# 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.

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## 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.

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## 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.

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## 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.

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## 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.

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## 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"

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## 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

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## 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](https://en.wikipedia.org/wiki/Free_energy_principle) - Wikipedia
- [The free-energy principle: a unified brain theory?](https://www.fil.ion.ucl.ac.uk/~karl/NRN.pdf) - Friston, FIL UCL
- [The free-energy principle: a rough guide to the brain?](https://www.fil.ion.ucl.ac.uk/~karl/The%20free-energy%20principle%20-%20a%20rough%20guide%20to%20the%20brain.pdf) - Friston
- [Problem of time](https://en.wikipedia.org/wiki/Problem_of_time) - Wikipedia
- [Speed of light](https://en.wikipedia.org/wiki/Speed_of_light) - Wikipedia
- [Planck units](https://en.wikipedia.org/wiki/Planck_units) - Wikipedia
- [Minimal Length Scale Scenarios for Quantum Gravity](https://pmc.ncbi.nlm.nih.gov/articles/PMC5255898/) - PMC
- [Laws beyond spacetime](https://pmc.ncbi.nlm.nih.gov/articles/PMC10449986/) - PMC


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


## North

```yaml[id=simulation-without-a-basement-north-fence-0]
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

```yaml[id=simulation-without-a-basement-east-fence-0]
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

```yaml[id=simulation-without-a-basement-south-fence-0]
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

```yaml[id=simulation-without-a-basement-west-fence-0]
slots:
- context:
  - Formal spine alongside informal wisdom
  slug: grandmothers-teatime

```