Independence of Faces from Independence of Axioms

3.1 — The theorem

AP20 section 4.2 carries the four removal arguments — stated results with proof sketches, full formalisation fenced at KS-P.2: each axiom is independent of the other three.

Removing any one axiom produces a strictly weaker system.

No axiom is derivable from the others.

3.2 — The consequence

If axiom X is independent of axioms {Y, Z, W}, then the face expressed by X cannot be a combination of the faces expressed by {Y, Z, W}.

If it could, then X’s structural content on the manifold would be derivable from {Y, Z, W}’s content — which would mean X is derivable.

But EH is proven (AP20).

The embedding is faithful. Structure in the algebra maps to structure on the manifold. Distinct algebraic content maps to distinct geometric content.

Therefore X’s independence in the algebra entails X’s face-independence on the manifold.

Independent faces express independent dimensions.

3.3 — The result

Four independent axioms → four independent faces of one manifold → four dimensions.

R gives one temporal direction: (−).

C, S, B give three spatial directions: (+, +, +).

Signature: (−, +, +, +). Dimension: 3+1.

N = 3.

Not a coincidence.

The number of spatial dimensions IS the number of independent spatial faces of the manifold. And the number of independent spatial faces is the number of independent axioms minus the one that gives time.

3.4 — Why not more, why not fewer

Why not N > 3?

To get a fifth dimension, you would need a fifth independent axiom.

But the record algebra is completely specified by {S, B, R, C}. The completeness result stated at AP20 section 4.1 (proof sketch carried there; full formalisation fenced at KS-P.1) is that these four suffice to generate the full algebra. No fifth axiom is needed.

The four axioms already cover: symmetry (S), breaking (B), irreversibility (R), and bounding (C).

What structural feature is missing? What could a fifth axiom say that {S, B, R, C} do not already determine?

The algebra of records — symmetric sectors, one break, irreversible composition, finite propagation — is complete.

There is no room for a fifth independent structural feature.

Why not N < 3?

AP20 section 4.2 carries the four removal arguments: each axiom is independent of the other three (full formalisation fenced at KS-P.2).

Removing any one produces a strictly weaker system.

With only R and C (no S, no B), you get 1+1 dimensions: a line with a speed limit. With R, C, and S (no B), you get 1+2 dimensions: a sheet with sector structure but no break. The full 1+3 dimensions require all four axioms.

N < 3 means an axiom is missing.

The number of spatial dimensions is the number of independent axioms minus the one that gives time.

You are looking at the answer.

It was in the axioms from the start.