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Station Five · Axiom 3

Truth is resolved asymmetry

👁 Asymmetry 👁A ⇄ 👁B → condition → licence → 👁A ⇒ 👁B → Truth Closed FOL

Axiom 3 — Truth is resolved asymmetry

Symmetry preserves possibility… asymmetry decides. Truth lives where a single licensed direction survives on the pair. Write it as 👁A ⇒ 👁B under its condition. The return is the decision… truth is the label on its code.

Why asymmetry matters

With 👁A ⇄ 👁B framed by a condition, equal licensing keeps orientation suspended. A tiny tilt selects a side. That tilt… the one-sided licence… makes direction available.

From symmetry to decision

Balance carries possibility. A grain of fit tips the frame. Once a single read opens and the reverse does not survive, orientation is fixed for that condition.

Return as the survivor

A return is the one-way survivor: 👁A ⇒ 👁B. It condenses the whole evaluation of ⟨👁A, 👁B⟩ into a single settled line.

Where truth attaches

Truth attaches to a code that names the return… not to the raw licence. The code inherits the settled orientation and carries it onward.

Dependency chain

  • Contrast on the pair 👁A ⇄ 👁B.
  • Condition frames ⟨👁A, 👁B⟩ and tests fit.
  • Licence opens one direction.
  • Single survivor yields the return 👁A ⇒ 👁B.
  • Truth applies to codes of that return.

Station schema

Contrast (👁A ⇄ 👁B) → Condition on ⟨👁A, 👁B⟩ → Licence (one way) → Return (👁A ⇒ 👁B) → Truth on the code of that return.

Axiom 3 · Truth as resolved asymmetry · FOL

Ladder (schema)

\[ \boxed{\ \text{Contrast }(D)\ }\ \Longrightarrow\ \boxed{\ \text{Distinction }(x\neq y)\ }\ \Longrightarrow\ \boxed{\ \text{Condition }C\ \&\ \text{Licence }L\ }\ \Longrightarrow\ \boxed{\ \text{Asymmetry }(\oplus)\ }\ \xRightarrow{\ \text{uncontested}\ }\ \boxed{\ \text{Return }Ret\ }. \]

Signature additions

  • Function: Sig(x) coding map.
  • Predicates: Ind(w,c), Part(w,x,y), Proof(c), Decep(c), Order(c_u,c_1,c_2), Ontic(x).
  • From earlier: S(·), Truth(·), False(·), D, C, L, Ret. Stipulate S(Sig(x)).

Definition · Asymmetry

\[ \mathrm{Asym}(c;x,y)\ \leftrightarrow\ C(c,x,y)\ \land\bigl(L(c,x,y)\ \oplus\ L(c,y,x)\bigr). \]

Contrast and condition · recap

∀x ¬D(x,x) \;,\; ∀x ∀y ( D(x,y) ↔ x ≠ y ) [A3.C]
∀c ∀x ∀y ( C(c,x,y) → D(x,y) ) [A3.C→D]

Ordering discipline · strict selector

∀c_u ∀c ¬Order(c_u,c,c) [A3.Ord0]
∀c_u ∀c₁ ∀c₂ ( Order(c_u,c₁,c₂) → c₁ ≠ c₂ ∧ ¬Order(c_u,c₂,c₁) ) [A3.Ord1]

Selector only. Order never alters L or Ret.

Laws and consequences

Participation

∀w ∀x ∀y ( Part(w,x,y) ↔ ∃c ( Ind(w,c) ∧ C(c,x,y) ) ) [A3.P0]
∀x ∀y ( Ret(x,y) → ∃w Part(w,x,y) ) [A3.P1]

Read and return · strong law

∀c ∀x ∀y ( L(c,x,y) → C(c,x,y) ) [A3.R0]
∀x ∀y ( Ret(x,y) ↔ (∃c [ C(c,x,y) ∧ L(c,x,y) ]) ∧ ¬∃d L(d,y,x) ) [A3.R≡]
∀x ∀y ¬( Ret(x,y) ∧ Ret(y,x) ) [A3.E0]

Neutrality and opposed licences

∀x ∀y ( (∀c ( C(c,x,y) → ¬L(c,x,y) ∧ ¬L(c,y,x) )) → ¬Ret(x,y) ∧ ¬Ret(y,x) ) [A3.N0]
∀c₁ ∀c₂ ∀x ∀y ( L(c₁,x,y) ∧ L(c₂,y,x) → ¬Ret(x,y) ∧ ¬Ret(y,x) ) [A3.N1]

Formation and validation

∀p ( Truth(p) → S(p) ) ∧ ∀p ( False(p) → S(p) ) [A3.V1]
∀x S(Sig(x)) [A3.V2]
∀p ( Truth(p) → ¬False(p) ) ∧ ∀p ( False(p) → ¬Truth(p) ) [A3.Vexcl]

Ontic return and semantic validation

∀x ∀y ( Ret(x,y) → Ontic(x) ) [A3.O0]
∀c ∀x ∀y ( Proof(c) ∧ C(c,x,y) ∧ L(c,x,y) ∧ ¬∃d L(d,y,x) → Truth(Sig(x)) ∧ False(Sig(y)) ) [A3.V0]
∀c ∀x ∀y ( Decep(c) ∧ C(c,x,y) ∧ L(c,x,y) ∧ ¬∃d L(d,y,x) → False(Sig(x)) ∧ Truth(Sig(y)) ) [A3.V1′]

Truth as resolved asymmetry

∀x ( Truth(Sig(x)) → ∃y ∃c [ C(c,x,y) ∧ L(c,x,y) ] ∧ ¬∃d L(d,y,x) ) [A3.T0]
∀c ∀x ∀y ( Asym(c;x,y) ∧ L(c,x,y) ∧ ¬∃d L(d,y,x) → Ret(x,y) ) [A3.T1+]
∀x ∀y ( ¬∃c Asym(c;x,y) → ¬Ret(x,y) ∧ ¬Ret(y,x) ) [A3.T2]

Consistency witness

Domain \(\{a,b,w\} \cup S\) with \(S \neq \varnothing\). Interpret \(D(x,y) \leftrightarrow x \neq y\). Take all of \(C,L,Ret,Ind,Part,Proof,Decep,Order,Ontic\) empty. Let \(S(p) \leftrightarrow p \in S\). \(\Truth\) and \(\False\) everywhere false. \(Sig\) maps into \(S\). All Axiom 3 clauses hold.

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