Station Seven · Axiom 4 · Coherence as Bounded Compatible Asymmetry Skip to content

Station Seven · Axiom 4

Coherence - where many directed truths must fit within one bounded frame.

👁 Coherence Contrast → Distinction → Condition → Licence → Return → Coherence → Truth Closed FOL

Axiom 4 - Coherence as Bounded Compatible Asymmetry

Coherence tests whether many truths can hold their shape when joined. One proof gives one direction. Many proofs must still fit together. Coherence measures that fit. It asks whether distinct asymmetries can share a frame without breaking each other’s hold.

From asymmetry to fit

Proof introduces asymmetry. It leans a relation in one direction. Counter-falsification can strike it down, returning the pair to unresolved symmetry. When several proofs meet, coherence decides which can live together. It names the range where directed truths remain compatible.

The compass example

If a magnetic field is present then the compass needle aligns. That is one asymmetry - field to needle, cause to orientation.
Add another field at an angle. The returns compete. One pulls north, the other east. The needle trembles until the fields reach a stable resultant.
Too much contrast and the needle loses direction. Perfect symmetry and it cannot move.
Coherence lives between - where distinct asymmetries hold one direction in common.

Collision and convergence

When two returns collide, coherence fails. When they align, structure forms.
A coherent frame binds difference within bounds. It does not erase variation - it fits it.
Stability appears when directed relations share a common orientation.

Empirical reading

In experiment, two identical methods giving the same result add little strength - they repeat a symmetry.
Two distinct methods that reach the same result strengthen coherence.
Truth stabilises when contrasted paths converge on one return.
Reliability grows through contrastive convergence, not sameness.

Bounded compatibility

Coherence keeps truth from scattering. It allows many asymmetries to stand so long as they do not tear the frame.
Too much contrast disperses relation. Too little collapses it into identity.
Within those bounds, truth holds its shape - a living equilibrium of directed returns.

Station schema

Contrast (difference between relata) → Distinction (off the diagonal) → Condition (introduces asymmetry through proof) → Licence (directed permission) → Return (truth in one direction) → Coherence (bounded compatibility of many returns) → Truth.
The coherence window lies between symmetry collapse and dispersion - where distinct asymmetries converge without contradiction.

Formal layer - full axiom set

Signature (closed FOL)

  • Predicates: Frame(f) frame, ActsIn(c,f) condition active in f, InFrame(x,f) relatum in f, Order(cu,c1,c2) order witness, Compose(c*,c1,c2) composition, D(x,y) contrast, C(c,x,y) condition, L(c,x,y) licenced read, Ret(x,y) resolved return.

Base contrast

  1. ∀x ¬D(x,x)

    [A4.C0] Irreflexive.

  2. ∀x ∀y ( D(x,y) ↔ x ≠ y )

    [A4.C1] Contrast coincides with inequality.

Contrast-respecting constraints

  1. ∀c ∀x ∀y ( L(c,x,y) → C(c,x,y) )

    [A4.LC] Every licence rests on a condition.

  2. ∀c ∀x ∀y ( C(c,x,y) → D(x,y) )

    [A4.CD] Conditions apply only on contrasted pairs.

Abbreviations

Coherent(c;x,y) := D(x,y) ∧ C(c,x,y)
Asym(c;x,y) := C(c,x,y) ∧ ( L(c,x,y) ⊕ L(c,y,x) )
Bounded(c;x,y,f) := Bnd(c;x,y;f)
Compat(c1,c2;x,y) := ( L(c1,x,y) ∧ L(c2,x,y) ) ∨ ( L(c1,y,x) ∧ L(c2,y,x) )
CompC(c1,c2;x,y,f) := Coherent(c1;x,y) ∧ Coherent(c2;x,y) ∧ Bounded(c1;x,y,f) ∧ Bounded(c2;x,y,f) ∧ Compat(c1,c2;x,y)

Ordering discipline

Strict order witness, meta only. Order never alters L or Ret.

  1. ∀cu ∀c ¬Order(cu,c,c)

    [A4.Ord0] Irreflexive ordering.

  2. ∀cu ∀c1 ∀c2 ( Order(cu,c1,c2) → c1 ≠ c2 ∧ ¬Order(cu,c2,c1) )

    [A4.Ord1] Asymmetric witness.

Framed asymmetry and return

ADirf(x,y) := ∃c ( Bounded(c;x,y,f) ∧ C(c,x,y) ∧ L(c,x,y) )
  1. ∀x ∀y ( Ret(x,y) ↔ ( ∃f ADirf(x,y) ) ∧ Unopp(y,x) )

    [A4.R≡] Return equals framed one-way licence with a global guard.

  2. ∀x ∀y ¬( Ret(x,y) ∧ Ret(y,x) )

    [A4.RA] No opposing returns.

Neutrality and opposition

  1. ∀x ∀y ∀f ( Frame(f) ∧ InFrame(x,f) ∧ InFrame(y,f) ∧ ∀c ( (ActsIn(c,f) ∧ C(c,x,y)) → ¬L(c,x,y) ∧ ¬L(c,y,x) ) ) → ¬Ret(x,y) ∧ ¬Ret(y,x)

    [A4.N0] In a frame with only neutral behaviour there is no return.

  2. ∀c1 ∀c2 ∀x ∀y ( L(c1,x,y) ∧ L(c2,y,x) ) → ¬Ret(x,y) ∧ ¬Ret(y,x)

    [A4.N1] Opposed licences block resolution.

Compatible composition in-frame

  1. ∀c1 ∀c2 ∀x ∀y ∀f ( CompC(c1,c2;x,y,f) → ∃c* ( Compose(c*,c1,c2) ∧ Bounded(c*;x,y,f) ∧ Asym(c*;x,y) ) )

    [A4.Comp] Compatible, bounded, coherent conditions compose to an asymmetric composite.

Order witness - optional

  1. ∀cu ∀c1 ∀c2 ∀x ∀y ∀f ( Order(cu,c1,c2) ∧ CompC(c1,c2;x,y,f) → ∃c* ( Compose(c*,c1,c2) ∧ Bounded(c*;x,y,f) ∧ Asym(c*;x,y) ) )

    [A4.OrdW] Optional global order can witness a composite with asymmetry.

Derivations and checks

Theorem - Compatible meet yields asymmetric composite

∀c1 ∀c2 ∀x ∀y ∀f ( CompC(c1,c2;x,y,f) → ∃c* ( Compose(c*,c1,c2) ∧ Asym(c*;x,y) ) )

Immediate from [A4.Comp] by projection.

Corollary - Composite resolves under global guard

∀c1 ∀c2 ∀x ∀y ∀f ( CompC(c1,c2;x,y,f) ∧ Unopp(y,x) → Ret(x,y) )

Use [A4.Comp] to obtain an asymmetric composite in-frame, then apply [A4.R≡].

Theorem - Return is directional

∀x ∀y ( Ret(x,y) → D(x,y) ∧ ¬Ret(y,x) )

Proof

From [A4.R≡] obtain a frame f and a condition c with C(c,x,y) and a licence L(c,x,y). By [A4.CD] we get D(x,y). The guard Unopp(y,x) blocks Ret(y,x). QED.

Consistency witness - model 𝓜

Universe. U = {a,b} ∪ 𝓒 ∪ {f0}.

Non-empty primitives in 𝓜.

Frame(f0), InFrame(a,f0), InFrame(b,f0), choose c* ∈ 𝓒 with ActsIn(c*,f0), C(c*,a,b), L(c*,a,b)

Everything else false. All other instances of C, L, Ret, Compose, Order are false in 𝓜. For contrast use [A4.C1]: D(x,y) ↔ x ≠ y.

Verification in 𝓜

  • Contrast. Since a ≠ b we have D(a,b) by [A4.C1]. Also D(x,x) fails for every x as required.
  • Condition and licence. By construction C(c*,a,b) and L(c*,a,b) hold and there is no c with C(c,b,a) or L(c,b,a).
  • Return. Using [A4.R≡] we get Ret(a,b) ↔ (∃c [C(c,a,b) ∧ L(c,a,b)]) ∧ ¬∃d L(d,b,a), which is satisfied in 𝓜 by the chosen c*. By [A4.RA] the reverse is forbidden, so Ret(b,a) fails.
  • Clean slate elsewhere. No other C, L, Ret instances hold by stipulation and Compose, Order are everywhere false. All axioms referenced are satisfied in 𝓜.

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