Derivation 5.6A — Local covariance and scoped sector realizations

Keywords

theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology

Purpose

This appendix supports Chapter 5.6. It states the clean true-frame/bound-frame transform, derives its local light and ranging condition, and tests that transform in three controlled settings:

an adiabatic hydrogenic electromagnetic proxy;
an adiabatic Newtonian bound orbit;
linear nonrelativistic matter growth at a prescribed background history.

The calculations are realizations of a covariance target, not a proof that every interaction sector follows one universal law. Strong and weak closure remain open, and exact local indistinguishability is not established.

Role, regime, premises, and status

Role: conditional local-covariance derivation and scoped sector correspondence.
Regime: homogeneous slowly varying yield; clean branch; adiabatic bound systems; hydrogenic EM proxy; Newtonian/quasi-static gravity; linear nonrelativistic perturbations.
Premises: bosic-slot \(\chi^2\) kinetics, fixed content constants, co-scaling rods, matched clocks, \(e^2\propto\chi\) in the EM realization, and \(G_{\mathrm{eff}}\propto\chi\) in the gravity realization.

Notation ledger

\(X\equiv\chi/\chi_o\) — normalized yield, with \(X_o=1\).
\(a_B\equiv X^{-1}\) — clean-branch bound-frame scale factor.
\(x_T,x_B\) — true-frame and bound-frame spatial coordinates.
\(t\) — common history parameter and clean-branch clock time.
\(\ell_T\) — local true-frame rod length.
\(p_f=m_0c\) — fixed fermic content scale.
\(\hbar,c,m_0\) — fixed content constants in the proxy calculations.
\(e^2(X)\) — Coulomb coupling in a convention with \(\alpha_X=e^2/(\hbar cX)\).
\(G_{\mathrm{eff}}(X)\) — effective Newtonian coupling on the true-frame chart.
\(\rho_T,\rho_B\) — true-frame and bound-frame matter densities.
\(\delta\) — density contrast, invariant under the homogeneous volume conversion.
\(H=\dot a_B/a_B=-\dot X/X\) — prescribed background rate.

The formulas below use \(X\) to keep normalization explicit. In Chapter 5 notation with \(\chi_o=1\), replace \(X\) by \(\chi\).

Clean true-frame/bound-frame transform

The clean branch takes local true-frame rods to co-scale linearly with the normalized yield:

\[ \boxed{ \ell_T(X)=\ell_oX }. \]

A local observer reports a true-frame interval by dividing by that rod standard. The corresponding bound-frame coordinate is

\[ \boxed{ x_B=\frac{x_T}{X} }, \]

and the bound-frame scale factor is

\[ \boxed{ a_B=\frac{1}{X}=\frac{\chi_o}{\chi} }. \]

A bound length that co-scales as \(R_T=R_oX\) is therefore constant locally:

\[ R_B=\frac{R_T}{X}=R_o. \]

A fixed true-frame separation instead appears as \(D_B=a_BD_T\). This transform is bookkeeping between two descriptions; it does not by itself supply the sector forces that make a bound length track \(X\).

The clean branch also takes local tick frequencies to be fixed relative to \(t\):

\[ \boxed{ \nu_T(X)=\nu_o }. \]

Thus \(dt_B=dt\) in the local comparisons used below.

Local light and radar-ranging condition

Lightlike motion in the true frame is

\[ \dot x_T=cX. \]

During one local tick \(\nu_o^{-1}\), the true-frame light distance is

\[ \ell_{\mathrm{tick},T} = \frac{cX}{\nu_o}. \]

Measured in local rods,

\[ \frac{\ell_{\mathrm{tick},T}}{\ell_T} = \frac{cX/\nu_o}{\ell_oX} = \frac{c}{\nu_o\ell_o}, \]

which is independent of \(X\). Local observers therefore report a fixed light speed even though both the true-frame light distance per tick and the rod length co-scale.

Radar ranging gives the same condition. For a co-scaled local radius

\[ r_T=r_oX, \]

the true-frame round-trip time is

\[ \Delta t_T = \frac{2r_T}{cX} = \frac{2r_o}{c}. \]

The local tick count \(\nu_o\Delta t_T\) is constant. This is the operational local-covariance condition used by the sector checks. It is not a general proof that every probe is insensitive to \(X\).

Hydrogenic electromagnetic proxy

Premises

Use the nonrelativistic limit of the bosic-slot generator after the rest phase is removed:

\[ H_{\mathrm{kin}} = \frac{X^2\mathbf p^2}{2m_e}. \]

Take

\[ \hbar=\text{constant}, \qquad m_e=\text{constant}, \qquad e^2(X)=e_o^2X. \]

The last relation is a scoped coupling realization, not a change in a conserved content constant. The hydrogenic proxy Hamiltonian is

\[ \boxed{ H_{\mathrm{H}}(X) = \frac{X^2\mathbf p^2}{2m_e} - \frac{e_o^2X}{r} }. \]

The history is assumed adiabatic on atomic timescales, so the bound state follows the instantaneous spectrum.

Rod scale

For a state of characteristic radius \(r\), uncertainty gives \(p\sim\hbar/r\). The energy scale is therefore

\[ E(r) \sim \frac{X^2\hbar^2}{2m_er^2} - \frac{e_o^2X}{r}. \]

Stationarity requires

\[ \frac{dE}{dr} \sim - \frac{X^2\hbar^2}{m_er^3} + \frac{e_o^2X}{r^2} =0. \]

Hence the characteristic Bohr scale is

\[ \boxed{ a_0(X) \sim \frac{X^2\hbar^2}{m_ee^2(X)} = X\frac{\hbar^2}{m_ee_o^2} = Xa_{0,o} }. \]

The atomic proxy rod co-scales exactly as required by the clean transform.

Spectral scale

The bound-state energies inherit the hydrogenic form with the dressed kinetic coefficient:

\[ E_n(X) = - \frac{m_ee^4(X)} {2X^2\hbar^2n^2}. \]

Since \(e^2(X)=e_o^2X\), one has \(e^4(X)=e_o^4X^2\), so

\[ \boxed{ E_n(X) = - \frac{m_ee_o^4}{2\hbar^2n^2} }. \]

Transition energies and frequencies are therefore independent of \(X\):

\[ \Delta E\propto X^0, \qquad \nu_{\mathrm{atom}}=\frac{\Delta E}{h}\propto X^0. \]

The proxy supplies the matched-clock condition used in Derivation 5.4A.

Fixed fine-structure and velocity checks

The operative dimensionless coupling is

\[ \alpha_X \equiv \frac{e^2(X)}{\hbar cX}. \]

With \(e^2(X)=e_o^2X\),

\[ \boxed{ \alpha_X = \frac{e_o^2}{\hbar c} = \text{constant} }. \]

The characteristic canonical momentum scales as

\[ p_{\mathrm{atom}}\sim\frac{\hbar}{a_0}\propto X^{-1}. \]

Hamilton’s equation gives the true-frame orbital velocity

\[ v_T \sim \frac{X^2p_{\mathrm{atom}}}{m_e} \propto X. \]

Thus \(v_T/(cX)\) is constant, and the velocity reported using co-scaling rods and fixed clocks is also constant.

These results establish a hydrogenic, adiabatic proxy only. They do not derive the scaling of QED corrections, molecular structure, hyperfine standards, nuclear scales, material rods, or strong/weak quantities.

Newtonian bound-orbit realization

Premises and Hamiltonian

Consider a test mass \(m\) in a quasi-static Newtonian potential generated by a fixed source content \(M_s\). Use

\[ G_{\mathrm{eff}}(X)=G_NX \]

and the nonrelativistic Hamiltonian

\[ \boxed{ H_{\mathrm{orb}} = \frac{X^2p^2}{2m} - \frac{G_{\mathrm{eff}}(X)M_sm}{r} }. \]

The orbit is assumed adiabatic, so its angular action

\[ J=pr \]

is conserved.

Orbit scale

For a circular orbit, substitute \(p=J/r\):

\[ H_{\mathrm{orb}}(r) = \frac{X^2J^2}{2mr^2} - \frac{G_{\mathrm{eff}}M_sm}{r}. \]

The stationary-radius condition is

\[ \frac{dH_{\mathrm{orb}}}{dr} = - \frac{X^2J^2}{mr^3} + \frac{G_{\mathrm{eff}}M_sm}{r^2} =0. \]

Therefore

\[ r = \frac{X^2J^2} {m^2G_{\mathrm{eff}}M_s}. \]

For fixed \(J,m,M_s\),

\[ r\propto\frac{X^2}{G_{\mathrm{eff}}}. \]

Demanding the clean rod scaling \(r\propto X\) fixes

\[ \boxed{ G_{\mathrm{eff}}(X)=G_NX }. \]

Conversely, that coupling law makes the bound orbital radius co-scale with local rods:

\[ \boxed{ r(X)=Xr_o }. \]

Orbital velocity and timing

Hamilton’s equation gives

\[ v_T = \frac{\partial H_{\mathrm{orb}}}{\partial p} = \frac{X^2p}{m}. \]

With \(p=J/r\propto X^{-1}\),

\[ \boxed{ v_T\propto X }. \]

The true-frame orbital period scales as

\[ T_T\sim\frac{r}{v_T}\propto X^0. \]

Fixed local clocks therefore count a constant number of ticks per orbit. The turning combination is

\[ v_T^2r = X^2G_{\mathrm{eff}}M_s \propto X^3, \]

where two powers come from the dressed kinetic relation and one from \(G_{\mathrm{eff}}\propto X\).

This is a Newtonian, adiabatic orbit realization. It does not derive a relativistic gravitational field equation, strong-field dynamics, lensing, or the cosmological gravitational source weight.

Conditional linear-growth correspondence

The same kinetic and coupling powers can be checked dynamically in the linear, quasi-static matter regime.

Premises

The homogeneous yield history \(X(t)\) is prescribed, with \(a_B=1/X\) and \(H=-\dot X/X\).
The true-frame chart is static for the homogeneous matter background, so \(\bar\rho_T\) is constant.
Bound-frame volume conversion gives \[ \bar\rho_B=X^3\bar\rho_T. \]
The density contrast \(\delta\) is invariant under the homogeneous conversion.
Matter is nonrelativistic and pressureless; perturbations are linear.
Gravity is quasi-static and obeys \[ \Delta_TU_T = 4\pi G_NX\,\bar\rho_T\delta. \]
The content-velocity variable \(\mathbf u\equiv\mathbf p/m\) obeys \[ \dot{\mathbf x}_T=X^2\mathbf u, \qquad \dot{\mathbf u}=-\boldsymbol\nabla_TU_T. \]

Derivation

Linear continuity on the static true-frame grid is

\[ \dot\delta = -X^2\boldsymbol\nabla_T\!\cdot\!\mathbf u. \]

Differentiate:

\[ \ddot\delta = -2X\dot X\boldsymbol\nabla_T\!\cdot\!\mathbf u - X^2\boldsymbol\nabla_T\!\cdot\!\dot{\mathbf u}. \]

The continuity equation gives

\[ -2X\dot X\boldsymbol\nabla_T\!\cdot\!\mathbf u = \frac{2\dot X}{X}\dot\delta = -2H\dot\delta. \]

Euler and Poisson give

\[ -X^2\boldsymbol\nabla_T\!\cdot\!\dot{\mathbf u} = X^2\Delta_TU_T = 4\pi G_NX^3\bar\rho_T\delta. \]

Since \(X^3\bar\rho_T=\bar\rho_B\),

\[ \boxed{ \ddot\delta+2H\dot\delta = 4\pi G_N\bar\rho_B\delta }. \]

This is the standard linear matter-growth equation at the same prescribed \(H(t)\) and \(\bar\rho_B(t)\). The friction coefficient comes from differentiating the \(X^2\) kinetic dressing. The source contains the same two kinetic powers plus the one coupling power in \(G_{\mathrm{eff}}=G_NX\).

The equality is a conditional correspondence, not a background prediction. It does not determine \(H(z)\) and does not establish relativistic perturbations, radiation-era transfer, nonlinear growth, or lensing.

Realization ledger

Sector or check	Result in this appendix	Regime	Status beyond the regime
Local rods, clocks, and ranging	\(x_B=x_T/X\), \(\ell_T\propto X\), \(\nu_T\propto X^0\), local light/radar ratios fixed	clean homogeneous branch	transform and target only
Electromagnetism	\(e^2=e_o^2X\), \(a_0\propto X\), \(\Delta E\propto X^0\), \(\alpha_X\) fixed	hydrogenic, adiabatic proxy	QED, molecular, hyperfine, and material closure open
Gravity-bound systems	\(G_{\mathrm{eff}}=G_NX\), \(r\propto X\), \(v_T\propto X\)	Newtonian, adiabatic, fixed source content	relativistic and strong-field closure open
Matter growth	standard linear equation at fixed \(H(z)\)	pressureless, linear, quasi-static	radiation, horizon-scale, nonlinear, and lensing sectors open
Strong sector	no realization derived	—	confinement and nuclear scaling open
Weak sector	no realization derived	—	transition-rate scaling open

The broader statement that all space-mediated couplings scale as \(X\) is therefore a hypothesis supported here only by the EM proxy and Newtonian gravity realization. It is not an established all-sector principle.

Local-drift falsifiers and scope

The clean realizations predict no drift in the dimensionless quantities they actually close:

\(\alpha_X\) and hydrogenic electronic spectral ratios remain fixed;
local light/radar ranging ratios remain fixed;
Newtonian orbital periods remain fixed relative to the matched clock;
co-scaled atomic and orbital lengths remain fixed relative to the same rods.

A confirmed nonzero drift that survives reduction to a dimensionless local comparison would falsify the corresponding clean realization. Relevant channels include cross-family clock ratios, \(\Delta\alpha/\alpha\), recoil-to-spectral ratios, radar-to-atomic length ratios, orbital-to-atomic timing, and composition-dependent residuals.

Because strong and weak realization is not derived here, an anomaly involving nuclear, confinement, or weak-transition physics would expose an open closure problem before it could be called a falsification of the \(\chi^2\) generator itself. Likewise, failure of \(G_{\mathrm{eff}}\propto X\) in the Newtonian mapping would reject that scoped gravity realization without undoing the free-carrier theorem of Derivation 5.2A.

Result and boundaries

This appendix establishes a mutually consistent clean-branch package under explicit premises:

\[ \boxed{ x_B=\frac{x_T}{X}, \qquad a_B=\frac{1}{X}, \qquad \ell_T\propto X, \qquad \nu_T\propto X^0 }, \]

\[ \boxed{ e^2\propto X, \qquad a_0\propto X, \qquad \alpha_X=\text{constant} } \quad \text{(hydrogenic proxy),} \]

and

\[ \boxed{ G_{\mathrm{eff}}\propto X, \qquad r_{\mathrm{orbit}}\propto X } \quad \text{(Newtonian orbit).} \]

With the additional linear and quasi-static premises, it also gives

\[ \boxed{ \ddot\delta+2H\dot\delta = 4\pi G_N\bar\rho_B\delta } \]

at a fixed background history.

It does not establish exact local indistinguishability, all-sector coupling co-scaling, Friedmann dynamics, a dressed gravitational source law, strong/weak closure, radiation physics, or a complete observational cosmology. Chapter 5.6 may cite the scoped EM, gravity, and linear-growth results only with these regime boundaries attached.