DFT-14: Spatial vs. Temporal Nonlocality as Projection Effects

MWells · Post by **MWells** » Wed Dec 03, 2025 8:08 pm

How nonlocality arises not from exotic physics, but from the geometry of projection itself.

Nonlocality is one of the most confusing features of modern physics. In quantum mechanics it appears as entanglement and interference. In relativity it appears as disagreements about simultaneity. In the Reciprocal System (RS), Larson sometimes spoke of effects that seem instantaneous or “distributed in time,” but he never gave a clean geometric account of how such behavior arises from scalar motion.

Dual-Frame Theory (DFT) offers a unified explanation. Once we see scalar progression through the two projections:

S-frame: geometric trajectory in a Minkowski-like spacetime,
T-frame: rotational or phase trajectory on a compact manifold,

it becomes clear that nonlocality is not a process happening "in the world", but a projection artifact. It arises because we are mapping a single scalar process into two limited representational frames.

In this post, we will:

Set up the projection picture in a simple, explicit way.
Explain spatial nonlocality as an S-frame artifact of compact T-frame structure.
Explain temporal nonlocality as a T-frame artifact of S-frame expansion.
Clarify how conflicts about simultaneity arise from the difference between RS “clock-time” and S-frame coordinate time.

This post stays at the geometric/interpretive level.
Explicit Bell-inequality calculations, EPR examples, and interference patterns are deferred to later work; here we are isolating the projection mechanisms that make such phenomena possible in DFT.

1. Scalar Progression and the Two Projections

In the Natural Reference System (NRS), we have a scalar progression: a one-parameter process with no geometry and no phase. We write this as a mapping from an intrinsic parameter 𝜆 to a scalar state 𝜎(𝜆):

$\sigma : \lambda \mapsto \sigma(\lambda)$

There is no notion of “near in space” or “near in phase” yet. Those arise only when we project this scalar progression into our two interpretive frames:

S-frame projection:

$F_S : \sigma(\lambda) \mapsto x^\mu(\lambda)$

$\mu = 0,1,2,3$

Here, 𝑥⁰ is the S-frame coordinate time, and 𝑥¹,𝑥²,𝑥³ are the spatial coordinates.

T-frame projection:

$F_T : \sigma(\lambda) \mapsto \theta^i(\lambda)$

$i = 1,2,3$

Here, the three components 𝜃^𝑖 represent phase or rotational angles.

Crucially:

The S-frame has only three spatial dimensions plus one coordinate time.
The T-frame has only three independent phase angles.

But the scalar progression σ(λ) can have more structural detail than either frame can fully resolve. This mismatch means:

Multiple scalar configurations can share the same S-frame representation.
Multiple scalar configurations can share the same T-frame representation.

Whenever this happens, we get projection ambiguity. That ambiguity is the geometric root of what looks, in our usual physical language, like nonlocal behavior.

2. Spatial Nonlocality: When Phase Adjacency Collapses into the Same S-Frame Pattern

2.1 Compact T-Frame Geometry

The T-frame lives on a compact three-dimensional torus, which we can write as:

$T^3 = S^1 \times S^1 \times S^1.$

Each phase coordinate is an angle on a circle, defined modulo 2𝜋. That means:

Two phase configurations can be “close” in the T-frame, even if the corresponding S-frame configurations are very different.
Phase adjacency is defined by small changes in the angles 𝜃^𝑖, not by geometric distance.

Symbolically, we can have scalar parameters λ and λ+Δλ such that

$F_T(\sigma(\lambda + \Delta\lambda)) \approx F_T(\sigma(\lambda)),$

but at the same time

$F_S(\sigma(\lambda + \Delta\lambda)) \neq F_S(\sigma(\lambda)).$

So, two states that are adjacent in phase might be far apart in space.

$\text{Phase adjacency:}\qquad \|\Delta\theta\| \ll 1 \qquad\Rightarrow\qquad \|\Delta x\| \text{ may be large}$

From the S-frame’s point of view, it then looks as if:

widely separated configurations are “linked,”
or “information” is shared across distance without passing through intermediate points.

This is what we call spatial nonlocality. But in DFT, this is simply the result of representing a compact phase structure (the T-frame) in a lower-dimensional geometric container (the S-frame). It is a many-to-one mapping: multiple phase-adjacent configurations project to S-frame patterns that look disconnected.

No mysterious physical influence is needed. The apparent nonlocality is a shadow of phase adjacency.

Comment on compactness:
The choice of a product of circles 𝑇³ matches DFT’s use of three independent angular coordinates and winding numbers. Other manifolds (e.g. 𝑆3, 𝑆𝑂(3), 𝑆𝑈(2)) are possible rotation spaces mathematically, but the torus is the minimal structure that supports independent integer windings and periodic phases in three channels, consistent with the RS rotational triplets.

2.2 The Motion Budget and the T-Dominant Regime

The motion budget, introduced in earlier posts, governs how much structural complexity can appear in each frame. Roughly, we can think of a balance:

$\text{(S-frame structural complexity)} + \text{(T-frame structural complexity)} = \text{(fixed budget determined by } \sigma(\lambda) \text{)}.$

When the T-frame “wins” this competition—meaning rotational or phase effects consume most of the budget—small changes in λ can produce large phase changes and minimal geometric changes.

Geometrically,

$\Delta \theta^i \; \text{large}, \quad \Delta x^\mu \; \text{small}.$

The S-frame then lacks the capacity to represent all of the fine-grained T-frame distinctions; many different phase configurations get mapped to the same or nearly the same geometric configuration.

This is exactly the regime in which quantum-like coherence and interference arise. Two “paths” that are distinct in phase but overlapping in geometry can interfere, not because anything travels faster than light, but because the S-frame is blind to part of the T-frame structure.

Spatial nonlocality, in this view, is:

the S-frame’s inability to distinguish certain phase-different but geometry-similar states,
combined with the T-frame’s ability to carry phase information across what looks like spatial separation.

In later work, this same structure will be used to discuss EPR/Bell-type setups: the correlations live in the T-frame phase structure; the S-frame only sees widely separated events that nevertheless share a phase-adjacent origin.

3. Temporal Nonlocality: When Large S-Frame Changes Look Phase-Adjacent

Now consider the opposite case, where the S-frame dominates the motion budget: large geometric displacements per unit scalar progression, and relatively small changes in phase.

We can write the local phase evolution schematically as:

$\Delta \theta^i \propto \omega^i \, \Delta\lambda,$

where 𝜔^𝑖 is a local phase-wind rate. If the S-frame is already consuming most of the available structural complexity, then the magnitude of these phase changes can be quite small:

$\Delta \theta^i \to 0 \quad \text{while} \quad \Delta x^\mu \text{ is large}.$

In this situation, large S-frame separations can correspond to small or negligible T-frame separations. That means:

$\|\Delta x\| \gg 0 \qquad\Rightarrow\qquad \|\Delta\theta\| \approx 0$

Two events that look far apart in both space and time can still be phase-near.
The T-frame “remembers” a connection that the S-frame representation has stretched out.

From the T-frame’s perspective, the same scalar trajectory may pass through configurations that are “almost the same phase” even if the S-frame shows them as occurring at very different times or locations. This creates the appearance of:

long-range temporal correlations,
phase-coherence effects that persist across distance,
what RS sometimes described heuristically as effects “distributed in time.”

Once again, nothing truly nonlocal is happening. The underlying scalar process is local. The T-frame just collapses what the S-frame has expanded.

This is temporal nonlocality: the appearance of connections across time, produced by the projection from a highly expanded S-frame geometry into a relatively compressed phase description.

4. Simultaneity: RS Clock-Time vs. S-Frame Coordinate Time

A further source of confusion arises from the difference between:

RS clock-time (scalar progression in the NRS), and
S-frame coordinate time (the time coordinate in Minkowski-like geometry).

In the NRS, the simplest notion of time is just:

$\Delta \tau = \Delta \lambda.$

Each unit of scalar progression is one unit of this intrinsic “clock-time.” It is not tied to any particular spatial coordinate system.

In the S-frame, however, time appears as the zeroth coordinate in a spacetime metric:

$x^\mu(\lambda) = F_S(\sigma(\lambda)), \quad \mu = 0,1,2,3,$

with a spacetime interval

$\Delta s^2 = c^2 (\Delta x^0)^2 - (\Delta x^1)^2 - (\Delta x^2)^2 - (\Delta x^3)^2.$

To maintain Lorentz invariance, the mapping from scalar progression to S-frame coordinates cannot simply identify:

$x^0 = \lambda.$

Instead, the relationship between 𝜆 and the S-frame coordinate time 𝑥⁰ depends on the local balance between translational and rotational components, gravitational context, and the motion budget. Two equal increments of 𝜆:

$\Delta \lambda_1 = \Delta \lambda_2$

need not correspond to equal increments of coordinate time:

$\Delta x^0_1 \neq \Delta x^0_2.$

In DFT-13, we saw that proper time Δ𝜏 arises from how much of the scalar increment is allocated to T-frame participation. Here we are simply noting that the coordinate time 𝑥⁰ is a projection variable chosen to keep S-frame physics Lorentz-invariant, not a direct copy of 𝜆.

This misalignment is the deep reason why:

different inertial observers disagree about simultaneity,
RS’s scalar “clock-time” does not line up trivially with S-frame time,
certain RS descriptions of “time distribution” can be reinterpreted precisely as projection effects.

In DFT, the conflict about simultaneity is not a paradox. It is simply the statement that:

scalar progression is absolute, but
coordinate time is a projection-dependent parameter designed to make the S-frame obey Lorentz symmetry.

When we confuse these two, we experience temporal nonlocality as a kind of conceptual whiplash. When we distinguish them, the geometry is straightforward.

5. Nonlocality as Projection Ambiguity, Not Exotic Physics

We can now summarize the two main forms of nonlocality in DFT language:

5.1 Spatial Nonlocality (S-Frame)

Root cause: compact phase structure in the T-frame (the T³ manifold).
Mechanism: many-to-one mapping from T-frame to S-frame:

$F_T(\sigma_1) \approx F_T(\sigma_2), \quad F_S(\sigma_1) \text{ and } F_S(\sigma_2) \text{ are far apart.}$

Result: phase-adjacent states appear spatially disconnected; the S-frame sees “mysterious” coherence or correlations.

This is the qualitative geometric mechanism underlying entanglement-like correlations in DFT. Whether those correlations match quantum Bell-violating statistics is a quantitative question left for later posts; here we are clarifying why “long-distance ties” need not imply superluminal influences.

5.2 Temporal Nonlocality (T-Frame)

Root cause: expansion of scalar trajectories in S-frame geometry.
Mechanism: many-to-one mapping from S-frame to T-frame when the S-frame dominates the motion budget:

$F_S(\sigma_1) \text{ and } F_S(\sigma_2) \text{ far apart}, \quad F_T(\sigma_1) \approx F_T(\sigma_2).$

Result: events that are widely separated in coordinate time and space can still be phase-near; the T-frame sees them as related.

This is the geometric backdrop for long-range temporal correlations, quantum memory-like effects, and RS’s “distributed in time” behavior—without requiring any literal stretching of processes across time.

5.3 Simultaneity

Clock-time in RS is just scalar progression:

$\Delta \tau = \Delta \lambda.$

S-frame coordinate time 𝑥⁰ is defined via a Lorentzian embedding and need not track lambda linearly. Differences in simultaneity are thus differences in projection, not contradictions in the underlying motion.

In all three cases, the message is the same:

The underlying scalar progression is strictly local.
Nonlocality appears when we project that progression into limited frames that must compress or overlap parts of the structure.

6. Looking Ahead: From Nonlocality to Lorentz Symmetry

In this post we have seen how spatial and temporal nonlocality follow naturally from the geometry of projection. The S-frame and T-frame are not independent worlds; they are two views of the same scalar process, each forced to discard some information.

This has several consequences:

It gives an interpretive handle on entanglement-like and interference-like behavior: correlations live in T-frame phase adjacency, not in superluminal influences.
It suggests a geometric basis for the quantum–classical transition via the motion budget (T-dominant vs S-dominant regimes).
It clarifies why simultaneity disputes are projection-level issues, not failures of an underlying “absolute time.”

However, this post has not yet:

Shown how DFT handles Bell’s theorem quantitatively,
Derived specific EPR-Bohm correlations,
Computed interference patterns or decoherence rates.

Those are essential tests. DFT must ultimately demonstrate that its projection geometry can reproduce:

Bell inequality violations,
EPR correlations,
double-slit interference,
and realistic decoherence behavior.

The present post is about making the geometry of nonlocality explicit. It sets the stage for such calculations.

In the next post, DFT-15 — Emergent Lorentz Invariance and Inertial Frames, we will flip the perspective:

Instead of asking how projection creates apparent nonlocality,
we will ask how projection forces strict relativistic limits.

We will see that Lorentz symmetry and the constancy of the speed of light are not postulates but consequences of how scalar motion can be embedded consistently into the S-frame while remaining compatible with the T-frame.