Canonical Axiom Foundation
Seven-Level Hierarchy
15-Point Iron Chain
Lean: 0 Axiom Declarations
The Canonical Axiom Foundation
Truth below the system. Primitives and presuppositions at the floor. Conceptual axioms above them. Theorems after that.
Pre-System: Truth
PRE-SYSTEM — NOT AN AXIOM
Truth
Truth is not inside the hierarchy. The system exists because Truth is already there.
Truth is not a primitive term, not a presupposition, and not an axiom. Any definition, proof, denial, or assertion already depends on truth being operative. The framework can point to this floor, but it cannot place Truth inside the system without circularity. This is why the canonical file says Truth is below the floor.
The Seven-Level Hierarchy
What belongs at each level
| Level | Name | Role | Canonical Content |
|---|---|---|---|
| Pre | Truth | Ground of truth-evaluation | Below the system; not an axiom |
| 0 | Primitive Terms | Undefined anchors | Existence, Distinction, Information |
| 1 | Presuppositions | What must hold before reasoning begins | Something exists; distinguishable states are real; the system can be reasoned about consistently |
| 2 | Conceptual Axioms | Load-bearing commitments | The public Iron Chain compresses these into 15 conceptual axioms |
| 3 | Definitions | Vocabulary built from prior levels | Coherence, good, grace operator, domain terms |
| 4 | Theorems | Derived claims | Observer, consciousness, laws, sign mechanics, collapse results |
| 5 | Corollaries and Lemmas | Supporting results | Standard logical furniture |
| 6 | Predictions and Empirical Claims | Where the system touches experiment | Defeasible, falsifiable, domain-facing claims |
Level 0: Primitive Terms
LEVEL 0 — THREE PRIMITIVES
Existence, Distinction, Information
These are undefined anchor concepts used by the framework but not defined from inside it.
Existence means something rather than nothing. Distinction means one thing can differ from another. Information means distinguishable states exist. These are not laboratory predictions and not Lean axioms; they are the vocabulary floor required before the rest of the chain can even be stated.
Level 1: Presuppositions
LEVEL 1 — THREE PRESUPPOSITIONS
The Floor of Reasoning
Something exists; distinguishable states are real; and the system can be reasoned about consistently.
These are the minimum commitments needed before the framework starts. They are not presented as new discoveries. They are the operating floor under any rational system: there is something to discuss, there are distinctions to track, and reasoning is not meaningless.
Level 2: The 15-Point Iron Chain
The public conceptual axiom spine
The public Iron Chain contains 15 conceptual axioms, grouped into five movements. This is the short load-bearing spine beneath the larger 188-item structural presentation.
| Axioms | Movement | Core claim |
|---|---|---|
| 1–3 | Existence | Something distinguishable exists |
| 4–6 | Grounding | Reality is grounded in a self-sustaining Logos field |
| 7–9 | Observation | Actualization requires observers; the chain terminates in God |
| 10–12 | Moral Physics | Humans have fixed moral sign and cannot self-flip |
| 13–14 | Spiritual Reality | Souls persist and spiritual warfare is real |
| 15 | Conclusion | External grace is the only mechanism for sign-flip |
The Seven Derived Chain Claims
What the chain derives
| # | Derived claim | Status in this layer |
|---|---|---|
| 1 | Reality requires a self-grounding information substrate, identified theologically as Logos | Derived chain claim |
| 2 | Actualization requires a terminal infinite observer, identified theologically as God | Derived chain claim |
| 3 | Humans have conserved moral sign, corresponding to the Fall layer | Derived chain claim |
| 4 | Self-transformation cannot flip moral sign | Derived chain claim |
| 5 | External intervention is required; this is the grace requirement | Derived chain claim |
| 6 | Souls persist beyond physical death | Derived chain claim |
| 7 | Spiritual warfare is real inside the framework's moral-relation model | Derived chain claim |
Lean 4 Connection
Lean does not certify these conceptual foundations by declaring them as Lean axioms. Instead, the current packet formalizes the parts that can be represented cleanly as types, definitions, structures, inductives, positive theorems, and adversarial negative tests. The right public claim is therefore: the formalized packet builds without Lean axiom declarations or source sorry/admit, while the conceptual axiom hierarchy remains the paper-level foundation.