The Character of Each Direction

One direction has opposite metric sign to the others — the Lorentzian signature. Its structural source is in the axioms themselves: exactly one axiom is intrinsically irreversible, and irreversibility read as metric character is the (−) sign; the formal metric-level derivation is the open obligation fenced at KS-D.4.

This section identifies which axiom produces which character.

4.1 — Time is R

The (−) direction is the one in which records accumulate.

R is the only axiom that introduces irreversibility — the monoid has no inverse.

All other axioms are compatible with reversibility in their contributed direction.

C gives a symmetric speed limit (propagation at c in either spatial direction).

S gives a symmetric involution (σ maps ℒ → 𝓟 and 𝓟 → ℒ equally).

B gives a definite break direction but does not prohibit the reverse spatial direction.

You cannot go backward in time because Axiom R has no inverse.

Only R contributes a direction that is intrinsically asymmetric — that has a “forward” but no “backward.”

The (−) in the signature.

4.2 — Space is {C, S, B}

The three (+) directions are the ones in which records can exist at either end — where forward and backward are structurally symmetric.

Spatial directions admit motion in both directions.

Temporal direction does not admit reversal of records.

This is the signature distinction.

C gives propagation distance: symmetric. You can travel in either direction along the propagation axis, up to speed c.

S gives sector-crossing: symmetric. σ maps in both directions by definition — it is an involution, σ² = identity.

B gives the break direction: the now advances in a specific direction, but the spatial axis itself admits both directions. The asymmetry of the break is temporal (the now writes records irreversibly, via R), not spatial.

The signature (−, +, +, +) is the axiom structure (R, C, S, B) read as metric character.