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---
## How a spatial structure manifests mass
ASD holds that one spatial structure carries rest energy, translational inertia, a gravitational source and response to external fields. The final manuscript defines the four masses separately and requires common matching in the relevant experimental range. Equality is a relation the same model must realize, rather than a merger of four definitions into one symbol.
ASD treats stable matter as a persistent organization of a core and surrounding fields. It stores energy at rest, changes the whole field state when translating, acts as a source of a longitudinal background, and responds to backgrounds produced by other sources. Mass connects four separately calculable relations rather than assigning an external parameter to the core.
Chapter 4 denotes orientation by $Q$, amplitude by $\chi$, compression-positive local density by $\eta$, and long-range gravitational stretch by dimensionless $\varphi$. Together they form a core–field system. Local density, stretch potential and the coefficient of translational inertia have distinct definitions.
This page maps Chapter 4’s long-range stretch $\varphi_g$ to Chapter 5’s $\varphi$, with $\Phi_N=-c^2\varphi$ and $\mathbf g=c^2\nabla\varphi$. Local density $\eta$ and the other Chapter 2 scale functions remain distinct.
## Complete rest energy and normalization
Select an isolated static configuration, or one with controlled internal periodic motion, and use the same distant background as reference. Define rest energy and energy mass by
$\mathcal H$ is the complete Hamiltonian energy density. The integral includes transverse shear, orientation gradients, stabilization, local volume state, the long-range scale field and coupling. $\mathcal H_{\rm vac}$ is the vacuum energy density of the same model. $E_{\rm rest}$ has energy units, $m_E$ mass units, and $c$ is the propagation-speed normalization of the corresponding background.
Exterior energy cannot be omitted or counted twice. If a microscopic field has already been eliminated into an effective potential or coupling, energy must be computed from that effective Hamiltonian. Adding the same pre-elimination energy again duplicates it. Velocity couplings also require the complete Legendre transform; a static expression is not automatically the energy for every moving state.
The definition $m_E=E_{\rm rest}/c^2$ expresses energy in mass units. Predicting an electron or proton mass requires fixed parameters and a finite-energy solution to calculate $E_{\rm rest}$. A measured mass used to set a model scale is calibration, not an independent prediction.
## Inertia from slow translation
Let $\mathbf X(t)$ be the center and begin with the rigid-translation approximation $\Phi_*(\mathbf x-\mathbf X(t))$. The chain rule gives $\partial_t\Phi_*=-\dot X_i\partial_i\Phi_*$. Inserting this into the kinetic part of the same action organizes spatial gradient integrals into an inertia tensor:
$M_{ij}$ has mass units and $\dot X_i$ is the center velocity. Only an isotropic low-speed limit permits $m_I\delta_{ij}$. For a static representative without velocity cross terms, Chapter 5, (5.6.27), gives
$$
M_{ij}=\int d^3x\left[Z_u\,\partial_i u_{*,a}\partial_j u_{*,a}
+I_\Theta\sum_A\partial_i\Theta_{*A}\partial_j\Theta_{*A}\right]+M_{ij}^{\rm add}.
$$
$Z_u$ is the displacement kinetic coefficient in mass-density units; $I_\Theta$ is the coefficient for dimensionless order parameters in energy-times-time-squared per volume. Index $a$ denotes displacement components, $A$ order-parameter components, and $M_{ij}^{\rm add}$ other retained fields and contributions. The inner product and normalization must agree with the master action, and all integrals must converge.
This explains inertia through the motion-induced field kinetic energy of the whole structure. An actual moving family may deform with speed; internal periodic motion and cross couplings can add contributions requiring self-consistent solution. Discrete pinning can remove an exact translation zero mode. The rigid low-speed formula cannot be extended to all-speed electron dynamics without those checks.
## Active sources and passive response
Active gravitational mass is identified from the scale potential produced far away. Passive mass is identified from the energy response to a weak external field. With the Chapter 2 sign convention, $\Phi_g^{\rm ext}=-c^2\varphi^{\rm ext}$ is a dimensional potential in energy-per-mass units:
Here $r$ is distance in the isolated-source far region and $G$ the weak-field coupling normalization. $m_G$ corresponds to the $1/r$ coefficient. $U_{\rm ext}$ is the leading interaction energy in a slowly varying weak background, and its slope defines $m_P$. The displayed force follows by taking a spatial gradient while treating $m_P$ as constant in the considered range.
Chapter 4 separates local source and long-range response using $s_\rho(r)=s_c f(r/L_\rho)$, with finite $f$ at the origin and convergent $\int_0^\infty f(x)x^2dx$. If $\nabla^2\varphi=-\gamma_\rho s_\rho$, the exterior is $\varphi\simeq\gamma_\rho\mathcal Q_\rho/(4\pi r)$, where $\mathcal Q_\rho=\int s_\rho d^3x$. A source can arise from density–shear coupling rather than being the direct deficit of spatial elements throughout space.
Equation (4.38) writes an exact exterior $1/r$ potential. This requires a spherical source that actually vanishes in the claimed exterior: compact source support inside $R_{\rm core}$, with zero potential at infinity. An integrable source tail, $\int |f(x)|x^2dx<\infty$, guarantees only the leading far-field $1/r$ asymptotic form, not exact equality at every finite “exterior” radius. The webpage states this source condition explicitly while preserving the manuscript equation number and recording the source discrepancy internally.
Chapter 5 instead describes an independent stretch field using source $\mathcal J_g$ and kinetic normalization $Z_\varphi$. Their units are energy density and energy-times-time-squared per volume, respectively. Under the specified static weak-field equation and decaying boundary, the candidate matching is
The symbols $M_E,M_g,E_0$ correspond here to $m_E,m_G,E_{\rm rest}$; this $E_0$ is complete rest energy. If the source integral equals rest energy and the passive energy slope uses the same coupling, a common-mass normalization can be matched. Choosing $Z_\varphi$ using measured $G$ is effective matching. Calculating these from independent microscopic parameters is required for an origin calculation of $G$.
## Long-range fields and finite mass
For $\varphi\sim1/r$, distant gradient energy converges as $\int_R^\infty r^{-2}dr$. Assigning the same $1/r$ profile directly to $\eta$ while retaining a constant-coefficient compression energy $\int\eta^2d^3x$ instead gives the divergent $\int_R^\infty dr$. These energy kernels are not interchangeable.
The local displacement–amplitude coupling of Chapter 5 can drive volume response without supplying the Chapter 2 stretch far field by itself. Retaining both independent stretch and displacement-divergence models requires constraints, coupling or scale separation to prevent duplicate description. This interface connects to [longitudinal fields](/en/theory/space-dynamics/longitudinal-field) and [source coupling](/en/open-problems/space-dynamics/source-coupling). An arbitrary exterior cutoff does not solve isolated particle mass.
## Matching masses within one system
The equality signs express the final manuscript’s common matching requirement (4.36), rather than a proof obtained from definitions alone. The manuscript aims for consistent rest energy, inertia, active source and passive response from the same fields, parameters and normalization in the relevant experimental range. Defining them to be the same number does not replace that calculation.
Upstream [dynamics](/en/theory/dynamics) uses $m_0$ for a joint core–field system in a uniform background with a fixed internal state and no net radiation, with a separate rest-energy normalization. [Macroscopic gravity](/en/theory/gravity#gravity-scale-field) uses a specified source mass $M$ and a test-body mass parameter normalized through universal coupling. This page asks how a microscopic configuration produces those effective parameters: calculate $m_E,m_I,m_G,m_P$ separately and check their matching in a common low-speed, weak-external-field range, then use a self-consistent moving family to connect to the upstream speed relations. A mass already supplied to a macroscopic calculation is not an independently calculated particle mass on this page.
## Material periods, local action and photon energy
Section 4.5 places the basic periods of open propagation and closed matter under a common action normalization. For a stable matter branch with fixed intrinsic mass $m_0$, let $E_0=m_0c^2$, $T_*$ be its phase-refresh period and $\lambda_*=cT_*$. Equation (4.86) gives
$$
h_{\rm eff}=E_0T_*=m_0c\lambda_*,\qquad
\nu_*=\frac{m_0c^2}{h_{\rm eff}}.
$$
The phase length $\lambda_*$ is not the core radius. A full period corresponds to $h$ and unit angular phase to $\hbar=h/(2\pi)$. This energy–period scale is not the arbitrary nonlinear-trajectory action $\int L\,dt$; a general closed orbit requires its own $J=\oint p\,dq$ calculation.
ASD holds that basic excitations in one local state share $h_{\rm eff}$, while different scale states obey $h'_{\rm eff}/h_{\rm eff}=\lambda'_*/\lambda_*=K$. This uses the fixed-$m_0,c$ stable branch and requires the effective shear coefficient, area, displacement amplitude and shape factor to satisfy the constitutive response (4.90). The fixed SI defining value $h_{\rm SI}$ is distinct from the model action scale $h_{\rm eff}$ calibrated in a reference state. Their physical comparison uses dimensionless ratios and differential readouts, rather than arbitrary changes to an SI number.
Different photons in one local state share $h_{\rm eff}$ but have different $E_\gamma=h_{\rm eff}\nu_\gamma$. Tracking the same free photon through a static background without real energy exchange instead uses $\lambda'=K\lambda$, $\nu'=\nu/K$, $h'=Kh$, preserving its intrinsic energy $E_\gamma^*=h\nu$. These are different comparisons. A nuclear transition and propagation mode can also respond differently: with common local normalization, (4.96) gives $\nu_\gamma(B)/\nu_A(B)=\Gamma_A/\Gamma_\gamma$. Fully common rescaling produces no detuning.
## How the nucleus and environment enter rotor readouts
For a stationary source and moving absorber, ASD assigns motion-induced detuning to the absorber’s differential lattice–electron–nuclear response. The incident resonant energy $E_A^{\rm res}(v,M)$ is expressed in one stationary-background calibration and includes recoil, material and constraints. In the second-order response model, $[E_A^{\rm res}(v,M)-E_0]/E_0=-k_Mv^2/c^2$. Averaging the common scale term and the squared direction-odd term over internal channels gives
$$
k_M=\frac{a_M}{2}+\frac{b_M}{3}.
$$
The factor $1/2$ comes from the low-speed scale expansion and $1/3$ from isotropic orientation averaging; $a_M,b_M$ are material differential responses. With $a_M=1$, the manuscript’s cited report $k_{\rm rep}=0.69\pm0.02$ corresponds to $b_M=0.57\pm0.06$. This is an effective constraint inferred from that report, with no material-model uncertainty included in the quoted propagation. It is neither a parameter-free prediction nor a loss of 19% of photon energy. Independently fixing the coupling and predicting material, orientation and apparatus differences would provide a new quantitative discriminator. This interface connects [matter waves](/en/theory/quantum/matter-wave) and the [test registry](/en/tests).
[OP-MAT-003](/en/open-problems/matter/mass-generation) tracks the joint mass calculation. The [mass-spectrum problem](/en/open-problems/matter/mass-spectrum) compares particles with shared parameters rather than calibrating each candidate to its own measured mass.