Spacetime Coherence, Rebuilt: Matter as Stable Mode, and Nothing Else

1. The Two Ideas

Assumption A1  

(B1: no position without momentum). No time slice of spacetime is privileged. A description that fixes “where” on one slice while saying nothing about “which slice” has silently chosen a rest frame, and there is none to choose. Position and momentum are therefore not independently specifiable.

Assumption A2  

(B2: matter is a stable mode, not a primitive). There are no particles at the bottom. There is a field on spacetime, and what we call a particle is a stable, localised, self-sustaining pattern of that field. The pattern’s properties are its properties as a pattern.

B1 is the same assumption as A1 of the companion paper on reference frame democracy, stated from the side of the time slice rather than the side of the boost. B2 is new, and it is the idea this paper is about.

The earlier paper attached to these two ideas a coherence field ${\Psi }_c$, a Lagrangian with a curvature coupling, a “crystallisation threshold,” a quantum number n, three fitted constants, and eleven sections of consequences. None of that is in B1 or B2. I rebuild from B1 and B2 alone and mark each point where a further assumption enters.

2. What B1 Buys, and Why It Is Not This Paper

The earlier paper tried to derive $[\widehat{x},\widehat{p}]=i\hslash$ from “non-commuting projection operators” ${\mathcal{P}}_{\mathrm{space}}$, ${\mathcal{P}}_{\mathrm{time}}$ on the four-vector $X^{\mu }$. That was not a derivation: the ADM projectors $h_{\mu \nu }$ and $-n_{\mu }{n}_{\nu }$ are orthogonal and commute, and the commutator written in the appendix has no meaning as an operator identity.

What B1 actually buys is settled by group theory and is set out in the companion paper: once descriptions carry phases, the Galilei (or Poincaré) group acts projectively, Bargmann’s classification fixes the boost–translation phase up to one constant, and that phase in infinitesimal form is $[\widehat{x},\widehat{p}]=i\hslash$, with the uncertainty relation as its Fourier expression. B1 gives quantum kinematics. It gives no field, no matter, no mass, no spectrum. I therefore do not repeat it, and everything below rests on B2 plus B1’s kinematic output.

3. What B2 Buys Exactly

3.1. Mass Is Rest Frequency

Take a field on Minkowski spacetime and suppose it has a stable mode that, in some frame, is stationary: $\Phi (x,t)=f\left(x\right)e^{-i{\omega }_{0}t}$. B1 forbids that frame from being privileged, so the same mode must be describable from every frame, and by the companion paper’s Section 7 the frame-democratic descriptions of a Poincaré-covariant field are classified by Wigner [1]: they carry a mass and a spin. Under a boost of rapidity $\eta$ the phase $e^{-i{\omega }_{0}t}$ becomes $e^{-i(Et-px)/\hslash }$ with

$$E=\hslash {\omega }_{0}cosh\eta ,pc=\hslash {\omega }_{0}sinh\eta ,E^2-p^2{c}^2={(\hslash {\omega }_0)}^2.$$

(1)

This is the dispersion relation of a particle of mass

$$m={\displaystyle \frac{\hslash {\omega }_0}{c^2}}.$$

(2)

Proposition 1 

(Matter is crystallised time). A stable stationary mode of a Poincaré-covariant field, described frame-democratically, is a particle, and its mass is its rest-frame frequency in units of $\hslash /c^2$.

This is the precise content of the earlier paper’s slogan. It is not new physics; it is de Broglie’s 1924 identification [2] read through Wigner. But it is exactly what B2 promises and exactly what B2 delivers: the particle is nothing over and above the mode, and mass is nothing over and above frequency.

3.2. The Coherence Length Is the Compton Wavelength

The earlier paper introduced a “coherence length” $l_c=2.42\times {10}^{-12}$ m as a parameter. It is not a parameter. From (2), the mode’s rest frequency defines a length,

$${\lambda }_C={\displaystyle \frac{2\pi c}{{\omega }_0}}={\displaystyle \frac{h}{mc}}=2.426\times {10}^{-12}\mathrm{m}\mathrm{for}\mathrm{the}\mathrm{electron},$$

(3)

which is the Compton wavelength. The earlier paper’s $l_c$ is ${\lambda }_C\left(m_e\right)$ to three figures. So the claim “$m_e=2\pi \hslash c/\left(l_c{c}^2\right)$” is an identity, not a prediction, and the coherence length carries no information beyond the electron mass.

What the Compton wavelength does carry is a physical meaning that B2 explains: it is the scale below which a single stable mode cannot be localised without exciting further modes (pair creation). Under B2 that is not a mystery about particles; it is the statement that a pattern cannot be squeezed below the wavelength that defines it.

3.3. The Threshold Energy, Corrected

The earlier paper set $E_{\mathrm{crit}}=\hslash c/l_c=8.19\times {10}^{19}$ GeV. The arithmetic is wrong by twenty-four orders of magnitude:

$${\displaystyle \frac{\hslash c}{l_c}}={\displaystyle \frac{\hslash c}{h/\left(m_{e}c\right)}}={\displaystyle \frac{m_e{c}^2}{2\pi }}=81.3\mathrm{keV}.$$

(4)

There is no Planck-scale threshold in B2. The only energy scale a stable mode defines is its own rest energy.

4. What B2 Can Buy with One More Assumption

B2 says a particle is a stable localised pattern. A free linear field has no localised stable patterns: every wavepacket disperses. So “crystallisation” requires a nonlinearity, and that is a third assumption. I state what is then known, because the earlier paper’s “crystallisation mechanism” has a real counterpart and it is worth naming precisely.

4.0.0.1. Derrick’s obstruction.

For a real scalar field in three spatial dimensions with a standard kinetic term and any potential bounded below, there are no static, finite-energy, localised solutions [3]. A static “crystal” of a single real coherence field does not exist. The earlier paper’s Gaussian profile $|{\Psi }_c^{\left(n\right)}{|}^2\propto exp(-r^2/r_c^2)$ is not a solution of any of the equations written there.

4.0.0.2. The way out: time dependence and charge.

Derrick’s argument is evaded by modes that are stationary but not static: $\Phi =f\left(r\right)e^{-i\omega t}$ with f real and $\Phi$ complex. These require a conserved $U\left(1\right)$ charge Q and a potential $U\left(\right|\Phi \left|\right)$ for which $U/{\left|\Phi \right|}^2$ has a minimum away from the origin. Coleman [4] showed that such non-topological solitons (Q-balls) exist and are stable when their energy per unit charge is below the free-particle mass. This is the honest form of the earlier paper’s stability condition ${\partial }^2{E}_c/\partial {\left|{\Psi }_c\right|}^2>0$.

4.0.0.3. What their spectrum looks like.

The mass of a Q-ball is a function of its charge, $M\left(Q\right)$, continuous in Q in the classical theory, with $M\left(Q\right)<Q{m}_{\mathrm{free}}$ for stability. The spectrum is set by the shape of U. Nothing in it is a ladder in an integer n with spacing $n^2$. To get a discrete spectrum one quantises the charge; to get specific mass ratios one must specify U and the couplings. B2 does not specify U. So B2 plus nonlinearity establishes that stable localised modes can exist and what stabilises them, and leaves their masses to the potential, which is where the Standard Model leaves them too.

4.0.0.4. Topological alternatives.

Kinks, vortices and monopoles are localised stable solutions protected by topology rather than charge; their masses are again fixed by couplings and by the topological sector, not by a quantum number of the kind the earlier paper used.

5. Checking the Earlier Paper’S Numbers

I list each quantitative claim, what it rests on, and what happens when it is recomputed. All constants are CODATA; the script is a dozen lines and is available on request.

Claim	Basis in earlier paper	Recomputed
$m_{\mu }/m_e=206.77$ from $1+{\alpha }_c(2^2-1)+{\beta }_s$	${\beta }_s$ “accounts for spin coupling”	${\alpha }_c(2^2-1)=0.0219$; so ${\beta }_s=205.7$. The parameter is the answer.
$m_{\tau }/m_e=3477.2$ from $1+{\alpha }_c(3^2-1)+{\gamma }_s$	${\gamma }_s$ “enhanced coupling”	${\alpha }_c(3^2-1)=0.0584$; so ${\gamma }_s=3476$. Same.
$n_{max}\approx 3.3$ from $n^7<2.4\times {10}^3$	${\rho }_{\mathrm{coh}}\propto n^7$ vs Planck density	The displayed expression evaluates to $3.8\times {10}^{75}$, giving $n_{max}\approx 6\times {10}^{10}$. The $n^7$ scaling itself has no derivation.
$E_{\mathrm{crit}}=\hslash c/l_c=8.19\times {10}^{19}$ GeV	Arithmetic	$81.3$ keV.
${\tau }_{\mu }=(\hslash /m_{\mu }{c}^2)exp[(m_{\mu }-m_e)/\left(m_e{\alpha }_c\right)]$	Coherence lifetime	Exponent is $2.8\times {10}^4$; the expression is $\sim {10}^{12000}$ s.
${\tau }_{\mu }=2.197\mu$s from $G_F^2{m}_{\mu }^5/192{\pi }^3\times 0.9998$	Fermi theory	This is Fermi theory. The factor $0.9998$ is unexplained and within the neglected radiative corrections.
CKM entries from $exp[-{(n_i-n_j)}^2/2{\sigma }^2]$, $\sigma =0.76$	Gaussian overlap, fitted to $|V_{us}|$	With $\sigma =0.76$ the formula gives $|V_{us}|=e^{-1/2{\sigma }^2}=0.42$, not $0.225$; matching $0.225$ needs $\sigma =0.58$, which then gives $|V_{ub}|=e^{-2/{\sigma }^2}=0.0026$ and $|V_{cb}|=|V_{us}|$, contradicting the table. The table does not follow from the formula.
$m_{\mathrm{DM}}\approx {10}^{-22}$ eV	“estimating $|{\Psi }_c{|}^2\approx {10}^{-34}{m}_p^2/\lambda$”	The ${10}^{-34}$ is chosen to produce the answer; the dimensional chain from a field amplitude to a particle mass is not valid.
${\rho }_{\mathrm{DE}}={10}^{-47}$ GeV4	$\langle |\delta {\Psi }_c{|}^2\rangle \approx {10}^{-123}$	This is the cosmological constant problem restated as an input.
$m_H=125$ GeV from $\sqrt{2\lambda }v$	$\lambda \approx 0.13$ “by self-consistency”	Standard Model with $\lambda$ taken from the measured Higgs mass.
Hierarchy protection	Exponential cutoff inserted into the loop integral	The cutoff is the hypothesis, not a consequence.
$m_e:m_{\mu }:m_{\tau }$ from Kaluza–Klein $n\hslash c/R_c$	$n=1,2,3$	Gives $1:2:3$, not $1:207:3477$. The two formulas in that subsection contradict each other and the lepton formula above.

Claim	Basis in earlier paper	Recomputed
$m_{\mu }/m_e=206.77$ from $1+{\alpha }_c(2^2-1)+{\beta }_s$	${\beta }_s$ “accounts for spin coupling”	${\alpha }_c(2^2-1)=0.0219$; so ${\beta }_s=205.7$. The parameter is the answer.
$m_{\tau }/m_e=3477.2$ from $1+{\alpha }_c(3^2-1)+{\gamma }_s$	${\gamma }_s$ “enhanced coupling”	${\alpha }_c(3^2-1)=0.0584$; so ${\gamma }_s=3476$. Same.
$n_{max}\approx 3.3$ from $n^7<2.4\times {10}^3$	${\rho }_{\mathrm{coh}}\propto n^7$ vs Planck density	The displayed expression evaluates to $3.8\times {10}^{75}$, giving $n_{max}\approx 6\times {10}^{10}$. The $n^7$ scaling itself has no derivation.
$E_{\mathrm{crit}}=\hslash c/l_c=8.19\times {10}^{19}$ GeV	Arithmetic	$81.3$ keV.
${\tau }_{\mu }=(\hslash /m_{\mu }{c}^2)exp[(m_{\mu }-m_e)/\left(m_e{\alpha }_c\right)]$	Coherence lifetime	Exponent is $2.8\times {10}^4$; the expression is $\sim {10}^{12000}$ s.
${\tau }_{\mu }=2.197\mu$s from $G_F^2{m}_{\mu }^5/192{\pi }^3\times 0.9998$	Fermi theory	This is Fermi theory. The factor $0.9998$ is unexplained and within the neglected radiative corrections.
CKM entries from $exp[-{(n_i-n_j)}^2/2{\sigma }^2]$, $\sigma =0.76$	Gaussian overlap, fitted to $|V_{us}|$	With $\sigma =0.76$ the formula gives $|V_{us}|=e^{-1/2{\sigma }^2}=0.42$, not $0.225$; matching $0.225$ needs $\sigma =0.58$, which then gives $|V_{ub}|=e^{-2/{\sigma }^2}=0.0026$ and $|V_{cb}|=|V_{us}|$, contradicting the table. The table does not follow from the formula.
$m_{\mathrm{DM}}\approx {10}^{-22}$ eV	“estimating $|{\Psi }_c{|}^2\approx {10}^{-34}{m}_p^2/\lambda$”	The ${10}^{-34}$ is chosen to produce the answer; the dimensional chain from a field amplitude to a particle mass is not valid.
${\rho }_{\mathrm{DE}}={10}^{-47}$ GeV4	$\langle |\delta {\Psi }_c{|}^2\rangle \approx {10}^{-123}$	This is the cosmological constant problem restated as an input.
$m_H=125$ GeV from $\sqrt{2\lambda }v$	$\lambda \approx 0.13$ “by self-consistency”	Standard Model with $\lambda$ taken from the measured Higgs mass.
Hierarchy protection	Exponential cutoff inserted into the loop integral	The cutoff is the hypothesis, not a consequence.
$m_e:m_{\mu }:m_{\tau }$ from Kaluza–Klein $n\hslash c/R_c$	$n=1,2,3$	Gives $1:2:3$, not $1:207:3477$. The two formulas in that subsection contradict each other and the lepton formula above.

The string-theory section asserted that string theory “reduced particles to zero-dimensional vibrating points” and thereby “confirmed” that time is irreducible. String theory does the opposite: it replaces points with extended objects, and the point-particle action displayed there is not a result of string theory but its low-energy limit. The section is removed. The PostMath appendix contains no operation with a defined domain, codomain or rule of computation, so it makes no claim that can be kept or dropped; it is removed.

6. What Was Dropped

Everything in Section 4–12 and Appendices A–C of the earlier paper: the gauge-group “derivation” (a list of homogeneous spaces with no map to the field), Einstein’s equations (obtained by writing the Einstein–Hilbert action and varying it, which assumes them), the Schrödinger limit (assumed in the form of the expansion), the dark sector, the lepton, neutrino and quark spectra, CP violation, the hierarchy problem, every experimental protocol, the falsification matrix, and the “already verified” list. None of it descends from B1 or B2.

7. What the Two Ideas Establish

1.

B1 is quantum kinematics, by Bargmann, as set out in the companion paper. It says nothing about matter.

2.

B2, with B1, gives one exact identity: a stable stationary mode of a covariant field is a particle of mass $\hslash {\omega }_0/c^2$ (Wigner, de Broglie). Matter is crystallised time in precisely this sense and no other.

3.

The earlier paper’s coherence length is the Compton wavelength; its threshold energy is $m_e{c}^2/2\pi$; neither is a new scale.

4.

“Crystallisation” has a real meaning once a nonlinearity is assumed: non-topological solitons exist, require a conserved charge and a suitable potential, and evade Derrick’s theorem by being stationary rather than static. Their spectrum is set by the potential, not by an integer.

5.

No mass ratio, mixing angle, dark-sector density or experimental signature follows from B1 and B2, with or without the nonlinearity.

The residue of the earlier paper is therefore one sentence, and I think it is a good one: a particle is a stable, localised, frame-democratic mode of a field, and its mass is the frequency of that mode in its own frame. What sets the frequencies is the question the Standard Model also leaves open, and B2 does not close it.