📘 Collapse-Theoretic Resolution of the Black Hole Information Paradox (BHIP) via AK-HDPST v13.0

🌌 What is the Black Hole Information Paradox (BHIP)?

The Black Hole Information Paradox is a foundational inconsistency between general relativity and quantum mechanics.
According to Hawking radiation, black holes emit thermal radiation and may eventually evaporate completely.
But this process seems to irreversibly destroy quantum information, violating the unitarity of quantum evolution.

In short:

• Quantum mechanics: pure states must evolve into pure states (information preserved).
• Hawking radiation: emits thermal (mixed) states with no memory of the infalling matter.
• ⇒ Contradiction arises: information appears lost.

This paradox threatens the foundations of modern physics.

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🧩 What is AK High-Dimensional Projection Structural Theory (AK-HDPST)?

AK-HDPST is a mathematically rigorous framework for eliminating obstructions across geometry, algebra, and information systems.
It is built on three core components:

1. High-dimensional projections — structural simplification via functors.
2. Collapse structures — processes that reduce complexity while encoding trace information.
3. Type-theoretic formalism — safety guarantees and logical closure via typing systems.

AK-HDPST v13.0 supports machine-verifiable collapse diagnostics using Coq / Lean formalization.(https://github.com/Kobayashi2501/AK-High-Dimensional-Projection-Structural-Theory)

🔓 How does this theory resolve the BHIP?

AK-HDPST resolves BHIP by reinterpreting information loss as a structural typing failure —
not as a violation of physics, but as the breakdown of projectability across a categorical boundary.

Key concepts:

• Black holes are modeled as collapse-typable structures: Type_Collapse(X) := (PH1, Ext1, ICM)

• PH1: persistent topological structure

• Ext1: categorical entanglement (nontrivial extensions)

• ICM: information compression measure

• The event horizon is a collapse boundary where:

• Typing degenerates

• Collapse functor becomes undefined

• Functorial information projection fails

• Hawking radiation emerges as partial projection of collapse-typable shell regions:

• Only high-ICM, low-Ext1 structures escape

• Core information remains unrecoverable

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🧠 What does this theory allow us to say concretely?

✅ Black Hole Interior Structure

The Bekenstein–Hawking black hole (BKH) interior is stratified:

• Core Layer:
• PH1 ≠ 0 (topological cycles remain)
• Ext1 ≠ 0 (entanglement persists)
• ICM: high (informational redundancy)
• Shell Layer:
• PH1 → 0, Ext1 → 0
• ICM: moderate
• Horizon:
• Typing becomes undefined
• Collapse is non-functorial

X_core → X_shell → X_horizon → Collapse(X) (observable)

✅ Collapse Q.E.D. (Epistemic Closure)

Collapse is irreversible:

• Collapse_QED(X) := PH1 = 0, Ext1 = 0, ICM > 0, KL_div > 0, and no inverse map exists.
• Entropy arises not from thermal noise, but from non-invertible projection.
• Formalized and verifiable in Coq.

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🧬 Structural Results

• 🔹 Collapse = functorial, structure-reducing projection (not invertible)
• 🔹 KL divergence used to measure structural entropy
• 🔹 Ext1 models entanglement as categorical obstruction
• 🔹 Firewall = total failure of collapse typing at the horizon
• 🔹 AdS/CFT and ER=EPR reinterpreted as projection-compatible approximations
• 🔹 Collapse Q.E.D. = formal endpoint of gravitational information encoding

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🧪 Simulation and Observation Possibilities (Appendix L)

• Gravitational wave ringdown analysis:
  • Use persistent homology to detect topological collapse
• AdS/CFT simulations (MERA, HaPPY):
  • Detect Ext1 collapse via entanglement wedge failure
• Quantum circuits:
  • Simulate collapse failure via tensor noise and recovery thresholds

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📂 Repository Contents

• main.tex: Full LaTeX manuscript
• appendix/: All appendix sections A–Z⁺
• README.md: (this file)

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📌 Citation (Zenodo DOI)

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🧠 Keywords

Black Hole Information Paradox, Collapse Functor, Category Theory, Entropy, Ext1, Persistent Homology, AK-HDPST, Coq Formalization, Type-Theoretic Closure

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📜 License

This work is released under the MIT License.