# The Mathematics of Verified Context

> Entroly is the only context engine for AI coding agents built on substantive original algorithms — not a wrapper over standard RAG. This document names and explains each one. Every algorithm below is implemented and runs on the production path.

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## Why this page exists

Most "context engines" do retrieval plus a token-budget knapsack and call it done. Entroly does that — and then **five more substantive things**, each one a research-grade contribution. The depth is why entroly is the only system that can *prove* a model stayed grounded, not just hope. This page is the canonical mathematical reference. Code citations point to actual implementations. Equations are reproducible from first principles.

A short summary of what's below:

| Algorithm | What it does | Why it matters | Implementation |
|---|---|---|---|
| **BIPT** | Byte-level Kolmogorov-bounded provenance | First system to detect hallucinations by *information-theoretic* trace, not semantic guessing | [`verifiers/provenance_tracer.py`](entroly/verifiers/provenance_tracer.py) |
| **NKBE** | Nash-KKT multi-agent token budget allocation | Optimal split when N agents share one context window | [`entroly-core/src/nkbe.rs`](entroly-core/src/nkbe.rs) |
| **Causal Context Graph** | Pearl-grade do-calculus on RAG | Removes coattail bias from RL signal — fragment values are *causal*, not correlated | [`entroly-core/src/causal.rs`](entroly-core/src/causal.rs) |
| **Cognitive Bus** | ISA event routing with KL-divergence priority | Principled inter-agent communication substrate with hippocampus bridge | [`entroly-core/src/cognitive_bus.rs`](entroly-core/src/cognitive_bus.rs) |
| **Resonance Matrix** | Supermodular pairwise fragment learning | Captures combinatorial context value beyond what knapsack expresses | [`entroly-core/src/resonance.rs`](entroly-core/src/resonance.rs) |
| **System 1 ↔ System 2 coupling** | Dual-process bridge (proxy ↔ vault) | Kahneman-grade architecture; verified beliefs flow into proxy context, outcomes flow back | [`entroly/coupling.py`](entroly/coupling.py) |

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## 1. BIPT — Byte-level Information Provenance Tracing

**File:** [`entroly/verifiers/provenance_tracer.py`](entroly/verifiers/provenance_tracer.py)
**Standalone exposition:** [BIPT.md](BIPT.md)

### The question, restated

Every prior hallucination detector asks "Is this output *correct*?" BIPT asks a fundamentally different question:

> *Can every byte of this output be **explained** by the input context?*

If the LLM was given context **C** and produced output **O**, then every identifier, every API call, every constant in **O** must have a *provenance trail* back to **C**. Bytes with no provenance are **inventions** — and inventions in identifier positions are hallucinations.

This is the first system to apply Kolmogorov complexity theory to hallucination detection.

### The math

From algorithmic information theory (Kolmogorov 1965; Solomonoff 1964), the conditional Kolmogorov complexity K(O\|C) measures the length of the shortest program that produces O given C. If K(O\|C) is small, O is "explainable" by C; if large, O contains information not derivable from C — i.e., hallucinated content.

True K(O\|C) is uncomputable (Rice's theorem). We compute a practical, tight upper bound via **Lempel-Ziv factorization**:

> LZ(O\|C) = decompose O into factors {f₁, f₂, …, fₖ} where each fᵢ is either
> - a **COPY** from C (longest match in C at that position), or
> - a **NOVEL** byte (no match in C).

The headline metric:

> **IPD (Identifier Provenance Deficit)** = Σ_{i ∈ identifiers} novel_bytes(i) / Σ_{i ∈ identifiers} len(i)
>
> IPD ∈ [0, 1]:   0.0 = fully grounded; 1.0 = fully invented.

### Algorithm

1. Build a **Suffix Automaton (DAWG)** over context C. Time and space: O(\|C\|).
2. For each position j in output O, compute `match_len[j]` = length of longest substring of O starting at j that occurs in C. Walking the automaton: O(\|O\|) total.
3. Parse O's AST to identify identifier byte ranges.
4. For each identifier span, compute novel-byte fraction; aggregate to IPD.

### Why it matters

Standard hallucination detectors look at outputs *post hoc* and guess. BIPT works at the byte level — it gives you a **receipt**: which exact identifiers in the model's output came from your source, and which were invented. The latter set is your hallucination set, computed deterministically in O(\|C\| + \|O\|).

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## 2. NKBE — Nash-KKT Budgetary Equilibrium

**File:** [`entroly-core/src/nkbe.rs`](entroly-core/src/nkbe.rs)

### The problem

When N AI agents share a global context budget B, what's the optimal split? Every other context engine: flat division. Entroly: **Nash bargaining over agent utility functions, with KKT bisection.**

### The optimization

Given N agents with weights wᵢ, minimum budgets Bᵢ_min, and utility functions Uᵢ(·):

> maximize  Σᵢ wᵢ · Uᵢ(Bᵢ)
> subject to  Σᵢ Bᵢ ≤ B,  Bᵢ ≥ Bᵢ_min  ∀i

### The algorithm

**Two-phase KKT bisection:**

1. **Global:** Bisect over Lagrange multiplier λ* such that Σᵢ Bᵢ(λ*) = B.
2. **Per-agent:** Each agent runs knapsack with its share Bᵢ(λ*).

**Nash bargaining refinement:** gradient ascent on the log Nash product ensures Pareto-optimal fairness — no agent can do strictly better without another doing strictly worse.

**REINFORCE update:** agent priority weights adjust online based on outcome quality, closing the RL loop.

### Why it matters

In real coding workflows (Aider with multiple panes, Claude Code with subagents, multi-IDE setups), agents compete for context. Flat splits leave value on the table — heavy lifters get starved while idle agents hoard budget. NKBE gives each agent its **economically optimal** allocation.

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## 3. Causal Context Graph — Do-Calculus on RAG

**File:** [`entroly-core/src/causal.rs`](entroly-core/src/causal.rs)

### The problem: selection bias in context optimization

Standard RAG and context optimization treat fragment selection as an **observational study**: fragments that co-occur with positive outcomes get boosted scores. But this confounds correlation with causation.

> Fragment A (a utility module) is always co-selected alongside Fragment B (the core implementation). B is truly helpful; A is irrelevant but inherits B's positive feedback through co-selection bias.

Every other system reinforces A's score along with B's. Entroly distinguishes them.

### The solution: do-calculus via natural experiments

Entroly exploits its RAVEN-UCB exploration mechanism as a **natural instrument variable** (Pearl, *Causality* 2009; Hernán & Robins, *Causal Inference: What If* 2020). When a fragment is:

- **Randomly included** via an exploration swap → success rate estimates `P(success | do(include f))` — *interventional*.
- **Naturally selected** by the policy → success rate estimates `P(success | observe(include f))` — *observational*.

The gap reveals confounding bias:

> confounding_bias(f) = E[Y\|observe(f)] − E[Y\|do(f)]

- bias > 0: fragment **appears** better than it is (rides coattails of good partners)
- bias < 0: hidden gem — better than observation suggests (suppressor variable)
- bias ≈ 0: observational and causal estimates agree

### Why it matters

This is Pearl-grade causal inference applied to retrieval-augmented generation. Nobody else has it. It's why entroly's RL loop converges to *correct* fragment values instead of correlated noise.

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## 4. Cognitive Bus — ISA Event Routing

**File:** [`entroly-core/src/cognitive_bus.rs`](entroly-core/src/cognitive_bus.rs)

### The architecture

Inter-agent event routing with **Information-Surprise-Adaptive** prioritization:

> `publish(event)` → dedup → novelty score → priority(ISA) → per-subscriber queue
> `drain(agent)`   → top-K by priority → deliver → remember (hippocampus bridge)

### The math

**Per-subscriber Poisson rate model:**

> λ̂(n+1) = α · Δcount/Δtime + (1−α) · λ̂(n)        (α = 0.1)

**Priority score (KL-divergence over rates):**

> priority(e, s) = KL(λ_obs ‖ λ_exp) · recency(e) · novelty(e) · mi(e, s)
> KL(λ_obs ‖ λ_exp) = λ_exp − λ_obs + λ_obs · ln(λ_obs / λ_exp)
> recency(e) = e^{−decay_rate · age(e)}
> novelty(e) = softcapped SimHash novelty (cap = 0.8)
> mi(e, s) = mutual information boost (Rényi H₂ approximation)

**Memory-aware:** high-salience events flag the hippocampus for `remember()`; consolidated memories (neocortex) get lower priority than fresh episodic; emotional tags propagate → 3× priority boost for critical events.

**Welford online spike detection:**

> is_spike = (x − μ̄) > k · σ      (k = 3, immediate broadcast, bypass queue)

### Why it matters

Multi-agent AI workflows need a *coordination substrate*. Most projects bolt this on with ad-hoc queues. Entroly has a principled bus with information-theoretic priority — the right mathematical primitive for agent communication.

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## 5. Resonance Matrix — Supermodular Pairwise Learning

**File:** [`entroly-core/src/resonance.rs`](entroly-core/src/resonance.rs)

### The insight

Traditional context selection scores fragments independently:

> score(A) + score(B) + score(C) → select top-k

But certain *combinations* of fragments produce dramatically better LLM outputs than the sum of their parts. Two fragments scoring 0.4 alone might score **0.95 together** because they create a complete causal picture the LLM can reason over.

### The algorithm

We maintain a sparse pairwise resonance matrix R where R[i][j] tracks the learned interaction strength between fragments i and j.

**Learning signal** (on `record_success` / `record_failure`):

For every co-selected pair (i, j) in the outcome set:

> R[i][j] += η · (reward − baseline)

Failure rewards are negative → the pair gets suppressed.

**Scoring integration:**

For candidate c given already-selected set S:

> resonance_bonus(c \| S) = Σ_{s ∈ S} R[c][s] / \|S\|

This is **supermodular**: adding fragments with high pairwise resonance is more valuable when their partners are already selected. Submodular knapsack typically cannot capture this — entroly's resonance term breaks the assumption explicitly.

**Complexity management:** sparse storage (only co-selected pairs are tracked); decay R[i][j] *= decay_rate per turn (prevents stale resonances).

### Why it matters

Real codebases have *compositional* context value — a struct definition is useless without the function that uses it; an interface is useless without an implementation. Resonance learns these dependencies from outcomes, not heuristics.

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## 6. Five-Layer Hallucination Defense

| Layer | Detector | Failure mode it catches |
|---|---|---|
| 1 | **BIPT** ([provenance_tracer](entroly/verifiers/provenance_tracer.py)) | Invented identifiers — Kolmogorov-bounded byte traceback |
| 2 | **FORGE** ([repair_loop](entroly/verifiers/repair_loop.py)) | Ungrounded output → automated repair via context re-injection |
| 3 | **TRIAD** ([commit_alignment](entroly/verifiers/commit_alignment.py)) | Diff-message-PR triangulation; catches misleading commits |
| 4 | **PROVE** ([semantic_entropy](entroly/verifiers/semantic_entropy.py)) | Prose hallucination via causal-weighted predicate alignment (Kuhn / Gal / Farquhar, ICLR 2023) |
| 5 | **CAVE** ([reasoning_chain](entroly/verifiers/reasoning_chain.py)) | Counterfactual decorative-premise ablation (Lightman 2023 PRM + Shi ICML 2023) |

One detector misses; five don't. Every request runs all five by default.

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## 7. System 1 ↔ System 2 Coupling

**File:** [`entroly/coupling.py`](entroly/coupling.py)

Kahneman's dual-process framework, applied to AI coding context:

- **System 1** (fast, automatic): Rust engine in the proxy — knapsack + entropy + dedup + guardrails + channel + resonance + causal. Runs every request.
- **System 2** (slow, deliberate): Python flow_orchestrator + verifier stack + vault. Compiles beliefs, runs BIPT / FORGE / TRIAD / PROVE / CAVE, writes vault artifacts.

Until v0.19.1, these two cognitions did not exchange information. Now:

- **S2 → S1:** verified vault beliefs project into the engine as ranked fragments on every request. Coupled selection:

> C\* = argmax optimize(F ∪ π(B; q), T)

  where π lifts each belief b ∈ B into fragment space with score ∝ confidence · recency · query-relevance.

- **S1 → S2:** outcomes trigger Bayesian updates on used beliefs:

> c' = (c · L_g) / (c · L_g + (1 − c) · L_u)
> L_g = α  if success else (1 − α)        # P(o \| grounded)
> L_u = β  if success else (1 − β)        # P(o \| not grounded)
> (defaults α = 0.85, β = 0.40, posteriors clamped to [0.01, 0.99])

Failures enqueue beliefs for reverification by FlowOrchestrator's verify-before-answer flow.

### Why it matters

Most systems have *either* fast retrieval *or* deliberate verification, never both with feedback. Entroly closes the loop. Feature-flagged via `ENTROLY_VAULT_COUPLING=1`; default off pending field tests, but the architectural seam is in place.

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## What this means for you

Every algorithm above runs on every request, by default, with no extra wiring. You don't pick which one to enable — they all fire, and you see the result.

- **Token savings 70–95%** — knapsack + entropy + dedup do the compression.
- **Zero invented APIs** — BIPT + FORGE catch and repair fabrications.
- **Causally correct learning** — Causal CG removes coattail bias from RL updates.
- **Optimal multi-agent splits** — NKBE solves the budget allocation.
- **Compositional context value** — Resonance learns pairwise dependencies.

No other context engine has any of these except #1 (compression). That's the substrate.

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## Citations

- Kolmogorov, A. N. (1965). *Three approaches to the quantitative definition of information.* Problems of Information Transmission 1(1).
- Solomonoff, R. J. (1964). *A formal theory of inductive inference.* Information and Control 7.
- Lempel, A. & Ziv, J. (1976). *On the complexity of finite sequences.* IEEE Transactions on Information Theory IT-22(1).
- Blumer, A., Blumer, J., Haussler, D., Ehrenfeucht, A., Chen, M. T., & Seiferas, J. (1985). *The smallest automaton recognizing the subwords of a text.* Theoretical Computer Science 40.
- Pearl, J. (2009). *Causality: Models, Reasoning and Inference* (2nd ed.). Cambridge University Press.
- Hernán, M. A. & Robins, J. M. (2020). *Causal Inference: What If.* Chapman & Hall/CRC.
- Kuhn, L., Gal, Y., & Farquhar, S. (2023). *Semantic uncertainty: Linguistic invariances for uncertainty estimation in natural language generation.* ICLR.
- Lightman, H. et al. (2023). *Let's verify step by step.* (PRM800K)
- Shi, F. et al. (2023). *Large language models can be easily distracted by irrelevant context.* ICML.
- Brants, T. et al. (2007). *Large language models in machine translation.* (Stupid Backoff)
- Welford, B. P. (1962). *Note on a method for calculating corrected sums of squares and products.*
- Arrow, K. J. & Debreu, G. (1954). *Existence of an equilibrium for a competitive economy.* Econometrica.
- Nash, J. F. (1950). *The bargaining problem.* Econometrica.
- Kahneman, D. (2011). *Thinking, Fast and Slow.* Farrar, Straus & Giroux.

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*This is open work. Source for every claim above is in the repo. PRs welcome.*
