Notation by work
Physics: origin locking
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205 symbol spellings in 236 mathematical passages across 2 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
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=Introduction passage · 98 passages
Search aliases: =
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
AIntroduction passage · 7 passages
Search aliases: A
Introduction passage · origin_locking.tex, line 62
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
\tau=At
aIntroduction passage · 51 passages
Search aliases: a
Introduction passage · origin_locking.tex, line 62
ber of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological scale factor is 10^-6a.
Exact mathematical passage
a=1
A^4Introduction passage · 1 passage
Search aliases: A^4
Introduction passage · origin_locking.tex, line 62
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
A^4
a_\starIntroduction passage · 2 passages
Search aliases: a_⋆ star
Introduction passage · origin_locking.tex, line 62
tion y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological scale factor is 10^-6a.
Exact mathematical passage
a_\star=10^{-6}
BIntroduction passage · 21 passages
Search aliases: B
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
B_LIntroduction passage · 19 passages
Search aliases: B_L
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
dIntroduction passage · 2 passages
Search aliases: d
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
d_hIntroduction passage · 2 passages
Search aliases: d_h
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
\ddIntroduction passage · 6 passages
Search aliases: dd d mathrm
Introduction passage · origin_locking.tex, line 62
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
\dd/\dd\tau
HIntroduction passage · 28 passages
Search aliases: H
Introduction passage · origin_locking.tex, line 57
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
H
hIntroduction passage · 8 passages
Search aliases: h
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
LIntroduction passage · 28 passages
Search aliases: L
Introduction passage · origin_locking.tex, line 57
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
L
tIntroduction passage · 5 passages
Search aliases: t
Introduction passage · origin_locking.tex, line 62
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
\tau=At
xIntroduction passage · 1 passage
Search aliases: x
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
yIntroduction passage · 18 passages
Search aliases: y
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
y_hIntroduction passage · 2 passages
Search aliases: y_h
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
y_h^4Introduction passage · 1 passage
Search aliases: y_h^4
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
\piIntroduction passage · 4 passages
Search aliases: π pi
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
\pi^2Introduction passage · 4 passages
Search aliases: π^2 pi
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
\tauIntroduction passage · 7 passages
Search aliases: τ tau
Introduction passage · origin_locking.tex, line 62
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
\tau=At
\sumIntroduction passage · 9 passages
Search aliases: ∑ sum
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
\sum_{h=1}^{L}Introduction passage · 1 passage
Search aliases: ∑_h=1^L sum
Introduction passage · origin_locking.tex, line 58
Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological s…
Exact mathematical passage
y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
\starIntroduction passage · 2 passages
Search aliases: ⋆ star
Introduction passage · origin_locking.tex, line 62
tion y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the cosmological scale factor is 10^-6a.
Exact mathematical passage
a_\star=10^{-6}
*Use in source · 6 passages
Search aliases: *
Use in source · origin_locking.tex, line 143
[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s],
Exact mathematical passage
q=\pm\sqrt{s_*}
+Use in source · 44 passages
Search aliases: +
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
a^{-2}Use in source · 1 passage
Search aliases: a^-2
Use in source · origin_locking.tex, line 103
Write a nonzero trace summand as T_j(a)=w_jy_j^2f_j a^-2(k_j^2+a^2y_j^2)^-1/2. Then frac dlog T_j dlog a=-2-fraca^2y_j^2k_j^2+a^2y_j^2. This is strictly negative and lies in [-3,-2]. The finite sum is continuous, strictly decreases, and tends to zero. The intermediate value theorem gives existence and strict monotonicity gives uniqueness of the threshold crossing.
Exact mathematical passage
T_j(a)=w_jy_j^2f_j\,a^{-2}(k_j^2+a^2y_j^2)^{-1/2}.
a^{-3}Use in source · 3 passages
Search aliases: a^-3
Use in source · origin_locking.tex, line 79
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ_χ(a) · =a^-3∑_jw_jE_j(a)f_j, p_χ(a) · =a^-3∑_jw_jfrack_j^2f_j3a^2E_j(a), T(a) · =a^-3∑_jw_jfracy_j^2f_jE_j(a). These definitions describe a condition…
Exact mathematical passage
E_j(a)&=\sqrt{k_j^2/a^2+y_j^2},\\ \rho_\chi(a)&=a^{-3}\sum_jw_jE_j(a)f_j,\\ p_\chi(a)&=a^{-3}\sum_jw_j\frac{k_j^2f_j}{3a^2E_j(a)},\\ \Tr(a)&=a^{-3}\sum_jw_j\frac{y_j^2f_j}{E_j(a)}.
a^{-4}Use in source · 1 passage
Search aliases: a^-4
Use in source · origin_locking.tex, line 209
· +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for the physical conversion. The radiation term has density ρ_r=ρ_r0a^-4. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in U.
Exact mathematical passage
\rho_r=\rho_{r0}a^{-4}
a^{-5}Use in source · 1 passage
Search aliases: a^-5
Use in source · origin_locking.tex, line 120
For a≥ a_i the slope bounds also give T(a_i)(a_i/a)^3≤ T(a)≤ T(a_i)(a_i/a)^2. Thus the threshold lies between the cube-root and square-root dilution estimates based on the initial trace. For a finite set of massive modes the cold limit is T(a)=a^-3∑_jw_jy_jf_j+O(a^-5). The pressure correction therefore matters before the distribution becomes cold.
Exact mathematical passage
\Tr(a)=a^{-3}\sum_jw_jy_jf_j+O(a^{-5}).\label{eq:cold}
a^2Use in source · 11 passages
Search aliases: a^2
Use in source · origin_locking.tex, line 79
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ_χ(a) · =a^-3∑_jw_jE_j(a)f_j, p_χ(a) · =a^-3∑_jw_jfrack_j^2f_j3a^2E_j(a), T(a) · =a^-3∑_jw_jfracy_j^2f_jE_j(a). These definitions describe a condition…
Exact mathematical passage
E_j(a)&=\sqrt{k_j^2/a^2+y_j^2},\\ \rho_\chi(a)&=a^{-3}\sum_jw_jE_j(a)f_j,\\ p_\chi(a)&=a^{-3}\sum_jw_j\frac{k_j^2f_j}{3a^2E_j(a)},\\ \Tr(a)&=a^{-3}\sum_jw_j\frac{y_j^2f_j}{E_j(a)}.
a^3Use in source · 11 passages
Search aliases: a^3
Use in source · origin_locking.tex, line 127
Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full coherent calculation.
Exact mathematical passage
\kappa_{\rm coh}=\frac{1}{2a^3}\sum_jw_j\frac{y_j^2}{E_j}u_j
a_0Use in source · 2 passages
Search aliases: a_0
Use in source · origin_locking.tex, line 379
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
U(s)=a_0+b_0s+c_0s^2+B_LG(s).
A_hUse in source · 2 passages
Search aliases: A_h
Use in source · origin_locking.tex, line 298
The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient used in the solver is Z(a,q)=C+∑_h=1^Lfracd_hy_h^248π^2 fracq^21+q^2(1-A_h). It is positive. Its q and a derivatives enter both [source reference] and…
Exact mathematical passage
K_h^2(a)=y_hz_{\max}^2/a^2,\quad A_h=\left[\frac{K_h^2}{K_h^2+y_h^2(1+q^2)}\right]^{3/2}.
a_iUse in source · 3 passages
Search aliases: a_i
Use in source · origin_locking.tex, line 96
_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j wit…
Exact mathematical passage
\Tr(a_i)>4D/S
a_{\rm rel}Use in source · 1 passage
Search aliases: a_rm rel rm
Use in source · origin_locking.tex, line 181
[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel
Exact mathematical passage
a_{\rm rel}
bUse in source · 5 passages
Search aliases: b
Use in source · origin_locking.tex, line 379
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
U(s)=a_0+b_0s+c_0s^2+B_LG(s).
b_0Use in source · 5 passages
Search aliases: b_0
Use in source · origin_locking.tex, line 379
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
U(s)=a_0+b_0s+c_0s^2+B_LG(s).
B_{12}Use in source · 1 passage
Search aliases: B_12
Use in source · origin_locking.tex, line 423
For an explicit comparison, truncating to L=12 retains about 20.0724% of the old Gaussian production energy. Recalibrating that stipulated family gives E_0=43.8579 and B_12K(S)=9.02826. Choosing Δ=0.3E_0=13.1574 satisfies the window. This changes the species content, physical mass scale, and background matching. It is a counterfactual model, not a recovery of the original cosmology.
Exact mathematical passage
B_{12}K(S)=9.02826
B_{15}Use in source · 1 passage
Search aliases: B_15
Use in source · origin_locking.tex, line 74
D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
B_{15}=3086.94039766
CUse in source · 5 passages
Search aliases: C
Use in source · origin_locking.tex, line 71
D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
C=2S_E/\epsilon,\quad \Ezero=C/2,\quad S_E=2.18498795564.
cUse in source · 8 passages
Search aliases: c
Use in source · origin_locking.tex, line 127
Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full coherent calculation.
Exact mathematical passage
\kappa_{\rm coh}=\frac{1}{2a^3}\sum_jw_j\frac{y_j^2}{E_j}u_j
c_0Use in source · 6 passages
Search aliases: c_0
Use in source · origin_locking.tex, line 379
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
U(s)=a_0+b_0s+c_0s^2+B_LG(s).
\cosUse in source · 1 passage
Search aliases: cos
Use in source · origin_locking.tex, line 132
For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet stability theorem or a derivation of cosmological decoherence.
Exact mathematical passage
u_j(t)=u_j(0)\cos(2E_jt)-v_j(0)\sin(2E_jt)
DUse in source · 12 passages
Search aliases: D
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
\ddot fUse in source · 1 passage
Search aliases: ddot f ddot
Use in source · origin_locking.tex, line 214
After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12,
Exact mathematical passage
\ddot f_j+3H\dot f_j+\Omega_j^2f_j=0,\qquad H=\dot a/a,
\ddot f_jUse in source · 1 passage
Search aliases: ddot f_j ddot
Use in source · origin_locking.tex, line 214
After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12,
Exact mathematical passage
\ddot f_j+3H\dot f_j+\Omega_j^2f_j=0,\qquad H=\dot a/a,
\ddot qUse in source · 2 passages
Search aliases: ddot q ddot
Use in source · origin_locking.tex, line 246
F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect to a before fixing the metric. It is not optional when the retained kinetic coefficient depends on the scale factor.
Exact mathematical passage
Z\ddot q+H(3Z+aZ_a)\dot q+\frac12Z_q\dot q^2+U_q+\mathcal F=0.\label{eq:background}
\dot aUse in source · 2 passages
Search aliases: dot a dot
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
\dot a^2Use in source · 1 passage
Search aliases: dot a^2 dot
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
\dot fUse in source · 4 passages
Search aliases: dot f dot
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
\dot f_jUse in source · 4 passages
Search aliases: dot f_j dot
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
\dot f_j^*Use in source · 1 passage
Search aliases: dot f_j^* dot
Use in source · origin_locking.tex, line 218
After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12,
Exact mathematical passage
a^3(f_j\dot f_j^*-f_j^*\dot f_j)=i.
\dot HUse in source · 2 passages
Search aliases: dot H dot
Use in source · origin_locking.tex, line 266
dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_ju_j/2 and dotΩ_j=(y_j^2qdot q-Hk_j^2/a^2)/Ω_j. The terms multiplying Hn_j and Hu_j cancel against 3H(ρ_χ+p_χ) using [source reference]. The remaining term is dot q F.
Exact mathematical passage
\dot H=-(\rho+p)/(2M^2)
\dot nUse in source · 2 passages
Search aliases: dot n dot
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
\dot n_jUse in source · 2 passages
Search aliases: dot n_j dot
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
\dot qUse in source · 16 passages
Search aliases: dot q dot
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
\dot q^2Use in source · 7 passages
Search aliases: dot q^2 dot
Use in source · origin_locking.tex, line 246
F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect to a before fixing the metric. It is not optional when the retained kinetic coefficient depends on the scale factor.
Exact mathematical passage
Z\ddot q+H(3Z+aZ_a)\dot q+\frac12Z_q\dot q^2+U_q+\mathcal F=0.\label{eq:background}
\dot uUse in source · 1 passage
Search aliases: dot u dot
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
\dot u_jUse in source · 1 passage
Search aliases: dot u_j dot
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
\dot vUse in source · 1 passage
Search aliases: dot v dot
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
\dot v_jUse in source · 1 passage
Search aliases: dot v_j dot
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
\dot\rhoUse in source · 3 passages
Search aliases: dotρ dot rho
Use in source · origin_locking.tex, line 258
[Finite exchange and the gravitational constraint] On any regular solution of [source reference] and [source reference] with a>0 and Z>0, dotρ_χ+3H(ρ_χ+p_χ) · =dot q F, dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_j…
Exact mathematical passage
\dot\rho_\chi+3H(\rho_\chi+p_\chi)&=\dot q\,\mathcal F,\\ \dot\rho_q+3H(\rho_q+p_q)&=-\dot q\,\mathcal F.
\dot\rho_qUse in source · 2 passages
Search aliases: dotρ_q dot rho
Use in source · origin_locking.tex, line 258
[Finite exchange and the gravitational constraint] On any regular solution of [source reference] and [source reference] with a>0 and Z>0, dotρ_χ+3H(ρ_χ+p_χ) · =dot q F, dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_j…
Exact mathematical passage
\dot\rho_\chi+3H(\rho_\chi+p_\chi)&=\dot q\,\mathcal F,\\ \dot\rho_q+3H(\rho_q+p_q)&=-\dot q\,\mathcal F.
\dot\rho_\chiUse in source · 1 passage
Search aliases: dotρ_χ dot rho chi
Use in source · origin_locking.tex, line 258
[Finite exchange and the gravitational constraint] On any regular solution of [source reference] and [source reference] with a>0 and Z>0, dotρ_χ+3H(ρ_χ+p_χ) · =dot q F, dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_j…
Exact mathematical passage
\dot\rho_\chi+3H(\rho_\chi+p_\chi)&=\dot q\,\mathcal F,\\ \dot\rho_q+3H(\rho_q+p_q)&=-\dot q\,\mathcal F.
\dot\OmegaUse in source · 2 passages
Search aliases: dotΩ dot Omega
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
\dot\Omega_jUse in source · 2 passages
Search aliases: dotΩ_j dot Omega
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
EUse in source · 10 passages
Search aliases: E
Use in source · origin_locking.tex, line 71
D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
C=2S_E/\epsilon,\quad \Ezero=C/2,\quad S_E=2.18498795564.
eUse in source · 4 passages
Search aliases: e
Use in source · origin_locking.tex, line 93
[Local locking criterion and dilution] For the above gas, T=ρ_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter…
Exact mathematical passage
\boxed{\kappa_{\rm eff}(a)=\rho_\chi(a)-3p_\chi(a)-\frac{4D}{S}.}\label{eq:locking}
e^{2iE_jt}Use in source · 1 passage
Search aliases: e^2iE_jt
Use in source · origin_locking.tex, line 137
For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet stability theorem or a derivation of cosmological decoherence.
Exact mathematical passage
e^{2iE_jt}
E_jUse in source · 7 passages
Search aliases: E_j
Use in source · origin_locking.tex, line 79
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ_χ(a) · =a^-3∑_jw_jE_j(a)f_j, p_χ(a) · =a^-3∑_jw_jfrack_j^2f_j3a^2E_j(a), T(a) · =a^-3∑_jw_jfracy_j^2f_jE_j(a). These definitions describe a condition…
Exact mathematical passage
E_j(a)&=\sqrt{k_j^2/a^2+y_j^2},\\ \rho_\chi(a)&=a^{-3}\sum_jw_jE_j(a)f_j,\\ p_\chi(a)&=a^{-3}\sum_jw_j\frac{k_j^2f_j}{3a^2E_j(a)},\\ \Tr(a)&=a^{-3}\sum_jw_j\frac{y_j^2f_j}{E_j(a)}.
E_j^2Use in source · 2 passages
Search aliases: E_j^2
Use in source · origin_locking.tex, line 99
oxedκ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j with [k_j^2/a^2+y_j^2(1+q^2)]^1/2. Its first q derivative vanishes at zero and its second derivative is y_j^2/E…
Exact mathematical passage
E_j^2-k_j^2/a^2=y_j^2
\EzeroUse in source · 12 passages
Search aliases: Ezero E_0
Use in source · origin_locking.tex, line 71
D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
C=2S_E/\epsilon,\quad \Ezero=C/2,\quad S_E=2.18498795564.
FUse in source · 3 passages
Search aliases: F
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
\mathcal FUse in source · 6 passages
Search aliases: F mathcal
Use in source · origin_locking.tex, line 239
Force and pressure. The same action gives ρ_χ · =a^-3∑_jw_jΩ_jn_j, p_χ · =a^-3∑_jw_j[frack_j^2n_j3a^2Ω_j -(fracΩ_j2-frack_j^26a^2Ω_j)u_j], F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect t…
Exact mathematical passage
\rho_\chi&=a^{-3}\sum_jw_j\Omega_jn_j,\label{eq:rhomode}\\ p_\chi&=a^{-3}\sum_jw_j\left[\frac{k_j^2n_j}{3a^2\Omega_j} -\left(\frac{\Omega_j}{2}-\frac{k_j^2}{6a^2\Omega_j}\right)u_j\right],\label{eq:pmode}\\ \mathcal F&=a^{-3}\sum_jw_j\Omega_{j,q}(n_j+u_j/2).\label{eq:force}
fUse in source · 11 passages
Search aliases: f
Use in source · origin_locking.tex, line 78
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,
Exact mathematical passage
f_j
F'Use in source · 1 passage
Search aliases: F'
Use in source · origin_locking.tex, line 161
For the current hierarchy D/B_L is extremely small. Expanding the stationary equation gives s_*∼sqrtfrac2DB_LS, 1-fracV_R(sqrts_*)D∼frac4s_*3S. These estimates are supported by simple bounds. With α=2D/(B_LS), the inequality s^2/(1+s)≤ F'(s)≤ s^2 implies that s_* lies between the positive roots of s^2+(α/S)s-α=0 and s^2-α s-α=0.
Exact mathematical passage
s^2/(1+s)\le F'(s)\le s^2
f_jUse in source · 9 passages
Search aliases: f_j
Use in source · origin_locking.tex, line 78
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,
Exact mathematical passage
f_j
f_j^*Use in source · 2 passages
Search aliases: f_j^*
Use in source · origin_locking.tex, line 218
After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12,
Exact mathematical passage
a^3(f_j\dot f_j^*-f_j^*\dot f_j)=i.
GUse in source · 7 passages
Search aliases: G
Use in source · origin_locking.tex, line 378
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
G(s)=(1+s)^2\log(1+s)
G'Use in source · 3 passages
Search aliases: G'
Use in source · origin_locking.tex, line 398
er, curvature, and origin restriction] Every such quartic-matched potential satisfies Δ=fracSκ8-B_LJ(S), U_qq(0)=frac4S[B_LK(S)-Δ]. Consequently, retaining an origin maximum and a stationary vacuum at sqrt S requires Δ>B_LK(S). Stationarity gives b_0=-2c_0S-B_LG'(S). Substitute this into Δ=-b_0S-c_0S^2+B_L[G(0)-G(S)] and κ=4S[2c_0+B_LG''(S)]. Eliminating c_0 gives the first identity. Substitution into U_qq(0)=2[b_0+B_LG'(0)] gives the second. An origin maximum requires negative curvature, yieldi…
Exact mathematical passage
b_0=-2c_0S-B_LG'(S)
G''Use in source · 2 passages
Search aliases: G''
Use in source · origin_locking.tex, line 400
potential satisfies Δ=fracSκ8-B_LJ(S), U_qq(0)=frac4S[B_LK(S)-Δ]. Consequently, retaining an origin maximum and a stationary vacuum at sqrt S requires Δ>B_LK(S). Stationarity gives b_0=-2c_0S-B_LG'(S). Substitute this into Δ=-b_0S-c_0S^2+B_L[G(0)-G(S)] and κ=4S[2c_0+B_LG''(S)]. Eliminating c_0 gives the first identity. Substitution into U_qq(0)=2[b_0+B_LG'(0)] gives the second. An origin maximum requires negative curvature, yielding the strict inequality.
Exact mathematical passage
\kappa=4S[2c_0+B_LG''(S)]
H^2Use in source · 6 passages
Search aliases: H^2
Use in source · origin_locking.tex, line 266
dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_ju_j/2 and dotΩ_j=(y_j^2qdot q-Hk_j^2/a^2)/Ω_j. The terms multiplying Hn_j and Hu_j cancel against 3H(ρ_χ+p_χ) using [source reference]. The remaining term is dot q F.
Exact mathematical passage
3M^2H^2=\rho
H_iUse in source · 2 passages
Search aliases: H_i
Use in source · origin_locking.tex, line 367
The actual microscopic expansion scale. With the stored cosmological conversion inputs, the origin branch has A≃49.2467 eV and fracH_iA=3.35456×10^-25. Here H_i in the numerator is the dimensionful physical rate. The dimensionless solver uses that ratio. Over 0≤τ≤12, the accumulated log a is about 4.03×10^-24. The largest background-position difference from the flat-space run is 3.57×10^-15.
Exact mathematical passage
\frac{H_i}{A}=3.35456\times10^{-25}.
H_i^2Use in source · 1 passage
Search aliases: H_i^2
Use in source · origin_locking.tex, line 330
The stress test sets the initial dimensionless Hubble rate to 0.02 by changing the gravitational coupling at fixed initial densities. It is deliberately not the inherited physical cosmology. A representative L=15, z_max=16, 128-node run gives max_τ|fraca^3ρ+W_pρ(0)-1| · =4.75×10^-14, max_τ|fracH^2H_i^2-fracρρ(0)| · =3.77×10^-14, where W_p=∫3Ha^3p dτ. The Friedmann constraint is evolved, not reset algebraically at each step.
Exact mathematical passage
\max_\tau\left|\frac{a^3\rho+W_p}{\rho(0)}-1\right|&=4.75\times10^{-14},\\ \max_\tau\left|\frac{H^2}{H_i^2}-\frac{\rho}{\rho(0)}\right|&=3.77\times10^{-14},
iUse in source · 8 passages
Search aliases: i
Use in source · origin_locking.tex, line 96
_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j wit…
Exact mathematical passage
\Tr(a_i)>4D/S
JUse in source · 4 passages
Search aliases: J
Use in source · origin_locking.tex, line 384
and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
J(S)=\tfrac12S^2-S+\log(1+S),\quad K(S)=\tfrac12S^2+S-(1+S)\log(1+S).
jUse in source · 40 passages
Search aliases: j
Use in source · origin_locking.tex, line 78
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,
Exact mathematical passage
w_j
KUse in source · 9 passages
Search aliases: K
Use in source · origin_locking.tex, line 181
[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel
Exact mathematical passage
K_q/\Ezero
kUse in source · 12 passages
Search aliases: k
Use in source · origin_locking.tex, line 78
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,
Exact mathematical passage
k_j
K_h^2Use in source · 1 passage
Search aliases: K_h^2
Use in source · origin_locking.tex, line 298
The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient used in the solver is Z(a,q)=C+∑_h=1^Lfracd_hy_h^248π^2 fracq^21+q^2(1-A_h). It is positive. Its q and a derivatives enter both [source reference] and…
Exact mathematical passage
K_h^2(a)=y_hz_{\max}^2/a^2,\quad A_h=\left[\frac{K_h^2}{K_h^2+y_h^2(1+q^2)}\right]^{3/2}.
k_jUse in source · 1 passage
Search aliases: k_j
Use in source · origin_locking.tex, line 78
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,
Exact mathematical passage
k_j
k_j^2Use in source · 10 passages
Search aliases: k_j^2
Use in source · origin_locking.tex, line 79
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ_χ(a) · =a^-3∑_jw_jE_j(a)f_j, p_χ(a) · =a^-3∑_jw_jfrack_j^2f_j3a^2E_j(a), T(a) · =a^-3∑_jw_jfracy_j^2f_jE_j(a). These definitions describe a condition…
Exact mathematical passage
E_j(a)&=\sqrt{k_j^2/a^2+y_j^2},\\ \rho_\chi(a)&=a^{-3}\sum_jw_jE_j(a)f_j,\\ p_\chi(a)&=a^{-3}\sum_jw_j\frac{k_j^2f_j}{3a^2E_j(a)},\\ \Tr(a)&=a^{-3}\sum_jw_j\frac{y_j^2f_j}{E_j(a)}.
K_qUse in source · 1 passage
Search aliases: K_q
Use in source · origin_locking.tex, line 181
[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel
Exact mathematical passage
K_q/\Ezero
k_{\rm phys}Use in source · 1 passage
Search aliases: k_rm phys rm
Use in source · origin_locking.tex, line 319
At large physical momentum and dot q=0, W≃2H. With the continuum measure k_rm phys^2 d k_rm phys/(2π^2), the leading contribution per species behaves as ρ^(2)∼ fracH^2K_rm phys^216π^2. This asymptotic diagnostic is not inserted as an isolated numerical subtraction. A consistent subtraction must also determine pressure, force, and gravitational terms. Full stress-tensor regularization requires the appropriate curvature and higher adiabatic contributions [source reference]. Initial-state regularit…
Exact mathematical passage
k_{\rm phys}^2\dd k_{\rm phys}/(2\pi^2)
K_{\rm phys}^2Use in source · 1 passage
Search aliases: K_rm phys^2 rm
Use in source · origin_locking.tex, line 320
At large physical momentum and dot q=0, W≃2H. With the continuum measure k_rm phys^2 d k_rm phys/(2π^2), the leading contribution per species behaves as ρ^(2)∼ fracH^2K_rm phys^216π^2. This asymptotic diagnostic is not inserted as an isolated numerical subtraction. A consistent subtraction must also determine pressure, force, and gravitational terms. Full stress-tensor regularization requires the appropriate curvature and higher adiabatic contributions [source reference]. Initial-state regularit…
Exact mathematical passage
\rho^{(2)}\sim \frac{H^2K_{\rm phys}^2}{16\pi^2}.
k_{\rm phys}^2Use in source · 1 passage
Search aliases: k_rm phys^2 rm
Use in source · origin_locking.tex, line 319
At large physical momentum and dot q=0, W≃2H. With the continuum measure k_rm phys^2 d k_rm phys/(2π^2), the leading contribution per species behaves as ρ^(2)∼ fracH^2K_rm phys^216π^2. This asymptotic diagnostic is not inserted as an isolated numerical subtraction. A consistent subtraction must also determine pressure, force, and gravitational terms. Full stress-tensor regularization requires the appropriate curvature and higher adiabatic contributions [source reference]. Initial-state regularit…
Exact mathematical passage
k_{\rm phys}^2\dd k_{\rm phys}/(2\pi^2)
lUse in source · 2 passages
Search aliases: l
Use in source · origin_locking.tex, line 181
[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel
Exact mathematical passage
a_{\rm rel}
\logUse in source · 6 passages
Search aliases: log
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
MUse in source · 6 passages
Search aliases: M
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
M^2Use in source · 6 passages
Search aliases: M^2
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
M_{\rm Pl}Use in source · 1 passage
Search aliases: M_rm Pl rm
Use in source · origin_locking.tex, line 209
· +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for the physical conversion. The radiation term has density ρ_r=ρ_r0a^-4. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in U.
Exact mathematical passage
M^2=(M_{\rm Pl}/A)^2
\maxUse in source · 4 passages
Search aliases: max
Use in source · origin_locking.tex, line 298
The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient used in the solver is Z(a,q)=C+∑_h=1^Lfracd_hy_h^248π^2 fracq^21+q^2(1-A_h). It is positive. Its q and a derivatives enter both [source reference] and…
Exact mathematical passage
K_h^2(a)=y_hz_{\max}^2/a^2,\quad A_h=\left[\frac{K_h^2}{K_h^2+y_h^2(1+q^2)}\right]^{3/2}.
\max_\tauUse in source · 1 passage
Search aliases: max_τ max tau
Use in source · origin_locking.tex, line 330
The stress test sets the initial dimensionless Hubble rate to 0.02 by changing the gravitational coupling at fixed initial densities. It is deliberately not the inherited physical cosmology. A representative L=15, z_max=16, 128-node run gives max_τ|fraca^3ρ+W_pρ(0)-1| · =4.75×10^-14, max_τ|fracH^2H_i^2-fracρρ(0)| · =3.77×10^-14, where W_p=∫3Ha^3p dτ. The Friedmann constraint is evolved, not reset algebraically at each step.
Exact mathematical passage
\max_\tau\left|\frac{a^3\rho+W_p}{\rho(0)}-1\right|&=4.75\times10^{-14},\\ \max_\tau\left|\frac{H^2}{H_i^2}-\frac{\rho}{\rho(0)}\right|&=3.77\times10^{-14},
NUse in source · 3 passages
Search aliases: N
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
nUse in source · 12 passages
Search aliases: n
Use in source · origin_locking.tex, line 126
Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full coherent calculation.
Exact mathematical passage
n_j+u_j/2
n_jUse in source · 11 passages
Search aliases: n_j
Use in source · origin_locking.tex, line 126
Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full coherent calculation.
Exact mathematical passage
n_j+u_j/2
OUse in source · 2 passages
Search aliases: O
Use in source · origin_locking.tex, line 100
een -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j with [k_j^2/a^2+y_j^2(1+q^2)]^1/2. Its first q derivative vanishes at zero and its second derivative is y_j^2/E_j. Also F(q^2)=q^6/3+O(q^8), so the matched determinant contributes no quadratic term at the origin. Differentiating the tree term gives -4D/S, proving [source…
Exact mathematical passage
F(q^2)=q^6/3+O(q^8)
oUse in source · 2 passages
Search aliases: o
Use in source · origin_locking.tex, line 127
Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full coherent calculation.
Exact mathematical passage
\kappa_{\rm coh}=\frac{1}{2a^3}\sum_jw_j\frac{y_j^2}{E_j}u_j
PUse in source · 1 passage
Search aliases: P
Use in source · origin_locking.tex, line 209
· +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for the physical conversion. The radiation term has density ρ_r=ρ_r0a^-4. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in U.
Exact mathematical passage
M^2=(M_{\rm Pl}/A)^2
pUse in source · 18 passages
Search aliases: p
Use in source · origin_locking.tex, line 38
ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussian potential retains that condition, but its empty-sector minima release only about 1.14×10^-12 of the reference vacuum energy in the tested tower. A f…
Exact mathematical passage
\rho_\chi-3p_\chi
p_qUse in source · 3 passages
Search aliases: p_q
Use in source · origin_locking.tex, line 250
F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect to a before fixing the metric. It is not optional when the retained kinetic coefficient depends on the scale factor.
Exact mathematical passage
\rho_q=\frac12Z\dot q^2+U,\qquad p_q=\frac12Z\dot q^2-U+\frac16aZ_a\dot q^2.\label{eq:scalarstress}
p_\chiUse in source · 8 passages
Search aliases: p_χ chi
Use in source · origin_locking.tex, line 38
ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussian potential retains that condition, but its empty-sector minima release only about 1.14×10^-12 of the reference vacuum energy in the tested tower. A f…
Exact mathematical passage
\rho_\chi-3p_\chi
qUse in source · 41 passages
Search aliases: q
Use in source · origin_locking.tex, line 45
The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are different physical questions. Accessibility of q=4 does not decide whether matter can stabilize the origin.
Exact mathematical passage
q=\sqrt\gamma\,\phi/v,\qquad \gamma=S=16,
q^2Use in source · 9 passages
Search aliases: q^2
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
q^4Use in source · 1 passage
Search aliases: q^4
Use in source · origin_locking.tex, line 378
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
q^4
q^6Use in source · 1 passage
Search aliases: q^6
Use in source · origin_locking.tex, line 100
een -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j with [k_j^2/a^2+y_j^2(1+q^2)]^1/2. Its first q derivative vanishes at zero and its second derivative is y_j^2/E_j. Also F(q^2)=q^6/3+O(q^8), so the matched determinant contributes no quadratic term at the origin. Differentiating the tree term gives -4D/S, proving [source…
Exact mathematical passage
F(q^2)=q^6/3+O(q^8)
q^8Use in source · 1 passage
Search aliases: q^8
Use in source · origin_locking.tex, line 100
een -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j with [k_j^2/a^2+y_j^2(1+q^2)]^1/2. Its first q derivative vanishes at zero and its second derivative is y_j^2/E_j. Also F(q^2)=q^6/3+O(q^8), so the matched determinant contributes no quadratic term at the origin. Differentiating the tree term gives -4D/S, proving [source…
Exact mathematical passage
F(q^2)=q^6/3+O(q^8)
q_*Use in source · 1 passage
Search aliases: q_*
Use in source · origin_locking.tex, line 165
A 70-digit stationary-point calculation for L=15 gives q_* · =3.70470008878×10^-6, fracV_R(q_*)D · =0.9999999999988562664, fracD-V_R(q_*)D=1.14373356231×10^-12. Extra arithmetic digits do not express physical parameter accuracy. They prevent cancellation from obscuring the very small difference of two potential energies.
Exact mathematical passage
q_*&=3.70470008878\times10^{-6},\\ \frac{V_R(q_*)}{D}&=0.9999999999988562664,\\ \boxed{\frac{D-V_R(q_*)}{D}=1.14373356231\times10^{-12}.}\label{eq:drop}
RUse in source · 4 passages
Search aliases: R
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
rUse in source · 3 passages
Search aliases: r
Use in source · origin_locking.tex, line 181
[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel
Exact mathematical passage
a_{\rm rel}
R_SUse in source · 1 passage
Search aliases: R_S
Use in source · origin_locking.tex, line 407
A constructive but altered branch. The restriction also gives a constructive family. Let R_S(s) · =G(s)-G(S)-G'(S)(s-S)-tfrac12G''(S)(s-S)^2, U_Δ(s) · =fracΔ+B_LJ(S)S^2(s-S)^2+B_LR_S(s). Then U_Δ(S)=U_Δ,s(S)=0 and U_Δ(0)=Δ.
Exact mathematical passage
R_S(s)&=G(s)-G(S)-G'(S)(s-S)-\tfrac12G''(S)(s-S)^2,\\ U_\Delta(s)&=\frac{\Delta+B_LJ(S)}{S^2}(s-S)^2+B_LR_S(s).
\operatorname{Re}Use in source · 1 passage
Search aliases: Re operatorname
Use in source · origin_locking.tex, line 222
After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12, u_j · =a^3(Ω_j|f_j|^2-frac|dot f_j|^2Ω_j), v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j,
Exact mathematical passage
n_j&=\frac{a^3(|\dot f_j|^2+\Omega_j^2|f_j|^2)}{2\Omega_j}-\frac12,\\ u_j&=a^3\left(\Omega_j|f_j|^2-\frac{|\dot f_j|^2}{\Omega_j}\right),\\ v_j&=-2a^3\operatorname{Re}(f_j^*\dot f_j).
\rmUse in source · 8 passages
Search aliases: rm
Use in source · origin_locking.tex, line 93
[Local locking criterion and dilution] For the above gas, T=ρ_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter…
Exact mathematical passage
\boxed{\kappa_{\rm eff}(a)=\rho_\chi(a)-3p_\chi(a)-\frac{4D}{S}.}\label{eq:locking}
SUse in source · 32 passages
Search aliases: S
Use in source · origin_locking.tex, line 45
The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are different physical questions. Accessibility of q=4 does not decide whether matter can stabilize the origin.
Exact mathematical passage
q=\sqrt\gamma\,\phi/v,\qquad \gamma=S=16,
sUse in source · 29 passages
Search aliases: s
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
S^2Use in source · 5 passages
Search aliases: S^2
Use in source · origin_locking.tex, line 147
[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s], V_ss · =frac2DS^2+2B_Llog(1+s)>0. Hence V_s is strictly increasing, starts negative, and tends to infinity. It has a unique zero. The potential decreases before that zero and increases afterward. It also tends to infinity. This proves the…
Exact mathematical passage
V_s&=-\frac{2D}{S}+\frac{2Ds}{S^2} +2B_L[(1+s)\log(1+s)-s],\\ V_{ss}&=\frac{2D}{S^2}+2B_L\log(1+s)>0.
s^2Use in source · 5 passages
Search aliases: s^2
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
s_*Use in source · 3 passages
Search aliases: s_*
Use in source · origin_locking.tex, line 143
[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s],
Exact mathematical passage
q=\pm\sqrt{s_*}
S_EUse in source · 1 passage
Search aliases: S_E
Use in source · origin_locking.tex, line 71
D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
C=2S_E/\epsilon,\quad \Ezero=C/2,\quad S_E=2.18498795564.
\sinUse in source · 1 passage
Search aliases: sin
Use in source · origin_locking.tex, line 132
For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet stability theorem or a derivation of cosmological decoherence.
Exact mathematical passage
u_j(t)=u_j(0)\cos(2E_jt)-v_j(0)\sin(2E_jt)
TUse in source · 3 passages
Search aliases: T
Use in source · origin_locking.tex, line 103
Write a nonzero trace summand as T_j(a)=w_jy_j^2f_j a^-2(k_j^2+a^2y_j^2)^-1/2. Then frac dlog T_j dlog a=-2-fraca^2y_j^2k_j^2+a^2y_j^2. This is strictly negative and lies in [-3,-2]. The finite sum is continuous, strictly decreases, and tends to zero. The intermediate value theorem gives existence and strict monotonicity gives uniqueness of the threshold crossing.
Exact mathematical passage
T_j(a)=w_jy_j^2f_j\,a^{-2}(k_j^2+a^2y_j^2)^{-1/2}.
T_jUse in source · 2 passages
Search aliases: T_j
Use in source · origin_locking.tex, line 103
Write a nonzero trace summand as T_j(a)=w_jy_j^2f_j a^-2(k_j^2+a^2y_j^2)^-1/2. Then frac dlog T_j dlog a=-2-fraca^2y_j^2k_j^2+a^2y_j^2. This is strictly negative and lies in [-3,-2]. The finite sum is continuous, strictly decreases, and tends to zero. The intermediate value theorem gives existence and strict monotonicity gives uniqueness of the threshold crossing.
Exact mathematical passage
T_j(a)=w_jy_j^2f_j\,a^{-2}(k_j^2+a^2y_j^2)^{-1/2}.
\TrUse in source · 7 passages
Search aliases: Tr T mathcal
Use in source · origin_locking.tex, line 79
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ_χ(a) · =a^-3∑_jw_jE_j(a)f_j, p_χ(a) · =a^-3∑_jw_jfrack_j^2f_j3a^2E_j(a), T(a) · =a^-3∑_jw_jfracy_j^2f_jE_j(a). These definitions describe a condition…
Exact mathematical passage
E_j(a)&=\sqrt{k_j^2/a^2+y_j^2},\\ \rho_\chi(a)&=a^{-3}\sum_jw_jE_j(a)f_j,\\ p_\chi(a)&=a^{-3}\sum_jw_j\frac{k_j^2f_j}{3a^2E_j(a)},\\ \Tr(a)&=a^{-3}\sum_jw_j\frac{y_j^2f_j}{E_j(a)}.
UUse in source · 24 passages
Search aliases: U
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
uUse in source · 12 passages
Search aliases: u
Use in source · origin_locking.tex, line 126
Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full coherent calculation.
Exact mathematical passage
n_j+u_j/2
u^2Use in source · 1 passage
Search aliases: u^2
Use in source · origin_locking.tex, line 431
The first failed gate is momentum-cutoff acceptance in the expansion stress test. The second is absolute preservation of the mode hyperboloid in highly occupied altered-model runs. The tight altered-model run reaches occupation of order 10^5. Evaluating (1+2n)^2-u^2-v^2-1 then subtracts large terms to resolve a value near zero. Its maximum absolute residual is 6.84738×10^-4, above the declared 10^-5 gate. Tightening the integration tolerance did not repair this diagnostic. It remains a failure r…
Exact mathematical passage
(1+2n)^2-u^2-v^2-1
u_jUse in source · 10 passages
Search aliases: u_j
Use in source · origin_locking.tex, line 126
Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full coherent calculation.
Exact mathematical passage
n_j+u_j/2
u_j^2Use in source · 1 passage
Search aliases: u_j^2
Use in source · origin_locking.tex, line 290
The mode equations also preserve (1+2n_j)^2-u_j^2-v_j^2=1. For a normalized pure initial state on the positive sheet, continuity keeps 1+2n_j≥1, so n_j≥0. This proves positivity in the exact finite equations. Floating-point preservation is a separate numerical check.
Exact mathematical passage
(1+2n_j)^2-u_j^2-v_j^2=1
U_qUse in source · 3 passages
Search aliases: U_q
Use in source · origin_locking.tex, line 246
F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect to a before fixing the metric. It is not optional when the retained kinetic coefficient depends on the scale factor.
Exact mathematical passage
Z\ddot q+H(3Z+aZ_a)\dot q+\frac12Z_q\dot q^2+U_q+\mathcal F=0.\label{eq:background}
U_{qq}Use in source · 3 passages
Search aliases: U_qq
Use in source · origin_locking.tex, line 383
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
\kappa=U_{qq}(\sqrt S)
U_sUse in source · 5 passages
Search aliases: U_s
Use in source · origin_locking.tex, line 382
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
U_s(S)=0
U_{ss}Use in source · 2 passages
Search aliases: U_ss
Use in source · origin_locking.tex, line 418
[An accessible double well under changed matching] If B_LK(S)<Δ<E_0, U_Δ(q^2) is nonnegative, has its only global minima at q=±sqrt S, and has an origin maximum. The origin barrier lies within the declared budget. Theorem~[source reference] gives U_s(0)<0. Also U_s(S)=0 and U_ss(S)=2[Δ+B_LJ(S)]/S^2>0. The function U_s is strictly convex because U_sss=2B_L/(1+s)>0. On 0<s<S it lies below its chord between a negative value and zero, so it is negative. For s>S, U_ss increases from a positive value,…
Exact mathematical passage
U_{ss}(S)=2[\Delta+B_LJ(S)]/S^2>0
U_{sss}Use in source · 1 passage
Search aliases: U_sss
Use in source · origin_locking.tex, line 418
uble well under changed matching] If B_LK(S)<Δ<E_0, U_Δ(q^2) is nonnegative, has its only global minima at q=±sqrt S, and has an origin maximum. The origin barrier lies within the declared budget. Theorem~[source reference] gives U_s(0)<0. Also U_s(S)=0 and U_ss(S)=2[Δ+B_LJ(S)]/S^2>0. The function U_s is strictly convex because U_sss=2B_L/(1+s)>0. On 0<s<S it lies below its chord between a negative value and zero, so it is negative. For s>S, U_ss increases from a positive value, so U_s>0. The po…
Exact mathematical passage
U_{sss}=2B_L/(1+s)>0
U_\DeltaUse in source · 4 passages
Search aliases: U_Δ Delta
Use in source · origin_locking.tex, line 407
A constructive but altered branch. The restriction also gives a constructive family. Let R_S(s) · =G(s)-G(S)-G'(S)(s-S)-tfrac12G''(S)(s-S)^2, U_Δ(s) · =fracΔ+B_LJ(S)S^2(s-S)^2+B_LR_S(s). Then U_Δ(S)=U_Δ,s(S)=0 and U_Δ(0)=Δ.
Exact mathematical passage
R_S(s)&=G(s)-G(S)-G'(S)(s-S)-\tfrac12G''(S)(s-S)^2,\\ U_\Delta(s)&=\frac{\Delta+B_LJ(S)}{S^2}(s-S)^2+B_LR_S(s).
U_{\Delta,s}Use in source · 1 passage
Search aliases: U_Δ,s Delta
Use in source · origin_locking.tex, line 411
U_Δ(s) · =fracΔ+B_LJ(S)S^2(s-S)^2+B_LR_S(s). Then U_Δ(S)=U_Δ,s(S)=0 and U_Δ(0)=Δ.
Exact mathematical passage
U_\Delta(S)=U_{\Delta,s}(S)=0
VUse in source · 5 passages
Search aliases: V
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
vUse in source · 9 passages
Search aliases: v
Use in source · origin_locking.tex, line 44
The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are different physical questions. Accessibility of q=4 does not decide whether matter can stabilize the origin.
Exact mathematical passage
\phi=\pm v
v^2Use in source · 1 passage
Search aliases: v^2
Use in source · origin_locking.tex, line 431
The first failed gate is momentum-cutoff acceptance in the expansion stress test. The second is absolute preservation of the mode hyperboloid in highly occupied altered-model runs. The tight altered-model run reaches occupation of order 10^5. Evaluating (1+2n)^2-u^2-v^2-1 then subtracts large terms to resolve a value near zero. Its maximum absolute residual is 6.84738×10^-4, above the declared 10^-5 gate. Tightening the integration tolerance did not repair this diagnostic. It remains a failure r…
Exact mathematical passage
(1+2n)^2-u^2-v^2-1
v_jUse in source · 5 passages
Search aliases: v_j
Use in source · origin_locking.tex, line 132
For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet stability theorem or a derivation of cosmological decoherence.
Exact mathematical passage
u_j(t)=u_j(0)\cos(2E_jt)-v_j(0)\sin(2E_jt)
v_j^2Use in source · 1 passage
Search aliases: v_j^2
Use in source · origin_locking.tex, line 290
The mode equations also preserve (1+2n_j)^2-u_j^2-v_j^2=1. For a normalized pure initial state on the positive sheet, continuity keeps 1+2n_j≥1, so n_j≥0. This proves positivity in the exact finite equations. Floating-point preservation is a separate numerical check.
Exact mathematical passage
(1+2n_j)^2-u_j^2-v_j^2=1
V_RUse in source · 3 passages
Search aliases: V_R
Use in source · origin_locking.tex, line 65
The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\ F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\ D&=4.65190089919765\times10^{-18}.
V_sUse in source · 2 passages
Search aliases: V_s
Use in source · origin_locking.tex, line 147
[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s], V_ss · =frac2DS^2+2B_Llog(1+s)>0. Hence V_s is strictly increasing, starts negative, and tends to infinity. It has a unique zero. The potential decreases before that zero and increases afterward. It also tends to infinity. This proves the…
Exact mathematical passage
V_s&=-\frac{2D}{S}+\frac{2Ds}{S^2} +2B_L[(1+s)\log(1+s)-s],\\ V_{ss}&=\frac{2D}{S^2}+2B_L\log(1+s)>0.
V_{ss}Use in source · 1 passage
Search aliases: V_ss
Use in source · origin_locking.tex, line 147
[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s], V_ss · =frac2DS^2+2B_Llog(1+s)>0. Hence V_s is strictly increasing, starts negative, and tends to infinity. It has a unique zero. The potential decreases before that zero and increases afterward. It also tends to infinity. This proves the…
Exact mathematical passage
V_s&=-\frac{2D}{S}+\frac{2Ds}{S^2} +2B_L[(1+s)\log(1+s)-s],\\ V_{ss}&=\frac{2D}{S^2}+2B_L\log(1+s)>0.
WUse in source · 8 passages
Search aliases: W
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
wUse in source · 11 passages
Search aliases: w
Use in source · origin_locking.tex, line 78
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,
Exact mathematical passage
w_j
W_jUse in source · 3 passages
Search aliases: W_j
Use in source · origin_locking.tex, line 228
v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
Exact mathematical passage
\dot n_j&=\tfrac12W_ju_j,& \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,& \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\ W_j&=3H+\frac{\dot\Omega_j}{\Omega_j} =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
w_jUse in source · 11 passages
Search aliases: w_j
Use in source · origin_locking.tex, line 78
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,
Exact mathematical passage
w_j
W_j^2Use in source · 1 passage
Search aliases: W_j^2
Use in source · origin_locking.tex, line 311
A diagnostic for missing terms. The leading adiabatic response of [source reference] has v_j≃ W_j/(2Ω_j) and n_j≃ W_j^2/(16Ω_j^2), with u_j at higher derivative order. Thus the corresponding second-order mode density is ρ_χ^(2)≃frac116a^3∑_jfracw_jΩ_j [fracΩ_j,qΩ_jdot q+ (3-frack_j^2a^2Ω_j^2)H]^2. This contains terms proportional to dot q^2, Hdot q, and H^2. Equation~[source reference] retains only the flat-background dot q^2 response. Including its scale-factor derivative correctly preserves th…
Exact mathematical passage
n_j\simeq W_j^2/(16\Omega_j^2)
W_pUse in source · 2 passages
Search aliases: W_p
Use in source · origin_locking.tex, line 330
The stress test sets the initial dimensionless Hubble rate to 0.02 by changing the gravitational coupling at fixed initial densities. It is deliberately not the inherited physical cosmology. A representative L=15, z_max=16, 128-node run gives max_τ|fraca^3ρ+W_pρ(0)-1| · =4.75×10^-14, max_τ|fracH^2H_i^2-fracρρ(0)| · =3.77×10^-14, where W_p=∫3Ha^3p dτ. The Friedmann constraint is evolved, not reset algebraically at each step.
Exact mathematical passage
\max_\tau\left|\frac{a^3\rho+W_p}{\rho(0)}-1\right|&=4.75\times10^{-14},\\ \max_\tau\left|\frac{H^2}{H_i^2}-\frac{\rho}{\rho(0)}\right|&=3.77\times10^{-14},
y_h^2Use in source · 2 passages
Search aliases: y_h^2
Use in source · origin_locking.tex, line 298
The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient used in the solver is Z(a,q)=C+∑_h=1^Lfracd_hy_h^248π^2 fracq^21+q^2(1-A_h). It is positive. Its q and a derivatives enter both [source reference] and…
Exact mathematical passage
K_h^2(a)=y_hz_{\max}^2/a^2,\quad A_h=\left[\frac{K_h^2}{K_h^2+y_h^2(1+q^2)}\right]^{3/2}.
y_jUse in source · 2 passages
Search aliases: y_j
Use in source · origin_locking.tex, line 78
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,
Exact mathematical passage
y_j>0
y_j^2Use in source · 11 passages
Search aliases: y_j^2
Use in source · origin_locking.tex, line 79
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ_χ(a) · =a^-3∑_jw_jE_j(a)f_j, p_χ(a) · =a^-3∑_jw_jfrack_j^2f_j3a^2E_j(a), T(a) · =a^-3∑_jw_jfracy_j^2f_jE_j(a). These definitions describe a condition…
Exact mathematical passage
E_j(a)&=\sqrt{k_j^2/a^2+y_j^2},\\ \rho_\chi(a)&=a^{-3}\sum_jw_jE_j(a)f_j,\\ p_\chi(a)&=a^{-3}\sum_jw_j\frac{k_j^2f_j}{3a^2E_j(a)},\\ \Tr(a)&=a^{-3}\sum_jw_j\frac{y_j^2f_j}{E_j(a)}.
ZUse in source · 11 passages
Search aliases: Z
Use in source · origin_locking.tex, line 201
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber
Exact mathematical passage
Z(a,q)>0
zUse in source · 3 passages
Search aliases: z
Use in source · origin_locking.tex, line 298
The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient used in the solver is Z(a,q)=C+∑_h=1^Lfracd_hy_h^248π^2 fracq^21+q^2(1-A_h). It is positive. Its q and a derivatives enter both [source reference] and…
Exact mathematical passage
K_h^2(a)=y_hz_{\max}^2/a^2,\quad A_h=\left[\frac{K_h^2}{K_h^2+y_h^2(1+q^2)}\right]^{3/2}.
Z_aUse in source · 5 passages
Search aliases: Z_a
Use in source · origin_locking.tex, line 246
F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect to a before fixing the metric. It is not optional when the retained kinetic coefficient depends on the scale factor.
Exact mathematical passage
Z\ddot q+H(3Z+aZ_a)\dot q+\frac12Z_q\dot q^2+U_q+\mathcal F=0.\label{eq:background}
z_{\max}Use in source · 2 passages
Search aliases: z_max max
Use in source · origin_locking.tex, line 329
The stress test sets the initial dimensionless Hubble rate to 0.02 by changing the gravitational coupling at fixed initial densities. It is deliberately not the inherited physical cosmology. A representative L=15, z_max=16, 128-node run gives max_τ|fraca^3ρ+W_pρ(0)-1| · =4.75×10^-14,
Exact mathematical passage
z_{\max}=16
z_{\max}^2Use in source · 1 passage
Search aliases: z_max^2 max
Use in source · origin_locking.tex, line 298
The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient used in the solver is Z(a,q)=C+∑_h=1^Lfracd_hy_h^248π^2 fracq^21+q^2(1-A_h). It is positive. Its q and a derivatives enter both [source reference] and…
Exact mathematical passage
K_h^2(a)=y_hz_{\max}^2/a^2,\quad A_h=\left[\frac{K_h^2}{K_h^2+y_h^2(1+q^2)}\right]^{3/2}.
Z_qUse in source · 3 passages
Search aliases: Z_q
Use in source · origin_locking.tex, line 246
F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect to a before fixing the metric. It is not optional when the retained kinetic coefficient depends on the scale factor.
Exact mathematical passage
Z\ddot q+H(3Z+aZ_a)\dot q+\frac12Z_q\dot q^2+U_q+\mathcal F=0.\label{eq:background}
|Use in source · 4 passages
Search aliases: |
Use in source · origin_locking.tex, line 133
For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet stability theorem or a derivation of cosmological decoherence.
Exact mathematical passage
\left|\frac1T\int_0^T\kappa_{\rm coh}(t)\,\dd t\right| \le\frac{1}{2a^3T}\sum_jw_j\frac{y_j^2}{E_j^2}\sqrt{u_j(0)^2+v_j(0)^2}.
|^2Use in source · 2 passages
Search aliases: |^2
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
\pmUse in source · 4 passages
Search aliases: ± pm
Use in source · origin_locking.tex, line 44
The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are different physical questions. Accessibility of q=4 does not decide whether matter can stabilize the origin.
Exact mathematical passage
\phi=\pm v
\timesUse in source · 18 passages
Search aliases: × times
Use in source · origin_locking.tex, line 38
transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussian potential retains that condition, but its empty-sector minima release only about 1.14×10^-12 of the reference vacuum energy in the tested tower. A finite expanding-space action gives an exact pressure-work balance and a working numerical implementation. …
Exact mathematical passage
1.14\times10^{-12}
\alphaUse in source · 3 passages
Search aliases: α alpha
Use in source · origin_locking.tex, line 160
For the current hierarchy D/B_L is extremely small. Expanding the stationary equation gives s_*∼sqrtfrac2DB_LS, 1-fracV_R(sqrts_*)D∼frac4s_*3S. These estimates are supported by simple bounds. With α=2D/(B_LS), the inequality s^2/(1+s)≤ F'(s)≤ s^2 implies that s_* lies between the positive roots of s^2+(α/S)s-α=0 and s^2-α s-α=0.
Exact mathematical passage
\alpha=2D/(B_LS)
\gammaUse in source · 1 passage
Search aliases: γ gamma
Use in source · origin_locking.tex, line 45
The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are different physical questions. Accessibility of q=4 does not decide whether matter can stabilize the origin.
Exact mathematical passage
q=\sqrt\gamma\,\phi/v,\qquad \gamma=S=16,
\DeltaUse in source · 12 passages
Search aliases: Δ Delta
Use in source · origin_locking.tex, line 383
Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+S), K(S)=tfrac12S^2+S-(1+S)log(1+S).
Exact mathematical passage
\Delta=U(0)-U(S)
\epsilonUse in source · 3 passages
Search aliases: ϵ epsilon
Use in source · origin_locking.tex, line 71
D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fit.
Exact mathematical passage
C=2S_E/\epsilon,\quad \Ezero=C/2,\quad S_E=2.18498795564.
\kappaUse in source · 7 passages
Search aliases: κ kappa
Use in source · origin_locking.tex, line 93
[Local locking criterion and dilution] For the above gas, T=ρ_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter…
Exact mathematical passage
\boxed{\kappa_{\rm eff}(a)=\rho_\chi(a)-3p_\chi(a)-\frac{4D}{S}.}\label{eq:locking}
\kappa_{\rm coh}Use in source · 2 passages
Search aliases: κ_rm coh kappa rm
Use in source · origin_locking.tex, line 127
Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full coherent calculation.
Exact mathematical passage
\kappa_{\rm coh}=\frac{1}{2a^3}\sum_jw_j\frac{y_j^2}{E_j}u_j
\kappa_{\rm eff}Use in source · 2 passages
Search aliases: κ_rm eff kappa rm
Use in source · origin_locking.tex, line 93
[Local locking criterion and dilution] For the above gas, T=ρ_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter…
Exact mathematical passage
\boxed{\kappa_{\rm eff}(a)=\rho_\chi(a)-3p_\chi(a)-\frac{4D}{S}.}\label{eq:locking}
\lambdaUse in source · 1 passage
Search aliases: λ lambda
Use in source · origin_locking.tex, line 462
55--56 for conditional locking and release. bibitemenergy Komorebi P. Energy-Conserving Production: A finite-depth backreaction study. Companion research manuscript. bibitemmatched Komorebi P. Vacuum Matching and Dynamical Energy Transfer. Companion research manuscript and numerical source package. bibitemmolina C. Molina-Par is, P. R. Anderson, and S. A. Ramsey. One-loop λϕ^4 theory in Robertson--Walker spacetimes: adiabatic regularization and analytic approximation. Physical Review D 61, 12750…
Exact mathematical passage
\lambda\phi^4
\rhoUse in source · 24 passages
Search aliases: ρ rho
Use in source · origin_locking.tex, line 38
ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussian potential retains that condition, but its empty-sector minima release only about 1.14×10^-12 of the reference vacuum energy in the tested tower. A f…
Exact mathematical passage
\rho_\chi-3p_\chi
\rho^{(2)}Use in source · 1 passage
Search aliases: ρ^(2) rho
Use in source · origin_locking.tex, line 320
At large physical momentum and dot q=0, W≃2H. With the continuum measure k_rm phys^2 d k_rm phys/(2π^2), the leading contribution per species behaves as ρ^(2)∼ fracH^2K_rm phys^216π^2. This asymptotic diagnostic is not inserted as an isolated numerical subtraction. A consistent subtraction must also determine pressure, force, and gravitational terms. Full stress-tensor regularization requires the appropriate curvature and higher adiabatic contributions [source reference]. Initial-state regularit…
Exact mathematical passage
\rho^{(2)}\sim \frac{H^2K_{\rm phys}^2}{16\pi^2}.
\rho_qUse in source · 3 passages
Search aliases: ρ_q rho
Use in source · origin_locking.tex, line 250
F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect to a before fixing the metric. It is not optional when the retained kinetic coefficient depends on the scale factor.
Exact mathematical passage
\rho_q=\frac12Z\dot q^2+U,\qquad p_q=\frac12Z\dot q^2-U+\frac16aZ_a\dot q^2.\label{eq:scalarstress}
\rho_rUse in source · 1 passage
Search aliases: ρ_r rho
Use in source · origin_locking.tex, line 209
· +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for the physical conversion. The radiation term has density ρ_r=ρ_r0a^-4. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in U.
Exact mathematical passage
\rho_r=\rho_{r0}a^{-4}
\rho_{r0}Use in source · 2 passages
Search aliases: ρ_r0 rho
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
\rho_\chiUse in source · 10 passages
Search aliases: ρ_χ rho chi
Use in source · origin_locking.tex, line 38
ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussian potential retains that condition, but its empty-sector minima release only about 1.14×10^-12 of the reference vacuum energy in the tested tower. A f…
Exact mathematical passage
\rho_\chi-3p_\chi
\rho_{\chi,0}Use in source · 1 passage
Search aliases: ρ_χ,0 rho chi
Use in source · origin_locking.tex, line 181
[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel
Exact mathematical passage
\rho_{\chi,0}/D
\rho_\chi^{(2)}Use in source · 1 passage
Search aliases: ρ_χ^(2) rho chi
Use in source · origin_locking.tex, line 312
A diagnostic for missing terms. The leading adiabatic response of [source reference] has v_j≃ W_j/(2Ω_j) and n_j≃ W_j^2/(16Ω_j^2), with u_j at higher derivative order. Thus the corresponding second-order mode density is ρ_χ^(2)≃frac116a^3∑_jfracw_jΩ_j [fracΩ_j,qΩ_jdot q+ (3-frack_j^2a^2Ω_j^2)H]^2. This contains terms proportional to dot q^2, Hdot q, and H^2. Equation~[source reference] retains only the flat-background dot q^2 response. Including its scale-factor derivative correctly preserves th…
Exact mathematical passage
\rho_\chi^{(2)}\simeq\frac{1}{16a^3}\sum_j\frac{w_j}{\Omega_j} \left[\frac{\Omega_{j,q}}{\Omega_j}\dot q+ \left(3-\frac{k_j^2}{a^2\Omega_j^2}\right)H\right]^2.\label{eq:adiabatic}
\phiUse in source · 4 passages
Search aliases: ϕ phi
Use in source · origin_locking.tex, line 44
The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are different physical questions. Accessibility of q=4 does not decide whether matter can stabilize the origin.
Exact mathematical passage
\phi=0
\phi^4Use in source · 1 passage
Search aliases: ϕ^4 phi
Use in source · origin_locking.tex, line 462
55--56 for conditional locking and release. bibitemenergy Komorebi P. Energy-Conserving Production: A finite-depth backreaction study. Companion research manuscript. bibitemmatched Komorebi P. Vacuum Matching and Dynamical Energy Transfer. Companion research manuscript and numerical source package. bibitemmolina C. Molina-Par is, P. R. Anderson, and S. A. Ramsey. One-loop λϕ^4 theory in Robertson--Walker spacetimes: adiabatic regularization and analytic approximation. Physical Review D 61, 12750…
Exact mathematical passage
\lambda\phi^4
\chiUse in source · 12 passages
Search aliases: χ chi
Use in source · origin_locking.tex, line 38
ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussian potential retains that condition, but its empty-sector minima release only about 1.14×10^-12 of the reference vacuum energy in the tested tower. A f…
Exact mathematical passage
\rho_\chi-3p_\chi
\OmegaUse in source · 10 passages
Search aliases: Ω Omega
Use in source · origin_locking.tex, line 198
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber
Exact mathematical passage
\Omega_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2).
\Omega_jUse in source · 7 passages
Search aliases: Ω_j Omega
Use in source · origin_locking.tex, line 202
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber · +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for th…
Exact mathematical passage
L={}&-\frac{3M^2a\dot a^2}{N} +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right] -\frac{N\rho_{r0}}{a}\nonumber\\ &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2 -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
\Omega_{j,q}Use in source · 2 passages
Search aliases: Ω_j,q Omega
Use in source · origin_locking.tex, line 239
Force and pressure. The same action gives ρ_χ · =a^-3∑_jw_jΩ_jn_j, p_χ · =a^-3∑_jw_j[frack_j^2n_j3a^2Ω_j -(fracΩ_j2-frack_j^26a^2Ω_j)u_j], F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 with respect t…
Exact mathematical passage
\rho_\chi&=a^{-3}\sum_jw_j\Omega_jn_j,\label{eq:rhomode}\\ p_\chi&=a^{-3}\sum_jw_j\left[\frac{k_j^2n_j}{3a^2\Omega_j} -\left(\frac{\Omega_j}{2}-\frac{k_j^2}{6a^2\Omega_j}\right)u_j\right],\label{eq:pmode}\\ \mathcal F&=a^{-3}\sum_jw_j\Omega_{j,q}(n_j+u_j/2).\label{eq:force}
\Omega_j^2Use in source · 7 passages
Search aliases: Ω_j^2 Omega
Use in source · origin_locking.tex, line 198
An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(fracdot qN)^2-U(q)] -fracNρ_r0anonumber
Exact mathematical passage
\Omega_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2).
\sum_{h=1}^LUse in source · 1 passage
Search aliases: ∑_h=1^L sum
Use in source · origin_locking.tex, line 303
tained term The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient used in the solver is Z(a,q)=C+∑_h=1^Lfracd_hy_h^248π^2 fracq^21+q^2(1-A_h). It is positive. Its q and a derivatives enter both [source reference] and [source reference]. The explicit modes and this …
Exact mathematical passage
Z(a,q)=C+\sum_{h=1}^L\frac{d_hy_h^2}{48\pi^2} \frac{q^2}{1+q^2}(1-A_h).\label{eq:tail}
\sum_jUse in source · 7 passages
Search aliases: ∑_j sum
Use in source · origin_locking.tex, line 79
Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ_χ(a) · =a^-3∑_jw_jE_j(a)f_j, p_χ(a) · =a^-3∑_jw_jfrack_j^2f_j3a^2E_j(a), T(a) · =a^-3∑_jw_jfracy_j^2f_jE_j(a). These definitions describe a condition…
Exact mathematical passage
E_j(a)&=\sqrt{k_j^2/a^2+y_j^2},\\ \rho_\chi(a)&=a^{-3}\sum_jw_jE_j(a)f_j,\\ p_\chi(a)&=a^{-3}\sum_jw_j\frac{k_j^2f_j}{3a^2E_j(a)},\\ \Tr(a)&=a^{-3}\sum_jw_j\frac{y_j^2f_j}{E_j(a)}.
\intUse in source · 3 passages
Search aliases: ∫ int
Use in source · origin_locking.tex, line 133
For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet stability theorem or a derivation of cosmological decoherence.
Exact mathematical passage
\left|\frac1T\int_0^T\kappa_{\rm coh}(t)\,\dd t\right| \le\frac{1}{2a^3T}\sum_jw_j\frac{y_j^2}{E_j^2}\sqrt{u_j(0)^2+v_j(0)^2}.
\int_0^TUse in source · 1 passage
Search aliases: ∫_0^T int
Use in source · origin_locking.tex, line 133
For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet stability theorem or a derivation of cosmological decoherence.
Exact mathematical passage
\left|\frac1T\int_0^T\kappa_{\rm coh}(t)\,\dd t\right| \le\frac{1}{2a^3T}\sum_jw_j\frac{y_j^2}{E_j^2}\sqrt{u_j(0)^2+v_j(0)^2}.
\int_0^\tauUse in source · 1 passage
Search aliases: ∫_0^τ int tau
Use in source · origin_locking.tex, line 267
dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_ju_j/2 and dotΩ_j=(y_j^2qdot q-Hk_j^2/a^2)/Ω_j. The terms multiplying Hn_j and Hu_j cancel against 3H(ρ_χ+p_χ) using [source reference]. The remaining term is dot q F.
Exact mathematical passage
\boxed{a^3(\tau)\rho(\tau)+\int_0^\tau3Ha^3p\,\dd t=\rho(0)}\label{eq:work}
\simUse in source · 2 passages
Search aliases: ∼ sim
Use in source · origin_locking.tex, line 156
For the current hierarchy D/B_L is extremely small. Expanding the stationary equation gives s_*∼sqrtfrac2DB_LS, 1-fracV_R(sqrts_*)D∼frac4s_*3S. These estimates are supported by simple bounds. With α=2D/(B_LS), the inequality s^2/(1+s)≤ F'(s)≤ s^2 implies that s_* lies between the positive roots of s^2+(α/S)s-α=0 and s^2-α s-α=0.
Exact mathematical passage
s_*\sim\sqrt{\frac{2D}{B_LS}},\qquad 1-\frac{V_R(\sqrt{s_*})}{D}\sim\frac{4s_*}{3S}.
\simeqUse in source · 6 passages
Search aliases: ≃ simeq
Use in source · origin_locking.tex, line 311
A diagnostic for missing terms. The leading adiabatic response of [source reference] has v_j≃ W_j/(2Ω_j) and n_j≃ W_j^2/(16Ω_j^2), with u_j at higher derivative order. Thus the corresponding second-order mode density is ρ_χ^(2)≃frac116a^3∑_jfracw_jΩ_j [fracΩ_j,qΩ_jdot q+ (3-frack_j^2a^2Ω_j^2)H]^2. This contains terms proportional to dot q^2, Hdot q, and H^2. Equation~[source reference] retains only the flat-background dot q^2 response. Including its scale-factor derivative correctly preserves th…
Exact mathematical passage
v_j\simeq W_j/(2\Omega_j)
\leUse in source · 4 passages
Search aliases: ≤ le
Use in source · origin_locking.tex, line 116
For a≥ a_i the slope bounds also give T(a_i)(a_i/a)^3≤ T(a)≤ T(a_i)(a_i/a)^2. Thus the threshold lies between the cube-root and square-root dilution estimates based on the initial trace. For a finite set of massive modes the cold limit is T(a)=a^-3∑_jw_jy_jf_j+O(a^-5). The pressure correction therefore matters before the distribution becomes cold.
Exact mathematical passage
\Tr(a_i)(a_i/a)^3\le\Tr(a)\le\Tr(a_i)(a_i/a)^2.
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Use in source · origin_locking.tex, line 115
For a≥ a_i the slope bounds also give T(a_i)(a_i/a)^3≤ T(a)≤ T(a_i)(a_i/a)^2. Thus the threshold lies between the cube-root and square-root dilution estimates based on the initial trace. For a finite set of massive modes the cold limit is T(a)=a^-3∑_jw_jy_jf_j+O(a^-5). The pressure correction therefore matters before the distribution becomes cold.
Exact mathematical passage
a\ge a_i
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