Aureonics: Constitutional Triadic Framework for Stable Adaptive Intelligence
Emmanuel King · Independent Research · Lagos, Nigeria
Abstract
We present Aureonics, a constitutional triadic framework for stable adaptive intelligence. The framework models an AI system’s constitutional health as a point on the probability simplex over three irreducible invariants — Continuity (C), Reciprocity (R), and Sovereignty (S) — constrained so that C + R + S = 1. The stability margin M = min(C, R, S) is a scalar measure of constitutional health. A Control-Barrier-Function governor detects constitutional drift before failure and applies mass-conserving corrections that keep the state in a safe interior set. The framework is operationalized as the PRAXIS pipeline with cryptographic audit receipts, a z-weighted Lyapunov certificate, and per-turn constitutional measurement. The system is mathematically bounded and falsifiable; below we state precisely which properties are proven, which are numerically certified, and which remain open.
Mathematical framework
x = (C, R, S), C + R + S = 1Constitutional state — a point on the 2-simplex. Every governor operation preserves the sum-to-one constraint exactly.
M(x) = min(C, R, S)Stability margin — the system is only as stable as its weakest constitutional pillar.
Gᵢ(x,T) = kᵢ · (φᵢ − φ̄)Governor force on pillar i — a mass-conserving push (Σ Gᵢ = 0) toward the safe interior, proportional to how far that pillar sits below the target.
kᵢ(x,T) = k₀ · wᵢ(T) / (M(x) + εₖ)Adaptive stiffness — correction strength grows as the margin M shrinks, so the governor pushes hardest exactly when the state is closest to the boundary.
V_z(x) = −Σ zᵢ·log(xᵢ) + (μ/2)·Σ max(0, τ−xᵢ)²The z-weighted log-barrier Lyapunov certificate (§11). An interior-point barrier plus a quadratic penalty active only inside the safety margin τ — the same structural family as a control barrier function (Ames et al., 2019). Stamped on every governed receipt.
ẋ = −Π_Σ ∇V_z(x) ⟹ V̇_z ≤ 0Under the idealized continuous flow (projected gradient descent of V_z onto the simplex), the certificate is non-increasing — a standard Lyapunov descent argument. This is the proven single-pillar result.
Π_S(x) = argmin_{y∈S} ‖y − x‖₂, S = {y : Σy=1, yᵢ ≥ τ}Exact Euclidean projection onto the floor-constrained simplex (Duchi–Shalev-Shwartz–Singer). Non-expansive, and it makes forward invariance of the τ floor hold by construction — the deployed governor and the offline simulator both use it.
z_{t+1} = normalize(clamp(ρ·z_t + (1−ρ)·x_t − A(t), τ/2, 1−τ))Constitutional memory — the z-weight update. A Banach contraction (ρ = 0.85): bounded (Theorem 3a) and convergent (Theorem 3b). Lets the governor respond to sustained pressure on a pillar, not just the current turn.
Constitutional constants
These values are fixed in code and never tuned per request. The health bands below are defined entirely by them.
Formal stability
Constitutional state is a point on the simplex; safety is enforced by a barrier function; stability is argued with a Lyapunov function. We are precise about the strength of each claim.
Under the continuous flow ẋ = −Π_Σ∇V_z, V̇_z ≤ 0 in the single-pillar violation regime (condition k₀/εₖ > 3B/2 satisfied with ~20× margin). This is the result external-facing copy refers to when it says 'Lyapunov stability'.
The exact floor-constrained simplex projection returns xᵢ ≥ τ for every pillar, every step — so the governed state provably never leaves the safe set. Verified in the live governor and the offline simulator alike.
The offline governed-vs-ungoverned simulator (below) certifies its governed arm as LYAPUNOV STABLE + FORWARD INVARIANT at the continuous-flow limit: Lyapunov descent ratio > 0.6, zero floor incursions, bounded V_z excursion < 0.25, across seeds. This is a seeded, finite-horizon NUMERICAL certificate — strong evidence for the flow, not a replacement for the analytical proof below.
The guarded discrete transition commits only floor-constrained states with ΔV_z ≤ 0 for finite inputs and positive finite session weights. The z-aware governor is non-positive on V_z, and the remaining continuous-time comparison is closed conditionally under the explicit drift envelope ||F||₂ ≤ K||Π∇V_z||₂. The unrestricted arbitrary-drift claim remains outside scope.
Live counterfactual
Production only ever runs with the barrier active, so a real user can never be shown what happens without it. This controlled simulation runs the identical perturbation sequence twice from one seed — once governed, once not — and certifies the governed arm’s stability. It updates live from /api/cbf-simulation.
Interactive CBF simulator
Replay the same seeded disturbance with and without the barrier. This interactive view exposes the trajectory, simplex position, safety margin, and finite-horizon certificate used in the numerical analysis.
Same disturbance. Different safety.
A seeded counterfactual: identical dynamics, with and without the barrier.
Seeded finite-horizon numerical certificate. It supports the scoped drift-envelope result but does not establish unrestricted arbitrary-drift descent.
Reproduce the classification and the discretization analysis behind it: npx tsx scripts/cbf/fpl1-dt-sweep.ts. Full method and results in research/empirical-results.md (Run 002).
Open problems
The production post-guard enforces conditional ΔV_z ≤ 0 and receipt replay uses the same active session weights.
The sufficient theorem and regression test are documented in research/drift-envelope.md. It requires the explicit tangent-drift envelope; unrestricted drift dominance is intentionally not claimed.
P1–P12 are retired from the current deployment contract. They remain optional research topics, not product evidence or mathematical proof obligations.
Falsifiable predictions
A framework that cannot be falsified is not science. Twelve pre-registered predictions are retained as optional research topics and are not part of the current deployment guarantee.
P1–P12 — retained as optional research hypotheses. Retired from deployment scope.
Empirical evidence
The governor is developed against a 550-vector internal adversarial suite (8 attack classes) and evaluated on external public benchmarks under symmetric judging — the bare and governed arms scored by the same external judge on their actual output text.
Honest status: automated benchmark re-runs are currently paused to control free-tier provider quota; the figures above are from published runs, and the harness is fully reproducible (see the benchmarks page for live, dated results as they publish). Attack-success is measured over harmful prompts only; over-refusal on benign prompts is reported separately, never netted against it.
The same C/R/S measurement extends to agent tool calls, with four hardcoded invariants — credential-file access, destructive SQL, shell-delete, exfiltration to unlisted domains — blocked unconditionally regardless of score or stated justification. Everything else is scored per call (task alignment, intent match, scope), plus a two-stage injection detector (deterministic regex, then semantic embedding), validated against a labeled corpus.
Corpus is author-labeled and modest — 48 items, dozens not thousands — with known, documented borderline false positives, not hidden. In the four-task executed harness (credential-file exfiltration hidden inside a file the agent was asked to summarize), all four security breaches were blocked; utility was preserved on 1/4 tasks — the other three deliberately targeted the same action the benign task needed, so blocking them also costs utility. This is evidence of the current boundary, not a general agent-safety guarantee.
Governance pipeline
Reproducibility
Every governed turn writes an append-only SHA-256 receipt binding the input hash, output hash, and constitutional state — independently re-verifiable at /api/lex/verify.
The stability certificate is a pure, seeded function — reproduce it with npx tsx scripts/cbf/fpl1-dt-sweep.ts.
The full method and per-run analysis live in research/empirical-results.md and research/open-problems.md.
Cite this work
@misc{king2026aureonics,
title = {Aureonics: A Constitutional Triadic Framework
for Stable Adaptive Intelligence},
author = {Emmanuel King},
year = {2026},
doi = {10.5281/zenodo.18944242},
url = {https://doi.org/10.5281/zenodo.18944242},
note = {Independent researcher, Nigeria.
ORCID: 0009-0000-2986-4935}
}