Stack Theory (Bennett 2026)
Formally verified delegation bounds for multi-layer intelligence architectures, with novel bridge theorems connecting weakness ordering to Heyting nucleus algebras.
Formal Verification Certificate
Every theorem in this project has been machine-checked by the Lean 4 kernel. No axiom is assumed without proof. No gap exists in the verification chain.
Stack Theory (Bennett 2026) • Lean 4 + Mathlib • Apoth3osis Labs
The Central Question
Are there hard mathematical limits on what a multi-layer delegation architecture can achieve? If so, what do those limits look like — and what goes wrong when you try to push past them?
Bennett answers yes: the Law of the Stack imposes an exponential ceiling (2|Ext(πᵢ)|) on each layer's adaptability, and over-constraining the system to exceed this bound causes structural fragmentation (splintering). We machine-checked these claims and discovered a novel algebraic characterization: Bennett's sociological concept of “collective identity” is equivalent to a nontrivial meet in the Heyting fixed-point algebra.
Context & Discovery
Dyadic Exactness Refinement
Bennett's Corollary 5.1 uses Nat.log which introduces discretization. We proved that on dyadic continuation counts (powers of 2), the discrete ordering is exact. This tightness result is not in the original paper.
Bridge Theorems (2 Novel + 4 Infrastructure)
We connected Bennett's combinatorial framework to HeytingLean's nucleus operator infrastructure. The central new result: collective identity is equivalent to nontrivial meet in the Heyting fixed-point algebra Ω_R. Four supporting infrastructure lemmas (one-line, definitional) complete the bridge.
Hostile Audit Remediation
Internal adversarial audit (automated J-Group probes) identified 5 minimal-passing patterns. All remediated with genuine domain reasoning.
Formal–Empirical Boundary
Formally Proved
- • Delegation bounds (Law of the Stack, Theorem 5.1)
- • Free energy equivalence (Corollary 5.1)
- • Cancer-analogue splintering (Proposition 6.1)
- • Bridge: collective ↔ nontrivial meet in Ω_R
- • Dyadic exactness refinement
Engineering / Application
- • AgentHALO: toy 2-layer model demonstrating bounds apply
- • Mapping to real container-agent architectures
- • RLHF splintering interpretation (empirical analogy)
Implementation Phases
Primitives
Programs, vocabularies, truth sets, language, extensions, weakness, tasks, utility. ESM Lemmas 1–3.
Stack Architecture
Abstractors (ESM Lemma 4), multi-layer architecture, Law of the Stack (Theorem 5.1).
Free Energy + Exactness
Viability weakness, free energy, w-maxing = FE min, dyadic exactness, delegation bottleneck.
Collective Identity + Splintering
Collective policies, boundary conditions, over-constraint, cancer-analogue splintering (Prop 6.1).
Bridge Theorems
2 novel bridge theorems + 4 infrastructure lemmas connecting Bennett to Core.Nucleus: collective ↔ Ω_R meet (novel), abstractor through nucleus (novel), plus definitional monotonicity/closure lemmas.
Applications + Audit
AgentHALO concrete stack, weak boundary design, hostile audit with J-Group probes, remediation.