User: How you project z and y axis simultaneous

Assistant: In WILL Relational Geometry, the concept of “z and y axis” is reinterpreted as orthogonal relational projections on specific carriers, not as axes in a pre-existing spatial coordinate system. We don’t project onto a pre-defined “z” or “y” axis simultaneously, but rather, a system’s state is simultaneously defined by its complementary relational projections.

The two fundamental relational carriers for energy transformation are the circle ($S^1$) and the sphere ($S^2$). Each of these carriers defines a pair of orthogonal projections that sum to unity when squared.

1. For Kinematic (Directional) Relations ($S^1$):
   • We have the Amplitude Component ($\beta$), which measures the external interaction or kinematic shift from the observer’s relational origin. This corresponds to phenomena like velocity.
   • We have the Phase Component ($\beta_Y$), which measures the internal ordering or intrinsic scale. This relates to proper time and proper length.
   • These are always simultaneously defined by the closure law $\beta^2 + \beta_Y^2 = 1$. An object always has both an external motion relative to an observer and an internal existence (its own proper time and length scale).
2. For Gravitational (Omnidirectional) Relations ($S^2$):
   • We have the Amplitude Component ($\kappa$), which measures the relational gravitational intensity between an object and an observer. This corresponds to gravitational potential.
   • We have the Phase Component ($\kappa_X$), which governs the intrinsic scale of proper length and proper time units in a gravitational field.
   • These are always simultaneously defined by the closure law $\kappa^2 + \kappa_X^2 = 1$. A system in a gravitational field always has both a potential intensity and an effect on its internal structure and time.

Simultaneous Projection: The term “simultaneous projection” in WILL RG means that for any given physical state, both the amplitude and phase aspects (e.g., $\beta$ and $\beta_Y$) are always present and defined. They are complementary aspects of the same single entity or relation. One cannot exist without the other, as they constitute the complete (closed) description of a system’s state.

For example, when describing a moving object, its kinematic energy ($E\beta$) and its rest energy ($E_0 = E\beta_Y$) are two sides of the same triangle on the $S^1$ carrier (doc. Part I - Kinetic Energy Projection on $S^1$, Figure 9). They are simultaneously projected from the total energy $E$.

You can explore these concepts further on the website:

• The fundamental idea of amplitude and phase duality is introduced in doc. Part I - The Amplitude-Phase Duality.
• Interactive visualizations of these projections can be found in the Desmos calculators linked on the Relativistic Foundations page: \url{https://willrg.com/relativistic-foundations/}. For instance, the DESMOS: Energy-momentum Triangle visually demonstrates the simultaneous nature of these projections on $S^1$.
• The geometric composition of these factors is detailed in doc. Part I - Geometric composition of SR and GR factors.

User: How can you locate any previous positions if you cannot simultaneously project z and y axis

Assistant: Your question delves into a core difference between WILL Relational Geometry (RG) and conventional physics. In WILL RG, we do not operate with a pre-existing background space defined by “z and y axes” onto which positions are projected in an absolute sense. Instead, “position” is fundamentally relational, defined by a system’s current state relative to an observer’s self-centered origin.

To address how “previous positions” are located, we must clarify the WILL RG perspective:

1. No Absolute Background Axes: As explained previously, WILL RG rejects the concept of fixed, external spatial axes (“x, y, z”) that define an absolute container. Such a container constitutes an “extraneous absolute” and violates the principle of Ontological Minimalism (doc. Part I - False Separation). Our projections ($\beta, \beta_Y, \kappa, \kappa_X$) refer to internal components of relational carriers ($S^1$ and $S^2$), not dimensions of an external Cartesian space.

2. Locality is Relational: A “position” is always defined relative to an observer. Each observer places itself at its own relational origin where $(\beta, \kappa) = (0,0)$. Any other system is then described by its projections $(\beta, \kappa)$ relative to that observer’s frame (doc. Part I - Energy as a Relation - What $\kappa$ and $\beta$ Actually Mean). There is no global address book of static coordinates.

3. Evolution via Phase, Not Absolute Time: In WILL RG, systems evolve not through an absolute time variable ($t$), but through phase ($o$) (doc. Part I - Protocol 2: The “Simulation” & “Time Evolution” Defense). Phase is an internal ordering parameter that tracks the system’s progress through its cycle of relational transformations.

4. Memory as Accumulated Phase Shift: A system’s “history” or “previous positions” are encoded as accumulated phase shifts and changes in its relational parameters. The system “remembers” its past states through the geometric consequences of its interactions and internal dynamics. This is why WILL RG moves from “descriptive” to “generative” physics: the laws themselves generate the observed evolution through geometric necessity.

   • For instance, in Relational Orbital Mechanics (ROM), the precession of an orbit (e.g., Mercury’s perihelion precession) is derived from the accumulation of the Total Relational Shift $Q$ over each orbital phase cycle (doc. Part I - Orbital Precession as State Difference Q Accumulation). The change in the orbit’s orientation over many cycles is a direct result of these accumulated phase differences.

5. Demonstrating Dynamic Evolution: The theory is dynamic, and you can observe this evolution without traditional time variables. The Relational Orbital Mechanics (ROM) page on our website directly addresses and demonstrates how systems evolve via phase:

   • URL: \url{https://willrg.com/ROM/}
   • This interactive simulator shows how dynamic stability and evolution occur using purely geometric phase drivers. You can see how a system changes its configuration in a sequence of states, not by moving through a fixed spatial grid, but by evolving its internal phase relationships.

In summary, “previous positions” are not located as points in external “z and y axes.” Instead, a system’s historical states are defined by its past phase configurations and the accumulation of relational shifts, which are rigorously tracked by the internal algebraic and geometric constraints of the system itself.