diff bc2b13b..72e56a3 hard_sphere_boltzmann_fluctuations_v2.txt
diff --git a/hard_sphere_boltzmann_fluctuations_v2.txt b/hard_sphere_boltzmann_fluctuations_v2.txt
index 3093658..59af6df 100644
--- a/hard_sphere_boltzmann_fluctuations_v2.txt
+++ b/hard_sphere_boltzmann_fluctuations_v2.txt
@@ -29,8 +29,13 @@ random force and no resampling. The Gaussian limit is emergent, produced by
the random initial configuration together with the extreme sensitivity of
which pairs collide. Both effects must be tracked, not averaged over.
-What was known. Lanford proved short-time validity of the
-Boltzmann equation. Van Beijeren--Lanford--Lebowitz--Spohn and Spohn
+What was known. The derivation descends from the BBGKY hierarchy of
+Bogolyubov and Kirkwood, from the Boltzmann--Grad limit for rarefied gases,
+and from the Hilbert and Chapman--Enskog expansions that organize the
+hydrodynamic corrections. Lanford proved short-time validity of the
+Boltzmann equation. Uchiyama gave a different derivation, and
+Gallagher--Saint-Raymond--Texier later recast the proof so that the
+geometric and probabilistic content separate cleanly. Van Beijeren--Lanford--Lebowitz--Spohn and Spohn
computed the fluctuation covariance. Pulvirenti--Simonella and
Bodineau--Gallagher--Saint-Raymond--Simonella built the quantitative
short-time fluctuation theory, the latter with process-level convergence,
@@ -59,29 +64,37 @@ cannot overlap; at the scale $\sqrt\mu=\varepsilon^{-1}$ that difference survive
so the model dynamics must match the gas with probability better than
$O(\varepsilon)$.
-A map of the symbols. Most notation is fixed once, in the opening
-paragraphs of the main text; this table lets a first-time reader proceed
-without hunting.
+Notation is fixed once and used without variation.
$N$ : number of spheres; $N/\mu$ stays near the mass $\int f_0 d z$.
+
$f_0(z)$, $f_t(z)$ : initial one-particle density and the regular Boltzmann
solution, both in $z=(x,v)$; regularity on $[0,T]$ is
-Assumption 1.
+the kinetic assumption.
+
$\|\cdot\|_{Bol,\beta}$ : sup norm with a Gaussian velocity weight
$e^{\beta|v|^2}$, summed over unit spatial cells so mass is controlled.
+
$B$, $\Delta\phi$ : collision kernel $[(v-v_*)\cdot\omega]_+$ and the change
$\Delta\phi$ of a test across one collision.
+
$A_t$, $C_t$ : linearized Boltzmann drift and collision-noise covariance:
the two coefficients of the fluctuating equation.
+
$\zeta_t^\varepsilon$, $\zeta_t$ : the centered, $\sqrt\mu$-scaled fluctuation
(microscopic) and its Gaussian limit.
+
$W$ : the Gaussian noise driving $\zeta_t$, with independent increments.
+
$g_i$, $f_{i\tau}$ : one-root mean at interface $i$ (the independent input)
and the Boltzmann density $f_{i\tau}$ at the same time.
+
$H_i,\Lambda_i,\Gamma$ : collision bookkeeping of the Deng--Hani--Ma
machinery: root count, particle-size cap, and cluster-excess cap.
+
$L, \tau=T/L$, $K$, $\rho$ : number of slabs, slab length, the complexity
cutoff, and the complexity of a history.
+
$c_h^\varepsilon$ : the $h$-th factorial cumulant density; its target size is the
sharp scale $\mu^{1-h}$.
@@ -149,7 +162,7 @@ resampling: the only randomness is the initial configuration.
Assumption. $f_0\ge0$ is a smooth probability density with
$\|f_0\|_{Bol,2\beta}+\|\nabla_xf_0\|_{Bol,2\beta}<\infty$ for some
$\beta>0$. There is a nonnegative classical solution $f$ of
-(4) on a prescribed interval $[0,T]$ with
+the Boltzmann equation on a prescribed interval $[0,T]$ with
$$M:=\sup_{0\le t\le T}\sup_{x,v}e^{2\beta|v|^2}f(t,x,v)<\infty$$
@@ -163,7 +176,7 @@ $$\pi_t^\varepsilon(\phi)=\mu^{-1}\sum_{i=1}^N\phi(z_i(t)),
\zeta_t^\varepsilon(\phi)=\sqrt\mu [\pi_t^\varepsilon(\phi)-E_\varepsilon\pi_t^\varepsilon(\phi)]$$
The expectation is the exact microscopic one. This is essential; see
-Remark 1.
+the centering remark.
Define the pair difference and the two limiting operators
@@ -184,7 +197,7 @@ no extra factor $1/2$ because of the pre/post substitution together with
$\omega\mapsto-\omega$ (the collision normalization, below). The quadratic form
$C_t$ is the collision noise, nonnegative because $f_t\ge0$.
-Theorem (Finite-dimensional fluctuation limit). Under Assumption 1, the centered Gaussian field $\zeta$ solving the
+Theorem (Finite-dimensional fluctuation limit). Under the kinetic assumption, the centered Gaussian field $\zeta$ solving the
weak fluctuating Boltzmann equation
$$d \zeta_t(\phi)=\zeta_t(A_t\phi) d t+ d W_t(\phi)$$
@@ -204,7 +217,7 @@ $$(\zeta_{t_1}^\varepsilon(\phi_1),...,\zeta_{t_m}^\varepsilon(\phi_m))
(\zeta_{t_1}(\phi_1),...,\zeta_{t_m}(\phi_m))
(\varepsilon\downarrow0)$$
-Remark. The initial covariance (11) has no subtraction of
+Remark (The initial covariance). The initial covariance has no subtraction of
$(\int f_0\phi)(\int f_0\psi)$: the particle number is random. More
importantly, microscopic centering is not the same as centering by the
Boltzmann mean. With $\varrho_0(x)=\int f_0(x,v) d v$,
@@ -218,15 +231,15 @@ leave a nonzero deterministic drift already at time zero. This is why
the proof must compare exact expectations, not merely laws.
The layered collision-history framework of Deng--Hani--Ma is kept, stated as the
-black box Theorem 2. Its role is to remove large
+black box the analytic package. Its role is to remove large
complexity: histories with many entangled collisions. What remains must
be controlled at the fluctuation scale, and that is where the new work
lies.
1. Initial ensemble and Gaussian target
(the initial ensemble, below). Exact initial cumulants, the initial CLT,
-the shift Remark 1, and construction of the Gaussian
-field of Theorem 1.
+the shift, the initial-covariance remark, and construction of the Gaussian
+field of the fluctuation limit theorem.
2. Connected expansion with records
(the connected expansion, below). Factorial cumulants are written as a
@@ -235,7 +248,7 @@ keeping finitely many observation records. Large complexity is dropped
by the Deng--Hani--Ma package.
3. Sharp correlation bounds (the sharp bounds, below,
-Theorem 4). The $h$-th cumulant is $O(\mu^{1-h})$,
+the sharp connected bounds). The $h$-th cumulant is $O(\mu^{1-h})$,
with the leading $h=1,2$ terms identified as signed sums of trees
weighted by $f_{i\tau}$, contact flux, and $\pm1$ signs. This is proved
by chronological merger coordinates: each collision that merges two
@@ -249,7 +262,7 @@ ordinary cumulants are computed; second cumulant converges, higher ones
vanish, giving the Gaussian law with the correct covariance.
5. Accuracy of the model dynamics (the cutoff comparison, below,
-Proposition 6). The pasted and true dynamics agree with
+the pasted accuracy bound). The pasted and true dynamics agree with
probability $O(\varepsilon^{1+\eta})$, $\eta=1/20$. This is a positive
geometric estimate on terminal witnesses, carried out with local
gains, a common birth-energy bound, and a causal time-order sum free of
@@ -263,7 +276,7 @@ what transfers the limit with exact centering.
This stage is independent of the dynamics. It does two things: prove the
fluctuation limit at time zero (a pure equilibrium statement about the
grand-canonical hard-sphere measure), and construct the Gaussian field
-that Theorem 1 claims to be the limit. It also records the initial
+that the fluctuation limit theorem claims to be the limit. It also records the initial
correlation bounds that later parts use.
Let $F_{h,0}^{\varepsilon}$ be the density of the ordered factorial measure
@@ -331,7 +344,7 @@ Proof.
(i) Factorial bound and moments. For fixed $Z_h$, let
$\mathcal Z_\varepsilon[Z_h]$ be the partition function of the remaining
particles, still mutually excluding and additionally avoiding the balls
-$B(x_j,\varepsilon)$. Directly from (5),
+$B(x_j,\varepsilon)$. Directly from the grand-canonical law,
$$F_{h,0}^{\varepsilon}(Z_h)= 1 _{\mathcal D_h^\varepsilon}(Z_h)\prod_{j=1}^hf_0(z_j)
\frac{\mathcal Z_\varepsilon[Z_h]}{\mathcal Z_\varepsilon}$$
@@ -364,7 +377,7 @@ connected graphs, and summing one interval gives
$$
\prod_{e\in\mathcal T}(-\chi_e)\prod_{e\in R(\mathcal T)\setminus\mathcal T}(1-\chi_e),
$$
-whose second factor lies in $[0,1]$, proving (22).
+whose second factor lies in $[0,1]$, proving the tree bound.
Penrose's scheme, with this precise tree bound, is reproduced in
Bodineau--Gallagher--Saint-Raymond--Simonella 2022, Proposition 2.3.3;
@@ -383,7 +396,7 @@ $$\log\mathcal Z_\varepsilon[a]=\sum_{n\ge1}\frac{\mu^n}{n!}\int\mathcal U_n(Z_n
\prod_{j=1}^nf_0(z_j)a(z_j) d Z_n$$
From the partition function to the cumulants. Apply
-(24) with $a=1+u$. The log of
+the logarithmic partition formula with $a=1+u$. The log of
$E\prod_i(1+u(z_i))$ has coefficients $\mu^hc_{h,0}^{\varepsilon}/h!$, so
differentiation gives the rooted Mayer series
@@ -399,7 +412,7 @@ $n!\ge(n/e)^n$, $(1+h/n)^n\le e^h$, $(h+n)^h\le h^he^n$,
$$\frac{(h+n)^{h+n-2}}{n!}\le(eh)^he^{2n}$$
-and the $n=0$ count is $\le(Ch)^{Ch}$. Thus (25)
+and the $n=0$ count is $\le(Ch)^{Ch}$. Thus the rooted Mayer series
converges absolutely for $\varepsilon\le\varepsilon_0$ with $\varepsilon_0$ independent of
$h$.
@@ -410,7 +423,7 @@ vertex. One auxiliary vertex touches at most $27$ root components: pick
one adjacent root from each; their mutual distances exceed $\varepsilon$, and
their disjoint balls of radius $\varepsilon/2$ lie in a ball of radius
$3\varepsilon/2$. Hence $q\le27n$ for a contributing tree with $n$ auxiliary
-vertices. Using (26) and $d_\varepsilon=O(\varepsilon)$,
+vertices. Using the counting inequality and $d_\varepsilon=O(\varepsilon)$,
$$
\frac{|c_{h,0}^{\varepsilon}(Z_h)|}{\prod_jf_0(z_j)}
\le(Ch)^{Ch}\varepsilon^{q/27}\le(Ch)^{Ch}\varepsilon^{(q-1)/81} (q\ge2),
@@ -418,31 +431,31 @@ $$
and for $q=1$ the same argument without the restriction on $n$ gives
$(Ch)^{Ch}$. A tree of the roots using intra-component and $q-1$
inter-component edges has weight at least $\varepsilon^{(q-1)/81}$ in
-(17). This proves (17).
+the factorial bound. This proves the factorial lower bound.
Sharp link bound. For fixed $h,D$, truncate
-(25) at $n_0+1\ge D(h-1)$. The tail is
+the rooted Mayer series at $n_0+1\ge D(h-1)$. The tail is
$C_{D,h}\varepsilon^{D(h-1)}\prod_jf_0(z_j)$. In any term with $n\le n_0$, all
roots are connected by paths of $\le h+n_0-1$ links of length $\le\varepsilon$,
-so their diameter is $\le(h+n_0)\varepsilon$; on that set (18)
-holds with $C_{D,h}$ chosen large. This proves (18).
-Integrating (25) and using (23)
+so their diameter is $\le(h+n_0)\varepsilon$; on that set the sharp link bound
+holds with $C_{D,h}$ chosen large. This proves the sharp link bound.
+Integrating the rooted Mayer series and using the rooted tree integral bound
instead gives
$$
\|c_{h,0}^{\varepsilon}\|_1\le b_\varepsilon^{h-1}\sum_{n\ge0}\frac{(h+n)^{h+n-2}}{n!}d_\varepsilon^n
\le C_h\varepsilon^{3(h-1)},
$$
-which is (19).
+which is the L1 bound.
(iii) Initial CLT and the shift. For real bounded $\phi$ put
$S_\phi=\sum_i\phi(z_i)$ and let $\kappa_k$ denote ordinary cumulants.
-Substituting $a=e^{u\phi}$ in (24) and
+Substituting $a=e^{u\phi}$ in the logarithmic partition formula and
differentiating $k$ times at $u=0$,
$$
\kappa_k(S_\phi)=\sum_{n\ge1}\frac{\mu^n}{n!}\int f_0^{\otimes n}\mathcal U_n
(\sum_{j=1}^n\phi(z_j))^k,
$$
-and (23) gives $\kappa_k(S_\phi)=\mu\int f_0\phi^k+
+and the rooted tree integral bound gives $\kappa_k(S_\phi)=\mu\int f_0\phi^k+
O_{k,\phi}(\mu\varepsilon)$. After exact centering and division by $\sqrt\mu$,
the second cumulant tends to $\int f_0\phi^2$ and every higher one to
zero. The moment-cumulant identity then gives every Gaussian moment; for
@@ -453,22 +466,21 @@ $\limsup$ is at most $2|u|^{2n}\sigma^{2n}/(2^n n!)\to0$. Joint
convergence follows on each linear combination. This includes zero
variance.
-For the shift, the $n=1$ term of (25) with $h=1$
+For the shift, the $n=1$ term of the rooted Mayer series with $h=1$
gives, uniformly in $z$,
$$
g_0(z)=f_0(z)[1-\mu\int_{|x-x_*|\le\varepsilon}\varrho_0(x_*) d x_*+O(\varepsilon^2)].
$$
The bounded gradient of $\varrho_0$ gives
$\int_{|x-x_*|\le\varepsilon}\varrho_0=\tfrac{4\pi}{3}\varepsilon^3\varrho_0(x)+O(\varepsilon^4)$,
-and $\sqrt\mu=\varepsilon^{-1}$ converts this into (20).
+and $\sqrt\mu=\varepsilon^{-1}$ converts this into the centering shift.
-Remark. The shift (20) is generally nonzero, so exact
+Remark (The centering shift). The shift is generally nonzero, so exact
microscopic centering is essential already at time zero. Meanwhile
$\mu\|c_{2,0}^{\varepsilon}\|_1=O(\varepsilon)$, so the limiting initial covariance has
only the same-particle term $\int f_0\phi\psi$.
-Write $M=\sup_{t\le T,x,v}e^{2\beta|v|^2}f_t(x,v)$. The noise form
-(9) needs global velocity integrability, which does not
+Write $M=\sup_{t\le T,x,v}e^{2\beta|v|^2}f_t(x,v)$. The noise form needs global velocity integrability, which does not
follow from a spatially uniform bound alone.
Lemma (Global weighted integrability). For every $0<q<2\beta$,
@@ -499,7 +511,7 @@ $\|P_tg\|_{X_q}\le CM(1+\delta^{-1/2})\|g\|_{X_{q+\delta}}$ uniformly for
exponents in a compact subinterval of $(0,2\beta)$. Since $f_0\in X_{q_0}$
for $q_0<2\beta$, distributing the exponent over the gain series and using
$n^{n/2}/n!$ bounds its $X_q$ norm uniformly for $t\le T$; integrating the
-pointwise majorant proves (27).
+pointwise majorant proves the velocity moment bound.
For $a\ge0$ define the test spaces
@@ -530,7 +542,7 @@ $$|K_th|\le CM\|h\|_a(1+|v|)e^{a|v|^2},
using $|\Delta_h|\le4\|h\|_ae^{a(|v|^2+|v_*|^2)}$ and
$\int B d \omega=\pi|v-v_*|$. Spatial tails vanish by replacing $\|h\|_a$
with $\eta_h(x)=\sup_ve^{-a|v|^2}|h|$, and $K_th$ is continuous in
-$E_b^0$. Iterating (29) bounds the $n$-th
+$E_b^0$. Iterating the backward mild equation bounds the $n$-th
time-ordered term by
$$\frac{(CM(t-r))^n}{n!}(1+\sqrt{n/(b-a)})^n\|h\|_a$$
@@ -540,10 +552,10 @@ an intermediate time gives composition, and the same estimate on an
iterated remainder gives uniqueness with a positive exponent allowance.
For compactly supported smooth $\phi$, differentiability of the series at
its upper end (its final transport gives $U(r,t)(v\cdot\nabla_x\phi)$;
-the simplex faces give $U(r,t)K_t\phi$) converges with (32),
-giving $\partial_tU(r,t)\phi=U(r,t)A_t\phi$ and hence (30).
+the simplex faces give $U(r,t)K_t\phi$) converges with the Dyson bound,
+giving $\partial_tU(r,t)\phi=U(r,t)A_t\phi$ and hence the upper-end identity.
-Proposition (Gaussian target). The equation (10) has a centered Gaussian solution on each
+Proposition (Gaussian target). The fluctuating equation has a centered Gaussian solution on each
test scale $E_a^0$, $0\le a<\beta$, mapping continuously and linearly into
$L^2$ uniformly on $[0,T]$, with covariance
@@ -553,11 +565,11 @@ $$E[\zeta_t(\phi)\zeta_s(\psi)]
$$+\int_0^{\min(s,t)}C_r(U(r,t)\phi,U(r,s)\psi) d r$$
It is unique in the class of solutions extending to these scales with
-uniformly bounded second-moment norms. Thus the law in Theorem 1
+uniformly bounded second-moment norms. Thus the law in the fluctuation limit theorem
is well defined.
Proof.
-For $b<\beta$, energy conservation and Lemma 1 give
+For $b<\beta$, energy conservation and the weighted integrability bound give
$$|C_t(h,k)|\le CM\|h\|_b\|k\|_b\int f_t(1+|v|)e^{2b|v|^2} d x d v\le C_b\|h\|_b\|k\|_b$$
@@ -573,19 +585,19 @@ $$\zeta_t(h)=I_0(U(0,t)h)+I( 1 _{r\le t}\Delta_{U(r,t)h}),
W_t(h)=I( 1 _{r\le t}\Delta_h)$$
For $h\in E_a^0$ propagate to $b\in(a,\beta)$ and apply
-(34): the maps are well defined and
+the isonormal construction: the maps are well defined and
$$\sup_{t\le T}\|\zeta_t(h)\|_{L^2}\le C_a\|h\|_a$$
-They are jointly Gaussian and linear with covariance (33);
+They are jointly Gaussian and linear with covariance the target covariance;
orthogonality of disjoint time intervals in $L^2(\nu)$ gives independent
increments of $W$, and $I_0\perp I$ gives initial independence.
-Weak equation. For $\phi\in C_c^\infty$, using (30)
+Weak equation. For $\phi\in C_c^\infty$, using the upper-end identity
and boundedness of the isonormal maps against Bochner integrals,
$\int_0^t\zeta_s(A_s\phi) d s=I_0(U(0,t)\phi-\phi)+
I( 1 _{r\le t}\Delta_{U(r,t)\phi-\phi})$, which together with
-(35) is $\zeta_t(\phi)=\zeta_0(\phi)+\int_0^t\zeta_s(A_s\phi)
+the isonormal construction is $\zeta_t(\phi)=\zeta_0(\phi)+\int_0^t\zeta_s(A_s\phi)
d s+W_t(\phi)$.
Uniqueness. On any candidate's probability space, construct the
@@ -598,10 +610,10 @@ $C_b\|\phi\|_a(CT)^n/n!(1+\sqrt{n/(b-a)})^n\to0$, proving uniqueness in
the class and uniqueness in law.
This is a Gaussian generalized field in the mean-square sense of
-(36); it does not claim samplewise membership in the
+the field bound; it does not claim samplewise membership in the
dual of $E_a^0$. Its restriction to $C_c^\infty$ does have a random
distribution version: expanding locally in Fourier functions times a
-compact cutoff, variances bounded by (36) with $a=0$
+compact cutoff, variances bounded by the field bound with $a=0$
make the expected local $H^{-s}$ norm finite for $s>3$, and a cube
exhaustion gives consistent local distributions. No path-space assertion
is needed.
@@ -645,15 +657,14 @@ not a semigroup (the prehistory resets at each slab), which is why
it is treated as a separate object to be compared with the true flow in
the cutoff comparison, below.
-Lemma (Kinetic envelopes). Under Assumption 1,
+Lemma (Kinetic envelopes). Under the kinetic assumption,
$$\sup_{0\le t\le T}(\|f_t\|_{Bol,\beta}+\|\nabla_xf_t\|_{Bol,\beta})<\infty$$
Proof.
Apply Deng--Hani--Ma, Proposition A.1 with decreasing exponent
-$\beta_*(t)=2\beta-\beta t/T$. Its hypothesis is (6)
-and its initial summable norms (14)-(1);
-since $\beta_*(t)\ge\beta$, its conclusion gives (39).
+$\beta_*(t)=2\beta-\beta t/T$. Its hypothesis is the initial Gaussian bound, and its two initial summable spatial norms are that same initial bound;
+since $\beta_*(t)\ge\beta$, its conclusion gives the propagated envelope.
This is Deng--Hani--Ma, Eq. (1.29).
Definition (Layered C/O history). A history consists of finitely many oriented particle lines and binary
@@ -720,7 +731,7 @@ $$\Delta_a^{\mathrm O}
=\delta(|X_1-X_2|-\varepsilon)\mathfrak b
\prod_{j=1}^2\delta(x_{j'}-x_j)\delta(v_{j'}-v_j)$$
-Reverse $\omega$ to match (2); surface measure is
+Reverse $\omega$ to match the scattering convention; surface measure is
unnormalized. With a set $E_*$ of integrated bonds and free ends, the
positive measure of a cut history $M$ with kernel $Q$ is
@@ -747,8 +758,7 @@ $\mathcal T_i$. The one-root signed tree sum contains every one-root
particle-line tree of size $<\Lambda_i/3$, with C/O choices and sign
$(-1)^{\#\mathrm O}$.
-Theorem (Deng--Hani--Ma analytic package). Assume Assumption 1, the hierarchy (37)--
-(38), and a common Gaussian $Bol$ bound on all birth
+Theorem (Deng--Hani--Ma analytic package). Assume the kinetic assumption, the cutoff hierarchy, the exponent hierarchy, and a common Gaussian $Bol$ bound on all birth
inputs. For $L$ sufficiently large (fixed) and $\varepsilon$ sufficiently small:
(a) For admissible dependent histories with prescribed labeled roots,
@@ -785,7 +795,7 @@ bounds each full edge integral by
$\tau^{b/9}\varepsilon^{-2h+2\upsilon/5+c_*\rho}|\log\varepsilon|^{C_*\rho}$.
Multiplication by $\mu^{-h}=\varepsilon^{2h}$ cancels the root power; weakening
$1/9\to1/10$ and discarding the extra positive power gives
-(46). These are the estimates inside the proof of Deng--Hani--Ma, Proposition 6.2, Section 9.2, not only its displayed conclusion.
+the coarse bound. These are the estimates inside the proof of Deng--Hani--Ma, Proposition 6.2, Section 9.2, not only its displayed conclusion.
The only use of a birth density is the envelope decomposition Deng--Hani--Ma, Eqs. (9.58)--(9.63): from (1) choose a summable
sequence $(m_a)$ bounding the weighted supremum on unit balls; bounded
@@ -800,7 +810,7 @@ For (b): the one-root integral is a function of a fixed endpoint
Deng--Hani--Ma, Eq. (14.2); the C/O involution Deng--Hani--Ma, Proposition 14.2 cancels
noncausal trees; the causal remainder satisfies the Gaussian $Bol$
estimate Deng--Hani--Ma, Proposition 14.3, Eq. (14.8); comparison to the short-time
-Boltzmann expansion, using Lemma 3, is Deng--Hani--Ma, Proposition 6.1, Section 14. To pass from $f_{(i-1)\tau}$ to $g$, expand each birth factor as
+Boltzmann expansion, using the kinetic envelope bound, is Deng--Hani--Ma, Proposition 6.1, Section 14. To pass from $f_{(i-1)\tau}$ to $g$, expand each birth factor as
$f_{(i-1)\tau}+(g-f_{(i-1)\tau})$; each residual costs
$\varepsilon^{\theta_{i-1}}$ by hypothesis, and the multilinear tree bound sums
all choices; since $\theta_{i-1}>\theta_i$ the residuals are
@@ -862,7 +872,7 @@ per continuation: the dummy activity $\mu$ cancels the extra root factor
$\mu^{-1}$, its straight line satisfies continuation in every slab, and
the new links are pendant so the initial forest is preserved. This is a
positive kernel inequality, valid even where $f_0$ vanishes; the envelope
-extension of Theorem 2 admits $q_0$, and records are
+extension of the analytic package admits $q_0$, and records are
untouched.
This stage is pure algebra plus the Deng--Hani--Ma coarse bound. Its job: write the
@@ -870,7 +880,7 @@ factorial cumulants of the pasted process as a finite sum of
globally connected diagrams, each with the correct label weight,
while carrying along finitely many observation records (running
sums of test values on designated particles). The high-complexity part is
-dropped with Theorem 2(a); what remains is what
+dropped with the analytic package(a); what remains is what
the sharp bounds, below controls at the fluctuation scale. The
connected coefficients are genuine factorial cumulants of the pasted
process, not a factorization assumption.
@@ -963,7 +973,7 @@ tree $T$, let $Q(T)$ be the edges later than the whole $T$-path; then
connected graphs on $B$ are exactly $T\subseteq G\subseteq T\cup Q(T)$, so
$u(B)=(-1)^{|B|-1}\sum_T\prod_{e\in T}a_e\prod_{e\in Q(T)}(1-a_e)$; since
the second product lies in $[0,1]$, dropping it proves
-(53).
+the autonomous partition.
Idea. The autonomous partition plus the forest identity become a
single linear operator on the factorial hierarchy. Its ``retained'' part
@@ -999,13 +1009,13 @@ where $\mathfrak F^{stop}_S(\mathcal C)$ are forests spanning
$\mathcal C$ with every component meeting $S$, exactly one of size
$[M,2M-2]$, the rest smaller, no unrooted isolated cluster; keep
$\prod_C\kappa_C$ and the root-output map. Define $\mathcal E_sF$ by
-(56) with this kernel and $j$ up to
+the retained operator with this kernel and $j$ up to
$$j\le(s+1)(M-1)\Lambda-s$$
Lemma (One-slab identity and variation remainder). For a positive input law with factorial hierarchy $F$, let $F'$ be its
hierarchy after the truncated slab. If factorial measures up to
-(57) are finite, then
+the fresh-label bound are finite, then
$$F'_s=\mathcal L_sF+R_s, |R_s|\le\mathcal E_sF$$
@@ -1013,10 +1023,10 @@ The record version holds, and $\mathcal E_sF$ is a positive domination of
$R_s$, not a claim that $R_s$ has this local kernel.
Proof.
-Insert (50) and Lemma 6 into
-(54). On a completed branch, fix the reached labels
+Insert the one-slab identity and the rooted forest stopping lemma into
+the branch recursion. On a completed branch, fix the reached labels
$V\supset S$, $|V|=s+j$; the unreached cluster partitions sum to one
-pointwise by (49), using no independence of the
+pointwise by the stopped bound, using no independence of the
input, and
$\frac1{(n-s)!}\binom{n-s}{j}=\frac1{j!(n-s-j)!}$ turns the remaining sum
into $\mu^{s+j}F_{s+j}$; the output normalization $\mu^{-s}$ leaves
@@ -1024,7 +1034,7 @@ $\mu^j/j!$. Stopped terms keep only the unreached compatibility product and
drop the rest, so positivity of the input and the multiplicity bound give
$|R_s(H)|\le(\mathcal E_sF)(|H|)$; there are $\le s$ rooted components and
only one of size up to $2M-2$ clusters, giving
-(57). Records are untouched by the spectator
+the fresh-label bound. Records are untouched by the spectator
summation and travel with the root-output map.
For a signed family $\Delta$ (uniformly in records and $s\le A_i/2$),
@@ -1045,7 +1055,7 @@ a finite signed measure using the full retained one-root block. It
approximates the Boltzmann density:
Proposition (Independent factors). $g_i$ has a Gaussian $Bol$-bounded density representative and, with the
-exponents of Theorem 2,
+exponents of the analytic package,
$$\|g_i-f_{i\tau}\|_{Bol,\beta_i}\le\varepsilon^{\theta_i},
\sup_{0\le i\le L}\|g_i\|_{Bol,\beta_i}\le C$$
@@ -1074,7 +1084,7 @@ the leading tree coefficient is exactly $(-1)^{\#\mathrm O}$ with the
labelled factorial $\mu^{n-1}/(n-1)!$.
Proof.
-Apply (53) to the single retained block and refine
+Apply the autonomous partition to the single retained block and refine
each autonomous C-cluster by its ordered accepted collisions; the internal
indicator is at most the prescribed-C-contact indicator. Each chosen
cluster-tree edge demands a prospective accepted encounter between two
@@ -1094,11 +1104,11 @@ cluster-pair contact) inserts an O-atom that changes no velocity, giving
(iii). Conversely every small unrestricted tree with no further encounter
is admitted with the displayed sign, so no symmetry factor is dropped.
-Proof (Proof of Proposition 3).
-Proposition 1 gives $\|g_0-f_0\|_{Bol,\beta_0}\le C\varepsilon$, which is
+Proof of the independent-factor construction.
+the initial ensemble gives $\|g_0-f_0\|_{Bol,\beta_0}\le C\varepsilon$, which is
the initial smallness for $\theta_0<1$. Assume the bound at the preceding
-interface. Lemma 8 plus the fixed-endpoint part of
-Theorem 2 bounds the three error families and compares the
+interface. the one-root bridge plus the fixed-endpoint part of
+the analytic package bounds the three error families and compares the
signed small-tree sum with $f_{i\tau}$, under the preceding
$Bol,\beta_{i-1}$ error, to $o(\varepsilon^{\theta_i})$, giving the induction and
the uniform $Bol$ bound. The estimate is a fixed-endpoint $Bol$ one, not
@@ -1126,28 +1136,28 @@ in slab $i$ contains a root of the interface above; a block is retained if
it has $\ge2$ such roots or meets a dependent input label below; all other
blocks resum to the independent factor $g_i$ on their single root; every
record-bearing line continues to time zero. Forgetting a record commutes
-with (62).
+with the projection identity.
Proof.
At time zero expand the hierarchy into exact initial cumulant blocks; put
every label of a nontrivial block and of $R$ in $H$, unmarked singletons
into $g_0$ on the complement, marked singletons into $H$ with their
initial record. Insert the preceding expansion into
-(56); blocks with one new root and disjoint from the
+the retained operator; blocks with one new root and disjoint from the
dependent set $H_-$ have independent $g_{i-1}$ inputs and sum to
-(60) (carrying no record since $R\subseteq H_-$);
+the one-root definition (carrying no record since $R\subseteq H_-$);
keep the rest and call their new root set $H$. This gives
-(63) and the rule; record labels stay in $H_-$ at
+the dependent-subset expansion and the rule; record labels stay in $H_-$ at
every step, so their lines continue. Fresh labels on distinct blocks are
disjoint with multinomial coefficients, and
$\frac1{(j_1+j_2)!}\binom{j_1+j_2}{j_1}=\frac1{j_1!j_2!}$ permits the
separate sums; everything is finite. Record forgetting commutes with the
-root-output kernel (55) (record-free weight) and with
+root-output kernel the root-output kernel (record-free weight) and with
the initial updates, so it commutes with the composition.
Regard each nontrivial initial cumulant block as a connection. A block on
$k$ labels is dominated by $\prod f_0$, $(Ck)^{Ck}$, and a root tree with
-edge kernel (40); the sharper kernel
+edge kernel the edge kernel; the sharper kernel
$ 1 _{|x-x'|\le C_{D,k}\varepsilon}+\varepsilon^D$ holds for bounded $k$. These dominate
the exact blocks; artificial links for marked lines are not connections.
@@ -1175,12 +1185,12 @@ size and excess caps are componentwise; every global component contains a
designated root (each slab block meets a root above). Distributing $j$ fresh
labels with sizes $j_1,...,j_q$ gives coefficient
$\frac1{j!}\frac{j!}{j_1!... j_q!}=\prod_a\frac1{j_a!}$, so everything
-factors; group components to get (64), and finite
+factors; group components to get the component expansion, and finite
partition inversion gives the cumulant statement. A one-root resummation
contracts a history attached through one line and cannot split or merge
components. For counting, the only non-fresh labels in a slab are
boundedly many roots/continued labels, giving the first part of
-(65) and then, via $n!/(n-d)!\le n^d$, the second.
+the crossing-factorial bound and then, via $n!/(n-d)!\le n^d$, the second.
There are $2^b$ C/O choices; once the ordered C/O graph is fixed its
C-components are the clusters and its O-forest the blocks, so no extra
partition enters; the exploration has one branch per selected forest.
@@ -1199,19 +1209,19 @@ Write $\Delta_i=F_i-\widehat F_i$. Then
$\Delta_i=R_i+\mathcal L_i\Delta_{i-1}$, $|R_i|\le\mathcal E_iF_{i-1}$,
$\Delta_0=0$. Substitute $F_{i-1}=\widehat F_{i-1}+\Delta_{i-1}$ in the
positive stopped bound and expand the retained part as in
-Lemma 9; refining autonomous histories into C graphs
+the resummation with records; refining autonomous histories into C graphs
and picking one incoming contact per incompatibility gives exactly the
positive C/O histories of the coarse package with previous-interface
continuations. The only top-slab difference, a stopped component with
$M_i\le\cdot<2M_i-2<\Lambda_i$ clusters, is in the same coarse family with
atom count $\ge M_i-1$. Bookkeeping: the lower independent inputs are
already resummed into $g$; the top stopped forest is not resummed. Top
-components failing continuation are paid by Lemma 4(i)
+components failing continuation are paid by the coarse boundary modifications, part (i)
($\varepsilon^{-C_*|S|}$), and bottom continuation of a record line with no
-initial link uses Lemma 4(ii)--(iii) (add a unit-mass Gaussian
+initial link uses the coarse boundary modifications, part (ii)--(iii) (add a unit-mass Gaussian
Bol dummy and an initial link $\ge\varepsilon^\upsilon$, cost $\varepsilon^{-\upsilon}$,
bounded many). Remaining label and initial-block factors are the cutoff
-powers of Lemma 10. Choosing $L$ so
+powers of the label bookkeeping. Choosing $L$ so
$C\tau^{c_0}<1$ and using $\Lambda_i=A_i^{30}$, $A_i\ge|\log\varepsilon|$ makes the
top loss absorb into the exponential size gain,
@@ -1221,7 +1231,7 @@ $$\|\mathcal E_i\widehat F_{i-1}\|_{TV}\le
Input orders are at most $(|S|+1)\Lambda_i^2/2$, which is below
$A_{i-1}/2$; the $\Delta_{i-1}$
-terms are bounded by (59), and
+terms are bounded by the stopped input bound, and
$A_i\Lambda_i^2\log(1/\varepsilon)=o(\Lambda_{i-1})=\Lambda_i^{30}$ absorbs them.
This is the induction; restricting the last slab preserves every bound.
@@ -1229,10 +1239,10 @@ Theorem (Marked component expansion). Choose $\Gamma$ then $L$ as above, indepen
observation list. On this single pasted dynamics:
(i) The factorial hierarchy with any fixed finite record list has the
-finite componentwise expansion (62) with variation
-error (66); independent inputs are the $g_i$
-of Proposition 3; label weights, continuations, and record
-projections are those of Lemma 9, Lemma 10.
+finite componentwise expansion the projection identity with variation
+error the variation error; independent inputs are the $g_i$
+of the independent-factor construction; label weights, continuations, and record
+projections are those of the resummation with records, the label bookkeeping.
(ii) At fixed final order $h$, the actual normalized marked cumulant
$c_h^\varepsilon$ is the sum of retained globally connected diagrams plus an
@@ -1241,27 +1251,27 @@ blocks count as connections; dummy links do not.
(iii) For any accuracy power $P$, one can choose $K$ (after the record list
and $h$) so that all retained diagrams with $\rho>K$ have summed absolute
-variation $\le\varepsilon^P$; at $\rho\le K$ (65) holds.
+variation $\le\varepsilon^P$; at $\rho\le K$ the crossing-factorial bound holds.
Increasing $K$ or $h$ never changes the exponential base per ordinary atom.
Proof.
(i) combines the preceding results. For (ii), invert both the actual and the
-retained hierarchies: (64) gives the connected sum for
+retained hierarchies: the component expansion gives the connected sum for
the retained one, and every term in the difference of the two finite
partition polynomials carries a factor $\Delta_{L,B}$. The actual
normalized factorial measures have mass $\le1$ (initially by the insertion
-bound of Proposition 1, and conserved since $N$ is preserved), and
+bound of the initial ensemble, and conserved since $N$ is preserved), and
the retained ones have variation $\le1+e^{-c\Lambda_L}$ at fixed orders by
-Lemma 11. For (iii) apply the coarse bound to each
+the stopped variation bound. For (iii) apply the coarse bound to each
resummed core with the same fixed record and top-root losses; independent
-top singletons are bounded by Proposition 3; for small $\varepsilon$,
+top singletons are bounded by the independent-factor construction; for small $\varepsilon$,
$|\log\varepsilon|^{C_*\rho}\varepsilon^{c_*\rho}\le\varepsilon^{c_*\rho/2}$, so after summing
the geometric atom factor the $\rho>K$ part is
$\le C_{h,m}\varepsilon^{-C_*h-C_m}\varepsilon^{c_*K/2}$; raise $K$ to make it
$\le\varepsilon^P$. Dummy lines change complexity by a fixed amount and are
removed before low-complexity asymptotics. All root-order and observation
dependence sits in cutoff powers or fixed polynomial factors of
-(65); the exponential atom base depends only on the
+the crossing-factorial bound; the exponential atom base depends only on the
fixed graph counting, the $Bol$ bounds, and $\Gamma$, so the chosen $L$
works for every subsequently fixed order.
@@ -1285,7 +1295,7 @@ low-complexity witness integrals keep their realizability restrictions.
Proof.
Since $n\le3\Lambda_i<A_{i-1}/2$, insert the preceding expansion from
-Theorem 3 into (68).
+the marked component expansion into the terminal-witness functional.
Every lower component has a root leading above and the final roots enter
the connected witness (attachment before discarding any initial link). The
variation error contributes at most
@@ -1300,7 +1310,7 @@ error and the argument starts from the initial cumulant expansion. No
comparison of true and pasted evolutions is made.
This is the central new estimate. The Deng--Hani--Ma package already removed large
-complexity (Theorem 3(iii)); the remaining issue is what
+complexity (the marked component expansion, part (iii)); the remaining issue is what
survives at the sharp fluctuation scale $\mu^{1-h}$ for the $h$-th
factorial cumulant. The mechanism is a change of relative
translations: every collision that merges two clusters of particles
@@ -1321,7 +1331,7 @@ points along each designated trajectory. A record is kept only along a
line on which it is required; forgetting part of a record means pushing
the measure forward under the corresponding coordinate projection, which
commutes with the finite hierarchy before regrouping by
-Theorem 3.
+the marked component expansion.
Let $c_h^\varepsilon$ be the ordered normalized factorial cumulant measure of the
pasted process, with records on its $h$ designated particles; thus $c_1^\varepsilon$
@@ -1329,7 +1339,7 @@ is the one-particle measure and the normalization at order $h$ is
$\mu^{-h}$. Constants $C_K$ may depend on a fixed complexity bound $K$, on
the fixed observations, and on $L$; the exponential base $C^n$ never does.
-The independent factors of Proposition 3 satisfy, for some $b>0$ and
+The independent factors of the independent-factor construction satisfy, for some $b>0$ and
$B_0<\infty$,
$$\sup_{0<\varepsilon\le\varepsilon_0, i}\|g_i\|_{Bol,b}\le B_0,
@@ -1338,7 +1348,7 @@ $$\sup_{0<\varepsilon\le\varepsilon_0, i}\|g_i\|_{Bol,b}\le B_0,
The exponent $b$ is decreased a fixed number of times when reserving Gaussian weights;
this does not change the microscopic dynamics.
-Theorem (Sharp connected bounds). Choose $\Gamma,L$ as in Theorem 3, increasing $L$ if
+Theorem (Sharp connected bounds). Choose $\Gamma,L$ as in the marked component expansion, increasing $L$ if
necessary by an amount depending only on the Boltzmann bounds and on $T$.
This single choice works for every fixed factorial order $h$ and every
fixed bounded observation list. For each fixed $h$,
@@ -1365,7 +1375,7 @@ while newly integrated particles do. This convention is essential for
counting translations and activity factors.
Work with a term of complexity $\le K$. Replace each initial cumulant by
-its bound from Proposition 1,
+its bound from the initial ensemble,
$$C_K\sum_{\mathcal L forest}\prod_{(p,q)\in\mathcal L}
( 1 _{|x_p(0)-x_q(0)|\le R_{K,D}\varepsilon}+\varepsilon^D)
@@ -1431,7 +1441,7 @@ $\dot d=v-v_*$, and $\omega$ opposite the outward normal, so the incoming
condition is $(v-v_*)\cdot\omega>0$. The determinant of
$(t,\omega)\mapsto X-X_*$ on the sphere is
$\varepsilon^2|(v-v_*)\cdot\omega|$, and the incoming convention selects the sign
-in (75). A scattering reflection or O-continuation
+in the coordinate formula. A scattering reflection or O-continuation
updates velocities; the two groups now share one translation.
If the next specified contact is redundant, its relative displacement and
@@ -1454,7 +1464,7 @@ ties are null in merger coordinates unless retained as a redundant-contact
indicator, where no change of variables is made. Work first on compact
subsets of transversal branches and exhaust.
-Multiplying (74) by the merger Jacobians and the
+Multiplying the activity bookkeeping factor by the merger Jacobians and the
physical-link volumes,
$$\mu^{m-h}\varepsilon^{2(m-c-p)}\varepsilon^{3p}=\mu^{c-h}\varepsilon^{p}$$
@@ -1463,9 +1473,9 @@ All acceptance indicators can be carried through the change of variables;
for an upper bound they may be dropped, except that a redundant scattering
determines its branch velocities when it occurs (extend by zero where it
does not). For a connected diagram $c=1$, and a physical initial block has
-$p\ge1$ so gains a factor $\varepsilon$. A term of (71)
+$p\ge1$ so gains a factor $\varepsilon$. A term of the sharp scaling
using a tail factor: erase that link; erasing $\le K$ links raises $c$ by
-$\le K$ and costs $\le\varepsilon^{-2K}$ in (76); choosing
+$\le K$ and costs $\le\varepsilon^{-2K}$ in the activity counting; choosing
$D>2K+M$ makes such terms $O(\varepsilon^M\mu^{1-h})$, with $M$ arbitrary and
chosen after the moment order.
@@ -1520,7 +1530,7 @@ $$\sum_{p<q}|v_p-v_q|\le(n_i-1)\sum_p|v_p|\le n_i^{3/2}\sqrt P$$
Applied to a discrete tree of successive pair choices at fixed merger
times, normals, and birth velocities, where earlier velocities and
intervening specified redundant contacts are already determined by
-Lemma 12, induction bounds the product of fluxes by
+the merger coordinates, induction bounds the product of fluxes by
$$\prod_i(n_i^{3/2}\sqrt P)^{a_i}$$
@@ -1530,7 +1540,7 @@ within a slab, leaving $\prod_i\tau^{a_i}/a_i!$.
A common translation remains per component. For fixed birth velocities and
merger coordinates, write the input spatial arguments as $X+d_p$,
integrate one chosen input in $X$, and use the spatial supremum of the
-others; (69) gives $\le C^m\exp(-bE_0)$ uniformly
+others; the Gaussian suppression gives $\le C^m\exp(-bE_0)$ uniformly
in the displacements. To prescribe an observation position, solve
$X=x-d_{obs}$ with Jacobian one, same bound uniformly in $x$.
@@ -1542,7 +1552,7 @@ lowest bin uses $P^{a/2}=1$ and costs only $C^m$). With $a=\sum_ia_i$,
$$\sum_{P=1,2,4,...}e^{-b_1P/2}P^{a/2}\le C^{a+1}(1+a)^{a/2}$$
by comparison with $\int e^{-b_1u/4}(1+u)^{a/2}$ and Stirling.
-Combining with Lemma 13, the remainder is a fixed polynomial
+Combining with the fresh-label factorials, the remainder is a fixed polynomial
in $1+n$ times
$$C^{n+a}\tau^a(1+a)^{a/2}\prod_i\frac{n_i^{3a_i/2}}{a_i!n_i!}$$
@@ -1557,15 +1567,15 @@ $(1+a)^{a/2}\le C(1+n)^{C_K}n^{n/2}$, and convexity of $u\log u$ gives
$$\prod_in_i^{n_i}\ge(n/L)^n$$
-Substitution proves (78). For weights: the sum of
+Substitution proves the Stirling bound. For weights: the sum of
squared designated velocities is $\le E_0$ at a fixed time and
$\le q_0E_0$ at $q_0$ record times, so a weight exponent below
$b_1/(2\max(1,q_0))$ is absorbed by the reserved Gaussian fraction; the
exponent shrinks with the record list but the exponential base does not.
-Since $\tau=T/L$, the base in (78) is $CT/\sqrt L$;
+Since $\tau=T/L$, the base in the Stirling bound is $CT/\sqrt L$;
increase $L$ once so it is $<1$. There are polynomially many vectors
-$(n_i)$ of total size $n$ (fixed $L$), so (78) gives
+$(n_i)$ of total size $n$ (fixed $L$), so the Stirling bound gives
uniform summability and vanishing size tails, with polynomial degrees that
may grow with $h$ but never need another enlargement of $L$.
@@ -1608,7 +1618,7 @@ $p_0,...,p_k$, $k\ge2$, with edge times $s_1,...,s_k$ gives an endpoint
displacement built from $(v_{p_{j-1}}-v_{p_j})s_j$; the intermediate
velocity $v_{p_j}$ ($1\le j<k$) occurs with coefficient
$s_{j+1}-s_j\ne0$, so varying it off the endpoint velocity direction
-disproves (84).
+disproves the null-set conclusion.
Neighbors meeting at time $a$: after $a$ with no later velocity change the
relative flight is the single separating flight, which has no second
@@ -1638,27 +1648,27 @@ times, proving (ii)--(iii) including contacts whose times vary with $\varepsilon
For a diagram with a cycle, keep its first redundant-contact indicator. All
prior contacts form a forest, and the tested relative trajectory is
determined by that forest prefix (no later redundant update enters). By
-Lemma 15 it tends to zero for a.e. earlier parameters. Truncate
+the no-extra-contact lemma it tends to zero for a.e. earlier parameters. Truncate
size, velocities, and translations; on the compact pieces the post-change
integrands have an integrable bound (from the finitely many speed factors
-or from the proof of Lemma 14), extend unrealizable branches
+or from the proof of the uniform summability bound), extend unrealizable branches
by zero, and apply dominated convergence. Remove velocity/spatial
truncations with the Gaussian bounds and the spatial tightness from
-(69), then the size truncation with
-(78). This order needs no size-uniform rate and proves
+the Gaussian suppression, then the size truncation with
+the Stirling bound. This order needs no size-uniform rate and proves
the asserted $o$ for cycles and nontrivial initial cumulants.
For a fixed tree all prescribed C-clusters and O-blocks fall below the caps
-once $\varepsilon$ is small; by Lemma 15 almost every merger parameter
+once $\varepsilon$ is small; by the no-extra-contact lemma almost every merger parameter
set has only prescribed incoming contacts for all small $\varepsilon$. Thus every
acceptance restriction holds, counting an encounter or its existence is the
same, and the connected compatibility coefficient equals its tree sign
$(-1)^{\#\mathrm O}$. Replacing $g_i$ by $f_{i\tau}$ in a fixed-size term is
-justified by a telescoping product and (69), with
-Lemma 14 (one factor replaced by its $Bol$-difference)
+justified by a telescoping product and the Gaussian suppression, with
+the uniform summability bound (one factor replaced by its $Bol$-difference)
giving convergence to zero. Dominated convergence in space (checked first
on bounded velocities/size, then removing truncations with
-(78)) defines the limiting signed tree measures.
+the Stirling bound) defines the limiting signed tree measures.
Lemma (Spatial traces of tree measures). Let $\mathcal T$ be one of the summable tree measures above, with a line
and a time at which its position is required. It has a disintegration
@@ -1680,9 +1690,9 @@ Proof.
Choose the free representative translation to be the designated position
$x$: every birth position is $x+d_p$, every recorded position likewise,
with displacements independent of $x$ and derivative the identity. This
-gives (85) as an integral formula, with both variation
+gives the expression for the limiting measure as an integral formula, with both variation
bounds from the fixed-position and integrated versions of
-Lemma 14. After size/velocity truncation the test integrand
+the uniform summability bound. After size/velocity truncation the test integrand
is continuous in $x$ with a uniform integrable majorant, and the tails
converge uniformly in $x$ by the fixed-position estimate; a stronger
reserved Gaussian handles Gaussian growth. For two components with
@@ -1723,20 +1733,20 @@ $$d\le2u$$
A lower component attached at one line is either absorbed (first
paragraph) or continues a required record, contributing at most the fixed
number of recorded lines. Starting from the fixed final roots and
-iterating (86) through $L$ slabs bounds all interface
+iterating the completeness recursion through $L$ slabs bounds all interface
roots and recorded singletons: this is $K_0$ (no cycle or initial-link
contribution in a leading tree). Regrouping an unmarked one-root component
is the defining finite sum of its factor with the original fresh-label
coefficient, attaching at one line, so it neither splits nor merges
-components; Theorem 3 shows record forgetting and
-regrouping commute before a limit, and Lemma 14 carries this
+components; the marked component expansion shows record forgetting and
+regrouping commute before a limit, and the uniform summability bound carries this
to the limits, preserving weights and projections.
Lemma (Last-contact products). In the leading tree sums, deleting a last contact of the highest slab that
meets a designated root splits its global tree into the product of the two
component tree measures, incident lines designated before the contact;
their product at the collision position is the conditional product of
-Lemma 16. Conversely joining two disjoint component
+the spatial-trace construction. Conversely joining two disjoint component
histories at that contact gives the corresponding retained tree with its
original factorial weight, and unused record portions project out. Valid
for bounded continuous record tests and, with a smaller exponent, for
@@ -1761,38 +1771,38 @@ $\binom{j_1+j_2}{j_1}\frac1{(j_1+j_2)!}=\frac1{j_1!j_2!}$ splits the
remaining labels; a contact between two designated particles uses one
unordered pair, no fresh label; the deleted contact contributes its C/O
sign, other signs multiply. Letting $N\to\infty$ passes through deletion
-and joining by Lemma 14, Lemma 16, the uniform
+and joining by the uniform summability bound, the spatial-trace construction, the uniform
conditional Gaussian bound controlling the joining flux with a fixed
exponent loss; records project because record forgetting commutes.
-Proof (Proof of Theorem 4).
+Proof of the sharp connected bounds.
Weighted high-complexity tail. Take the birth exponent $b>0$ of
-(69) independent of $L$, and $L$ large enough for
+the Gaussian suppression independent of $L$, and $L$ large enough for
the coarse package at the smaller exponent $3b/4$. For a fixed record list
let $J$ count designated endpoint and record velocity slots; on every
retained history the sum of their squared velocities is $\le JE_0$. Choose
$\alpha>0$ with $\alpha J\le b/4$; the weight is dominated by
$e^{bE_0/4}$, absorbed by tilting every birth input to
$|g_i(x,v)|e^{b|v|^2/4}$ (and $f_0$ likewise), which has a common
-$Bol,3b/4$ bound. The positive coarse bound Theorem 2(a)
+$Bol,3b/4$ bound. The positive coarse bound the analytic package(a)
needs only this envelope; apply it to the tilted inputs with the same
initial-link and record bounds. The geometric complexity sum makes the
weighted retained tail smaller than any power of $\varepsilon$ after choosing $K$.
The Gaussian fraction reserved is fixed, so its atom base is
independent of $J,h,K$.
-Assembling the scale. Use Theorem 3 at accuracy
+Assembling the scale. Use the marked component expansion at accuracy
$\varepsilon^M$, $M>2(h-1)$, with $K$ large enough for that accuracy and for
-Lemma 17. The expansion error and the high-complexity
+the tree-class completeness. The expansion error and the high-complexity
tail are negligible after multiplication by $\mu^{h-1}$ (they need only the
unweighted variation bound, since observation tests are bounded).
-For the remaining connected diagrams, (76) and
-Lemma 14 prove the uniform bound at scale $\mu^{1-h}$.
+For the remaining connected diagrams, the activity bookkeeping and
+the uniform summability bound prove the uniform bound at scale $\mu^{1-h}$.
Physical initial links gain $\varepsilon$; nonphysical initial tails gain any
-power after choosing $D$; cycles are negligible by Lemma 15 plus
+power after choosing $D$; cycles are negligible by the no-extra-contact lemma plus
finite-size dominated convergence and the uniform tails; the survivors are
-exactly the leading trees by Lemma 17. Their fluxes,
+exactly the leading trees by the tree-class completeness. Their fluxes,
signs, input limits, and record tests are evaluated above, and the same
summable majorant yields the asserted Gaussian variation bounds for their
retained sums and limits, including the one-root and two-root limits of the
@@ -1802,7 +1812,7 @@ the $\Gamma,L$ choice remains the single choice made before the moment
order.
The central limit theorem is now run on the single pasted dynamics fixed by
-Theorem 3, Theorem 4. Its $\Gamma,L$ do not
+the marked component expansion, the sharp connected bounds. Its $\Gamma,L$ do not
depend on the observation list or the moment order. The method: attach a
mark that accumulates test values along each particle, so that any
fixed multitime linear combination becomes one scalar. Its ordinary
@@ -1823,25 +1833,25 @@ no randomness. On either dynamics,
$$\sum_{i=1}^Ny_i(T)=\sum_j\sum_{i=1}^N\phi_j(z_i(t_j))$$
Use a superscript $\varepsilon$ for the normalized factorial cumulants of the
-pasted marked process. At any observation-free time Theorem 4
+pasted marked process. At any observation-free time the sharp connected bounds
provides limits
$$c_1^\varepsilon\longrightarrow F_t, \mu c_2^\varepsilon\longrightarrow J_t$$
against bounded continuous tests of the records; $F_t$ is a one-root
measure and $J_t$ a symmetric signed two-root measure, both with the
-Gaussian variation bounds of Theorem 4. Before the updates
-at time zero, Proposition 1 gives
+Gaussian variation bounds of the sharp connected bounds. Before the updates
+at time zero, the initial ensemble gives
$$F_0=f_0(x,v) d x d v \delta_0( d y), J_0=0$$
(since $\mu\|c_{2,0}^\varepsilon\|_1=O(\varepsilon)$). Forgetting records in the finite
marked hierarchy commutes with the one-root regroupings; by
-Theorem 3, Proposition 3, Lemma 17, the
+the marked component expansion, the independent-factor construction, the tree-class completeness, the
$(x,v)$ marginal of $F_t$ is $f_t(x,v) d x d v$ (also inside a slab, via
-the partial-slab version (47) of the one-root block).
+the partial-slab version the partial-slab one-root bound of the one-root block).
-As $F_t$ is positive, it disintegrates by Lemma 16 as
+As $F_t$ is positive, it disintegrates by the spatial-trace construction as
$$F_t= d x F_t(x, d v d y)$$
@@ -1868,7 +1878,7 @@ $$(H_t\Phi)(\xi)=\int_{R^3\times[-M_{obs},M_{obs}]\times\mathbb S^2}
B(v-v_*,\omega)\Delta\Phi F_t(x, d v_* d y_*) d \omega$$
In the formulas below, $\int BFF_*(...)$ is the simultaneous position
-integral from (91), with $ d x d \omega$.
+integral, with $ d x d \omega$.
Proposition (Evolution of the marked tree laws). Between updates,
@@ -1892,19 +1902,20 @@ equivalently integrate in the time of the last top-slab contact. Last
contacts meeting no tested root cancel between scattering and non-scattering
(same flux, opposite signs, test unchanged). When a tested root is incident
the change is its scattered-minus-free value; deleting that last edge splits
-the tree into two components, which by Lemma 18 are the
+the tree into two components, which by the last-contact products are the
component measures multiplied at the common position, with the inverse
joining and its fresh-label coefficients.
-Counts: (a) one root, partner chosen among $j$ fresh labels changes $1/j!$
+Three counts enter. (a) One root: the partner chosen among $j$ fresh labels changes $1/j!$
to $1/(j-1)!$ and splitting remaining labels uses
$\frac1{(j_1+j_2)!}\binom{j_1+j_2}{j_1}=\frac1{j_1!j_2!}$, giving
$\int BFF_*(\Phi(\xi')-\Phi(\xi))$, symmetrized by particle interchange
(including $\omega\mapsto-\omega$) into the collision term of
-(93); (b) two roots, a contact between them is one
-unordered pair, no fresh label, deleting it gives two one-root means and
-the last integral of (94) with no factor $1/2$; (c) a
-contact between the first root and a new partner: the second original root
+the pushforward identity.
+(b) Two roots: a contact between them is one
+unordered pair and no fresh label; deleting it gives two one-root means and
+the last integral of the pushforward identity with no factor $1/2$.
+(c) A contact between the first root and a new partner: the second original root
lies with the first (contribution $J$ on it, $F$ on the partner, change
$\Phi(\xi')-\Phi(\xi)$) or with the partner (interchange; the $J$ component
carries $\Phi(\xi_*')-\Phi(\xi_*)$), summing to $J(H_t\Phi\otimes\Psi)$,
@@ -1915,14 +1926,14 @@ transport the test to the last-contact time $s$; no-contact term plus
$a<s<t$ contact integrals). The finite re-expansion, deletion, and joining
were bijections of labelled trees with their factorial weights; passing to
full sums and unbounded velocities is justified by
-Lemma 14, Lemma 16, Lemma 18. Interface
+the uniform summability bound, the spatial-trace construction, the last-contact products. Interface
concatenation is the empty-top-slab no-contact term, preserved by the
absolutely convergent tree sums. Updates are deterministic pushforwards,
-giving (95).
+giving the pushforward identity.
The pre/post substitution with $\omega\mapsto-\omega$ preserves
$B(v-v_*,\omega) d v d v_* d \omega$, as does particle interchange with
-the normal reversed. Hence the normalization in (4) gives
+the normal reversed. Hence the normalization in the Boltzmann equation gives
$$\int Q(f,f)\phi=\tfrac12\int Bff_*\Delta\phi,
\int[Q(g,f)+Q(f,g)]\phi=\int g K_t\phi$$
@@ -1934,7 +1945,7 @@ Restore the same-particle term by defining
$$G_t(\Phi,\Psi)=J_t(\Phi\otimes\Psi)+F_t(\Phi\Psi)$$
-Then Proposition 4 gives
+Then the tree-law evolution gives
$$\partial_tG_t(\Phi,\Psi)=G_t((D+H_t)\Phi,\Psi)+G_t(\Phi,(D+H_t)\Psi)
+\tfrac12\int BF_tF_{t,*}\Delta\Phi \Delta\Psi$$
@@ -1943,7 +1954,7 @@ Direct check of the source: with $P=\Phi+\Phi_*$, $R=\Psi+\Psi_*$, the
collision change of the diagonal products from the $F$ equation plus the
$J$ source, symmetrized, is $\tfrac12\int BFF_*\Delta(PR)$; the diagonal
drifts contribute $\tfrac12\int BFF_*(R\Delta P+P\Delta R)$; the difference
-is the last term of (98) since
+is the last term of the G evolution since
$\Delta(PR)=R\Delta P+P\Delta R+\Delta P \Delta R$. This uses neither
detailed balance nor equilibrium. The source is positive semidefinite
($F_t\ge0$). Initially, before updates,
@@ -1952,7 +1963,7 @@ $$G_0(\Phi,\Psi)=\int f_0(z)\Phi(z,0)\Psi(z,0) d z$$
and at an update $G$ pushes forward in both slots.
-(98) is needed on the spaces
+the G evolution is needed on the spaces
$$
\mathcal V_a=\{\Phi=c y+h:c\in R, h\in E_a^0\},
\|\Phi\|_{\mathcal V_a}=|c|+\|h\|_a .
@@ -1962,7 +1973,7 @@ Gaussian variation bounds for $F,J$, and choose $0<a<b<\beta$ with
$2b<\alpha$ (a positive margin for the flux; the factor $2$ also controls
the diagonal $F_t(hk)$). The mark is constant on a particle between updates
and conserved at a collision, so the marginal statement after
-(91) gives
+the collision-position integral gives
$$(D+H_t)(c y+h)=A_th,
\tfrac12\int BF_tF_{t,*}\Delta(c y+h)\Delta(d y+k)=C_t(h,k)$$
@@ -1973,7 +1984,7 @@ Use the backward propagator in this possibly nonsmooth class: the Gaussian
variation bounds make $G_t$ a uniformly bounded bilinear form on
$\mathcal V_b$, and the transport mild identity extends from smooth tests
by local uniform approximation with a small Gaussian margin; the unmarked
-collision operator is $K_t$, so (31) applies. Iterating
+collision operator is $K_t$, so the linearized operator estimate applies. Iterating
the bilinear mild identity chooses one of two slots per collision factor;
an $n$-fold remainder is bounded by
$\frac{(CT)^n}{n!}(1+\sqrt{n/\delta})^n$ for fixed allowance
@@ -1989,15 +2000,15 @@ $$h_r=\sum_{j: t_j\ge r}U(r,t_j)\phi_j$$
(The convention at $r=0$ includes observations at time zero; values at
positive pulse times do not affect a time integral.) Combining dual
-propagation, (99), and the pulse substitutions,
+propagation, the backward pulse test, and the pulse substitutions,
$$G_T(y,y)=\int f_0h_0^2+\int_0^TC_r(h_r,h_r) d r$$
After the last pulse the unmarked backward part is zero, so the last
observation time may replace $T$. By the explicit construction
-(35), $\sum_j\zeta_{t_j}(\phi_j)=I_0(h_0)+I(\Delta h_r)$,
-whose variance is exactly (102), which by expanding
-$h_r=\sum_{j:t_j\ge r}U(r,t_j)\phi_j$ equals (33). This
+the isonormal construction, $\sum_j\zeta_{t_j}(\phi_j)=I_0(h_0)+I(\Delta h_r)$,
+whose variance is exactly the variance formula, which by expanding
+$h_r=\sum_{j:t_j\ge r}U(r,t_j)\phi_j$ equals the target covariance. This
identifies all multi-time and repeated-observation correlations.
Proposition (Gaussian limit on the pasted dynamics). For the single pasted dynamics fixed above and every fixed observation list,
@@ -2019,12 +2030,12 @@ Directly: write
$\exp(\lambda S^\varepsilon)=\prod_i[1+(\exp(\lambda y_i)-1)]$, insert the factorial
series, and take the logarithm; each $\exp(\lambda y)-1$ must receive a
nonempty set of the $k$ derivatives, and distributing those sets gives
-exactly the partitions of (103), whose ordering
+exactly the partitions of the factorial-to-ordinary identity, whose ordering
cancels the factorial denominator. This is a finite formal identity through
degree $k$.
All $y$-powers are bounded on the mark interval (extend to bounded
-continuous functions on the mark line). By Theorem 4 every
+continuous functions on the mark line). By the sharp connected bounds every
term on the right is $O_k(\mu)$. For
$X^\varepsilon=(S^\varepsilon-E S^\varepsilon)/\sqrt\mu$, the first cumulant vanishes and
$$
@@ -2033,7 +2044,7 @@ F_T(y^2)+J_T(y\otimes y)=G_T(y,y),
$$
while for $k>2$, $\kappa_k(X^\varepsilon)=O_k(\mu^{1-k/2})\to0$. That these hold on
one dynamics is essential: the bases fixing $\Gamma,L$ in
-Theorem 3, Theorem 4 are independent of the fixed
+the marked component expansion, the sharp connected bounds are independent of the fixed
order; enlarging the auxiliary complexity cutoff or $D$ only changes the
error representation, not the random variable $X^\varepsilon$.
@@ -2049,7 +2060,7 @@ $$
$$
and $n\to\infty$ proves convergence for every real $u$, including
$\sigma=0$. This gives weak convergence with variance
-(102).
+the variance formula.
The pasted dynamics recolors every restarted slab, resets the collision
prehistory at each interface, and truncates components above the caps
@@ -2065,14 +2076,14 @@ would fail. The sharper bound is proved.
Proposition (Accuracy of the pasted dynamics). There is a large dimensional integer $\Gamma_0$ with the following
property. Fix $\Gamma\ge\Gamma_0$. There is
$L_0=L_0(\Gamma,f_0,f,T,\beta)$ such that for every fixed $L\ge L_0$, the
-true and pasted dynamics of Theorem 3, coupled from the
+true and pasted dynamics of the marked component expansion, coupled from the
same initial configuration, satisfy
$$\mathbf{P}\{the two trajectories differ at some time in [0,T]\}\le
C\varepsilon^{1+\eta}$$
for all small $\varepsilon$, with $\eta=1/20$. The size cutoffs are the
-$A_i,\Lambda_i$ of Theorem 2; $C$ and the upper bound on $\varepsilon$
+$A_i,\Lambda_i$ of the analytic package; $C$ and the upper bound on $\varepsilon$
may depend on the displayed fixed data.
The rest of this argument proves it. The plan:
@@ -2120,8 +2131,8 @@ rejection is exactly ``component size $>\Lambda_i$ or excess added $>\Gamma$''
within this block. Summing over possible blocks, each such rejection is
charged to a distinct terminal witness; summing charges gives the bound.
-Insert the terminal-witness functional (68)
-and apply Corollary 1. Begin with the previous pasted
+Insert the terminal-witness functional
+and apply the terminal-witness expansion. Begin with the previous pasted
factorial measure at interface $i-1$. Since $n\le3\Lambda_i<A_{i-1}/2$, the
variation error is below every power. The connected terminal witness is
itself a block at the top of the pasted hierarchy and enters through its
@@ -2137,7 +2148,7 @@ record; the latter are boundedly many (the records through this interface,
including continued empty lines).
Choose a complexity threshold $K$ after $L$ is fixed. By
-Corollary 1, Theorem 2, the sum of positive majorants
+the terminal-witness expansion, the analytic package, the sum of positive majorants
with $\rho>K$, including the activity cost of an artificial top root when
required, is $O(\varepsilon^3)$ for $K$ large enough. The terms left to estimate
have $\rho\le K$. Their interface crossings, initial-link vertices,
@@ -2155,7 +2166,7 @@ without new crossings and within the same realization family by
$\le C^b$ choices of the initial labelled $M$ and its geometric data.
The number of scattered velocities in any witness is at most $B_\varepsilon$, and
-initial data carry only $\le2\beta$ Gaussian under Assumption 1; a
+initial data carry only $\le2\beta$ Gaussian under the kinetic assumption; a
terminal witness has controllable total birth energy with Gaussian
cost $e^{-cP}$ for total energy $P$.
@@ -2237,7 +2248,7 @@ the surviving mass by $\prod_\ell H_\ell$, and intervening deterministic
substitutions have norm at most one on bounded functions. Nonnegative
Fubini represents $\nu$ as the product in $\vartheta$ times these
kernels, and majorizing the indicator of $E$ by
-$ 1 _{\mathcal D}(\vartheta)$ proves (109).
+$ 1 _{\mathcal D}(\vartheta)$ proves the extraction inequality.
Exhaustion removes the boxes. Equivalently, restrict ordinary relative
speeds away from zero and bound a rectangle by its product volume; cover a
compact containing set by products of rectangles and take the infimum;
@@ -2347,7 +2358,7 @@ at least $\varepsilon^\delta$ up to a fixed polylogarithmic factor.
Lemma (Amplified pure-collision cut). For large dimensional $\Gamma_0$, every connected realizable full scattering
graph with $\Gamma<r(M)\le2\Gamma$, $\Gamma\ge\Gamma_0$, decomposes into cut
sequences such that either it contains a freely cut pair estimated by
-(110) with no other activity loss, or its retained
+the gain balance with no other activity loss, or its retained
improved pieces $g$ and free singletons $s_4$ satisfy
$$\delta g-2s_4\ge6/5$$
@@ -2360,15 +2371,15 @@ The local powers must survive simultaneous integration without a
per-vertex logarithmic loss. Call a neutral type-three singleton
ordinary; all other pieces exceptional. Counting both vertices
of an exceptional pair, their number is bounded by a constant depending on
-$K,\Gamma$; the terminal part is Lemma 25, below it
-Lemma 23; the degree count
+$K,\Gamma$; the terminal part is the amplified cut, below it
+the local kernels; the degree count
$s_2+p_{33}=r(M)+s_4+p_{44}-1$ on the complete graph (or its version with
boundedly many crossing ends fixed) bounds the extra type-two pieces. With
$b$ the total atoms and $q$ the ordinary ones, $q=b-O_K(1)$.
Lemma (Total birth energy bounds the velocity domain). Fix a cutoff $E_{cut}=P_\varepsilon$ and, as in
-Lemma 20, use the local kernels at the scopes where
-Proposition 7 and Lemma 25 retain them. Then
+the energy tail, use the local kernels at the scopes where
+the cutting alternatives and the amplified cut retain them. Then
for a witness with ordinary projection within the containing set $\mathcal D$
(energy $\le P_\varepsilon$), there is a compact measurable set inside
$\mathcal D$, of volume
@@ -2407,7 +2418,7 @@ $$
$$
The identity enlargement handles small singular values without an inverse
Jacobian; a finite union of continuous images of compact parameter sets
-gives the compact containing set for Lemma 22.
+gives the compact containing set for the extraction lemma.
Lemma (Summation with finitely many exceptional pieces). Suppose a cut sequence has $q=b-O_K(1)$ ordinary atoms and boundedly many
exceptional pieces whose conditional mass bounds have product
@@ -2419,28 +2430,28 @@ $$\varepsilon^\alpha|\log\varepsilon|^{C_*}(C\tau\sqrt L)^q$$
The base $C$ depends on the fixed kinetic envelopes and dimension, not on
$K,L$. This is (Deng--Hani--Ma, Lemma 16.2).
-Proof (Proof of Proposition 6).
-Start from Lemma 19, Corollary 1, and fix the
+Proof of the pasted accuracy bound.
+Start from the first-disagreement bound, the terminal-witness expansion, and fix the
complexity cutoff $K$ large enough that the coarse part is $<\varepsilon^3$
-(Theorem 3(iii) after Corollary 1):
+(the marked component expansion, part (iii) after the terminal-witness expansion):
$\rho>K$ is discarded. The remaining witnesses are connected, full, realizable
graphs with $\rho\le K$, hence bounded crossing/initial/cycle counts, and
-$r(M)\le2\Gamma$. Localize by Lemma 20: outside the
+$r(M)\le2\Gamma$. Localize by the energy tail: outside the
polylogarithmic energy region the sum is smaller than any power.
-On the low-energy region, apply Proposition 7, Lemma 25
+On the low-energy region, apply the cutting alternatives, the amplified cut
to the top-slab graph. The freely cut primitive pair (if any) gives
-(110), product $\varepsilon^{13/10}|\log\varepsilon|^{C_*}$. Otherwise
-(111) holds: the retained improved pieces number $g$
+the gain balance, product $\varepsilon^{13/10}|\log\varepsilon|^{C_*}$. Otherwise
+the summation bound holds: the retained improved pieces number $g$
with a fixed bound except for the bounded seed, and gain $\varepsilon^{\delta g}$,
while each free type-four seed costs $\mu=\varepsilon^{-2}$; so the exceptional
product is at most $\varepsilon^{6/5}|\log\varepsilon|^{C_*}$ (using
$\delta g-2s_4\ge6/5$ in both the $\varepsilon^{13/10}$ and the $\varepsilon^{6/5}$
branches). All the extra variables inside the containing set are then
-removed by the coarea and time-order bounds: Lemma 21
+removed by the coarea and time-order bounds: the causal-order bound
gives $\prod_i\tau^{q_i}(2q_iP_\varepsilon)^{q_i/2}/q_i!$, and the ordinary
-velocity box by Lemma 26. Combining
-Lemma 27, Lemma 22, the low-complexity sum is
+velocity box by the birth-energy bound. Combining
+the exceptional-piece summation, the extraction lemma, the low-complexity sum is
bounded by
$$
\varepsilon^{6/5}|\log\varepsilon|^{C_*}(C\tau\sqrt L)^{b-O_K(1)} .
@@ -2448,7 +2459,7 @@ $$
Choose $L$ so large that $C\tau\sqrt L=C T/\sqrt L<1/2$, absorbing
constants depending on $K,L$ into the fixed prefactor. The exponential atom
counting ($C_1^b$ with $C_1$ independent of $K,L$, per
-Theorem 3(iii) and Corollary 1) together
+the marked component expansion, part (iii) and the terminal-witness expansion) together
with the bounded crossing labels, selected pairs, and initial coefficients
costs only a fixed-degree polynomial; the resulting sum over $b$ is a
convergent geometric series, with the $\Lambda_i$ cutoff removing the large-$b$
@@ -2457,27 +2468,27 @@ part. Thus the total sum is at most
$$C\varepsilon^{6/5}|\log\varepsilon|^{C_*}$$
Since $\tfrac{6}{5}=1.2>1+\tfrac{1}{20}=1.05$, this is
-$\le C\varepsilon^{1+1/20}$, proving (104). (The witnesses
+$\le C\varepsilon^{1+1/20}$, proving the discrepancy bound. (The witnesses
with terminal component larger than $\Lambda_i$ are smaller than any power
-by Lemma 20, Theorem 2, via the activity cost
+by the energy tail, the analytic package, via the activity cost
$\mu=\varepsilon^{-2}$ versus the Gaussian gain over $\Lambda_i\ge|\log\varepsilon|^{30}$
births, summing geometrically as in the coarse step.)
Remark (Order of choices). The local exponent $\delta$ and the finite hierarchy of
-Lemma 25 fix $\Gamma_0$ from dimensional constants. Then
+the amplified cut fix $\Gamma_0$ from dimensional constants. Then
choose $\Gamma$, then $L$ for the imported estimates and the common
exponential base; only afterward choose $K$, the localization powers, and the
small upper bound on $\varepsilon$. No observation number or moment order enters the
first two choices. This is what allows a single pasted dynamics to be fixed in
the Gaussian limit of the model dynamics, below.
-The pasted process of Proposition 5 is centered by its own
+The pasted process of the pasted Gaussian limit is centered by its own
microscopic mean. The real gas is centered by a slightly different mean,
since the pasted and true laws charge the discrepancy event (probability
-$\le C\varepsilon^{1+\eta}$, Proposition 6) with different mass. At the
+$\le C\varepsilon^{1+\eta}$, the pasted accuracy bound) with different mass. At the
fluctuation scale $\sqrt\mu=\varepsilon^{-1}$ this difference of means must go to
-zero, and that is exactly the sharp estimate of Proposition 6. This is made
-quantitative, then Theorem 1 is assembled.
+zero, and that is exactly the sharp estimate of the pasted accuracy bound. This is made
+quantitative, then the fluctuation limit theorem is assembled.
Lemma (Discrepancy and centering). Couple the true and pasted dynamics from the same initial configuration.
For every fixed time $t$ and bounded continuous test $\phi$, with
@@ -2495,7 +2506,7 @@ $$\zeta_t^{\varepsilon,true}(\phi)
Proof.
Let $A=\{the two trajectories agree on [0,T]\}$, so
$\mathbf{P}(A^c)\le C\varepsilon^{1+\eta}$, and both $N$ and $N/\mu$ have bounded second
-moments (Proposition 1). On $A$ the empirical measures agree for every
+moments (the initial ensemble). On $A$ the empirical measures agree for every
$t$, hence so do $\pi_t^\varepsilon(\phi)$ and the running records. Write
$\pi=(1/\mu)\sum_i\phi$. Off $A$, $|\pi|\le\|\phi\|_\infty N/\mu$ on either
side, so $|\pi^{true}-\pi^{pasted}|
@@ -2506,13 +2517,13 @@ $$
\le C_\phi \varepsilon^{(1+\eta)/2},
$$
and multiplying by $\sqrt\mu=\varepsilon^{-1}$ gives
-$\varepsilon^{(\eta-1)/2}\to0$, proving (116).
+$\varepsilon^{(\eta-1)/2}\to0$, proving the mean comparison.
-For (117), insert the centered definitions:
+For the centered-difference estimate, insert the centered definitions:
$\zeta^{true}-\zeta^{pasted}
=\sqrt\mu[(\pi^{true}-\pi^{pasted})
-(E_{true}\pi-E_{pasted}\pi)]$.
-The second bracket is bounded by (116). For the first,
+The second bracket is bounded by the mean comparison. For the first,
using the $A^c$ indicator,
$$
E\sqrt\mu |\pi^{true}-\pi^{pasted}|
@@ -2523,33 +2534,33 @@ and $\sqrt\mu \varepsilon^{(1+\eta)/2}=\varepsilon^{-1}\varepsilon^{(1+\eta)/2}
=\varepsilon^{(\eta-1)/2}\to0$. So the $L^1$ norm of the difference of the
centered fields tends to zero.
-Proof (Proof of Theorem 1).
-Existence and uniqueness of the target. Proposition 2
+Proof of the fluctuation limit theorem.
+Existence and uniqueness of the target. the Gaussian target
constructs the centered generalized Gaussian field $\zeta$ with covariance
-(33), initial field $\zeta_0$ of covariance
-(11) independent of the noise $W$, and weak equation
-(10); uniqueness in law holds in the class of
+the target covariance, initial field $\zeta_0$ of covariance
+the initial covariance independent of the noise $W$, and weak equation
+the fluctuating equation; uniqueness in law holds in the class of
$L^2$-continuous fields there.
-Initial fluctuation limit. Proposition 1(iii) gives, for any
+Initial fluctuation limit. the initial ensemble, part (iii) gives, for any
fixed list of tests, joint convergence of
$(\zeta_0^\varepsilon(\phi_1),...,\zeta_0^\varepsilon(\phi_m))$ to the centered Gaussian
vector with covariance $\int f_0\phi_i\phi_j$.
Dynamical limit on the pasted dynamics. Fix a single pasted
dynamics as in the Gaussian limit of the model dynamics, below ($\Gamma,L$ independent of the
-observation list and order). Proposition 5 gives, for every fixed
+observation list and order). the pasted Gaussian limit gives, for every fixed
list of times and tests, convergence of the centered combination
$\varepsilon\sum_i y_i(T)$ to $\sum_j\zeta_{t_j}(\phi_j)$, with convergence of all
-centered moments. By eq:factorial-ordinary and
-Theorem 4, this is the finite-dimensional Gaussian limit
-with the multi-time covariance (33).
+centered moments. By the factorial-to-ordinary identity and
+the sharp connected bounds, this is the finite-dimensional Gaussian limit
+with the multi-time covariance the target covariance.
-Transfer to the true flow. By Lemma 28,
+Transfer to the true flow. By the discrepancy and centering bound,
$\zeta_t^{\varepsilon,true}(\phi)-\zeta_t^{\varepsilon,pasted}(\phi)
=o_{L^1}(1)$, while
$|\zeta_t^\varepsilon(\phi)|\le\|\phi\|_\infty(\varepsilon N+\varepsilon E N)$ has uniformly
-bounded moments because $E(N/\mu)^p<\infty$ (Proposition 1). Hence the
+bounded moments because $E(N/\mu)^p<\infty$ (the initial ensemble). Hence the
$p$-moments of the difference vanish for every fixed $p$ by interpolation of
the $L^1$ bound with the uniform $L^{p+1}$ bound, and, for any bounded
continuous functional of a fixed finite collection
@@ -2561,21 +2572,21 @@ $$
one, then the moment control of the difference off the compact). Therefore
the true centered fields converge in distribution to the same Gaussian
vector. Sending the finite list over a countable dense family and using the
-variance bound (36) upgrades this to the map
+variance bound the field bound upgrades this to the map
$t\mapsto\zeta_t(\phi)$ in joint law.
-Weak equation. The limiting field inherits (10) from the
-construction (35), so its law is the one prescribed. The
-covariances (11) and (12) are identified in
-Proposition 2 and via (102). This is
-(13).
+Weak equation. The limiting field inherits the fluctuating equation from the
+isonormal construction, so its law is the one prescribed. The
+initial and noise covariances are identified in
+the Gaussian target and via the variance formula. This is
+the convergence assertion of the fluctuation limit theorem.
-Remark (What is exact and what is asymptotic). The mean subtracted in (7) is the exact microscopic
+Remark (What is exact and what is asymptotic). The subtracted mean is the exact microscopic
expectation, not a Boltzmann approximation; this is essential because of
-Remark 1. The Gaussian limit and its covariance are exact
+the centering remark. The Gaussian limit and its covariance are exact
in the limit; on the way, only the discrepancy probability, not the law
-itself, is estimated with a power gain. The single imported analytic package
-is Theorem 2, used solely to remove large complexity and to
+itself, is estimated with a power gain. The single imported analytic input is the
+Deng--Hani--Ma package, used solely to remove large complexity and to
compare signed small trees with the Boltzmann solution; every connected
correlation, the sharp tree limits, the covariance, the Gaussian central
limit, and the exact-centering transfer are proved.