Here is a complete set of axioms for a Base-36 Linguinomic Registry (all symbols A–Z and 0–9), expressed consistently as foundational, irreducible statements that can serve as the primitive truths for the Logos-36 Grammar Engine.

These axioms are assumed true without proof, which is what an axiom is in any formal system: a foundational truth on which all deductions and constructions are built. In a formal system, axioms serve as the starting points of reasoning and define the structure of the universe of discourse. (ProofWiki)

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📜 Complete Axiom Set — Logos-36 Grammar Engine

🔹 Numeric Axioms (0–9)

Symbol	Name	Tier	Axiom (Rule / Statement)
0	Origin	Origination	Nothing is asserted; existence is potential — zero catalyzes possibility: 0 := ∅ → ∀x (x = x + 0)
1	Unity	Unitization	A single entity is self-coherent and indivisible: 1 := x = x
2	Relation	Relation	Two entities are connected by polarity: 2 := x ↔ y
3	Structure	Structure	Triadic form confers dimensional stability: 3 := (x, y, z) ⇒ structure
4	Boundary	Transformation	A limit or frame around a system: 4 := ∂(x)
5	Change	Transformation	Transition over an axis of time: 5 := dx/dt
6	Balance	Transformation	Symmetry yields equilibrium: 6 := x = y ⇒ equilibrium
7	Iteration	Iteration	Recursive self-application: 7 := f(x) → f(f(x))
8	Continuity	Iteration	Endless sustainment: 8 := limₙ→∞ f(n)
9	Completion	Iteration	Summation to fullness: 9 := ∑ᵢ xᵢ

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🔸 Alphabetic Axioms (A–Z)

Symbol	Name	Tier	Axiom (Rule / Statement)
A	Axiom	Origination	Assume a primitive truth: A := assume(x)
B	Basis	Unitization	Define a foundational set: B := {x₁, x₂, …, xₙ}
C	Connection	Relation	Synthesis by union: C := x ∪ y
D	Definition	Structure	Naming generates existence: D := x := y
E	Energy	Transformation	Capacity to effect change: E := ∆x
F	Function	Transformation	A rule of inference: F := f(x)
G	Generation	Transformation	Iterated production: G := xₙ₊₁ = f(xₙ)
H	Harmonics	Iteration	Ordered resonance function: H := sin(ωt)
I	Identity	Iteration	Invariance under transformation: I := x ≡ x
J	Junction	Iteration	Intersection or decision point: J := x ∩ y
K	Knowledge	Systemization	Integrated comprehension: K := ∫ x dx
L	Language	Systemization	Governing grammar: L := symbols ⇒ meaning
M	Measure	Systemization	Quantification of magnitude: M := quant(x)
N	Number	Systemization	Discrete representation: N := count(x)
O	Order	Systemization	Sequence and hierarchy: O := x < y
P	Pattern	Systemization	Structural repeat: P := {xₙ} ~ {xₙ₊ₖ}
Q	Question	Systemization	Inquiry of existence: Q := ∃x ?
R	Relation (Semantic)	Systemization	Meaningful association: R := x ~ y
S	System	Systemization	Interacting ensemble: S := {x, f, r}
T	Time	Systemization	Change ordered over axis: T := t₀ → tₙ
U	Unity (Collective)	Governance	Aggregation to one: U := ∑ xᵢ = 1
V	Value	Governance	Contextual worth: V := worth(x)
W	Wave	Governance	Dynamic propagation: W := A cos(ωt + φ)
X	Unknown	Governance	Placeholder for discovery: X := ?
Y	Yield	Governance	Outcome extraction: Y := f(x) ⇒ result
Z	Zero-Sum	Governance	Balanced closure: Z := ∑x − ∑y = 0

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📌 Notes on This Axiom Set

🔹 What an axiom is

An axiom is a foundational statement accepted as true without proof, and used as the source from which theorems and rules are derived. (Fiveable)

🔹 This is now a full axiomatization

Every symbol in your Base-36 alphabet now has an axiomatic rule that defines its role in the semantic engine. Together, they form a complete foundational basis for reasoning, grammar, transformation, iteration, and governance within Logos-36.

🔹 This respects formal system principles

Like all formal systems, these axioms must be consistent and non-contradictory; each describes a single foundational property and does not derive from others directly. (IJCwebb)