slow drift across adjacent lock bands.

6. Quantum Descents: Atomic & Lattice Analogues

Atomic spectra (Bohr/Rydberg) present discrete shells. While standard QM derives these from Coulomb potentials and angular momentum quantization, the AOL view interprets them as emergent closure locks on a nested hex substrate. Electron probability densities (e.g., $s,p,d,f$ orbitals) can be projected onto triangular/hex tilings, where nodal patterns align with motif junctions. In solids, honeycomb and kagome lattices (graphene-like systems) already sit directly on the AOL’s Bravais class; Dirac cones and flat bands emerge from motif symmetries that the MDLN reproduces at multiple nested scales.

7. Predictions & Observational Programs

7.1 Black Hole Mass Bands

For chosen $s$, SALI predicts mass bands where $r_s$ aligns closely with shell indices (low $\Delta r$). Surveys (gravitational waves, AGN/SBH catalogs) should show over-representation at these bands. The band spacing scales linearly with $s$.

7.2 Ring & Gap Statistics

Planetary rings and debris disks should exhibit gap/ring locations near shell boundaries or motif junctions, modulated by migration and turbulence.

7.3 Resonance Chains

Multi-moon resonance chains should correspond to sequences of shell indices with small $\Delta r$ when mapped by an effective $s$ for the system.

7.4 Quantum Lattice Features

Graphene-family systems at specific fillings/strains reveal plateau-like phenomena where motif-driven locks appear at nested scales (MDLN prediction: recurrence across scales under geometric rescaling).

8. Discussion: Relation to Discrete Spacetime Approaches

Loop Quantum Gravity, causal dynamical triangulations, and lattice field theories discretize spacetime for quantization or computation, but do not prescribe a π-quantized hexagonal physical substrate with fractal nesting. PFT’s AOL/MDLN does, and it couples that substrate to closure laws (AFCL) and Resolutions of Scale™ that operationalize smoothing. SALI is a practical, falsifiable tool derived from that substrate: map radii to nodes; measure $\Delta r$; look for statistical excess where locks are tight.

9. Conclusions & Next Work

We introduced SALI as a cosmological quantization tool. It converts continuous Schwarzschild radii into discrete lattice invariants and extends naturally to orbital radii and quantum shells. Immediate work: generate lock maps for black-hole catalogs; test ring/gap alignments; simulate resonance chains under lattice-aware dissipation; and explore quantum materials under nested motif rescaling. If the cosmic template is correct, the same closure mathematics should appear wherever we look, from Planck to superclusters.

Appendix: Formal Definitions & Proof Sketches

A.1 AOL Basis, Hex Distance, and Shell Count

A.2 SALI

A.3 Resolution Equivalence

A.4 Key Locking Points

Key locks minimize a motif’s loss over a shell neighborhood. Prime-anchored closures predicted by AFCL generate recurrent low-error motifs at nested scales. Existence follows from density of rational approximants in the nested hex tiling; stability from AFCL energetic minima arguments.

A.5 Planetary/Lunar Mapping

Given a semi-major axis $a$, define rough index $n\approx a/s$. For a resonance chain with ratios $p:q$, the mapped indices exhibit near-integer differences, and gap edges correspond to shell boundaries where AFCL predicts closure tension transitions.

A.6 Quantum Analogues

Project atomic orbital nodes onto triangular tilings; identify motif junctions that match nodal surfaces. In solids, honeycomb/kagome sit on the AOL Bravais class; nested motifs predict recurrence of Dirac points under geometric rescaling.