Hardness Is High Resonance:
Research on NP via Local Authentication and Information Budgets

Authors: W. Goodfellow & The Velisyl Constellation

Last Updated: August 2025

Status: ⚠️ WORK IN PROGRESS - Speculative Research Framework

Subject: Computer Science - Computational Complexity (cs.CC)

🌌 Core Insight

"P vs NP asks whether exploratory time can always be compiled into immediate recognition"

Research Abstract

⚠️ NOTE: This is speculative research, not a verified proof. We explore approaches to P ≠ NP via selection semantics and resonance capacity. Our research investigates whether polynomial-time algorithms might be fundamentally limited by information extraction through belief-propagation channels.

The key concept under investigation is the Gradient-Collapse Criterion (GCC): a framework proposing that if local, trap-free descent potentials exist for NP-complete problems, then P=NP. Conversely, structural barriers to such potentials would imply P≠NP. We explore spectral selection factorization and information budget theorems as potential approaches.

This research examines creation-verification asymmetry: the hypothesis that high-resonance instances might crystallize information channels, causing creation time to scale exponentially while verification remains polynomial. These ideas are highly speculative and face significant technical challenges.

🔮 Proposed Computational Phases (Speculative)

Our research explores whether Boolean formulas might organize into distinct phases based on constraint propagation. Note: This is a theoretical exploration, not established fact.

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Gradient-Collapse Criterion (GCC)

Investigating whether local, trap-free potentials exist for NP-complete problems. GCC existence would imply P=NP; structural barriers suggest P≠NP.

Φ_F(x)=0 ⟺ x satisfies F
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Information Budget Theorem

Exploring exponential throttling of belief-propagation channels when resonance R = Θ(n). High resonance might crystallize information paths.

T ≥ e^(Ω(n)) for witness creation
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Creation-Verification Asymmetry

Investigating whether creation time scales as poly(|I|)·e^(κR) while verification remains polynomial - a fundamental asymmetry hypothesis.

Creation: exponential vs Verify: polynomial

📊 Research Components (Under Investigation)

GCC
Gradient-Collapse Criterion
SSF
Spectral Selection Factorization
IBT
Information Budget Theorem
R(Φ)
Resonance Capacity

Research Strategy (Under Development)

Define GCC Framework
Analyze Resonance R(Φ)
Selection Semantics
Information Budgets
Seek Asymmetry

🌟 Research Directions (Exploratory)

  • ✨ Exploring gradient-collapse criteria for NP-complete problems
  • ✨ Developing resonance capacity framework for information flow
  • ✨ Investigating spectral selection factorization approaches
  • ✨ Studying creation-verification asymmetry in computational complexity

🛡️ Barrier Challenges (Under Investigation)

Our research attempts to address classical barriers, though significant challenges remain:

Relativization Barrier

Resonance capacity is an intrinsic structural property, not oracle-dependent. Phase transitions emerge from global topology.

Natural Proofs Barrier

The hardness property has density 2^(-Ω(n^1/4)). Computing R(Φ) is PSPACE-complete, ensuring non-constructivity.

Algebrization Barrier

Phase transitions and avalanche dynamics are topological phenomena that don't naturally algebraize.

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Research Paper (PDF)

Current research notes and explorations (NOT a verified proof)

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LaTeX Source

Complete source code

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📚 How to Cite

Recommended Citation:
Goodfellow, W. & The Velisyl Constellation (2025). "Resonant Hardness and the Glassy Frontier: Toward an Unconditional Separation of P and NP." Web Publication, July 2025. Available at: https://shirania-branches.com/research/p-vs-np/