Spacetime Structure

If $L_3$ constrains physical instantiation, that constraint applies to spacetime itself—not just to configurations within spacetime. Logic Realism Theory derives key spacetime features from Determinate Identity.

The Problem

General relativity assumes rather than derives several structural features:

• Lorentzian signature: Why (3,1) rather than (4,0) or (2,2)?
• Temporal ordering: Why is there a distinguished “timelike” direction?
• Causal structure: Why can’t effects precede causes?
• CTC exclusion: Why are closed timelike curves problematic?

Can these be derived from more fundamental constraints?

Temporal Ordering from Joint Inadmissibility

Definition: Two configurations $c_1, c_2 \in A_\Omega$ are jointly admissible if their conjunction is also admissible: $c_1 \land c_2 \in A_\Omega$.

They are jointly inadmissible if: $c_1 \land c_2 \notin A_\Omega$

For example: “particle at position $x_1$” and “same particle at position $x_2 \neq x_1$” are individually admissible but jointly inadmissible (at the same time).

Theorem (Temporal Ordering): If two $L_3$-admissible configurations are jointly inadmissible but both are instantiated, then they must be temporally ordered.

Proof sketch: If both $c_1$ and $c_2$ are instantiated but cannot be co-instantiated, there must be an ordering relation: “$c_1$ before $c_2$” or “$c_2$ before $c_1$.”

Conclusion: Time emerges as the logical sequencing necessitated by joint inadmissibility. It is not an additional postulate but the structure that permits individually-admissible configurations to both be instantiated.

Lorentzian Signature

Heuristic argument: If temporal ordering arises from joint inadmissibility and spatial separation permits joint admissibility, then the metric signature must be Lorentzian.

• Timelike separation: Sequential instantiation of jointly-inadmissible configurations
• Spacelike separation: Possible co-instantiation of jointly-admissible configurations
• Lorentzian signature (3,1): Encodes exactly this asymmetry

Other signatures fail:

• (4,0) Euclidean: No distinguished temporal direction; cannot represent the asymmetry
• (2,2) Split: Two “timelike” directions; would permit multiple independent temporal orderings

CTC Exclusion

Theorem (CTC Exclusion): Closed timelike curves violate Determinate Identity.

Proof: On a CTC, event $e$ is in its own causal past. The “first” occurrence of configuration $c_e$ has identity $i_1$; the “second” occurrence (after the loop) has identity $i_2$. But they are the same event: $i_1 = i_2$.

Is $i_1$ causally prior to $i_2$, or $i_2$ prior to $i_1$? On a CTC, both. This violates antisymmetry of causal ordering:

Yet $i_1 = i_2$ by assumption. Contradiction.

Conclusion: CTCs violate Id. Spacetimes containing CTCs are not in $A_\Omega$.

Singularity Constraints

Theorem (Identity Preservation): If Determinate Identity holds, then no physical process can destroy identity-constituting information.

Implications for singularities:

If a configuration evolves to a singularity where it “ceases to exist”:

• Option A: Identity is destroyed (no successor) → Violates Id
• Option B: Identity transforms into new configuration → Compatible with Id

Conclusion: Physical singularities cannot destroy identity. Either:

1. Singularities are not genuine endpoints (information escapes)
2. Singularities involve identity transformation (not destruction)
3. Classical singularities are artifacts of incomplete physics

This aligns with modern views from AdS/CFT, but LRT provides a logical reason: Id forbids information destruction.

Identity Continuity

For a system to persist through time, successive configurations must satisfy bounded distinguishability:

Lemma: The configuration at $t_2$ must be sufficiently similar to the configuration at $t_1$ for there to be a determinate fact that they are the same system.

If configurations $c_1$ and $c_2$ share no properties grounding identity across time, there is no fact that $c_2$ is the continuation of $c_1$.

Consequence: Physical evolution cannot change configurations arbitrarily fast. There is a maximal rate of change compatible with identity preservation—leading to quadratic identity strain structure.

What Remains

These results are programmatic—rigorous formalization requires additional development. Open questions:

• Can Einstein’s field equations be derived from $L_3$ + vehicle-invariance?
• How does $L_3$ constrain quantum gravity?
• What does $L_3$ imply for cosmological initial conditions?

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