5 Space Expansion and Stage Dressing
theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology
5.1 Space Expansion and the Conservation Problem
Cosmic expansion confronts M1 with an immediate tension. Momentum conservation is the framework’s founding principle. Yet the expansion of space appears to change the momentum expressed by freely travelling light. How can momentum remain conserved when the universe itself seems to produce a non-conservative result?
The question becomes especially sharp across cosmological distances. A photon may travel for billions of years before reaching us. When it arrives, its wavelength is longer, and the event that produced it appears to unfold more slowly. Something differs between emission and observation. But what has changed?
M1 separates two physical roles. Momentum is the actor of change; space is the stage on which that change becomes motion and distance. If a distant photon arrives stretched, there are therefore two places to look. The actor may have changed, or the stage may have changed how the actor’s momentum is expressed.
Suppose first that the actor changes. A photon might gradually surrender momentum to space itself. Nothing in logic forbids an exchange, but the word loss hides the hard part. Momentum is directed. It cannot simply fade away: something must receive its direction, transmit it onward, or balance it with an equal and opposite change. A drain into space would therefore require a physical receiver and a closed conservation ledger. Without them, the proposed loss only moves the problem elsewhere.
The expansion engine developed here takes the other route. In a homogeneous stage, spatial translation symmetry protects the carrier’s spatial momentum content. The directed momentum does not disappear, and the intrinsic scale \(p_f=m_0c\) remains fixed. What changes is how the stage turns those quantities into translation.
A simplified physical picture begins with an elongation of the particle’s internal bosic cycle. The carrier retains the same bosic momentum content, but that momentum is expressed through a longer cycle, shifting its frequency.
Picture a violin string resonating with a note. Now imagine the violinist slowly sliding her finger toward the scroll, lengthening the part of the string that is free to resonate. The pitch falls. In the same way, elongating the internal bosic cycle shifts its frequency without taking momentum from the carrier.
The directional asymmetry still belongs to the cycle as a whole. But as the internal cycle elongates, the same asymmetry completes fewer cycles in a given interval. Its translation yield in real space is therefore reduced: the carrier has not lost directed momentum, but the altered cycle produces less translation from it.
The quantum chapter prepared the mathematical form of this distinction. Its first-order core-momentum operator contains a fermic slot and a directed bosic slot. Expansion leaves the fermic slot fixed while dressing the bosic one. The carrier can therefore retain the same spatial momentum label even as the generator frequency and the motion produced from that label change with cosmic epoch.
The analogy gives the physical picture; the operator supplies the law. The second power of the stage factor in the massive translation equation is neither inferred from the string nor added by hand. It follows when the stage-dressed generator is differentiated with respect to directed momentum.
Light provides the cleanest first signal of the construction, but not its only test. One engine must preserve the lightlike propagation law, recover the correct history of freely moving massive bodies, connect wavelength redshift with duration stretching, and remain compatible with stable local rods, clocks, atoms, and bound orbits. It must also say clearly what it has not yet supplied.
This chapter builds that engine. It establishes homogeneous stage-dressed kinematics and several scoped local realizations. It does not derive the history of the stage, a cosmological field equation, or a complete observational model. Those are later burdens. The task here is narrower: to show how a changing stage can alter the expression of conserved momentum across epochs without turning momentum conservation into a metaphor.
5.2 Core Terms and Variables
The expansion engine adds one stage variable and two coordinate descriptions to momentum and operator notation already established. This section fixes their meanings and governing relations in one place. The mechanism and derivation begin in the next section.
5.2.1 Inherited momentum and operator terms
Fermic momentum \(p_f\) is the fixed intrinsic scale,
\[ p_f=m_0c. \]
The directed bosic momentum is \(\mathbf p\), with magnitude \(p=|\mathbf p|\). For an oriented unit direction \(\hat{k}\), its signed component is
\[ p_k=\mathbf p\mathbin{\cdot}\hat{k}. \]
The inertial core-momentum magnitude is
\[ M=\sqrt{p_f^2+p^2}. \]
The canonical momentum operator is \(\hat{\mathbf p}=-i\hbar\nabla\). The flat core-momentum operator is
\[ \hat M = \beta p_f+ \boldsymbol\alpha\!\cdot\!\hat{\mathbf p}. \]
The constant \(c\) retains its inherited role as the intrinsic local propagation scale.
5.2.2 Stage state and dressed generator
The dimensionless homogeneous stage state is \(\chi(t)>0\), with endpoint values
\[ \chi_e=\chi(t_e), \qquad \chi_o=\chi(t_o). \]
The observation epoch is the default reference, so \(\chi_o=1\) unless another normalization is stated. When the normalization remains explicit, the normalized stage ratio is
\[ X(t)=\frac{\chi(t)}{\chi_o}, \qquad X_o=1. \]
Compact operator formulas use \(\chi\) in the normalized convention; otherwise \(X\) takes its place.
The stage-dressed core-momentum operator and its positive instantaneous magnitude are
\[ \boxed{ \hat M_\chi = \beta p_f+ \chi(t)\,\boldsymbol\alpha\!\cdot\!\hat{\mathbf p} }, \qquad \boxed{ M_\chi= \sqrt{p_f^2+\chi^2p^2} }. \]
Here \(M\) is the inertial magnitude, while \(M_\chi\) is the instantaneous eigenvalue or principal-symbol magnitude of the dressed generator. The spatial momentum content \(\mathbf p\) remains a separate quantity.
Translation yield names the stage-conditioned relation between directed momentum and generated motion. Its governing law is
\[ \boxed{ \dot x_k = \chi^2c\,\frac{p_k}{M_\chi} }. \]
Derivation 5.2A supplies the operator algebra and group-velocity theorem.
5.2.3 True-frame and bound-frame variables
The common homogeneous history parameter is \(t\). The true-frame coordinate \(x_T\) records propagation on the homogeneous chart. The bound-frame coordinate \(x_B\) reports lengths using co-scaling local rods:
\[ x_B=\frac{x_T}{X}. \]
The associated clean-branch scale factor and fractional history rate are
\[ \boxed{ a_B=\frac1X=\frac{\chi_o}{\chi} }, \qquad \boxed{ H=\frac{\dot a_B}{a_B}=-\frac{\dot X}{X} }. \]
The clean branch uses matched local clocks, so \(t\) also serves as the local time variable in the comparisons below. The function \(H(t)\) is supplied rather than derived in this chapter.
5.2.4 Propagation and signal labels
For lightlike propagation, the history kernel is
\[ \mathcal D_\chi(t_e,t_o) = c\int_{t_e}^{t_o}\chi(t)\,dt. \]
It labels a path interval, not a complete observational distance.
For neighboring signal features, \(\delta t_e\) and \(\delta t_o\) denote endpoint separations in the common parameter \(t\). The symbols \(\Delta t_e\) and \(\Delta t_o\) denote durations reported by the emitter’s and observer’s local clocks.
Observed wavelength redshift is
\[ 1+z=\frac{\lambda_o}{\lambda_e}. \]
On the matched-standard branch developed below, the conditional endpoint relation is
\[ 1+z=\frac{\chi_e}{\chi_o}. \]
Momentum labels conserved spatial content, \(M_\chi\) labels the dressed generator value, \(X\) compares stage states, and bound-frame variables describe measurements made with local standards.
5.3 Stage Dressing and Massive Motion
Expansion acts on the part of the core-momentum generator that carries translation. The fermic scale \(p_f=m_0c\) continues to sustain the carrier’s intrinsic cycle, while the directed bosic term is expressed through the changing stage:
\[ \hat M_\chi = \beta p_f+ \chi(t)\,\boldsymbol\alpha\!\cdot\!\hat{\mathbf p}. \]
Because \(\chi(t)\) is homogeneous, it changes no spatial direction and introduces no spatial gradient. The carrier therefore keeps its momentum label \(\mathbf p\). What changes is the magnitude generated from that content. Squaring the operator gives
\[ M_\chi = \sqrt{p_f^2+\chi^2p^2}. \]
The bosic contribution now enters through \(\chi p\). As the internal bosic cycle elongates, the same directed content turns through that cycle at a different rate, and the generator magnitude changes with it.
5.3.1 From the dressed generator to motion
For a narrow positive-branch packet, motion along the selected direction \(k\) is determined by how the energy-unit generator \(cM_\chi\) changes with the signed momentum component \(p_k\):
\[ \begin{aligned} \dot x_k &= \frac{\partial(cM_\chi)}{\partial p_k} \\ &= c\frac{\partial}{\partial p_k} \sqrt{p_f^2+\chi^2p^2} \\ &= \chi^2c\,\frac{p_k}{M_\chi}. \end{aligned} \]
Thus
\[ \boxed{ \dot x_k = \chi^2c\,\frac{p_k}{M_\chi} }. \]
The two powers of \(\chi\) belong to one operation. Stage dressing places \(\chi\) on the bosic term, so that term contributes \(\chi^2p^2\) to the core-momentum magnitude. Differentiating that magnitude with respect to directed momentum brings the full factor into translation. The dressed generator itself therefore yields the \(\chi^2\) law.
The law immediately retains the expected directional behavior. At \(\chi=1\), it returns to inertial motion, \(\dot x_k=cp_k/M\). If \(p_k=0\), the carrier has no motion along \(\hat{k}\). Reversing the chosen orientation reverses both \(p_k\) and \(\dot x_k\), while \(M_\chi\) remains unchanged.
For light, \(p_f=0\) and \(M_\chi=\chi|p|\). One power then cancels from the ratio, leaving forward propagation at
\[ \dot x_T=\chi c. \]
Massive and lightlike motion are therefore two limits of the same stage-dressed generator.
5.3.2 Why the bosic slot carries the dressing
The two operator slots have different physical jobs. The fermic term carries the intrinsic scale tied to the carrier’s identity. The bosic term carries the directional asymmetry through which that carrier translates. Expansion changes the resonating length through which the bosic cycle is expressed, so its factor belongs on the bosic term.
This placement also joins the chapter’s two kinematic requirements. Keeping \(p_f\) fixed preserves the intrinsic fermic scale. Dressing the bosic term by one power of \(\chi\) gives the lightlike true-frame speed \(\chi c\). The first-order two-slot construction is therefore
\[ \hat M_\chi = \beta p_f+ \chi\,\boldsymbol\alpha\!\cdot\!\hat{\mathbf p}. \]
The massive law then follows from the same operator rather than from a separate rule for material particles.
5.3.3 Free massive motion in bound-frame units
The observation epoch now sets the normalization \(\chi_o=1\), so \(X=\chi\) and
\[ a_B=\frac1\chi. \]
A local observer uses rods that co-scale with \(X\). After the common recession of the bound-frame coordinate grid is removed, the carrier’s peculiar velocity is the local-motion term
\[ v_B=a_B\dot x_T. \]
Substituting the stage-dressed translation law gives
\[ \begin{aligned} v_B &= a_B\chi^2c\, \frac{p}{\sqrt{p_f^2+\chi^2p^2}} \\ &= c\frac{p/a_B} {\sqrt{p_f^2+(p/a_B)^2}}. \end{aligned} \]
Hence
\[ \boxed{ v_B = c\frac{p/a_B} {\sqrt{p_f^2+(p/a_B)^2}} }. \]
The conserved content \(p\) has not decayed. Local rods express it through the epoch-dependent quantity \(p/a_B\), so the measured velocity changes while the underlying spatial momentum label remains fixed.
This is exactly the special-relativistic free-particle velocity in a spatially flat FLRW description when
\[ \boxed{p\equiv q}, \]
where \(q\) is conserved comoving momentum. The correspondence identifies two descriptions of the same homogeneous free-carrier history: M1 keeps the spatial momentum content \(p\), while the bound frame reports its changing expression \(p/a_B\).
Differentiating the velocity gives
\[ \boxed{ \dot v_B = -Hv_B \left(1-\frac{v_B^2}{c^2}\right) }. \]
Slow massive motion therefore falls as \(a_B^{-1}\), while light remains at the local speed \(c\).
5.3.4 A massive-motion placement diagnostic
Massive motion also tests where the stage factor belongs. If one factor of \(\chi\) dresses the complete inertial magnitude instead of the bosic slot, the true-frame law is \(\dot x_T=\chi cp/M\). Bound-frame conversion then cancels the stage dependence:
\[ \boxed{ v_B^{\mathrm{whole}} = a_B\chi c\frac{p}{M} = c\frac{p}{M}, \qquad \dot v_B^{\mathrm{whole}}=0 }. \]
Both placements retain the lightlike speed, but only bosic-slot dressing produces the massive FLRW drag law. Together with the fixed fermic scale, this selects the bosic-slot form within the first-order two-slot multiplicative architecture.
The expansion engine therefore reaches the exact free-carrier history for every speed once the stage history is supplied. Derivation 5.2A carries the operator algebra and scoped placement result, and Derivation 5.3A gives the full bound-frame conversion and drag relation.
5.4 Light Propagation and Endpoint Comparison
Light follows the same dressed generator as a massive carrier. Setting \(p_f=0\) gives
\[ M_\chi=\chi|p|. \]
Substituting this magnitude into the translation law leaves
\[ \boxed{ \dot x_T=\chi(t)c }. \]
One power of \(\chi\) remains after the dressed magnitude cancels the other. The bosic-slot construction therefore carries massive and lightlike motion in one law: fermic content changes the massive response, while a purely bosic carrier translates at the stage-conditioned rate \(\chi c\).
5.4.1 A fixed spatial pattern with a changing phase rate
Write the lightlike spatial momentum as
\[ \mathbf p=\hbar\mathbf k. \]
The stage state depends on time but not position. It therefore changes no spatial phase gradient, and the external wavevector remains fixed during flight:
\[ \dot{\mathbf k}=0. \]
The generator frequency changes with the stage. For a lightlike carrier,
\[ \omega_\chi(t) = \frac{cM_\chi}{\hbar} = \chi(t)c|\mathbf k|. \]
The signal consequently carries two distinct features. Its external spatial pattern is set by the conserved wavevector, with true-frame coordinate wavelength
\[ \lambda_T=\frac{2\pi}{|\mathbf k|}, \]
while its phase turns at the changing rate \(\omega_\chi(t)\). The pattern remains fixed on the homogeneous true-frame chart as the bosic cycle changes its temporal expression. A wavelength redshift appears when that transported pattern is read against local rods at the two endpoints.
5.4.2 The path accumulated through the stage
A signal emitted at \(t_e\) and observed at \(t_o\) samples every stage state between them. Integrating its translation rate gives the true-frame path interval
\[ \boxed{ \mathcal D_\chi(t_e,t_o) = c\int_{t_e}^{t_o}\chi(t)\,dt }. \]
The integral measures the translation accumulated by light through the supplied history. Different histories with the same endpoint ratio can therefore produce different path intervals even though they produce the same matched-standard redshift.
The clean bound frame uses \(a_B=1/\chi\) in the normalized convention. Its conformal time satisfies
\[ \boxed{ d\eta=\frac{dt}{a_B}=\chi(t)\,dt }, \qquad \mathcal D_\chi=c\int d\eta. \]
If the observation normalization remains explicit, the exact relation is \(d\eta=Xdt\). The stage-dressed path kernel and the usual conformal propagation kernel are therefore the same path written in the two coordinate descriptions. Their conformal causal reach is identical: once the history is supplied, stage-dressed kinematics does not enlarge the light cone or solve the horizon problem.
Observable distance and brightness relations require the later transport layer built on this interval.
5.4.3 Neighboring features on the same path
Now consider two neighboring features emitted by the same source and received by the same observer. Over their short separation, take the source and observer to retain the same true-frame coordinate separation. The two features then traverse equal paths:
\[ c\int_{t_e}^{t_o}\chi(t)\,dt = c\int_{t_e+\delta t_e}^{t_o+\delta t_o}\chi(t)\,dt. \]
Subtracting the common part and keeping the first-order endpoint contributions gives
\[ \boxed{ \chi_o\delta t_o = \chi_e\delta t_e }. \]
The later feature must wait a different interval at observation because translation proceeds under a different stage state there. The propagation law therefore carries the endpoint ratio directly into the separation between neighboring features:
\[ \frac{\delta t_o}{\delta t_e} = \frac{\chi_e}{\chi_o}. \]
These intervals belong to the common history parameter. Local rods and clocks now turn them into reported wavelengths and durations.
5.4.4 Wavelength at the endpoints
On the clean branch, a local true-frame rod scales with \(X(t)=\chi(t)/\chi_o\). The transported wavelength \(\lambda_T\) is fixed, while its numerical reading in local rod units scales as \(1/X\). Comparing the same spatial pattern at emission and observation gives
\[ \boxed{ 1+z \equiv \frac{\lambda_o}{\lambda_e} = \frac{\chi_e}{\chi_o} }. \]
The signal arrives redshifted because the observer reads its conserved external pattern with a different local rod. Momentum has not drained from the carrier along the path; the endpoint standards express the same transported pattern differently.
5.4.5 Duration at the endpoints
Matched local clocks retain the same rate against the common history parameter. Their reported durations therefore inherit the neighboring-feature relation:
\[ \frac{\Delta t_o}{\Delta t_e} = \frac{\delta t_o}{\delta t_e} = \frac{\chi_e}{\chi_o}. \]
Combining this with the wavelength ratio gives
\[ \boxed{ \Delta t_o=(1+z)\Delta t_e }. \]
Wavelength redshift and duration stretching are thus two readings of one endpoint comparison. The stage history determines the path, equal paths determine the neighboring-feature separation, and co-scaling rods with matched clocks determine what emitter and observer report. Derivation 5.4A gives the full endpoint expansion and regime assumptions.
5.5 Bound Frames and Local Standards
A universal stage state cannot be interpreted by applying the free-propagation law indiscriminately to every atom, laboratory, planet, or galaxy. A bound system has an equilibrium set by its kinetic and interaction terms. The local question is therefore not whether cosmological motion is simply switched off, but whether the whole bound configuration and the standards used to measure it transform coherently.
5.5.1 The clean bound-frame construction
Normalize the stage state by
\[ X=\frac{\chi}{\chi_o}. \]
The clean construction takes a local true-frame rod to scale linearly with \(X\),
\[ \ell_T(X)=\ell_oX, \]
while local tick frequencies remain fixed relative to the common history parameter. A true-frame interval is then reported in local rod units through
\[ \boxed{ x_B=\frac{x_T}{X} }, \qquad \boxed{ a_B=\frac1X }. \]
A bound separation that co-scales as \(R_T=R_oX\) is locally constant:
\[ R_B=\frac{R_T}{X}=R_o. \]
A fixed true-frame separation between freely separated systems behaves differently. In bound-frame units it is multiplied by \(a_B\), and therefore appears to expand as \(X\) falls. The same transform can thus describe local stability and cosmological separation without declaring bound systems exempt from the stage.
5.5.2 Local light and radar measurements
True-frame lightlike motion is
\[ \dot x_T=cX. \]
During one fixed local tick, the true-frame light path scales as \(X\). The rod used to measure it scales by the same factor. Their ratio is therefore constant, so a local observer continues to report the intrinsic speed \(c\).
Radar ranging gives the same result over a sufficiently short round trip. If \(H\Delta t_T\ll1\), the stage ratio is effectively constant during the measurement. For a local radius that co-scales as \(r_T=r_oX\), the round-trip light time is, to leading adiabatic order,
\[ \Delta t_T = \frac{2r_T}{cX} = \frac{2r_o}{c}. \]
A fixed local clock counts the same number of ticks. Co-scaling rods, light paths, and bound separations can therefore remain stable in local measurement even while their true-frame lengths vary with the stage.
5.5.3 Adiabatic scope
For a bound system with characteristic internal time \(\tau_{\mathrm{dyn}}\), the natural regime is
\[ H\tau_{\mathrm{dyn}}\ll1. \]
The stage then changes little during one internal cycle, allowing the system to follow a sequence of near-equilibrium states. This condition does not force every interaction to co-scale correctly. It identifies the regime in which the proposed local transform can be tested without violent nonequilibrium driving.
The clean transform is therefore a target rather than a universal proof. Electromagnetic binding must produce the required rod and spectral behavior. Gravity must preserve bound orbital scales and timing. Different clock families must not acquire forbidden relative drifts, and the remaining interaction sectors must be compatible with the same standards.
The next section tests that target in a hydrogenic electromagnetic proxy and a Newtonian bound orbit, then checks linear matter growth at a supplied background history. These controlled cases make the construction concrete without closing every interaction sector.
5.6 Scoped Realizations and Correspondence Checks
The clean bound frame sets a definite physical target. Local rods must scale as \(X\), local clock frequencies must remain fixed, and a bound system must keep the same size and timing when measured by those standards. The low-speed bosic generator supplies two powers of \(X\) in the kinetic response. The question is then which interaction scaling lets an electromagnetic atom or a Newtonian orbit meet the target.
In both proxies, the answer is one power of \(X\). That exponent is not inserted merely because it works downstream; it is selected independently by the required bound scale in each sector.
5.6.1 A hydrogenic electromagnetic proxy
After the rest phase is removed, write the normalized dressed magnitude as \(M_X=\sqrt{p_f^2+X^2p^2}\), with \(p_f=m_ec\) for the electron. Its low-speed expansion is
\[ c(M_X-p_f) \simeq \frac{X^2\mathbf p^2}{2m_e}. \]
Let the Coulomb strength scale provisionally as
\[ e^2(X)=e_o^2X^n, \]
with \(\hbar\) and \(m_e\) fixed. For a hydrogenic state, \(p\sim\hbar/r\), so balancing the \(X^2\) kinetic contribution against the Coulomb attraction gives
\[ a_0(X) \propto \frac{X^2\hbar^2}{m_e e^2(X)} \propto X^{2-n}. \]
The clean-branch rod target is \(a_0(X)=Xa_{0,o}\). It therefore selects
\[ \boxed{ n=1, \qquad e^2(X)=e_o^2X }. \]
The same exponent fixes the spectral scale. Hydrogenic transition energies vary as
\[ \Delta E(X) \propto \frac{e^4(X)}{X^2} \propto X^{2n-2}, \]
so \(n=1\) gives
\[ \boxed{ a_0(X)=Xa_{0,o}, \qquad \Delta E(X)=\Delta E_o }. \]
Atomic lengths co-scale with local rods while electronic transition frequencies remain fixed. The dimensionless coupling read against the local light scale is likewise constant:
\[ \alpha_X = \frac{e^2(X)}{\hbar cX} = \frac{e_o^2}{\hbar c}. \]
The electromagnetic proxy therefore supplies the rod and clock behavior used in the endpoint comparison. Its first-power law is selected inside the hydrogenic adiabatic model; fuller electromagnetic and material standards require their own realization.
5.6.2 A Newtonian bound orbit
The gravitational proxy begins from the same \(X^2\) low-speed kinetic response. Write the effective Newtonian coupling provisionally as
\[ G_{\mathrm{eff}}(X)=G_NX^g. \]
For an adiabatic circular orbit, the angular action \(J=pr\) is conserved. Balancing the kinetic and attractive terms gives the orbital scale
\[ r_{\mathrm{orbit}}(X) \propto \frac{X^2J^2} {m^2G_{\mathrm{eff}}(X)M_s} \propto X^{2-g}. \]
The clean branch requires the orbit to co-scale with local rods, \(r_{\mathrm{orbit}}\propto X\). This independently selects
\[ \boxed{ g=1, \qquad G_{\mathrm{eff}}(X)=G_NX }. \]
The orbital radius and true-frame velocity then obey
\[ \boxed{ r_{\mathrm{orbit}}(X)=Xr_{\mathrm{orbit},o} }, \qquad v_T\propto X. \]
Their common factor cancels from the period:
\[ T_T\sim\frac{r_{\mathrm{orbit}}}{v_T}\propto X^0. \]
The Newtonian orbit therefore retains its size against local rods and its timing against the matched clock. As in the atomic case, the clean target selects the needed first power within the proxy rather than deriving a complete microscopic sector law.
5.6.3 Linear matter growth at a fixed history
The same powers reappear in the linear growth equation. In the pressureless, linear, quasi-static regime, take \(H(t)\) as supplied and use the selected Newtonian scaling \(G_{\mathrm{eff}}=G_NX\). The source acquires two powers from the kinetic response and one from the coupling:
\[ \boxed{ X^3 = \underbrace{X^2}_{\text{kinetic response}} \underbrace{X}_{\text{Newtonian coupling}} }. \]
That product is exactly the homogeneous volume conversion between true-frame and bound-frame matter density,
\[ \bar\rho_B=X^3\bar\rho_T. \]
The density contrast therefore obeys
\[ \boxed{ \ddot\delta+2H\dot\delta = 4\pi G_N\bar\rho_B\delta }. \]
This is the standard linear matter-growth equation at the same supplied \(H(t)\) and \(\bar\rho_B(t)\). The matching friction and source terms are consequences of the same kinetic and coupling powers that stabilize the local proxies.
5.6.4 What these checks establish
The clean-branch requirement selects first-power scaling independently in the hydrogenic and Newtonian models. With those powers, atomic lengths, electronic clock frequencies, orbital radii, orbital periods, and linear matter growth fit one common local-standard construction. Derivation 5.6A provides the detailed balances, growth derivation, and regime ledger.
The result is a coherent electromagnetic/Newtonian realization, not yet an all-sector law. Its value is sharper than an assumed fit: two different bound systems select the same coupling exponent, and that exponent then completes the \(X^3\) structure required by linear growth.
5.7 What the Expansion Engine Establishes
The expansion problem began with a conservation seam. Distant signals arrive stretched, yet directed momentum cannot simply vanish into an unnamed reservoir. Bosic-slot stage dressing resolves the homogeneous free-carrier kinematic seam by separating conserved spatial content from the generator that turns that content into phase evolution and motion. It does not yet close the full conservation ledger of stage and carrier together.
5.7.1 The established engine
In a homogeneous background, spatial translation symmetry preserves the carrier label \(\mathbf p\). Expansion leaves the intrinsic fermic scale \(p_f=m_0c\) fixed and dresses the directed bosic slot of the core-momentum generator. For the resulting magnitude \(M_\chi=\sqrt{p_f^2+\chi^2p^2}\), group velocity gives the chapter’s anchor law:
\[ \boxed{ \dot x_k = \chi^2c\frac{p_k}{M_\chi} }. \]
Given the bosic-slot ansatz, no second multiplier is attached downstream: the second power follows when the dressed generator is differentiated with respect to conserved momentum. The law recovers inertial motion at \(\chi=1\), respects directional reversal and zero projected momentum, and reduces to true-frame lightlike motion at speed \(\chi c\).
On the clean bound-frame branch, the same law reproduces the exact special-relativistic FLRW velocity and drag relations when M1’s conserved spatial content is identified with comoving momentum. Massive motion also distinguishes the placement: whole-generator dressing would leave peculiar velocity constant, while bosic-slot dressing produces the required drag.
Light follows the same engine. Its external wavevector remains fixed while its generator frequency changes with \(\chi\). The signal accumulates the standard conformal propagation interval, and neighboring features inherit the corresponding endpoint timing ratio. The causal reach is unchanged for a supplied history.
5.7.2 Conditional local realization
Turning propagation into measurements requires a local-standard dictionary. On the clean branch, true-frame rods co-scale with \(X=\chi/\chi_o\), bound-frame lengths are reported through \(x_B=x_T/X\), and matched clocks retain a fixed rate against the common history parameter. These premises give the endpoint relations \(1+z=\chi_e/\chi_o\) and \(\Delta t_o=(1+z)\Delta t_e\).
The same local target selects the interaction powers in two controlled bound-system checks. The hydrogenic proxy requires \(e^2(X)=e_o^2X\) to produce co-scaling atomic lengths and fixed electronic transition frequencies. The Newtonian orbit independently requires \(G_{\mathrm{eff}}(X)=G_NX\) to produce co-scaling orbital radii and fixed local timing. With that Newtonian power, the \(X^2\) kinetic response and the \(X\) coupling combine into the \(X^3\) density conversion that yields standard linear pressureless growth.
These are coherent realizations in their stated regimes. Full electromagnetic, material, relativistic-gravity, strong, and weak closure remains to be built.
5.7.3 Open dynamics and handoff
Once \(H(t)\) is supplied, the homogeneous scale history \(a_B(t)\), free-carrier motion, light paths, endpoint ratios, and—under the scoped Newtonian realization—linear matter growth are fixed. The load-bearing cosmological problem is therefore the law that determines \(H(t)\) itself.
This chapter derives no stage action or evolution equation for \(\chi(t)\), and it does not settle the gravitational source weight of the dressed generator. Conserved spatial content, instantaneous generator value, gravitational source weight, and any future stage-plus-carrier constraint remain distinct ledger questions; ADMC is unchanged here.
Radiation-era perturbations, nonlinear growth, lensing, complete signal transport, thermal history, and observational fits belong to the later cosmological layer. The deeper theory must first produce a stage history; Tier 3 can then turn that history and completed sector laws into distances, brightness, growth, lensing, degeneracy tests, and falsifiers.
The Tier 2 gain is concrete: one generator now governs massive and lightlike motion while preserving homogeneous spatial momentum content, its placement passes the massive-motion diagnostic, and its clean local branch selects a common first-power scaling in two distinct bound-system proxies. The remaining center of gravity has moved from kinematics to dynamics—from how a given history is expressed to why that history occurs.