Relativity and reciprocal reality

A block interpretation with explicit geometric commitments

Change remains inside the history

For a timelike worldline in inertial coordinates,

$$d\tau^2=dt^2-\|d\mathbf x\|^2/c^2.$$

Proper time measures physical processes along the worldline; it selects no simultaneous universe-wide state. An extended organization occupies a worldtube, transporting signals and using local clock conventions. Relativity fixes causal relations and proper durations, while spacelike simultaneity varies by inertial frame.

Eternalism takes actual events at their temporal locations to belong to reality without an additional activation process. Its economy is concrete: one organization explains change, clocks and records without another present selecting which region exists. Equation (4) supplies a target-relative version of this economy when its equivalences hold. It neither proves the selector nonexistent nor deletes relations definable from geometry.

A manifold represents events; tenseless existence is an interpretation of that representation. Relational becoming can use invariant past-definiteness without a preferred simultaneity surface. Myrvold extends such relations to regions and hypersurface-indexed state/chance descriptions while retaining quantum nonseparability [17,43,44]. This is a serious alternative. The block branch is defended for its economy and fit with one actual physical history.

Geometry, determinism, time-reversal symmetry and eternalism answer different questions. Determinism asks about uniqueness from adequate data. A complete actual history does not establish uniqueness. Time reversal transforms solutions. Change is difference along a history: propagation, memory writing and deliberation occur at ordered events without an outside clock. An indeterministic history law can still have one actual complete outcome history.

One reciprocal pair has a decisive consequence

Let $R(p,q)$ be one fixed event relation, conditionally interpreted as "q is co-real relative to p". It is not local access or an observer-indexed family. In real 1+1 Minkowski space, set $c=1$ and write the proper orthochronous boosts as

$$B_\ell(t,x)=\left(\frac{\ell+\ell^{-1}}2t+\frac{\ell-\ell^{-1}}2x, \frac{\ell-\ell^{-1}}2t+\frac{\ell+\ell^{-1}}2x\right),\qquad\ell\gt0.$$

Spatial reflection is $P(t,x)=(t,-x)$.

Theorem 3.1: reciprocal pair under relativistic symmetries. Assume transitivity and invariance under every translation. In real 1+1 dimensions assume invariance under every $B_\ell$ and reflection. In real 3+1 dimensions assume instead all proper spatial rotations and proper orthochronous boosts. If distinct $p_0,q_0$ obey both $R(p_0,q_0)$ and $R(q_0,p_0)$, then $R(p,q)$ for every pair. Reflexivity follows; dilation, global symmetry, continuity and measurability are unnecessary.

Proof mechanism. The mutual core $H=\{v:R(0,v)\land R(v,0)\}$ is an additive subgroup containing a nonzero displacement. In 1+1, reflection extracts a nonzero pure-axis vector. Differences of opposite boosts generate the whole other axis because $\ell-\ell^{-1}$ ranges over all reals. One fixed nonzero boost generates the first axis. Thus $H=\mathbb R^2$. In 3+1 the subgroup proof uses proper spatial half-turns and coordinate-axis boosts to generate the time axis and then all spatial axes. Translation yields universality. The full proof supplies every step.

This is a direct Stein-type consequence. Stein's fixed-length spacelike argument already removes the no-unit/dilation stipulation [44,p.150]; no general historical priority is claimed. The direct 1+1 and 3+1 statements have formal modules independent of the dilation-based classification. The 3+1 proof needs the coordinate-axis boosts and three proper half-turns in Appendix A. Full proper-orthochronous Lorentz invariance is a stronger sufficient package, not a new full 3+1 classification.

The separate six-way 1+1 classification has stronger hypotheses: reflexivity, transitivity, translation/boost/dilation/reflection invariance and an ordered field with positive square roots. Its outcomes are identity, chronological past or future with identity included, causal past or future, and universality. A fixed positive proper-interval threshold supplies an invariant transitive past relation outside that list when dilation is omitted. It has no distinct mutual pair, so does not refute the direct theorem.

The bridge to actuality is substantive

The theorem proves universality under its mathematical premises. Applying it to actuality needs independent reasons for the meaning of $R$, the invariances, transitivity and one distinct reciprocal witness. A process's duration, dependence on an earlier state, or two contributions constituting one transition do not themselves establish reciprocal actuality.

One may propose that actuality means a homogeneous mutual belonging, and motivate a reciprocal witness through one actual relation spanning events. The theorem makes the consequence exact. Yet taking a process as that witness already adopts the disputed reading of belonging. Its occurrence does not independently establish it. Reciprocal actuality is therefore not added as a universal Shadow Theory axiom.

A fixed relation's invariance also exceeds covariance of a family. For future-directed unit velocity $u$, define

$$R_u(p,q)\Longleftrightarrow u\cdot(q-p)=0.$$

Each fixed $R_u$ is a nonuniversal equivalence relation with reciprocal spacelike pairs. Transforming $u,p,q$ together gives covariance; holding $u$ fixed does not give boost invariance. The distinction applies to apparatus, observer and process-window relations. Generic covariance cannot license the theorem for access.

Countermodels isolate the premises. Without boosts, equal coordinate time is nonuniversal and reciprocal. Without transitivity, equality-or-spacelike separation is another. Without translations, equality of Minkowski norm relative to a chosen origin survives Lorentz symmetry. Without reflection in 1+1, a null-line equivalence survives every boost. Their details appear in Appendix A. Global co-reality never implies global observation.