Physics: origin locking

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12 \fancyhead[L]{\sffamily\scriptsize\color{teal}KOMOREBI PHYSICS}
13 \fancyhead[R]{\sffamily\scriptsize ORIGIN LOCKING AND ENERGY EXCHANGE}
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27 {\sffamily\small\color{teal}REALIZATION CALCULUS\hfill MATHEMATICAL PHYSICS\par}
28 \vspace{1.0in}
29 {\fontsize{29}{33}\selectfont\bfseries\color{navy}Origin Locking and\par Cosmological Energy\par Exchange\par}
30 \vspace{.25in}
31 {\Large Vacuum constraints and a finite expanding-space calculation\par}
32 \vspace{.25in}{\color{copper}\rule{1.2in}{1pt}\par}
33 \vspace{.2in}
34 \noindent A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model.\par
35 \vspace{.38in}
36 \noindent\textbf{Komorebi P.}\par
37 \vfill
38 \noindent\textbf{Results and scope.} The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by $\rho_\chi-3p_\chi$. The origin-matched Gaussian potential retains that condition, but its empty-sector minima release only about $1.14\times10^{-12}$ 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. Momentum-cutoff independence fails in the accelerated-expansion stress test. These findings do not recover the original cosmological production benchmark.\par
39 \vspace{.25in}
40 \noindent{\small Original prose and figures: CC BY 4.0. Original accompanying software: MIT. Mathematical proofs, finite numerical checks, and unestablished physical interpretations are identified separately. No new Lean certification is claimed.}
41 \clearpage
42 
43 \section{The physical question and its correct location}
44 The source cosmological model proposes density locking of the average scalar at $\phi=0$ \cite{bridge,cosmology}. The minima $\phi=\pm v$ belong instead to its empty-sector tree potential. With the dimensionless coordinate
45 \begin{equation}
46 q=\sqrt\gamma\,\phi/v,\qquad \gamma=S=16,
47 \end{equation}
48 the locked point is $q=0$ and the empty-sector tree vacua are $q=\pm4$. These are different physical questions. Accessibility of $q=4$ does not decide whether matter can stabilize the origin.
49 
50 The finite production studies established energy conservation for specified oscillator systems and showed why an imposed crossing spectrum cannot be treated as a closed event when its energy exceeds the background reservoir \cite{energy,matched}. They also found a large change in the potential after matching vacuum energy, mass, and quartic coupling at the origin. The resulting energy barrier at $q=4$ is a restriction on the old empty-sector potential. It is not a prohibition on origin locking.
51 
52 This distinction organizes the present study. First, the local locking condition is derived for a stated incoherent distribution and compared with the coherent mode dynamics. Second, the empty-sector fate of the same matched potential is determined. Third, the quantum modes and background are evolved with a scale factor, pressure, and a conserved Friedmann constraint. Finally, alternative matching data are examined as changes of theory, not as renormalization tricks that leave the physical model unchanged.
53 
54 The classical covariant action and its FLRW exchange identity already appear in the source bridge \cite[Sections 2.4--2.7]{bridge}. The new numerical work here extends the explicitly regulated quantum calculation. It does not claim that classical stress conservation was previously missing.
55 
56 \subsection{Fixed model and units}
57 Let $L$ denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter $H$. Define
58 \begin{equation}
59 y_h=x2^h,\qquad d_h=2^{h-1},\qquad x=10^{-4},\qquad
60 B_L=\frac{1}{64\pi^2}\sum_{h=1}^{L}d_hy_h^4.
61 \end{equation}
62 Time is $\tau=At$, energies are densities divided by $A^4$, and a dot below denotes $\dd/\dd\tau$ 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 $a_\star=10^{-6}$, so the cosmological scale factor is $10^{-6}a$.
63 
64 The inherited origin-matched potential is
65 \begin{align}
66 V_R(q)&=D(1-q^2/S)^2+B_LF(q^2),\label{eq:Vorigin}\\
67 F(s)&=(1+s)^2\log(1+s)-s-\tfrac32s^2,\label{eq:F}\\
68 D&=4.65190089919765\times10^{-18}.
69 \end{align}
70 The nominal Gaussian family sets
71 \begin{equation}
72 C=2S_E/\epsilon,\quad \Ezero=C/2,\quad S_E=2.18498795564.
73 \end{equation}
74 At $\epsilon=0.01$, $C=436.997591128$ and $\Ezero=218.498795564$. The parameter $\epsilon$ 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.
75 
76 \section{Origin locking and the stress trace}
77 \subsection{A conditional incoherent distribution}
78 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
79 \begin{align}
80 E_j(a)&=\sqrt{k_j^2/a^2+y_j^2},\\
81 \rho_\chi(a)&=a^{-3}\sum_jw_jE_j(a)f_j,\\
82 p_\chi(a)&=a^{-3}\sum_jw_j\frac{k_j^2f_j}{3a^2E_j(a)},\\
83 \Tr(a)&=a^{-3}\sum_jw_j\frac{y_j^2f_j}{E_j(a)}.
84 \end{align}
85 These definitions describe a conditional transport problem. They do not assert that an instantaneous occupation from a coherent production calculation is already a relic distribution.
86 
87 \begin{theorem}[Local locking criterion and dilution]\label{thm:trace}
88 For the above gas,
89 \begin{equation}
90 \Tr=\rho_\chi-3p_\chi.
91 \end{equation}
92 At fixed occupations and fixed $a$, the curvature of the effective background potential at the origin is
93 \begin{equation}
94 \boxed{\kappa_{\rm eff}(a)=\rho_\chi(a)-3p_\chi(a)-\frac{4D}{S}.}\label{eq:locking}
95 \end{equation}
96 If at least one $w_jf_j$ is positive, $\Tr(a)$ is strictly decreasing to zero. Each nonzero summand has logarithmic slope between $-3$ and $-2$. If $\Tr(a_i)>4D/S$, there is exactly one later $a$ at which the local curvature vanishes.
97 \end{theorem}
98 \begin{proof}
99 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
100 $[k_j^2/a^2+y_j^2(1+q^2)]^{1/2}$. 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 \eqref{eq:locking}.
101 
102 Write a nonzero trace summand as
103 \[
104 T_j(a)=w_jy_j^2f_j\,a^{-2}(k_j^2+a^2y_j^2)^{-1/2}.
105 \]
106 Then
107 \begin{equation}
108 \frac{\dd\log T_j}{\dd\log a}=-2-\frac{a^2y_j^2}{k_j^2+a^2y_j^2}.
109 \end{equation}
110 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.
111 \end{proof}
112 
113 Positive curvature is a local energetic criterion. It is not by itself a proof of capture from a passing trajectory, asymptotic attraction, or stability under a time-dependent coherent drive. At the origin the leading background kinetic coefficient is $C$, so the corresponding local squared frequency is $\kappa_{\rm eff}/C$.
114 
115 For $a\ge a_i$ the slope bounds also give
116 \begin{equation}
117 \Tr(a_i)(a_i/a)^3\le\Tr(a)\le\Tr(a_i)(a_i/a)^2.
118 \end{equation}
119 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
120 \begin{equation}
121 \Tr(a)=a^{-3}\sum_jw_jy_jf_j+O(a^{-5}).\label{eq:cold}
122 \end{equation}
123 The pressure correction therefore matters before the distribution becomes cold.
124 
125 \subsection{Why coherence cannot be omitted}
126 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
127 \begin{equation}
128 \kappa_{\rm coh}=\frac{1}{2a^3}\sum_jw_j\frac{y_j^2}{E_j}u_j
129 \end{equation}
130 to the occupation contribution. The trace criterion in Theorem~\ref{thm:trace} therefore does not substitute for the full coherent calculation.
131 
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
133 \begin{equation}
134 \left|\frac1T\int_0^T\kappa_{\rm coh}(t)\,\dd t\right|
135 \le\frac{1}{2a^3T}\sum_jw_j\frac{y_j^2}{E_j^2}\sqrt{u_j(0)^2+v_j(0)^2}.
136 \end{equation}
137 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.
138 
139 \section{The empty-sector fate of the matched potential}
140 Origin matching preserves the local quadratic condition but changes the potential away from the origin. The location and energy of its empty-sector minima can be determined without integrating the production equations.
141 
142 \begin{theorem}[Unique positive-radius vacuum]\label{thm:min}
143 For $D,B_L,S>0$, the potential \eqref{eq:Vorigin} has exactly one minimum in $s=q^2>0$. It is the global minimum on $s\ge0$. The two background minima are $q=\pm\sqrt{s_*}$.
144 \end{theorem}
145 \begin{proof}
146 As a function of $s$,
147 \begin{align}
148 V_s&=-\frac{2D}{S}+\frac{2Ds}{S^2}
149  +2B_L[(1+s)\log(1+s)-s],\\
150 V_{ss}&=\frac{2D}{S^2}+2B_L\log(1+s)>0.
151 \end{align}
152 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 assertion, including the two symmetric minima in $q$.
153 \end{proof}
154 
155 For the current hierarchy $D/B_L$ is extremely small. Expanding the stationary equation gives
156 \begin{equation}
157 s_*\sim\sqrt{\frac{2D}{B_LS}},\qquad
158 1-\frac{V_R(\sqrt{s_*})}{D}\sim\frac{4s_*}{3S}.
159 \end{equation}
160 These estimates are supported by simple bounds. With $\alpha=2D/(B_LS)$, the inequality
161 $s^2/(1+s)\le F'(s)\le s^2$ implies that $s_*$ lies between the positive roots of
162 $s^2+(\alpha/S)s-\alpha=0$ and $s^2-\alpha s-\alpha=0$.
163 
164 A 70-digit stationary-point calculation for $L=15$ gives
165 \begin{align}
166 q_*&=3.70470008878\times10^{-6},\\
167 \frac{V_R(q_*)}{D}&=0.9999999999988562664,\\
168 \boxed{\frac{D-V_R(q_*)}{D}=1.14373356231\times10^{-12}.}\label{eq:drop}
169 \end{align}
170 Extra arithmetic digits do not express physical parameter accuracy. They prevent cancellation from obscuring the very small difference of two potential energies.
171 
172 \fig{near_origin_vacuum.pdf}{The origin-matched empty-sector minima are very close to zero. The plotted vertical scale resolves a fractional vacuum-energy decrease of order $10^{-12}$, not a decrease to zero.}{fig:min}
173 
174 This separates two conclusions. Matter can still supply the positive curvature needed to stabilize the origin. But after that curvature disappears, the empty-sector vacuum of this specified theory does not release the original height $D$. A positive vacuum energy remains almost unchanged. Recovering the source model's intended post-release relaxation therefore needs additional physical matching data or a different completion, even if origin locking itself is achieved.
175 
176 \section{What inherited crossings can and cannot predict}
177 The saved flat-space trajectory in the preceding study contains 39 crossings. Three exact event states were selected without altering their occupations or their remaining background kinetic energy. At each state we asked a deliberately conditional question: what would happen if the instantaneous $n_j$ were now an incoherent, conserved comoving distribution?
178 
179 \begin{table}[htbp]\centering\small
180 \begin{tabular}{@{}rrrrr@{}}\toprule
181 Crossing & $\rho_\chi/\Ezero$ & $K_q/\Ezero$ & conditional $\rho_{\chi,0}/D$ & conditional $a_{\rm rel}$\\\midrule
182 1 & 0.0285398 & 0.9714602 & 0.961962 & 1.56701\\
183 20 & 0.459011 & 0.540989 & 13.4682 & 3.77680\\
184 39 & 0.699922 & 0.300078 & 21.8476 & 4.43767\\\bottomrule
185 \end{tabular}
186 \caption{Conditional transport only. The last two columns use $a_\star=10^{-6}$ and the threshold in \eqref{eq:locking}. They are not derived late-time abundances or release forecasts. Background kinetic energy is explicitly retained as unresolved.}
187 \end{table}
188 
189 The numerical trace identity $\rho_\chi-3p_\chi=\Tr$ agrees to a maximum scaled discrepancy of $3.17\times10^{-16}$ on the tested moments. That verifies moment arithmetic, not the physical applicability of the incoherent assumption. At the first crossing the dephased trace is approximately $3.3883$, whereas the coherent curvature correction is approximately $446.288$. Coherence is not negligible merely because the field passes through zero.
190 
191 \fig{conditional_locking.pdf}{Conditional trace histories obtained from three different saved crossings. Each curve assumes conserved incoherent occupations from that point onward. Their disagreement exposes the missing physical matching time and out-state prescription.}{fig:trace}
192 
193 The source cosmology's conditional release scale, approximately $1.14807$, came from its stipulated spectrum. It cannot be transferred unchanged to these different distributions. Conversely, none of the three new values is a replacement prediction. The calculation has not established when production ends, whether the field is captured, how coherence becomes irrelevant, or where the remaining scalar kinetic energy goes.
194 
195 \section{An expanding finite system from one action}
196 \subsection{Regulator, lapse, and subtraction}
197 Choose a finite comoving momentum quadrature with positive weights $w_j$. For each mode,
198 \begin{equation}
199 \Omega_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2).
200 \end{equation}
201 Let $Z(a,q)>0$ be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is
202 \begin{align}
203 L={}&-\frac{3M^2a\dot a^2}{N}
204  +Na^3\left[\frac{Z}{2}\left(\frac{\dot q}{N}\right)^2-U(q)\right]
205  -\frac{N\rho_{r0}}{a}\nonumber\\
206 &+\sum_jw_j\left[\frac{a^3}{2N}|\dot f_j|^2
207  -\frac{Na^3\Omega_j^2}{2}|f_j|^2+\frac{N\Omega_j}{2}\right].\label{eq:action}
208 \end{align}
209 Here $N$ is the lapse and $M^2=(M_{\rm Pl}/A)^2$ for the physical conversion. The radiation term has density $\rho_r=\rho_{r0}a^{-4}$. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in $U$.
210 
211 Equation~\eqref{eq:action} defines a finite, time-reparametrization-invariant homogeneous model. Its explicit regulator and $a$ dependence are not asserted to arise from a complete local covariant quantum effective action. In particular, a conserved finite reduced action is not a substitute for the curvature-dependent subtractions needed by a renormalized stress tensor \cite{molina,ferreiro}.
212 
213 After variation and $N=1$, a mode obeys
214 \begin{equation}
215 \ddot f_j+3H\dot f_j+\Omega_j^2f_j=0,\qquad H=\dot a/a,
216 \end{equation}
217 with preserved normalization
218 \begin{equation}
219 a^3(f_j\dot f_j^*-f_j^*\dot f_j)=i.
220 \end{equation}
221 Use real variables
222 \begin{align}
223 n_j&=\frac{a^3(|\dot f_j|^2+\Omega_j^2|f_j|^2)}{2\Omega_j}-\frac12,\\
224 u_j&=a^3\left(\Omega_j|f_j|^2-\frac{|\dot f_j|^2}{\Omega_j}\right),\\
225 v_j&=-2a^3\operatorname{Re}(f_j^*\dot f_j).
226 \end{align}
227 Direct differentiation yields
228 \begin{align}
229 \dot n_j&=\tfrac12W_ju_j,&
230 \dot u_j&=W_j(1+2n_j)-2\Omega_jv_j,&
231 \dot v_j&=2\Omega_ju_j,\label{eq:kinetic}\\
232 W_j&=3H+\frac{\dot\Omega_j}{\Omega_j}
233 =3H+\frac{y_j^2q\dot q-Hk_j^2/a^2}{\Omega_j^2}.\label{eq:W}
234 \end{align}
235 The $3H$ term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.
236 
237 \subsection{Force and pressure}
238 The same action gives
239 \begin{align}
240 \rho_\chi&=a^{-3}\sum_jw_j\Omega_jn_j,\label{eq:rhomode}\\
241 p_\chi&=a^{-3}\sum_jw_j\left[\frac{k_j^2n_j}{3a^2\Omega_j}
242 -\left(\frac{\Omega_j}{2}-\frac{k_j^2}{6a^2\Omega_j}\right)u_j\right],\label{eq:pmode}\\
243 \mathcal F&=a^{-3}\sum_jw_j\Omega_{j,q}(n_j+u_j/2).\label{eq:force}
244 \end{align}
245 The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is
246 \begin{equation}
247 Z\ddot q+H(3Z+aZ_a)\dot q+\frac12Z_q\dot q^2+U_q+\mathcal F=0.\label{eq:background}
248 \end{equation}
249 Its density and pressure are
250 \begin{equation}
251 \rho_q=\frac12Z\dot q^2+U,\qquad
252 p_q=\frac12Z\dot q^2-U+\frac16aZ_a\dot q^2.\label{eq:scalarstress}
253 \end{equation}
254 The last term follows by differentiating $a^3Z\dot q^2/2$ with respect to $a$ before fixing the metric. It is not optional when the retained kinetic coefficient depends on the scale factor.
255 
256 \begin{theorem}[Finite exchange and the gravitational constraint]\label{thm:flrw}
257 On any regular solution of \eqref{eq:kinetic} and \eqref{eq:background} with $a>0$ and $Z>0$,
258 \begin{align}
259 \dot\rho_\chi+3H(\rho_\chi+p_\chi)&=\dot q\,\mathcal F,\\
260 \dot\rho_q+3H(\rho_q+p_q)&=-\dot q\,\mathcal F.
261 \end{align}
262 After adding separately conserved radiation, the total density and pressure obey
263 \begin{equation}
264 \boxed{\dot\rho+3H(\rho+p)=0.}\label{eq:continuity}
265 \end{equation}
266 If $\dot H=-(\rho+p)/(2M^2)$ and $3M^2H^2=\rho$ initially, the Friedmann constraint remains satisfied. In addition,
267 \begin{equation}
268 \boxed{a^3(\tau)\rho(\tau)+\int_0^\tau3Ha^3p\,\dd t=\rho(0)}\label{eq:work}
269 \end{equation}
270 when $a(0)=1$.
271 \end{theorem}
272 \begin{proof}
273 Differentiate \eqref{eq:rhomode}. Insert $\dot n_j=W_ju_j/2$ and
274 $\dot\Omega_j=(y_j^2q\dot q-Hk_j^2/a^2)/\Omega_j$.
275 The terms multiplying $Hn_j$ and $Hu_j$ cancel against $3H(\rho_\chi+p_\chi)$ using \eqref{eq:pmode}. The remaining term is $\dot q\mathcal F$.
276 
277 For the background,
278 \[
279 \dot\rho_q=Z\dot q\ddot q+\tfrac12(Z_q\dot q+aHZ_a)\dot q^2+U_q\dot q.
280 \]
281 Substitute \eqref{eq:background}. The terms involving $Z_q$ and $U_q$ cancel, leaving
282 $-\dot q\mathcal F-3HZ\dot q^2-\tfrac12aHZ_a\dot q^2$.
283 This is $-\dot q\mathcal F-3H(\rho_q+p_q)$. Radiation adds zero to the total divergence.
284 
285 Differentiating $3M^2H^2-\rho$ now gives
286 $6M^2H\dot H+3H(\rho+p)=0$. Finally, multiply \eqref{eq:continuity} by $a^3$ and integrate. This gives \eqref{eq:work}.
287 \end{proof}
288 
289 The mode equations also preserve
290 $(1+2n_j)^2-u_j^2-v_j^2=1$.
291 For a normalized pure initial state on the positive sheet, continuity keeps $1+2n_j\ge1$, so $n_j\ge0$. This proves positivity in the exact finite equations. Floating-point preservation is a separate numerical check.
292 
293 There is no assertion of global existence for arbitrary expanding solutions in this theorem. It applies while the finite solution remains regular. Nor does positivity of the instantaneous mode energy make that energy an asymptotic particle yield.
294 
295 \section{The kinetic tail and what expansion adds}
296 \subsection{The retained term}
297 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
298 \begin{equation}
299 K_h^2(a)=y_hz_{\max}^2/a^2,\quad
300 A_h=\left[\frac{K_h^2}{K_h^2+y_h^2(1+q^2)}\right]^{3/2}.
301 \end{equation}
302 The coefficient used in the solver is
303 \begin{equation}
304 Z(a,q)=C+\sum_{h=1}^L\frac{d_hy_h^2}{48\pi^2}
305 \frac{q^2}{1+q^2}(1-A_h).\label{eq:tail}
306 \end{equation}
307 It is positive. Its $q$ and $a$ derivatives enter both \eqref{eq:background} and \eqref{eq:scalarstress}. The explicit modes and this omitted-momentum term do not overlap in their declared momentum ranges.
308 
309 \subsection{A diagnostic for missing terms}
310 The leading adiabatic response of \eqref{eq:kinetic} has
311 $v_j\simeq W_j/(2\Omega_j)$ and $n_j\simeq W_j^2/(16\Omega_j^2)$, with $u_j$ at higher derivative order. Thus the corresponding second-order mode density is
312 \begin{equation}
313 \rho_\chi^{(2)}\simeq\frac{1}{16a^3}\sum_j\frac{w_j}{\Omega_j}
314 \left[\frac{\Omega_{j,q}}{\Omega_j}\dot q+
315 \left(3-\frac{k_j^2}{a^2\Omega_j^2}\right)H\right]^2.\label{eq:adiabatic}
316 \end{equation}
317 This contains terms proportional to $\dot q^2$, $H\dot q$, and $H^2$. Equation~\eqref{eq:tail} retains only the flat-background $\dot q^2$ response. Including its scale-factor derivative correctly preserves the chosen finite action, but does not supply the other terms.
318 
319 At large physical momentum and $\dot q=0$, $W\simeq2H$. With the continuum measure $k_{\rm phys}^2\dd k_{\rm phys}/(2\pi^2)$, the leading contribution per species behaves as
320 \begin{equation}
321 \rho^{(2)}\sim \frac{H^2K_{\rm phys}^2}{16\pi^2}.
322 \end{equation}
323 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 \cite{ferreiro}. Initial-state regularity must be treated at the same time \cite{baacke}. The present test does not isolate all cutoff sensitivity into one cause.
324 
325 \section{Numerical comparisons and rejected controls}
326 \subsection{Independent variables and exact budgets}
327 The real kinetic solver was compared with an independently coded complex mode solver. In a finite $L=14$ expansion stress test, their maximum background-position difference is $1.34\times10^{-11}$ and their maximum mode-energy difference is $3.31\times10^{-9}\Ezero$. The complex representation's maximum Wronskian residual is $4.44\times10^{-12}$. These are cross-checks of the same finite model, not independent cosmological evidence.
328 
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
330 \begin{align}
331 \max_\tau\left|\frac{a^3\rho+W_p}{\rho(0)}-1\right|&=4.75\times10^{-14},\\
332 \max_\tau\left|\frac{H^2}{H_i^2}-\frac{\rho}{\rho(0)}\right|&=3.77\times10^{-14},
333 \end{align}
334 where $W_p=\int3Ha^3p\,\dd\tau$. The Friedmann constraint is evolved, not reset algebraically at each step.
335 
336 \fig{expansion_work.pdf}{Finite expansion stress test. Comoving energy alone is not constant. The pressure-work term completes the budget. The deliberately accelerated expansion is not an observational model.}{fig:work}
337 
338 Four incorrect versions were also integrated. They omit a term while retaining the rest of the closed finite system.
339 \begin{table}[htbp]\centering\small
340 \begin{tabular}{@{}lr@{}}\toprule
341 Incorrect modification & maximum relative work-balance defect\\\midrule
342 Omit coherent mode pressure & $1.34148\times10^{-2}$\\
343 Omit the volume-squeezing term in $W$ & $1.13805\times10^{-2}$\\
344 Omit the $aZ_a$ pressure term & $3.42649\times10^{-5}$\\
345 Omit coherence from the background force & $2.92396\times10^{-3}$\\\bottomrule
346 \end{tabular}
347 \caption{Intentional negative controls. Their failure detects inconsistent equations, not a failure of the correct finite identity.}
348 \end{table}
349 
350 \subsection{Quadrature converges, the expansion cutoff does not}
351 At fixed cutoff, increasing the quadrature from 64 to 128 nodes leaves the first-crossing mode-energy fraction stable to about $5.83\times10^{-13}$. Holding 128 nodes and doubling the momentum cutoff instead gives
352 \begin{center}
353 \begin{tabular}{@{}rrr@{}}\toprule
354 $z_{\max}$ & nodes per segment & first-crossing $\rho_\chi/\Ezero$\\\midrule
355 16 & 128 & 0.4198449952\\
356 32 & 128 & 2.1854778816\\\bottomrule
357 \end{tabular}
358 \end{center}
359 The relative shift is $4.20544$, or about $421\%$. The declared acceptance threshold of $10^{-3}$ is not met. This is a failed scientific gate, not a quadrature failure hidden by an energy check.
360 
361 \fig{expansion_cutoff_failure.pdf}{The two lower curves overlap after quadrature refinement at fixed cutoff. Increasing the cutoff changes the result strongly. Each correct finite system still satisfies its pressure-work balance.}{fig:cutoff}
362 
363 A mode density above the initial scalar budget is not by itself the original flat-space overdraw: total radiation and pressure work are explicitly present, and the subtracted coherent pressure need not satisfy a positive-pressure condition. It is also not validated cosmological production. The large cutoff response, incomplete metric subtraction, and chosen initial state prevent that interpretation.
364 
365 \subsection{The actual microscopic expansion scale}
366 With the stored cosmological conversion inputs, the origin branch has $A\simeq49.2467$ eV and
367 \begin{equation}
368 \frac{H_i}{A}=3.35456\times10^{-25}.
369 \end{equation}
370 Here $H_i$ in the numerator is the dimensionful physical rate. The dimensionless solver uses that ratio. Over $0\le\tau\le12$, the accumulated $\log a$ is about $4.03\times10^{-24}$. The largest background-position difference from the flat-space run is $3.57\times10^{-15}$.
371 
372 This finite comparison confirms that the inherited microscopic scale does not produce substantial Hubble damping during a few crossings. It does not permit extrapolation to cosmological times. The accelerated stress test deliberately makes expansion large enough to expose structural errors in the finite prescription.
373 
374 \section{Alternative vacuum matching is a different problem}
375 \subsection{An exact restriction on quartic counterterms}
376 The origin locking problem does not require the old empty-sector vacuum at $q=4$ to remain intact. Nevertheless, preserving that vacuum is a useful separate test of the intended late-time theory.
377 
378 Set $s=q^2$ and $G(s)=(1+s)^2\log(1+s)$. The most general static potential with this Gaussian determinant and polynomial counterterms through $q^4$ is
379 \begin{equation}
380 U(s)=a_0+b_0s+c_0s^2+B_LG(s).
381 \end{equation}
382 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
383 $\Delta=U(0)-U(S)$ and $\kappa=U_{qq}(\sqrt S)$. Define
384 \begin{equation}
385 J(S)=\tfrac12S^2-S+\log(1+S),\quad
386 K(S)=\tfrac12S^2+S-(1+S)\log(1+S).
387 \end{equation}
388 
389 \begin{theorem}[Barrier, curvature, and origin restriction]\label{thm:matching}
390 Every such quartic-matched potential satisfies
391 \begin{equation}
392 \boxed{\Delta=\frac{S\kappa}{8}-B_LJ(S),\qquad
393 U_{qq}(0)=\frac{4}{S}[B_LK(S)-\Delta].}\label{eq:invariants}
394 \end{equation}
395 Consequently, retaining an origin maximum and a stationary vacuum at $\sqrt S$ requires $\Delta>B_LK(S)$.
396 \end{theorem}
397 \begin{proof}
398 Stationarity gives $b_0=-2c_0S-B_LG'(S)$. Substitute this into
399 $\Delta=-b_0S-c_0S^2+B_L[G(0)-G(S)]$ and
400 $\kappa=4S[2c_0+B_LG''(S)]$. 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.
401 \end{proof}
402 
403 At $L=15$ and the inherited budget, $B_LK(S)/\Ezero\simeq1353.96$. Matching the old tiny vacuum curvature instead places the origin below the proposed vacuum by about $1622.36\Ezero$. Keeping the old barrier requires an empty-vacuum curvature roughly $7.62\times10^{22}$ times its tree value. These are different specified matching choices. None is a compensated change of convention that preserves all the original physical data.
404 
405 \subsection{A constructive but altered branch}
406 The restriction also gives a constructive family. Let
407 \begin{align}
408 R_S(s)&=G(s)-G(S)-G'(S)(s-S)-\tfrac12G''(S)(s-S)^2,\\
409 U_\Delta(s)&=\frac{\Delta+B_LJ(S)}{S^2}(s-S)^2+B_LR_S(s).
410 \end{align}
411 Then $U_\Delta(S)=U_{\Delta,s}(S)=0$ and $U_\Delta(0)=\Delta$.
412 
413 \begin{proposition}[An accessible double well under changed matching]
414 If $B_LK(S)<\Delta<\Ezero$, $U_\Delta(q^2)$ is nonnegative, has its only global minima at $q=\pm\sqrt S$, and has an origin maximum. The origin barrier lies within the declared budget.
415 \end{proposition}
416 \begin{proof}
417 Theorem~\ref{thm:matching} gives $U_s(0)<0$. Also $U_s(S)=0$ and
418 $U_{ss}(S)=2[\Delta+B_LJ(S)]/S^2>0$. 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 potential decreases to its zero at $S$ and then increases. The final budget statement follows from $U(0)=\Delta<\Ezero$.
419 \end{proof}
420 
421 \fig{matching_window.pdf}{Counterfactual tower changes. The original depth-15 content fails the displayed barrier window. A shallower theory can pass it, but removes most of the stipulated production weight and requires a new calibration. This is not selection of a physical cutoff.}{fig:window}
422 
423 For an explicit comparison, truncating to $L=12$ retains about $20.0724\%$ of the old Gaussian production energy. Recalibrating that stipulated family gives $\Ezero=43.8579$ and $B_{12}K(S)=9.02826$. Choosing $\Delta=0.3\Ezero=13.1574$ 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.
424 
425 Finite runs of this branch cross the origin. A tighter run through $\tau=300$ has 23 crossings and has not settled into a vacuum. Its total energy balance is excellent, but a separate absolute mode-normalization check fails at large occupation. The comparison is retained to show both the constructive matching possibility and its current numerical boundary.
426 
427 \section{Verification, failures, and the research boundary}
428 The delivered study contains 29 finite configurations. Twenty-five use the stated full equations and four are deliberate negative controls. The aggregate audit records \textbf{19 passing gates and two failed gates}. Four selected configurations were independently replayed in the same pinned numerical environment and reproduced their final background positions at stored precision.
429 
430 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
431 $(1+2n)^2-u^2-v^2-1$ then subtracts large terms to resolve a value near zero. Its maximum absolute residual is $6.84738\times10^{-4}$, above the declared $10^{-5}$ gate. Tightening the integration tolerance did not repair this diagnostic. It remains a failure rather than being converted into a pass by switching to a scaled norm. A structure-preserving or higher-precision representation is the appropriate next check for that secondary branch.
432 
433 The accepted origin calculations pass this absolute normalization criterion. All correct finite runs have maximum sampled work-balance defects below $10^{-8}$. The saved-state audit independently reconstructs density and pressure-work combinations from exact sampled full mode states. The supplied diagnostics do not constitute interval-certified error bounds.
434 
435 \subsection{What is established}
436 The finite study establishes the corrected distinction between origin locking and the empty-sector vacuum, proves a local trace criterion under explicit incoherent assumptions, locates the matched empty-sector minima, and constructs a pressure-consistent finite expanding model. It also proves matching restrictions within a stated Gaussian quartic class. Those are mathematical results at their stated scope.
437 
438 The computations cross-check those constructions, reproduce the flat limit, test the physical short-time scale, and expose a genuine ultraviolet failure under appreciable expansion. They do not establish a covariantly renormalized cosmology, an incoming state derived from a cosmic history, a captured late-time branch, a relic abundance, or the old post-release vacuum relaxation.
439 
440 \subsection{The next construction}
441 The continuation should keep the original target at the origin. The background force, energy density, pressure, and metric terms need a common curvature-dependent subtraction prescription through the order required for the stress tensor. The initial state must have compatible adiabatic regularity. The $H^2$ and mixed terms in \eqref{eq:adiabatic} identify missing structure but cannot be subtracted alone while leaving the rest of the action unchanged.
442 
443 After that construction, the same physical matching data should be held fixed while momentum and time resolution are varied. Only a controlled late-time state can justify replacing coherent dynamics by the trace transport law. Residual scalar energy must remain in the budget. Separately, the empty-sector potential must support the intended release of vacuum energy. The original energy excess is therefore not repaired by this study, but the remaining obligations are no longer confused with a barrier at the wrong locking point.
444 
445 \appendix
446 \section{Reproducibility and source correspondence}
447 The software modules are \texttt{matching.py}, \texttt{locking.py}, \texttt{expansion.py}, \texttt{audit.py}, and \texttt{verify.py}. Every finite configuration has a JSON settings record, dense saved diagnostic series, final modes, and 21 exact sampled full mode states. The sampled states are not interpolated. The full continuously sampled mode trajectory is not claimed to be stored. The inherited 39-crossing event data used by the conditional transport comparison are included separately and identified by hashes.
448 
449 \texttt{verify.py --replay} audits the delivered arrays and reruns four configurations. Failed scientific gates remain present in the machine-readable report. The original benchmark input has SHA-256
450 \begin{quote}\ttfamily\footnotesize
451 7a08c0964796a8cd607b2a59ed8b2bfb6cfb8e4df31b7b03627c8201bc2c796e.
452 \end{quote}
453 The reference numerical environment is Python 3.13.5, NumPy 2.3.5, SciPy 1.17.0, SymPy 1.14.0, and mpmath 1.3.0. Runtime settings and exact software versions are recorded with the verification results. No Lean compiler run was performed for this supplement.
454 
455 The source bridge's Sections 2.6 and 2.7 identify locking at the origin and give the classical exchange and stability laws. The source cosmology's Section 9.8 describes density locking and conditional release. The accompanying correction note explicitly repairs the earlier interpretation that confused this state with $q=4$. The preceding papers and main monograph are preserved rather than silently overwritten.
456 
457 \begin{thebibliography}{9}\small
458 \bibitem{bridge} Komorebi P. \emph{Covariant Refinement-Fiber Dynamics}. Source bridge, Sections 2.4--2.7 and Chapter 6. Original PDF pages 15--16 for the action, exchange identity, and locked origin.
459 \bibitem{cosmology} Komorebi P. \emph{Cosmology, Dark Sectors and Structure Formation}. Source Volume II, Section 9.8. Original PDF pages 55--56 for conditional locking and release.
460 \bibitem{energy} Komorebi P. \emph{Energy-Conserving Production: A finite-depth backreaction study}. Companion research manuscript.
461 \bibitem{matched} Komorebi P. \emph{Vacuum Matching and Dynamical Energy Transfer}. Companion research manuscript and numerical source package.
462 \bibitem{molina} C. Molina-Par\'is, P. R. Anderson, and S. A. Ramsey. One-loop $\lambda\phi^4$ theory in Robertson--Walker spacetimes: adiabatic regularization and analytic approximation. \emph{Physical Review D} \textbf{61}, 127501 (2000). arXiv:gr-qc/9908037.
463 \bibitem{ferreiro} A. Ferreiro, S. Monin, and F. Torrenti. Physical scale adiabatic regularization in cosmological spacetimes. arXiv:2311.08986, version 2 (2024).
464 \bibitem{baacke} J. Baacke, K. Heitmann, and C. P\"atzold. On the choice of initial states in nonequilibrium dynamics. \emph{Physical Review D} \textbf{57}, 6398--6405 (1998). arXiv:hep-th/9711144.
465 \end{thebibliography}
466 \end{document}

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