Conservation as Doubly Stochastic Matrix

The "laws of physics" aren't laws. They're accounting.

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The Core Insight

All conservation laws reduce to one statement: the routing graph is doubly stochastic.

A doubly stochastic matrix has rows and columns that each sum to 1. Flow in = flow out. No creation, no destruction, only routing.

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Spacetime Budget

You have a fixed routing budget per unit of proper time. Allocate it across channels:

         Space    Time    Total
-------------------------------
At rest: 0.0   +  1.0   =  1.0
Moving:  0.6   +  0.4   =  1.0
Photon:  1.0   +  0.0   =  1.0

That's special relativity. You're not "slowing down" - you're reallocating across channels. The matrix is doubly stochastic so the total is invariant.

Why c is the speed limit: At c, you've allocated 100% to space. Nothing left to reallocate. Can't exceed 1.0.

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The Cascade

Domain	Conservation Statement
Special Relativity	Spacetime interval invariant (ds² conserved)
Quantum Mechanics	Unitary evolution (probability conserved)
Energy/Momentum	Flow in = flow out at every node
Noether's Theorem	Symmetries = matrix structure preserved
Thermodynamics	Entropy = log of valid allocations

All the same thing. Doubly stochastic matrix. Accounting.

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Why Conservation is Universal

It's not that nature "obeys" conservation laws.

It's that non-conservation is incoherent. A matrix that doesn't conserve isn't a valid routing table. You can't have a row sum to 1.3 - where did the 0.3 come from?

The laws aren't imposed. They're the definition of valid routing.

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Time Dilation: Two Sources, Same Mechanism

Velocity (Special Relativity):

• Moving fast = spending hops on spatial traversal
• Fewer hops left for temporal processing
• Your clock slows

Mass (General Relativity):

• Mass = information density = routing congestion
• Congested region = operations take longer
• Your clock slows

Both are hop budget constraints:

• Velocity spends it internally
• Mass congests it externally
• Either way, less for your clock

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The Invariant

The spacetime interval:

ds² = c²dt² - dx² - dy² - dz²

This IS the doubly stochastic constraint. Total routing through spacetime is conserved. More space → less time. More time → less space.

At the horizon: infinite congestion → all budget consumed → no temporal routing possible → time stops.

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Connection to Horizons

Every horizon is where the matrix "runs out":

Horizon	What exhausts	Budget goes to
Black hole	Mass congestion	Infinite queue depth
Light speed	Velocity	100% spatial routing
P vs NP	Search paths	Exponential exploration
Decoherence	Environmental coupling	Route divergence

Same accounting. Different channels.

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The Punchline

Physics isn't laws that nature follows.

Physics is the statement that routing tables must be valid.

Conservation isn't a mystery. It's bookkeeping.

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North

slots:
- context:
  - Extends unification with conservation insight
  slug: computational-horizons-section-6
- context:
  - Linking conservation insight to unification section
  slug: computational-horizons-section-6

East

slots:
- slug: computational-horizons-paper
  context:
  - Supporting material for paper

Provenance

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