simulation-without-a-basement

Simulation Without a Basement

The formal spine of the streams-with-gaps thesis: no turtles all the way down

Summary

A mathematical formalization of the LOOKUP–FETCH–SPLICE–CONTINUE loop as the minimal engine for any causally-structured universe. The key insight: there's no simulation "basement" - just a DAG of resolved states and the dynamics that extend it.

1. Base Objects

Let there be:

A directed acyclic graph $G = (V, E)$
Each node $v \in V$ is a resolved state (a "materialized" fact)
Each edge $e = (u \to v) \in E$ is a causal link / free-energy transfer

There is no "underlying stuff" other than this graph; "reality" = this DAG plus its dynamics.

2. Tension, Thunks, and Lazy Evaluation

For any not-yet-resolved situation:

Thunk: A deferred computation $\tau$ that encodes:

A set of possible next nodes ${v_i}$
A probability measure $P(v_i \mid \text{context})$ over them
A graph-rewrite rule for adding any chosen $v_i$ and its edges

Tension Field: $T: V \to \mathbb{R}_{\ge 0}$ assigns to parts of the graph a scalar "free energy" or "mismatch" value.

Gap: A region where:

A thunk exists (future is under-specified)
Nonzero tension could be reduced by committing to a specific node/rewrite

Lazy Evaluation Rule: The universe only forces a thunk $\tau$ when some process needs its value to continue causal propagation. No global eager evaluation of all possibilities.

3. The Core Loop: LOOKUP–FETCH–SPLICE–CONTINUE

Define a tick as one execution of this loop:

LOOKUP (Attention)

Select a gap $\gamma$ in the graph: a thunk region where tension is nonzero and some process "depends" on a resolution. Attention picks a gap worth resolving.

FETCH (Context)

Gather relevant upstream context: a subgraph $C \subseteq G$ reachable to $\gamma$. This is the internal "model" / prior used to evaluate which resolutions are available.

RESOLVE (Free-energy Reduction)

Sample a specific outcome $v^*$ from the thunk's distribution $P(v \mid C)$, following whatever rule gives you the Born-like probabilities in QM and variational updates in FEP. Choose the outcome that (locally) reduces expected free energy according to some functional $F$.

SPLICE (Write Node)

Add $v^*$ to $V$ and edges from its causes in $C$ to $v^*$, updating the DAG:

$V' = V \cup {v^*}$
$E' = E \cup {(u \to v^*) : u \in C \text{ and causally relevant}}$

CONTINUE (Propagate)

Update the tension field $T$ in light of $v^*$; some tensions drop, others are induced downstream. The system is now at a new graph state $(G', T')$, ready for the next tick.

Definition (Time): The time order is the partial order induced by the sequence of ticks and edges; time is nothing over and above the ordered chain of LOOKUP–FETCH–SPLICE–CONTINUE operations.

4. Quantum Mechanics as Inside-View of the DAG

Superposition: A system "in superposition" corresponds to a thunk $\tau$ whose possible outcomes ${v_i}$ are not yet nodes in $V$. "The particle has no definite value" = no node exists yet; only $\tau$ and its probability structure exist.

Measurement: A measurement interaction is a tick that:

Includes both system and apparatus in the context $C$
Forces $\tau$ to a concrete $v^*$
Writes a node encoding "apparatus shows result $r$" plus the system's new state

Uncertainty: Conjugate observables are aspects of the same thunk; making one sharp (choosing a narrow subset of ${v_i}$) prunes the graph of branches where the other observable had other values.

Entanglement: Two "entangled particles" are multiple downstream paths querying the same upstream thunk/subgraph. Correlations come from shared ancestry in $G$; there is no signal across distance because, at the graph level, they are not two independent nodes until after certain splits.

5. Free Energy Principle ↔ Ticks

Let each system (subgraph with a Markov blanket) maintain an internal model $Q$ of its expected sensory inputs and dynamics.

Variational free energy $F(Q)$ is a scalar functional roughly upper-bounding surprisal w.r.t. the environment
A tick that involves that system is a joint update of:- Its internal model $Q$ (perceptual inference), and/or
Its actions on the environment (active inference)
Chosen such that $F$ tends to decrease on average

The Move:

A tick = one unit of free-energy-resolving graph rewrite
"Systems minimize free energy over time" becomes: "what we call time is just the ordered accumulation of these low-$F$ moves"

6. Attention, Agency, and Distance

Attention: Selection of which gap $\gamma$ to address next (which thunk to force).

Agency: The capacity of a subsystem to bias which outcome $v^*$ is realized, within the allowed distribution $P(v \mid C)$, by acting/shaping context.

Define a cost metric $d$ on the graph:

$d(x, y)$ = minimal aggregate cost (e.g., number of ticks, weighted by free-energy moved) along any path from node/region $x$ to $y$

Then:

Legal, social, institutional "distance" = large $d$ between states ("here" and "there") because many expensive ticks are needed to route tension/information between them
Physical "speed limit" (speed of light) = an upper bound on how fast tension (useful information) can move along physical edges per tick
Planck-scale discreteness = a lower bound on the granularity of distinct nodes

7. Bedrock Claim

Bedrock Algorithm Claim

Any universe that supports persistent structure, causality, and observers must be representable (up to homomorphism) as a context-accumulating DAG whose dynamics are some instance of the LOOKUP–FETCH–SPLICE–CONTINUE loop minimizing a free-energy-like tension functional over time.

That's the formal spine: DAG objects, thunks, ticks, free-energy functional, and a universality statement that "this loop, or something structurally equivalent, is the minimal possible engine for a world like ours."

References

Free energy principle - Wikipedia
The free-energy principle: a unified brain theory? - Friston, FIL UCL
The free-energy principle: a rough guide to the brain? - Friston
Problem of time - Wikipedia
Speed of light - Wikipedia
Planck units - Wikipedia
Minimal Length Scale Scenarios for Quantum Gravity - PMC
Laws beyond spacetime - PMC

Provenance

Source: Perplexity research synthesis, 2026-01-05
Context: Formalizing the streams-with-gaps thesis
Status: 🟡 Partially verified (mathematical formalization, references checked)

North

slots:
- context:
  - Formal mathematical specification supporting the bedrock claim
  slug: bedrock
- context:
  - The invariant formalized with mathematical notation
  slug: streams-with-gaps-invariant
- context:
  - Mathematical formalization flows up to the bedrock claim
  slug: bedrock

East

slots:
- context:
  - Sibling theoretical work on the same thesis
  slug: theoretical-foundations-streams-with-gaps
- context:
  - Prior art research validating novelty
  slug: research-prior-art-streams-with-gaps
- context:
  - Formalization alongside prior art research
  slug: research-prior-art-streams-with-gaps

South

slots:
- context:
  - CFR describes how consciousness operates on the DAG
  slug: capability-from-recognition
- context:
  - Consciousness as the evaluation context required by lazy-evaluated DAG
  slug: consciousness-as-maintenance-cost
- context:
  - Both deal with computational substrates underlying reality
  slug: wolfram-convergence

West

slots:
- context:
  - Formal spine alongside informal wisdom
  slug: grandmothers-teatime

↑ northbedrockstreams-with-gaps-invariant

↓ southcapability-from-recognitionconsciousness-as-maintenance-costwolfram-convergence

→ easttheoretical-foundations-streams-with-gapsresearch-prior-art-streams-with-gaps

← westgrandmothers-teatime