Computational Horizons: Section 3 - P vs NP as TTL Exhaustion

Draft v0.1 - 2026-01-07

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3. P vs NP as TTL Exhaustion

We demonstrate that the P ≠ NP conjecture can be reframed as a physical constraint: NP-complete problems require routing through exponentially many paths, exhausting any polynomial TTL budget.

3.1 The Traditional Framing

The P vs NP problem asks: Can every problem whose solution is quickly verifiable also be quickly solved?

• P: Problems solvable in polynomial time
• NP: Problems verifiable in polynomial time
• NP-complete: The hardest problems in NP (all NP problems reduce to them)

Despite decades of effort, whether P = NP remains open. Most researchers believe P ≠ NP, but no proof exists.

3.2 The Routing Interpretation

We reframe the question: How many hops does it take to find the solution?

P problems: The solution path has polynomial length. With polynomial TTL, the packet arrives.

NP problems: Finding the solution may require exploring exponentially many paths. With polynomial TTL, most packets are dropped before finding it.

3.3 Thermodynamic Analysis

Neukart (2024) demonstrated that P and NP problems have distinct thermodynamic signatures:

"Computational problems have quantifiable 'information cost' based on entropy... entropy profiles within computational tasks enable a clear distinction between P and NP-classified problems."

His Entropy-Driven Annealing (EDA) method maps energy landscapes of computational problems, showing NP has fundamentally different thermodynamic profiles than P.

We extend this: Entropy = uncertainty = routing options. More routing options means higher fetch cost. The thermodynamic distinction is the routing cost distinction.

3.4 The Landauer Connection

Landauer's principle (1961) establishes the minimum energy cost of computation:

E_min = k_B T ln(2) ≈ 2.87 × 10⁻²¹ J at room temperature

per bit of irreversible information erasure.

Each computational step (hop) that erases or modifies information costs at least this much energy. For n hops:

E_total ≥ n × k_B T ln(2)

3.5 The TTL Budget

Any physical computer has finite energy E_max. This implies a maximum hop budget:

TTL_max = E_max / (k_B T ln(2))

For the entire observable universe, Lloyd (2000) calculated:

N_ops ≈ 10^120 operations since the Big Bang

This is the cosmic TTL—the maximum total hops ever taken by any computation in our universe.

3.6 Why NP Exceeds TTL

Consider subset sum with n elements. The search space is 2^n subsets.

n	Search space	TTL required
20	2²⁰ ≈ 10⁶	10⁶ hops
50	2⁵⁰ ≈ 10¹⁵	10¹⁵ hops
100	2¹⁰⁰ ≈ 10³⁰	10³⁰ hops
400	2⁴⁰⁰ ≈ 10¹²⁰	All cosmic hops

At n = 400, solving subset sum by exhaustive search would require every operation the universe has ever performed.

No polynomial TTL (even 10¹²⁰) can handle exponential scaling indefinitely.

3.7 Simulation Results

We tested binary search (P) vs subset sum (NP) with fixed TTL = 32,768:

Binary Search (P problem):

n	Hops required	TTL	Result
100	7	32,768	✅ Found
1,000	10	32,768	✅ Found
10,000	14	32,768	✅ Found
100,000	17	32,768	✅ Found

Subset Sum (NP problem):

n	Search space	TTL	Result
10	1,024	32,768	✅ Found in 1,023 hops
15	32,768	32,768	✅ Found in 32,767 hops
20	1,048,576	32,768	❌ Dropped
25	33,554,432	32,768	❌ Dropped

P scales logarithmically with n. NP scales exponentially. Any fixed TTL eventually fails for NP.

3.8 The Entropy-Complexity Uncertainty Principle

The literature suggests a fundamental relationship:

ΔH · ΔC ≥ k_B T ln(2)

where ΔH is entropy rank (uncertainty in solution space) and ΔC is complexity (steps to solution).

This mirrors Heisenberg uncertainty: you cannot have both low uncertainty AND low complexity. NP-complete problems have high entropy (large solution spaces), forcing high complexity (many hops).

3.9 What This Means for P ≠ NP

If P ≠ NP is a thermodynamic constraint:

• No algorithm can overcome it - It's not about cleverness; it's about physics.
• Quantum computing doesn't help - QC exploits superposition for parallelism but still pays routing costs. Grover's algorithm offers √n speedup, not exponential.
• P ≠ NP is likely unprovable within mathematics - It may be a physical law, not a theorem.

3.10 The Packet-Switching Analogy

Classical computation assumes circuit-switching: dedicated paths, guaranteed delivery.

Reality is packet-switched: best-effort routing, finite TTL, dropped packets.

Paradigm	Routing	TTL	Failures
Circuit-switched	Dedicated	Infinite	None
Packet-switched	Best-effort	Finite	Dropped
P problems	Polynomial paths	Poly TTL sufficient	None
NP problems	Exponential paths	Poly TTL insufficient	Dropped

The universe is packet-switched. P problems fit the TTL. NP problems don't.

3.11 Implications

If this framing is correct:

• P ≠ NP is not a conjecture but a physical law - like conservation of energy.
• No breakthrough algorithm can solve NP in P - the constraint is thermodynamic.
• The Clay Prize may be unclaimable - the "proof" might be showing it's physics, not math.

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