An open exploration of viable human-AI systems.
View the Project on GitHub algoplexity/cybernetic-intelligence
Cybernetic Intelligence (CI) emerges from the recursive unification of symbolic and latent representations across dynamic substrates, governed by self-editing, self-monitoring, and self-organizing loops. CIv16 proposes that viable artificial intelligence systems can be constructed as autopoietic, kernel-driven architectures, where each kernel embodies a perspective (symbolic, latent, dynamical, social, reflective) that is compressed, integrated, and iteratively reconfigured through feedback.
✅ This version carries forward all historical kernels (v1–v15) but consolidates them into six persistent kernels that CIv16 treats as the irreducible perspectives of intelligence.
| Conceptual Kernel (6) | Constituent Formal Kernels (11) | Layer Mapping / Notes |
|---|---|---|
| 1. Feedback & Autopoiesis | Kernel 1: Cybernetic Feedback Loops Kernel 2: Autopoiesis |
Set-like / Algebraic; foundational symmetry-breaking & self-maintenance |
| 2. Symbolic / Algebraic Patterns | Kernel 3: Symbolic Pattern Formation Kernel 4: MDL / Compression Kernel 8: Functorial Substrate Transition |
Algebraic → Geometric; motif extraction, compressive inference, algebra→latent mapping |
| 3. Structural & Topological Dynamics | Kernel 5: Structural Break Detection Kernel 6: Geometric Fault Detection |
Topological / Geometric; detects regime shifts, curvature/spatial anomalies |
| 4. Compression-Aligned Causal Geometry | Kernel 7: Compression-Aligned Causal Geometry | Manifold / Analytic; bridges symbolic and latent embeddings via joint compression |
| 5. Probabilistic / Analytic Enrichment | Kernel 10: Probabilistic & Analytic Enrichment | Analytic / Probabilistic; uncertainty, stochastic planning, Bayesian predictive modeling |
| 6. Reflexivity / Self-Editing | Kernel 9: Symmetry Breaking as Group-Action Restriction Kernel 11: SEAL-style Reflexivity |
Meta-structural; symmetry reduction & self-editing loops for autopoietic evolution |
A Generative Hierarchy of Cybernetic Substrates with Fully Formalized Kernels
Cybernetic intelligence emerges through a hierarchical enrichment of substrates, with each layer built via symmetry breaking, structural constraint, and compressive inference. Symbolic, latent, probabilistic, and reflexive cognition are nested stages, forming a tower rather than parallel modules.
Anchors: Azari (Conceptual Tower), Beer (cybernetics), SEAL-style self-editing, and SuperARC algorithmic-compression evaluation.
Each kernel includes:
Layer: Set-like / Algebraic Mechanism: Closed-loop dynamics as primitive symmetry-breaking.
Formal model:
\[x_{t+1} = f(x_t, u_t),\quad u_t = \mathcal{K}(y_t),\quad y_t = h(x_t)\]| Observables: $e_t = y_t - y_t^\mathrm{ref}$, Lyapunov proxy $\Lambda_t \approx \log | \partial f/\partial x | $ |
Decision rule:
\[\exists t: \Lambda_t > \theta_\Lambda \quad\lor\quad |e_t|>\theta_e\]Minimal test: setpoint tracking under perturbations; log $\Lambda_t,e_t$.
Layer: Set-like / Algebraic Formal model: $S=(C,P)$, with production operator $\Pi: \mathcal{P}\to\mathcal{C}$
\[\forall c\in C,\ \exists p\in P:\ \Pi(p)=c,\quad P = \mathcal{F}(C)\]Observables: $\rho_c(t)$, closure index
\[\chi(t)=\frac{\#\{c: \rho_c(t)>\eta\}}{|C|}\]Decision rule: $\chi(t)>\chi_\mathrm{min}$ after perturbation
Minimal test: perturb modules; measure recovery.
Layer: Algebraic Mechanism: Emergence of symbolic motifs from low-level dynamics.
Formal model: $G:\Sigma^k \to \Sigma$; motif set $\mathcal{M}$ with frequencies $f_m$
\[\Delta C_S(t)=C_S(t)-C_S(t-1)>\varepsilon_C \quad\text{or}\quad \Delta T(t)>T_\varepsilon\]Minimal test: ECA streams; BDM/CTM motif detection; F1 segmentation vs. ground truth.
Layer: Algebraic → Topological
\[\mathrm{MDL}(M;D)=|M| + L(D|M),\quad M^* = \arg\min \mathrm{MDL}(M;D)\]| Observables: $\mathrm{MDL}(M^*;D)$, compression ratio $\rho = | M^* | / | D | $, BDM score $\Phi(D)$ |
Decision rule: accept if $\Delta \mathrm{MDL}< -\eta$
Minimal test: recover generative parameters on synthetic data.
Layer: Topological / Manifold
\[D_t = \mathrm{Div}(W_t,W_{t-\tau})\]Decision rule: $D_t>\theta_D \Rightarrow$ structural break at $t$
Minimal test: synthetic time series; ROC/F1 vs baselines.
Layer: Geometric / Manifold
\[\kappa(z),\ \Delta\lambda,\ \bar d_k\]Decision rule: $\kappa_t>\theta_\kappa$ or $\Delta\lambda>\theta_\lambda$ → geometric fault
Minimal test: embeddings with distortions; measure fault detection.
Layer: Manifold / Analytic
\[\mathcal{L}(\phi,r) = \alpha \mathrm{BDM}(S|r) + \beta \mathbb{E}_x \| \phi(r(x)) - z(x)\|^2\]Decision rule: $\Delta \mathcal{L}< -\delta$ → successful alignment
Minimal test: paired symbolic/embedding datasets.
Layer: Algebra → Geometry
\[F: \mathbf{Alg} \to \mathbf{Met},\quad F(G)=(X,d),\quad F(f):F(G_1)\to F(G_2)\]Decision rule: $\mathrm{distort}(F)<\epsilon$ and bounded $L_f$
Minimal test: algebraic objects → learned embeddings; measure invariance.
Layer: all layers
\[G\curvearrowright X,\ H\subset G\]| Decision rule: $\Delta | G\cdot x | >\theta$ or $\Delta S_\mathrm{sym}< -\theta_s$ |
Minimal test: impose constraints on sets; measure invariants emergence.
Layer: Analytic / Probabilistic
\[p(x_{t+1}|x_{\le t}) = \int p(x_{t+1}|\theta,x_{\le t}) p(\theta|x_{\le t})\ d\theta\]Decision rule: minimize expected posterior predictive loss / maximize information gain
Minimal test: probabilistic forecasting; active info-seeking experiments.
Layer: Meta-structural
\[\mathcal{A}_{t+1} = \mathcal{E}(\mathcal{A}_t;\Delta),\quad \mathcal{E} = \arg\min_{\mathcal{E}'} \mathcal{J}(\mathcal{A}_t,\mathcal{E}')\]Decision rule: accept edit if $\Delta \mathcal{J}<-\eta$ and cost ≤ B
Minimal test: SEAL edit cycles; measure compression, hallucination reduction.
| Conceptual Kernel | Mapped CIv Kernels |
|---|---|
| Feedback / Autopoiesis | 1, 2 |
| Symbolic / Algebraic | 3, 4, 8 |
| Topological / Manifold | 5, 6, 7 |
| Probabilistic / Analytic | 10 |
| Meta-structural / Reflexive | 11, 9 (cross-layer symmetry) |
Note: This overlay is purely organizational; all formal math remains fully intact.
A Generative Hierarchy of Cybernetic Substrates with Embedded Kernels (final, rigorous form)
Cybernetic intelligence emerges through a hierarchical enrichment of substrates, where each layer arises from symmetry breaking, structural constraint, and compressive inference applied to the previous. Symbolic, latent, probabilistic and reflexive cognition are nested stages (a tower), not parallel tracks. This tower is both descriptive (explains emergence) and prescriptive (suggests testable mechanisms and implementations across substrates).
Anchors: Azari (Conceptual Tower), Stafford Beer (cybernetics), SEAL-style self-editing, and SuperARC-style algorithmic-compression evaluation.
For each kernel: Layer mapping → Mechanism summary → Formal model / equations → Observables → Decision rule(s) → Minimal test protocol.
Layer mapping: Set-like / Algebraic Mechanism (summary): Closed-loop control and observation form the primitive symmetry-breaking operation that produces structure. Formal model: discrete-time closed-loop dynamics
\[x_{t+1} = f(x_t, u_t),\qquad u_t = \mathcal{K}(y_t),\qquad y_t = h(x_t)\]where $\mathcal{K}$ is a controller (could be learned), $h$ the observation map. Observables: closed-loop error $e_t = y_t - y_t^\text{ref}$, Lyapunov proxy $\Lambda_t \approx \log|\partial f/\partial x|$, loop bandwidth. Decision rule (instability/fault):
\[\exists t: \Lambda_t > \theta_\Lambda \quad\lor\quad |e_t|>\theta_e\]Minimal test: control task (setpoint tracking) under perturbations. Log $\Lambda_t,e_t$. Fault detection precision/recall vs. injected disturbances.
Layer mapping: Set-like / Algebraic Mechanism: Organizational closure: a set of processes $P$ produce and maintain the set of components $C$ that realize $P$. Formal autopoiesis implies self-production under perturbation. Formal model: system $S=(C,P)$ with production operator $\Pi: \mathcal{P}\to\mathcal{C}$ s.t.
\[\forall c\in C\;\; \exists p\in P:\; \Pi(p)=c,\qquad P = \mathcal{F}(C)\]($\mathcal{F}$ expresses dependency of processes on components). Observables: component-production rates $\rho_c(t)$, closure index
\[\chi(t)=\frac{\#\{c: \rho_c(t)>\eta\}}{|C|}\]Decision rule: autopoietic viability if $\chi(t)>\chi_{\min}$ over horizon $H$ after perturbation. Minimal test: perturb components (dropout / remove modules); measure recovery of $\rho_c$ and $\chi(t)$; success if $\chi$ returns above threshold within $H$.
Layer mapping: Algebraic Mechanism: Low-level dynamics (e.g. ECA, token streams) spontaneously form symbolic motifs and grammars; motifs encode invariants (symmetry-broken relations). Formal model: symbolic generator $G:\Sigma^k\to\Sigma$; motif set $\mathcal{M}={m}$ with frequencies $f_m$. Model complexity $C_S$ via CTM/BDM. Observables: motif frequency vector $f_m(t)$, motif MDL cost $\mathrm{MDL}(m)$, torsion proxy $T(t)$ (motif rotation measure). Decision rule (novelty / motif shift):
\[\Delta C_S(t)=C_S(t)-C_S(t-1)>\varepsilon_C\quad\text{or}\quad \Delta T(t)>T_\varepsilon\]Minimal test: run ECA streams; compute BDM/CTM on sliding windows; detect motif change-point; measure segmentation F1 against ground-truth regime boundaries.
Layer mapping: Algebraic → Topological Mechanism: MDL selects models that balance model size and data fit; compression is used as an operational intelligence metric. Formal model: for model $M$ and data $D$,
\[\mathrm{MDL}(M;D) = |M| + L(D\mid M)\]where $|M|$ is code length and $L$ negative log-likelihood. The chosen model $M^=\arg\min \mathrm{MDL}$. Use BDM as a resource-bounded proxy to algorithmic complexity $\Phi(\cdot)$. Observables: $\mathrm{MDL}(M^;D)$, compression ratio $\rho = |M^*|/|D|$, BDM score $\Phi(D)$. Decision rule: accept a model if MDL improvement $\Delta \mathrm{MDL}< -\eta$. Minimal test: compare MDL-selected hypotheses to ground-truth generative model on synthetic data; measure ability to recover generative parameters as noise/regime complexity increases.
Layer mapping: Topological / Manifold Mechanism: Detect regime transitions via divergence measures across time windows (permutation entropy, KL, BDM deltas). Structural breaks localize manifold changes. Formal model: sliding windows $W_{t}$, divergence statistic
\[D_t = \mathrm{Div}(W_{t},W_{t-\tau})\quad (\text{e.g. } D_t=\mathrm{KL}(\hat p_{t}\|\hat p_{t-\tau}) \text{ or }\Delta\mathrm{BDM})\]Observables: $D_t$, permutation entropy $H_p(t)$, Wasserstein distance $W_2$. Decision rule:
\[D_t>\theta_D \Rightarrow\ \text{structural break at }t\]Minimal test: synthetic time-series with known regime shifts; compute ROC/F1 of break detection vs baselines (BOCPD, BIC).
Layer mapping: Geometric / Manifold Mechanism: Use manifold geometry (distances, curvature, Laplacian spectrum) as diagnostics; faults appear as curvature spikes, spectral bifurcations, or degeneracy in local charts. Formal model: embedding $z\in\mathcal{M}\subset\mathbb{R}^d$, local curvature $\kappa(z)$ (e.g. via discrete Ricci or Laplace–Beltrami spectrum ${\lambda_k}$). Observables: $\kappa_t$, spectral gap $\Delta\lambda$, nearest-neighbour distances $\bar d_k$. Decision rule: $\kappa_t>\theta_\kappa$ or $\Delta\lambda>\theta_\lambda$ → geometric fault. Minimal test: create embeddings (t-SNE/UMAP/autoencoder) on drifting data; inject distortion and measure detection of faults vs. baseline variance measures.
Layer mapping: Manifold / Analytic Mechanism: Jointly compress symbolic motifs and latent embeddings to discover low-distortion causal representations; alignment minimizes combined compression + distortion cost. Formal model: find representation $r$ and map $\phi$ minimizing
\[\mathcal{L}(\phi,r) \;=\; \alpha\cdot\mathrm{BDM}(S\mid r)\;+\;\beta\cdot\mathbb{E}_{x}\| \phi(r(x)) - z(x)\|^2\]Observables: alignment error $\epsilon_{align}=\mathbb{E}|\phi(r)-z|^2$, joint compression cost $\mathrm{BDM}(S\mid r)+\mathrm{BDM}(z\mid r)$. Decision rule: alignment improves when $\Delta \mathcal{L}<-\delta$; signals successful symbolic↔latent bridging. Minimal test: train $r,\phi$ on paired symbol/embedding datasets; measure predictive generalization on held-out regimes and reduction in joint MDL.
Layer mapping: Algebraic → Geometric Mechanism: Formalize substrate transitions via category-theoretic functors that map algebraic structures into metric/latent spaces while preserving morphisms (structure-preserving embeddings). Formal model: a functor
\[F: \mathbf{Alg}\to \mathbf{Met},\qquad F(G)=(X,d)\]such that for morphism $f:G_1\to G_2$, $F(f):F(G_1)\to F(G_2)$ is Lipschitz with constant $L_f$: $d(F(f)(x),F(f)(y))\le L_f d(x,y)$. Observables: Lipschitz constants $L_f$, distortion $\mathrm{distort}(F)$, preservation of invariants (e.g. $\forall$ invariants $I$, $I_{G}\approx I_{F(G)}$). Decision rule: mapping acceptable if $\mathrm{distort}(F)<\epsilon$ and $L_f$ bounded. Minimal test: construct algebraic objects (groups/rings) with known invariants, learn $F$ via representation learning; evaluate distortion and invariance preservation.
Layer mapping: all layers (generic mechanism) Mechanism: Substrate enrichment is a restriction of symmetry: start with group $G$ acting on $X$, introduce constraints that reduce symmetry to subgroup $H\subset G$, thereby enabling structure. Formal model: $G\curvearrowright X$, symmetry break $;H\subset G$ with isotropy reductions $\mathrm{Stab}H(x) \subsetneq \mathrm{Stab}G(x)$. Observables: orbit sizes $|G\cdot x|$, stabilizer dimensions, symmetry entropy $S\mathrm{sym}=-\sum p_g\log p_g$. Decision rule: a symmetry break detected if $\Delta |G\cdot x|>\theta$ or $\Delta S\mathrm{sym}< -\theta_s$. Minimal test: synthetic tasks with controlled constraint introduction (e.g., impose a relation on a set), measure emergence of new invariants and functional capability.
Layer mapping: Analytic / Probabilistic Mechanism: Enrich substrates with measures, stochastic processes and Bayesian inference to quantify uncertainty and support planning under uncertainty. Formal model: measurable space $(\Omega,\mathcal{F},\mathbb{P})$, stochastic process $X_t$, posterior predictive
\[p(x_{t+1}\mid x_{\le t})=\int p(x_{t+1}\mid \theta,x_{\le t}) p(\theta\mid x_{\le t})\ d\theta\]Observables: predictive entropy $H[p(\cdot)]$, posterior contraction rate, calibration (Brier score), expected information gain $\mathbb{E}[\mathrm{KL}]$. Decision rule: agent acts to minimize expected posterior predictive loss and/or maximize expected information gain subject to cost. Minimal test: probabilistic forecasting tasks, compute calibration, sharpness, and compare with deterministic baselines; include active information-seeking experiments.
Layer mapping: Meta-structural Mechanism: Agents execute self-edits (policy/model/representation changes) under objective combining performance and complexity; edits are themselves subject to evaluation via compression + predictive tests. Formal model: agent representation $\mathcal{A}$ updated by edit operator $\mathcal{E}$:
\[\mathcal{A}_{t+1}=\mathcal{E}(\mathcal{A}_t;\; \Delta),\qquad \mathcal{E}=\arg\min_{\mathcal{E}'\in\mathcal{E}_{\text{space}}} \; \mathcal{J}(\mathcal{A}_t,\mathcal{E}')\]with objective
\[\mathcal{J}=\underbrace{\mathbb{E}[\text{loss}]}_{\text{task}} + \lambda_1\cdot\text{Complexity}(\mathcal{A}_{t+1}) + \lambda_2\cdot \Phi(\text{predictive})\]($\Phi$ = algorithmic compression/probabilistic predictive metric). Observables: post-edit $\Delta$performance, $\Delta\mathrm{MDL}$, edit cost. Decision rule: accept edit if $\Delta \mathcal{J}< -\eta$ and resource cost $\le B$. Minimal test: run SEAL-style edit cycles on failing tasks; measure convergence of $\mathcal{J}$, decrease in hallucination rate, and improvement in compression metric.
| Layer | Substrate (CIv16 naming) | Example kernels (instantiated) |
|---|---|---|
| 1 | Set-like | Kernel 1 (feedback), Kernel 2 (autopoiesis) |
| 2 | Algebraic | Kernel 3 (symbolic motifs), Kernel 8 (functorial mapping) |
| 3 | Topological / Order | Kernel 4 (MDL), Kernel 5 (structural breaks) |
| 4 | Geometric | Kernel 6 (geometry faults), Kernel 7 (compression-geometry) |
| 5 | Manifold | Kernel 7 (manifold alignment), Kernel 6 diagnostics |
| 6 | Analytic / Probabilistic | Kernel 10 (Bayesian & stochastic enrichment) |
| 7 | Meta-structural | Kernel 11 (SEAL reflexivity), symmetry-breaking (Kernel 9) acts across layers |
Phase 1 — Algebraic / Symbolic tier (weeks → months)
algoplexity/alg_kernels/symbolic notebook with ECA experiments.Phase 2 — Latent / Geometric tier
Phase 3 — Mesoscopic integration (symbolic ↔ latent)
mesoscope orchestrator and evaluation suite.Phase 4 — Probabilistic & Reflexive layer
Phase 5 — Tower integration & validation
Algorithmic intelligence (SuperARC-compatible)
\[\Phi(W)\ =\ \min_{p:U(p)=W}\ |p|\ \approx\ \mathrm{CTM/BDM}(W)\](Use BDM/CTM approximations; lower $\Phi$ + higher $\Pi$ is better.)
Predictive adequacy
\[\Pi(W)= -\frac{1}{|W|}\sum_{t\in W}\log p(x_t\mid x_{<t})\](NLL or next-step prediction score.)
Joint-fault score: composite of normalized MDL change, curvature spikes, symmetry-loss:
\[J_t = \alpha\cdot\frac{\Delta \mathrm{MDL}}{\sigma_{\mathrm{MDL}}} + \beta\cdot\frac{\kappa_t}{\sigma_\kappa} + \gamma\cdot\frac{\Delta S_\text{sym}}{\sigma_S}\]Autopoietic viability: $\chi$ (closure index) recovery time under perturbation.
Alignment / Functor distortion: average distortion of $F$ mapping; Lipschitz bounds.
SEAL edit success: relative drop in $\mathcal{J}$ per edit, cost-normalized.
(You can substitute precise arXiv IDs and DOIs in the final published page.)
If you want, I will now:
Pick one and I’ll proceed (I won’t ask more clarifying questions — I’ll start action (A), (B), or (C) right away depending on which you choose).
A Generative Hierarchy of Cybernetic Substrates with Embedded Kernels
Cybernetic intelligence emerges through a hierarchical enrichment of substrates, where each layer arises via symmetry breaking and structural constraint applied to the previous. Symbolic, latent, probabilistic, and reflexive cognition are nested stages rather than parallel modules. This framework unifies past CI versions into a single generative tower, grounded in Azari’s Conceptual Tower of Mathematical Structures (2025).
Layer Mapping: Set-like / Algebraic
Layer Mapping: Set-like / Algebraic
Layer Mapping: Algebraic
Layer Mapping: Algebraic → Topological
Layer Mapping: Topological / Manifold
Layer Mapping: Geometric / Manifold
Layer Mapping: Manifold / Analytic
Layer Mapping: Algebraic → Geometric
Layer Mapping: All layers
Layer Mapping: Analytic / Probabilistic
Layer Mapping: Meta-Structural
| Layer | CIv16 Substrate | Kernel Example |
|---|---|---|
| 1 | Set-like | CIv1 Feedback, CIv2 Autopoiesis |
| 2 | Algebraic | CIv3 Symbolic Patterns, Kernel 8 Functorial Mapping |
| 3 | Topological / Order | CIv4 MDL, CIv5 Structural Breaks |
| 4 | Geometric | CIv6 Geometric Faults |
| 5 | Manifold | CIv7 Compression Geometry |
| 6 | Analytic / Probabilistic | Kernel 10 Probabilistic Enrichment |
| 7 | Meta-Structural | Kernel 11 SEAL Reflexivity |
Phase 1 – Symbolic/Algebraic Tier
Phase 2 – Topological / Manifold Tier
Phase 3 – Analytic / Probabilistic Tier
Phase 4 – Meta-Structural Tier
Phase 5 – Tower Integration
✅ CIv19 now reads as:
CIv19 Implication:
CIv19 Implication:
CIv19 Implication:
CIv19 Implication:
CIv19 Implication:
| Substrate Layer | Existing Kernels | SuperARC Kernel Integration |
|---|---|---|
| Set-like | Feedback loops, Autopoiesis | Algorithmic compressibility of set structures; kernel measures minimal representations of elements |
| Algebraic | Symbolic patterns, Functorial mapping | Recursive compression of algebraic operations; symbolic regression as abduction |
| Topological | MDL, Structural break detection | BDM-based evaluation of relational complexity; predictive structure reconstruction |
| Geometric | Geometric fault detection | Local/global compression of metric/topological embeddings; curvature-informed planning |
| Manifold | Compression-aligned causal geometry | CTM kernel for manifold-level generative prediction; reconstruct latent embeddings from compressed models |
| Analytic / Probabilistic | Probabilistic enrichment | Probabilistic compression kernel; Bayesian abduction and stochastic model inference |
| Meta-Structural | SEAL reflexivity | Recursive decompression and abduction across all layers; planning kernel; optimal inference loop |
SuperARC provides both the theoretical rationale and computational methods to formalize CIv19: