Material Memory Makes Scalar-Gravity Mass Centers Carry Their Internal Histories

Authors: Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Arminjon derives the post-Newtonian motion of mass centers in the second version of a scalar gravity theory with a preferred frame. The source treats the result as a celestial-mechanics equation: a scalar pressure field determines acceleration, the physical metric, and the post-Newtonian corrections for separated weakly gravitating bodies. Pudding Theory reads the same result as a statement about retained structure. The mass center is not a structureless bookkeeping point. It is a moving boundary condition carrying the memory of internal binding, rotation, and pressure history. Material Memory is therefore not an added anomaly. It is visible in the source equation itself, where the internal parameters $\epsilon_a$, $T_a$, and especially $\xi_a$ enter the motion after volume integration. The reading says that matter stores repeated internal signal as a bias in later trajectory. If the dimensionless coefficient multiplying $\xi_a \omega_a \times v_a$ were measured to be zero in a resolved weak-field scalar-gravity analogue, this Postulate would be falsified.

Source Synopsis

The source paper derives equations of motion for the mass centers of weakly gravitating bodies in version two of Arminjon's scalar theory of gravity with a preferred frame. The theory interprets gravity as a pressure force in a micro-ether. Its scalar field $\beta$, or equivalently $\psi=-\log\beta$, determines both the gravitational acceleration and the relation between a flat background metric and a curved physical metric. In coordinates adapted to the preferred inertial frame, the physical metric has line element \[ ds^2=\beta^2(dx^0)^2-\beta^{-2}dx^i dx^i. \] The scalar potential $V=c^2\psi$ obeys a wave equation whose Newtonian limit matches the Poisson equation for the Newtonian potential.

The derivation uses an asymptotic post-Newtonian scheme. All fields, including matter fields, are expanded in a small field-strength parameter proportional to $1/c^2$. The zeroth-order equation is the Newtonian Euler equation for a perfect fluid. The first post-Newtonian equation adds corrections involving the active mass density, the scalar potential correction, pressure, kinetic energy, and time-dependent gravitational terms.

The paper then introduces a second asymptotic parameter for good separation of bodies. Each body has a Newtonian mass center, radius, translational velocity, and rigid rotational velocity. Integrating the local field equations over each body gives equations for the mass-center acceleration. The Newtonian term is the usual mutual attraction among bodies, but the post-Newtonian correction contains internal structure parameters. The parameters $\epsilon_a$ and $T_a$ enter through a mass redefinition. The parameter $\xi_a$, built from the radial density profile and self-potential gradient, remains as a spin-dependent self-acceleration term: \[ 3c^{-2}M_a^{-1}\xi_a\,\omega_a \times \dot a. \] The conclusion stresses that internal structure affects motion in this asymptotic scheme, and that the resulting equations differ from the Lorentz-Droste-Einstein-Infeld-Hoffmann equations.

Postulate Lens

The applicable Postulate is Material Memory. The source paper already exhibits the structure named by the Postulate: the future motion of a body depends on integrals over its internal past-formed material organization, not only on its instantaneous mass center and external field. In Arminjon's calculation, volume integration does not erase internal structure. It compresses structure into persistent parameters. The scalar theory therefore gives a clean mathematical site where matter remembers the organization of its own interior.

This is not a psychological claim. It is a reading of extended-body dynamics. The source begins with a pressure interpretation of gravity and ends with mass-center equations that retain internal rotational and self-binding data. Pudding Theory identifies that retention as physical memory. The repeated signal is the body's sustained internal circulation and pressure-supported cohesion. The trace is the structure parameter that survives coarse-graining. The bias is the extra acceleration term that remains after the mass renormalization has hidden the other internal parameters.

Pudding Theory Reading

Pudding Theory reads Arminjon's scalar-gravity mass center as a memory-bearing object. The ordinary point-particle ideal suppresses this fact. It replaces the body with a location, a mass, and a velocity. Arminjon's asymptotic construction shows why that suppression fails inside this scalar theory. The body is first a continuous medium. Its density, pressure, self-potential, and rotation are integrated only after the local post-Newtonian equations have been expanded. The mass center is therefore not primitive. It is the residue of a field history.

Material Memory says that matter retains the trace of repeated signals and that the trace biases future probability. In the source system the repeated signal is not an external intention or a laboratory command. It is the body's own recurrent internal motion: rigid rotation, pressure balance, and self-gravitating binding. Those signals are repeated over many internal periods. They become encoded in $\epsilon_a$, $T_a$, and $\xi_a$. The first two can be absorbed into a redefinition of Newtonian mass. That fact is important. Some memory becomes inertial bookkeeping. It changes what mass means operationally. The remaining $\xi_a$ term cannot be hidden in that way. It produces a direction-sensitive contribution proportional to $\omega_a \times \dot a$.

The source frames this as an internal-structure effect in an alternative gravitational theory. Pudding Theory sharpens it. The scalar pressure field does not merely push bodies according to external gradients. It reads the stored organization of the body. The mass center responds as a weighted boundary of the material history. A rotating extended body and a non-rotating body with the same Newtonian mass are not dynamically equivalent. Nor are two rotating bodies with the same spin but different radial density histories. The integral \[ \xi_a=-\frac{4\pi}{3}\int_0^{r_a}\rho_a(r)\frac{du_a}{dr}r^3dr \] is the memory register. It stores how self-gravity is distributed through the material volume.

The source's preferred-frame scalar theory thus becomes, under Pudding Theory, a theory in which matter keeps a record of its internal coherence and later spends that record as acceleration. The post-Newtonian correction is not a nuisance term. It is the visible conversion of stored internal pattern into center-of-mass motion. What standard point mechanics treats as background composition, this reading treats as signal-bearing structure.

Falsifiable Observable

The discriminating observable is the coefficient $C_\xi$ of the spin-memory term in a resolved weak-field analogue or numerical celestial-mechanics fit: \[ a_{\xi}=C_\xi\,c^{-2}M_a^{-1}\xi_a\,\omega_a \times v_a. \] Arminjon's scalar equation gives $C_\xi=3$. The Pudding Theory reading requires that the retained material trace appears as a nonzero, orientation-sensitive acceleration with this sign structure after mass renormalization has removed the $\epsilon_a$ and $T_a$ terms. If the dimensionless coefficient multiplying $\xi_a \omega_a \times v_a$ were measured to be zero in a resolved weak-field scalar-gravity analogue, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks renaming a standard finite-size effect. Arminjon integrates a perfect-fluid equation over an extended body. Of course internal integrals appear. That does not show that matter remembers anything. It shows that the point-particle approximation was not taken first.

Sterling: That objection would hold if the internal integrals merely corrected a multipole expansion and then vanished under a cleaner definition of the body. They do not all vanish. The paper explicitly separates what can be hidden by redefining mass from what remains as a spin-dependent self-acceleration. The memory claim concerns that remainder. It is not a word placed over any finite-size correction. It is the claim that repeated internal organization leaves a persistent dynamical register.

Tanaka: But the register is computed from density and self-potential at one time. That is not history.

Sterling: A stable radial density profile and a rigid rotation field are histories made stationary. The integral is instantaneous only because the repeated internal signal has reached a persistent form. Celestial mechanics often hides this by choosing point masses. Arminjon's asymptotic scheme does not. It lets the stored organization pass through the integration.

Tanaka: Then the burden is the coefficient.

Sterling: Yes. The coefficient and orientation are the hard constraint. If the $\xi_a$ term disappears under direct resolved tests, the reading fails.

Discussion

This reading buys a more concrete interpretation of Arminjon's internal-structure result. The source already states that the motion depends on $\epsilon_a$, $T_a$, and $\xi_a$, with only $\xi_a$ operationally surviving as a separate term. Pudding Theory explains why that survival matters. It says the body is not exhausted by mass and location. It carries a stored material pattern that enters motion after coarse-graining.

The limitation is clear. The reading depends on the scalar theory's asymptotic framework. It does not claim that the same coefficient appears in general relativity, nor that every finite-size correction is Material Memory. The claim is narrower and stronger: in this source system, the surviving spin-self term is the observable sign of stored internal organization. A direct comparison with ephemeris fits would be difficult because ordinary data reduction already assumes standard equations of motion. A cleaner test would use numerical bodies with controlled density profiles and rotation, or an analogue model whose pressure-scalar dynamics can be resolved before the mass-center reduction.

The conclusion would change if $\xi_a$ could be removed by a legitimate variable redefinition, or if resolved simulations found no orientation-sensitive acceleration after controlling for external multipoles. Until then, Arminjon's equation is best read as a mass-center dynamics with memory.

References

1. Mayeul Arminjon, “Equations of motion of the mass centers in a scalar theory of gravity with a preferred frame,” arXiv:2604.15397, DOI: doi:10.48550/arxiv.2604.15397, 2026.

2. Sterling Geisel, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab Working Papers, September 10, 2025.

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