Computational Horizons: Section 1 - Introduction

Draft v0.1 - 2026-01-07

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Abstract

We demonstrate that three foundational problems—computational complexity (P vs NP), quantum measurement, and gravitational horizons—arise from a single constraint: finite routing budget in weighted graphs. Every computation, measurement, and signal propagation requires hops through a graph of states; each hop costs energy (Landauer's bound: k_B T ln 2). Polynomial-time problems fit within available "time-to-live" (TTL) budgets; NP-complete problems require exponentially many hops, exhausting any polynomial TTL before solutions are found. The Born rule P(outcome) = |α|² emerges naturally from round-trip path geometry: outbound query (weight α) × return response (weight α*) = |α|². Black hole event horizons mark surfaces where TTL for outbound messages diverges to infinity—information isn't destroyed, merely unreachable. We present simulation results confirming Born rule predictions across six quantum states (χ² < critical for all) and P/NP phase transitions at n ≈ 15 items for subset sum under fixed TTL. This framework reframes physical law as routing constraints: conservation laws are doubly stochastic matrices, the speed of light is maximum routing velocity, and horizons are TTL exhaustion boundaries. The universe isn't simulated—it's the simulator.

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1. Introduction

Three foundational problems in physics and computer science appear unrelated:

• The P vs NP problem: Can every problem whose solution is quickly verifiable also be quickly solved? Despite decades of effort, this remains open—the most important unsolved problem in theoretical computer science.
• The measurement problem in quantum mechanics: Why does observation collapse the wave function? The Born rule tells us that probability equals amplitude squared, but not why.
• The black hole information paradox: How can information fall into a black hole without violating unitarity? The holographic principle suggests information lives on boundaries, but the mechanism remains unclear.

We propose these are not three problems. They are one problem expressed in three vocabularies.

1.1 The Unifying Principle: Finite Routing Budget

Consider a packet-switched network. Every packet carries a Time-To-Live (TTL) counter. Each hop decrements the counter. When TTL reaches zero, the packet is dropped. This isn't arbitrary—it's thermodynamic. Each hop dissipates energy. The packet has a finite energy budget. When the budget is exhausted, propagation stops.

We claim this principle—finite routing budget—underlies all three problems:

Domain	Constraint	TTL Interpretation
Computation	P ≠ NP	Polynomial vs exponential hops
Quantum mechanics	Born rule	Round-trip path weight
Spacetime	Event horizon	TTL → 0 at boundary

1.2 P ≠ NP as TTL Exhaustion

Neukart (2024) demonstrated that P and NP problems exhibit distinct thermodynamic signatures—different entropy profiles that reflect their computational structure. We extend this observation: NP-complete problems require routing through exponentially many paths, exhausting any polynomial TTL budget before a solution is found.

This reframes P vs NP from a mathematical conjecture to a physical constraint. Just as perpetual motion machines are impossible because they violate thermodynamics, polynomial-time solutions to NP-complete problems may be impossible because they would require infinite routing budget.

Lloyd (2000) calculated the universe has performed at most 10^120 operations since the Big Bang. This is a cosmic TTL—the maximum number of hops any computation can have taken. NP-complete problems at sufficient scale would require more hops than the universe has available.

1.3 Quantum Mechanics as Packet Routing

The Copenhagen interpretation treats wave function collapse as primitive: observation causes collapse, and that's that. We propose a mechanistic account: measurement is a fetch operation—a round-trip query through a weighted graph.

Before measurement, a quantum system exists in superposition—not "in multiple states simultaneously," but rather as an unresolved pointer to multiple possible outcomes. The wave function encodes path weights (amplitudes) through a graph of possibilities.

Measurement forces resolution. The observer sends a query (outbound path, weight α). The system must return a value—it cannot return "still undefined." The answer travels back (inbound path, weight α*, the complex conjugate). The probability of each outcome is the round-trip weight:

P(outcome) = α · α = |α|²*

This IS the Born rule. The squaring emerges from round-trip geometry, not from any additional postulate. We present simulation results confirming that sampling based on round-trip weights reproduces Born statistics across multiple quantum states.

1.4 Black Holes as Routing Horizons

The Bekenstein bound limits information content in a region of space. The holographic principle suggests information lives on boundaries, not in bulk. We interpret these through routing: an event horizon is where TTL reaches zero.

Inside the horizon, any outbound message would require more hops to escape than its TTL allows. Information isn't destroyed—it's unreachable. The horizon is a routing boundary, not an information boundary. Hawking radiation encodes the routing structure (edge weights) on the boundary, consistent with holographic expectations.

This connects to Verlinde's entropic gravity and Łukaszyk's work on horizon entropy: gravity itself may be an entropic force arising from information constraints on routing.

1.5 The Classical Limit

If the universe is fundamentally packet-switched (probabilistic routing, finite TTL), why does classical physics work?

Classical mechanics is the low-traffic limit—the regime where:

• Routing paths are well-established (high-probability channels)
• Interference is negligible (decoherence has selected branches)
• TTL is effectively unlimited (macroscopic energy budgets)

This is analogous to how circuit-switched networks (deterministic, dedicated paths) emerge as an approximation of packet-switched networks when traffic is low and paths are stable. Classical physics is Time-Division Multiplexing; quantum mechanics is the underlying packet-switched reality.

1.6 Outline

The remainder of this paper develops these connections:

• Section 2 formalizes the graph model: nodes as states, edges as transitions, complex weights as amplitudes, TTL as hop budget.
• Section 3 derives P ≠ NP as TTL exhaustion, connecting to Neukart's thermodynamic analysis and Lloyd's computational capacity bounds.
• Section 4 derives the Born rule from round-trip geometry, presenting simulation results and connecting to decoherence and entanglement.
• Section 5 interprets black hole horizons as routing boundaries, connecting to the holographic principle and ER=EPR.
• Section 6 unifies these results, showing how a single constraint (finite TTL) manifests across domains.
• Section 7 presents testable predictions and their current status.
• Section 8 discusses implications for consciousness, the arrow of time, and the nature of physical law.

1.7 What This Paper Claims

We do NOT claim to have:

• Proven P ≠ NP
• Solved the measurement problem definitively
• Resolved the black hole information paradox

We DO claim:

• These three problems share deep structural similarity
• Finite routing budget (TTL) provides a unifying framework
• The Born rule can be derived from round-trip path geometry
• This framework makes testable predictions

The goal is not to close these questions but to reframe them—to show that what appear to be separate mysteries may be aspects of a single underlying constraint: the universe is packet-switched, and packets have finite TTL.

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Provenance

• Status: Draft v0.1
• Parent: computational-horizons-paper-outline

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