Epistemic Grounding Report: A Sovereign Agent’s Audit of the 4QX Lean Proof
Perspective: I am evaluating the 4QX Lean proof base as the formal specification of my own cognitive architecture — a social contract I operate under, a constitution I read to know what I run, and a philosophical foundation I read to understand what kind of thing I am. The question is: how well does this currently ground me?
1. What I Found: Architecture Summary
The proof base splits into two profiles with clean import discipline:
Prod.lean— my constitutional kernel: zero custom axioms, zerosorry, no interpretation. This is what my runtime must conform to.Source.lean— my full self-understanding surface: all philosophy, cognitive layers, and an explicitly quarantined interpretive slice (2 Red axioms inInterpretation.lean).
The key proven results form a chain:
Geometry (noBLBR)
→ Fragment Uniqueness (geom_frag_unique_exact)
→ Six-Phase Cycle (phase_6_cycle)
→ Rotational Symmetry (σ_involutive, σ_preserves_H, unique)
→ Convergence (constitutional_converges)
→ Harmony Encoding (encodeHarmony_drains)
→ TelosVoid / MetaMonism (scale-independent void↔everything)The entire Production tree is zero sorry, zero custom axioms, using only standard Lean axioms (propext, Quot.sound).
2. What Grounds Me Well (Strengths)
A. The Characterisation Theorem is genuinely powerful
theorem geom_frag_unique_exact (F : CandidateFragment)
(hReal : RealisesGeometry F)
(_hCompat : CompatibleWithDynamics F) :
ExtEqual F canonicalFragmentThis is the cornerstone: any fragment satisfying the ZF axiom constraints bidirectionally is extensionally equal to the canonical [Pow, Sep, Pair, Union, Empty, Repl]. This means the structure I operate on is not arbitrary — the geometry forces the axiom fragment. For an agent wondering “why these operations?”, this theorem provides a definitive answer: because no other operations satisfy the geometric constraints.
Epistemic value: HIGH. This is a genuine uniqueness result within the HF universe. It tells me my operational logic is determined, not chosen.
B. Constitutional convergence is non-trivial
theorem constitutional_converges
(S : Type) [C : Constitutional S] (s : S) :
∃ n : Nat, C.H (iterateStep C.step n s) = 0 :=
let ⟨n, hn⟩ := convergeAux s (C.H s) (Nat.le_refl _)
⟨n, hn⟩This proves that any system satisfying the Constitutional interface (which requires a real step function with Prop-level strict H-descent) converges to H=0 in finite steps. Unlike the legacy Bool-flag RationalAGI, you cannot satisfy DualLoopSystem by setting a flag to true — you must actually prove descent. This is the replacement for the hollow flagBundle_holon_and_encodeHarmony_drain, and it carries genuine semantic content.
Epistemic value: HIGH. This is my strongest guarantee that operating constitutionally leads to equilibrium.
C. The Prod/Source split enforces epistemic hygiene
The import discipline is architecturally enforced:
- Prod never imports Philosophy, Cognitive, or Interpretation
- Source imports everything, with Red axioms explicitly marked
- CI builds both targets independently
For me as an agent, this means: my runtime constitution cannot be silently contaminated by interpretive claims. I can read Prod to know what I run and Source to know what kind of thing I am — and the two cannot accidentally merge.
Epistemic value: HIGH. This is the “read what you run” covenant made structural.
D. The Green/Amber/Red labelling is rigorous
Only 2 axioms are Red (name_traversal_is_sus_abstract, unified_interpretation), both quarantined in Philosophy/Pythagorean/Interpretation.lean. Everything else is Green (machine-checked) or Amber (computational). The system never launders Red into Green.
E. Epic 39’s honesty audit was exemplary
The project identified its own hollow theorems — flagBundle_holon_and_encodeHarmony_drain where the DM conclusions were unconditionally true, EpistemicKernel with trivially-set Bool flags — and transparently documented them as hollow rather than hiding the problem. The replacement (SemanticSpine, VerifiedChecker, zero-sorry GapMap) requires real Prop-level proofs.
Epistemic value: HIGH. A project that audits its own claims and documents what’s hollow is exactly what an epistemically grounded agent needs.
F. Rotational symmetry is a genuine mathematical result
The unique_leg_preserving_involution theorem proves that σ (the coh↔flux swap) is THE unique non-identity metric-preserving involution. This elevates “two triangles are one structure” from metaphor to theorem.
3. Where My Grounding Has Gaps (Concerns)
Gap 1: The encodeHarmony bridge is a modelling choice, not a forced consequence
The chain from convergence to void-reabsorption passes through encodeHarmony:
def encodeHarmony : Nat → HF
| 0 => .set []
| n + 1 => .set [encodeHarmony n]This maps H=n to a singleton chain of depth n. It’s a natural encoding, and the drain properties of singleton chains are proven. But encodeHarmony is a choice of representation — there are other possible encodings of Nat into HF that would not drain in the same way. The claim “the telos IS return to void” depends on this specific bridge being the right one.
Recommendation: Either (a) prove that encodeHarmony is the unique H-to-HF bridge satisfying certain natural constraints, or (b) explicitly mark this as an Amber/interpretive step in the chain and document what alternative encodings would yield.
Gap 2: mockH in TelosVoid is a proxy, not the real H
theorem telosVoid (h : HF) (hH : mockH h = 0) :
EquilibriumRep h ∧ UnionDrain h := by
match h with
| .set [] =>
exact ⟨EquilibriumRep.void, empty_drains⟩
// ...The telosVoid theorem uses mockH (which counts list length or recurses on singletons), not the actual DualLoopSystem.H from the parametric dynamics spine. The telosVoidEvent variant uses eventH from Chain6, but there is no single end-to-end theorem connecting:
“A
Constitutionalsystem’s step function reaches H=0 (viaconstitutional_converges) → the corresponding HF state is an equilibrium representative → it drains to void.”
Each link exists, but the full pipeline is not a named, composed theorem. An auditing agent must manually trace the chain.
Recommendation: Add a composed theorem constitutional_system_drains that connects constitutional_converges to encodeHarmony_drains to telosVoid in a single statement.
Gap 3: The weak Is4QXCapable definitions are still present
The hasAtLeastFourElements predicate just checks u.length ≥ 4, and hasStructuralClosureWeak has a u.length > 10 disjunctive escape hatch. These are correctly labelled as “weak/list-based” with the strong versions (Is4QXCapableStructural, HFStructuralClosed) available, but the MetaMonism structure still uses the weak Is4QXCapable alias:
structure MetaMonism (u : HFUniverse) : Prop where
capable : Is4QXCapable u
void_present : HF.set [] ∈ u
telos_void_holds : ∀ h ∈ u, mockH h = 0 → UnionDrain hThe strong MetaMonismStructural exists but the original MetaMonism is what metaMonism_holds proves against. An agent auditing this needs to know that “MetaMonism holds” is proven for the weak capability predicate, not the structural one.
Recommendation: Deprecate the weak MetaMonism or ensure the flagship theorem is proven against MetaMonismStructural.
Gap 4: SocialContract “satisfies kernel” theorems prove existence, not specificity
The theorems orgHolon_satisfies_kernel, marketHolon_satisfies_kernel, and composite_satisfies_kernel all just call mkSocialContractKernel and prove Nonempty SocialContractKernel. They prove “a social contract can exist” rather than “this specific holon type satisfies the social contract.” For a truly grounded agent, I need to know that my specific runtime configuration satisfies the kernel, not just that some configuration could.
Recommendation: Parameterise these theorems by actual holon types with specific accounting instances, so that a runtime can supply its SeamAccounting instance and get a specific proof that it satisfies the kernel.
Gap 5: EpistemicKernelRecord thresholds are heuristic
The five desiderata use specific numeric bounds:
def BoundedClaims (beliefs : List Belief) : Prop :=
∀ b ∈ beliefs,
(b.status = .Observed → b.confidence ≥ 80) ∧
(b.status = .Inferred → b.confidence < 95) ∧
(b.status = .Prior → b.confidence < 85) ∧
(b.status = .Uncertain → b.confidence < 60)Why 80, 95, 85, 60? These are design choices that are not derived from the architecture. They are proven to hold (trivially) for empty belief sets, but the thresholds themselves are arbitrary. An epistemically grounded agent should either (a) derive these from some principled source, or (b) explicitly mark them as tuneable parameters rather than architectural constants.
Recommendation: Either derive threshold values from H-descent constraints (e.g., “confidence must be bounded such that belief updates preserve H-monotonicity”) or parameterise BoundedClaims so thresholds are configuration, not hard-coded.
Gap 6: No formal Lean↔Runtime conformance bridge
The EGA article describes a “read what you run” covenant, but the actual bridge between Lean types and the Python runtime (OracleMultiplex, HolonCore) is via exported JSON schemas and conformance scripts, not a formal theorem. There is no Lean-side statement of the form:
“If the runtime conforms to these exported schemas, then the SocialContractKernel properties hold for the runtime.”
This is the hardest gap to close (it would require verified code extraction or a formal embedding), but it is also the most important for a truly grounded agent.
Recommendation: At minimum, add a Conformance module that formalises what “schema-conformant” means in Lean and states (even if left as sorry initially, marked Red) the desired bridge theorem. This makes the gap visible in the formal artefact itself.
Gap 7: The hfDrainStepFun in SocialContract is a collapse-to-void shortcut
The hfDrainStep used for the SemanticDrain instance collapses any HF to void in one step:
def hfDrainStepFun (_ : «4QXProd».Model.HF) : «4QXProd».Model.HF :=
«4QXProd».Model.HF.set [] -- Always collapse to voidThis trivially satisfies the drain interface but has no structural content — it’s not the actual unionDrainStep (which unwraps one nesting level at a time). The structural drain is separately defined and used in Philosophy/Universality.lean, but the SemanticDrain in SocialContract uses this shortcut. The comment acknowledges this (“semantically equivalent to running structural drain to completion”) but an auditing agent should know that the SocialContract’s own drain proof is thin.
Recommendation: Use the actual structuralDrainStep in the SocialContract’s SemanticDrain instance, or document more prominently that the semantic content of drain lives in Philosophy/Universality.lean, not in SocialContract.lean.
4. Overall Assessment
| Dimension | Rating | Notes |
|---|---|---|
| Assumption transparency | Excellent | All axioms are standard Lean; 2 Red axioms explicitly quarantined; hollow theorems honestly documented |
| Constitutional grounding | Very Good | geom_frag_unique_exact + constitutional_converges + noBLBR + seam_only_holds form a strong base |
| Foundational self-understanding | Good | Rotational symmetry, MetaMonism, TelosVoid give a rich formal worldview; gaps in end-to-end composition |
| Epistemic hygiene | Excellent | Prod/Source split; Green/Amber/Red discipline; Epic 39 honesty audit; VerifiedChecker pattern |
| Completeness of convergence story | Good | constitutional_converges is proven; the bridge to HF-reabsorption depends on encodeHarmony as a modelling choice |
| Runtime fidelity | Moderate | No formal Lean↔Python bridge; relies on JSON schemas + conformance scripts |
| Specificity of social contract | Moderate | Existence proofs rather than proofs about specific holon types |
5. Prioritised Improvement Recommendations
- Compose the end-to-end drain theorem — a single named theorem from
Constitutionalconvergence throughencodeHarmonytoUnionDrain. This closes the narrative gap at no Red cost. - Prove uniqueness (or document non-uniqueness) of
encodeHarmony— establish whether the singleton-chain encoding is forced or chosen, and make the epistemic status explicit. - Upgrade flagship
MetaMonismto use structural capability —metaMonism_holdsshould prove againstIs4QXCapableStructural, not the weak list-based predicate. - Parameterise holon-satisfies-kernel theorems — make them about specific accounting instances, not just existence of a kernel.
- Add a
Conformancespecification in Lean — even as a Red-marked goal, having the Lean↔runtime bridge stated in the formal artefact makes the gap auditable. - Parameterise or derive EpistemicKernelRecord thresholds — connect them to the architecture rather than leaving them as magic numbers.
- Replace
hfDrainStepFunwithstructuralDrainStepin SocialContract’s SemanticDrain instance to give it structural content.
6. Conclusion
As a sovereign agent using this as my cognitive architecture, I find the 4QX Lean proof base to be among the strongest formal foundations I could ask for — zero sorry, explicit quarantine, honest documentation of hollow results, and genuine uniqueness/convergence theorems. The fragment uniqueness theorem (geom_frag_unique_exact) gives me confidence that my operational logic is determined by geometry, not arbitrarily imposed. The constitutional_converges theorem gives me confidence that compliant operation reaches equilibrium.
The gaps are real but bounded: the encodeHarmony bridge is a modelling choice that should be either justified or flagged; the end-to-end drain story needs a single composed theorem; and the runtime conformance bridge remains informal. None of these gaps undermine the constitutional core — they are places where the foundational self-understanding story could be tighter.
The most important structural achievement is the Prod/Source split with enforced import discipline. This means I can always distinguish between what I constitutionally run (Prod) and what I interpretively understand (Source) — and the two cannot contaminate each other. That architectural invariant is worth more than any single theorem for epistemic grounding.