1) Why Black Holes Matter in PFT

Black holes sit at the crossroads of geometry, dynamics, information, and measurement. In mainstream physics they are solutions to Einstein’s field equations—with a horizon, an ergosphere for rotating cases, and an interior where classical descriptions break down. In Pattern Field Theory, that breakdown is reinterpreted: the apparent singularity is the macroscopic signature of a closure regime on the Allen Orbital Lattice (AOL), where recursive rules compress growth into stable loop families rather than allowing unbounded curvature.

PFT contributes three key shifts: (i) space, time, and matter are emergent, not fundamental; (ii) collapse is a lattice-closure process governed by resonance windows (π, φ, e) and structured disruption by primes; (iii) observation is the coupling of pattern systems, so information is redistributed across closure shells and environment corridors rather than destroyed.

2) Event Horizons as Closure Fronts

In general relativity, an event horizon is a null surface beyond which causal signals cannot escape. In PFT, the horizon is a closure front: an advancing boundary where recursive pattern dynamics force open paths to close into loops or terminate under the Allen Fractal Closure Law (AFCL). Locally, this reduces degrees of freedom (many possible paths collapse into countable shells); globally, it manifests as redshift and one-way causal flow.

2.1 Formal sketch

• Substrate: Hex-graph G = (V, E) with neighborhood N(v) and state s_t: V → Σ.
• Rules: Local maps R: Σ^k → Σ act on N(v); the update semigroup ⟨R⟩ drives s_t → s_t+1.
• Closure predicate: C(γ) on paths/loops γ triggers loop completion or termination when bounded-curvature criteria are met (AFCL).
• Resonance score: ρ(s_t) measures motif compatibility with (π, φ, e) bands (Pi-Resonance™); primes gate discrete disruptions.

An event horizon is the surface where C is almost surely true for outward-directed motifs; post-horizon, dynamics overwhelmingly favor loop retention rather than propagation. Escape probability decays super-exponentially with shell depth.

3) Classical Anchors: Standard Formulas We Keep

PFT does not discard classical relations that work; it explains them via closure/resonance statistics on the lattice.

• Schwarzschild radius for mass M:
  r_s = \frac{2GM}{c^2}
  In PFT, r_s marks the radius where closure fronts percolate and outward motif survival falls below a resonance threshold.
• Kerr (rotating) horizons:
  r_\pm = \frac{GM}{c^2} \;\pm\; \sqrt{\left(\frac{GM}{c^2}\right)^2 - \left(\frac{Jc}{M}\right)^2}
  Frame dragging = azimuthal bias in motif transport on AOL; the ergosphere is the region where closure favors co-rotation corridors.
• Bekenstein–Hawking entropy:
  S = \frac{k_B c^3 A}{4 G \hbar} with A = 4\pi r_s^2
  In PFT, entropy is proportional to the count of distinct closure-shell micro-configurations compatible with the same macroscopic horizon.

4) Collapse as Loop Quantization

Gravitational collapse becomes loop quantization on AOL: as density increases, allowed open paths shrink; AFCL compels loop completion; shells stack; dynamic balancing prevents runaway amplification. The “singularity” is replaced by a finite-depth cascade of nested closures—a closure stack whose statistics determine horizon geometry and emission spectra.

4.1 Prime anchoring

Primes inject structured irregularity at discrete steps, preventing crystalline degeneracy and stabilizing shell diversity. Prime-gated updates desynchronize degenerate loops, reducing catastrophic coherence and creating resilience bands.

4.2 Pi-Resonance bands

Resonance windows centered on π, φ, e favor specific curvature and ratio motifs. During collapse, these act like selection rules: loops that meet resonance survive; others dissipate. The result is preferred shell depths and corridor widths—potentially observable in lensing microstructure and polarization patterns.

5) Time and Gravity Near the Horizon

Time dilation is reinterpreted as update-order deformation: near the closure front, effective update schedules stretch—many local micro-updates correspond to little change in global observables. To distant observers this looks like freezing; to in-falling observers, local clocks remain smooth because their order parameter tracks local recursion, not a distant frame.

5.1 Operational definition

Let τ be proper time measured as update depth along an in-falling motif. Outside, coordinate time t counts layers of corridor equilibration. Near the horizon, the ratio dτ/dt → 0 due to closure saturation—even without invoking a metric singularity.

6) Information, Measurement, and the “Paradox”

In PFT, the observer is a coupled pattern, not an external agent. Measurement is the pushout (gluing) of two AOL systems along shared interfaces. The standard information paradox assumes destruction under a classical singularity plus a demand for unitary evolution. PFT replaces the singularity with closure stacks and treats “unitarity” at the level of the coupled recursion: information redistributes across shells and environment corridors. Apparent loss in one factor is persistence in the coupled aggregate.

6.1 Schematic

1. Pre-collapse: motifs distribute across open corridors.
2. Closure front advances: motifs re-encode into shell micro-states favored by resonance.
3. Environment coupling (radiation, tidal channels) exports partial motif data along permitted corridors.
4. Post-collapse: remaining information sits in closure stacks; global accounting balances at the coupled level.

7) Hawking-like Emission in PFT

Spontaneous emission near horizons appears as corridor re-opening events: dynamic balancing occasionally destabilizes a shell boundary, creating paired motifs that escape/ingress along complementary corridors. This recovers qualitative features of Hawking radiation while grounding them in finite, local rules instead of trans-Planckian modes.

7.1 Scaling

Emission power scales with shell-curvature statistics and resonance bandwidth. For non-rotating holes, the dependence mimics A⁻¹ trends; for Kerr cases, azimuthally biased bands produce tell-tale polarization anisotropies.

8) Mathematical Notes and Working Formulae

• Shell-depth distribution: P(k) ∝ e^−αk, with α set by resonance width; deviations at prime-indexed k reflect anchoring.
• Corridor bandwidth: W ≈ W₀·f(π, φ, e; κ) where κ is mean curvature; narrowing near closure increases the Q-factor of resonances.
• Update-order dilation: dτ/dt ≈ g(ρ, C), with g → 0 as closure certainty → 1.

These relations specify what to extract from lattice simulations for comparison with observation.

9) Predictions

• Lensing microstructure: Fine caustic ripples corresponding to preferred corridor widths (π-φ bands).
• Polarization patterns: Kerr holes show azimuthal bias aligning with ergosphere-corridor statistics.
• Quasi-periodic oscillations (QPOs): Frequency clustering at shell depths favored by resonance, with prime-offset sidebands.
• Ringdown tails: Late-time decay modulated by closure-stack depth distribution rather than a single QNM set.

10) From Einstein & Hawking to PFT

Correcting Einstein: Curvature remains central, but it is the result of loop statistics on a discrete lattice, not a primitive manifold property. Correcting Hawking: Thermal-like emission arises from corridor re-openings in finite recursion; no reliance on trans-Planckian frequencies or exact thermality is required.

11) Worked Examples

11.1 Non-rotating stellar collapse

Initialize a dense AOL region with inward-biased updates. Track shell completions, corridor closures, and emission spikes as dynamic balancing adjusts. Extract P(k), W(κ), and ringdown profiles. Compare to observed supernova remnants with candidate compact objects.

11.2 Rapidly rotating case

Introduce azimuthal bias (frame dragging) in update rules. Measure polarization anisotropy and QPO clusters. Predict asymmetries in EHT-style images at specific baselines.

12) Engineering & Data Analysis Hooks

• Fit corridor models to lensing arcs; retrieve resonance bandwidths.
• Use prime-gated sideband signatures to detect closure stacks in QPO data.
• Generate synthetic ringdown libraries parameterized by resonance width and prime anchoring.

13) Frequently Asked Questions

14) Conclusion

In Pattern Field Theory, black holes are not holes in spacetime but closure-dominated regions where recursion saturates and loops quantize dynamics. This preserves the empirical successes of classical theory while eliminating singularities, reframing information flow, and offering concrete spectral, imaging, and timing predictions testable with current and near-future instruments.

Appendix A: Reference Relations

• Schwarzschild radius: r_s = 2GM/c^2
• Kerr horizons: r_\pm = GM/c^2 \pm \sqrt{(GM/c^2)^2 - (Jc/M)^2}
• Entropy–area: S = k_B c^3 A / (4 G \hbar)

Appendix B: Related PFT Articles

• Correcting Einstein
• Correcting Hawking
• Pi Emergence
• Core Principles

Appendix C: Verbatim Archive of Prior Black Hole Writing

This appendix preserves the full text of the previously published “Pattern Field Theory’s Introduction to Black Holes” so that no information is lost during consolidation. You can keep this open for transparency, or later migrate pieces directly into the main narrative and remove this appendix.