Proof. Geometry and smooth convergence. Choose \(k_N\) by rounding \(N\int V\) to an integer and integrate (5). Then \(H_N(u+1)-H_N(u)=k_N\), so the map \(u\mapsto H_N(u)\bmod1\) is globally defined. The map \(C_N\) is a smooth torus diffeomorphism of Jacobian one, with inverse \((\theta',u',a')\mapsto(a'+H_N(\theta'),\theta',u')\). Direct substitution gives \[C_NT_NC_N^{-1}(\theta',u',a') =(\theta'+\sigma_N(u'),u',a').\] This proves the global assertions, except convergence. The integral identity \[H_N(u+\sigma_N(a))-H_N(u) =N\int_0^{\sigma_N(a)}(V(u+t)+\Delta_N)\,\mathrm dt =V(u)+O_{C^k}(N^{-1})\] holds for each fixed \(k\), with constants depending on the fixed \(c,J\) and \(\phi\). Since \(c\) is an integer, addition of \(V\) and of \(\phi\) agree on \(\mathbb T\). The inverse of (6) is \[(\theta,u,a)\longmapsto (\theta-H_N(u)+H_N(u-\sigma_N(a)),u-\sigma_N(a),a),\] and the same estimate proves inverse convergence. Composing with a fixed physical chart and its inverse preserves convergence in every fixed smooth norm. No bound uniform over future coordinate charts is being asserted.

Fourier coefficients. Work on a compact subarc of \(I\) containing all old coefficient supports in its interior. For \(s=j/N\) the phase in (10), divided by \(N\), has critical equation \[ m(V(u)+\Delta_N)=s \tag{14}\] and second derivative \(m\phi'(u)\). It has at most one critical point on this arc. With \(u_*=u_*(s-m\Delta_N)\) on that branch, set \[L_{m,j}(a)= \frac{A(u_*,a)}{\sqrt{N|m\phi'(u_*)|}} \mathop{\mathrm{e}}\left(mH_N(u_*)-ju_*+\frac{\operatorname{sgn}(m)}8\right),\] and set this to zero wherever the amplitude at the critical point is zero or there is no point on the branch. Uniformly for \(j/N\) in any fixed compact interval and \(a\in\mathbb T\), \[ C_{m,j}(a)-L_{m,j}(a)=o(N^{-1/2}). \tag{15}\]

For general stationary-phase background see (Hörmander 2003); we give the compact-support one-dimensional argument needed here. Enlarge the support arc twice inside \(I\). Critical points near the original support then range over a fixed compact interval on which the absolute second derivative is bounded below. A smooth change of variable, depending smoothly on the critical point, reduces the divided phase to its critical value plus \(\operatorname{sgn}(m)y^2/2\); its Jacobian at zero is \(|m\phi'(u_*)|^{-1/2}\). After putting \(v=\sqrt N y\), the integral on \(|v|\le D\) converges uniformly to the constant amplitude times the quadratic exponential integral there. The remaining local tails are \(O(N^{-1/2}/D)\), uniformly in the parameters, by integrating against \(\mathop{\mathrm{e}}(\operatorname{sgn}(m)v^2/2)\): the rescaled amplitudes have uniformly bounded suprema and total variation. The part away from the critical point is \(O(N^{-1})\) by integration by parts. Finally \[\lim_{D\to\infty}\int_{-D}^D\mathop{\mathrm{e}}(\pm v^2/2)\,\mathrm dv =\exp(\pm i\pi/4);\] this follows by inserting a Gaussian factor, evaluating the Gaussian integral, and removing the factor using the same tail bound. Taking first \(N\to\infty\), then \(D\to\infty\), proves (15). When the potential critical point is outside the enlarged support interval, the integral is uniformly nonstationary. This also covers zero amplitudes and the ends of the branch.

There are \(O(N)\) indices in a fixed bounded \(j/N\) range. Outside a sufficiently large such range, repeated integration by parts gives, for any fixed positive integer \(q\), \(|C_{m,j}(a)|\le C_q|j|^{-q}\) uniformly in \(a\) and \(N\). Indeed the phase derivative has size comparable to \(|j|\), and its higher derivatives are \(O(N)\) with \(N=O(|j|)\) there. Thus (15) implies \[\sum_j\int_\mathbb T|C_{m,j}-L_{m,j}|^2\,\mathrm da=o(1).\] The \(L^2\) norm of the leading array is bounded by a Riemann sum. Cauchy–Schwarz therefore gives the useful mass estimate \[ \sum_j\int_\mathbb T \left||C_{m,j}(a)|^2 -\frac1N W(j/N-m\Delta_N,a)\right|\,\mathrm da=o(1). \tag{16}\]

Cutoffs and transport. The map \((u,a)\mapsto(mV(u),a)\) is injective on \(I\times\mathbb T\). It takes disjoint compact supports with the same \(m\) to disjoint compact sets. Our choice of \(c\) separates the images for different \(m\). The stated cutoffs \(\chi_\alpha\) therefore exist. For large \(N\) they equal one on the support of the corresponding leading array, since its \(s\) coordinate is shifted by only \(m_\alpha\Delta_N\). Because each cutoff is one on the shifted leading-array support, the deleted exact coefficient mass is bounded by the summed squared \(C_{m,j}-L_{m,j}\) error, which tends to zero by the preceding estimate and Parseval’s identity. The same argument is valid after multiplication by \(\psi\), proving (11) and (16) for the retained arrays with the factor \(\psi^2\) on the right. The cutoffs make separation exact, not merely asymptotic. Different old \(m\) never contribute to the same retained \(j\), and at a shared \(j\) the remaining supports in \(u'=a\) are disjoint. The factor \(\mathop{\mathrm{e}}(ma')\) in (9) proves the asserted independence of the new spare-coordinate modulus, also after combining packets at one index. Linearity proves the statement about attached constants.

Fine temporal density. We first remove rounding before any fine-scale sampling. Smoothness and compact support of the fixed \(W\) give \[ \frac1N\sum_j\int_\mathbb T |W(j/N-m\Delta_N,a)-W(j/N,a)|\,\mathrm da=o(1). \tag{17}\] For example, a uniform bound on \(\partial_s W\) and \(O(N)\) nonzero summands give an \(O(|m|/N)\) bound. The same holds with \(\psi^2\) and for any residue subset. Pushforward contracts total variation, so (16) and (17) allow us to replace the retained squared coefficient measures by \(N^{-1}W(j/N,a)\psi(a)^2\,\mathrm da\), at total error \(o(1)\), before pushing to temporal frequency. In particular, the actual lattice \(j/N\) is never translated by \(m\Delta_N\).

On the retained arc, the change of variable \(y=sJ(a)\) gives \[ K(s,y)=\frac{W(s,a)\psi(a)^2}{|s|J'(a)}, \qquad a=J|_{I'}^{-1}(y/s), \tag{18}\] with zero extension. The compact support of \(\psi\) inside \(I'\) makes this extension continuous (in fact smooth away from \(s=0\)). The real temporal frequency of a new angular index \(j\) is \[ t=j\sigma_N(a)=j/N+(j/N)J(a)/N. \tag{19}\] The density of the pushforward of \(N^{-1}W(j/N,a)\psi(a)^2\,\mathrm da\) under this map is exactly \(K(j/N,Nt-j)\). The factor \(N\) in the density change of variable cancels the coefficient mass \(N^{-1}\).

Choose \(Y\) so that \(K(s,y)=0\) for \(|y|>Y\). For fixed \(t\), at most \(2Y+2\) integers contribute to either \(\sum_j K(j/N,Nt-j)\) or \(\sum_jK(t,Nt-j)\); on every contributing term \(|j/N-t|\le Y/N\). Uniform continuity of \(K\), and a fixed bounded interval containing the \(t\) supports for large \(N\), prove that replacing \(j/N\) by \(t\) costs \(o(1)\) in \(L^1(\mathbb R)\). The same proof works after restriction to \(j\equiv h\pmod P\). This is (13). Folding modulo one contracts the \(L^1\) norm and finishes the proof. ◻

Proof. Partition the \(t\) axis into intervals of length \(P/N\) on each of which \(Nt+\beta_N\) traverses one period. On each interval replace the slow argument \(t\) of \(G\) by an endpoint. Uniform continuity on the fixed compact support bounds the integrated error by a constant times its modulus of continuity at \(P/N\), with an additional \(O(N^{-1})\) boundary error. The resulting averages are Riemann sums for the right side of (21). This proves the first assertion, uniformly in the offset. Apply it to the continuous absolute difference in (22). The integral over one period of that difference is bounded by \[\sum_k\int_0^P |s|^{-1} |\rho((y+kP)/s)-\widetilde\rho((y+kP)/s)|\,\mathrm dy =\|\rho-\widetilde\rho\|_1.\] Finally, on a compact set of nonzero \(s\), quadratic tail bounds give a uniform summable bound \(C'/(1+k^2)\) on the periodizing summands, for \(0\le y\le P\). Uniform convergence proves joint continuity. Compactly supported densities are handled in the same way with only finitely many nonzero summands on that compact set. ◻

Proof. Let \(F(y)=\int_{-\infty}^y\rho(v)\,\mathrm dv\). Strict positivity makes \(F\) a smooth diffeomorphism from \(\mathbb R\) to \((0,1)\). Choose \(0<\delta<\min(1/8,\varepsilon/4)\) and a smooth function \(\xi\) compactly supported in \((\delta/2,1-\delta/2)\) and equal to one on \([\delta,1-\delta]\). In the circle coordinate \(0<a<1\), set \(J(a)=\xi(a)F^{-1}(a)\) and extend it by zero near the identified endpoints. It is a smooth bounded circle function, and on \(I'=(\delta,1-\delta)\) it has derivative \(1/\rho(F^{-1}(a))>0\). Choose \(\psi\in C_c^\infty(I')\) with \(0\le\psi\le1\) and \(\psi=1\) on \([2\delta,1-2\delta]\). Change of variables gives \[\widetilde\rho(y)=\rho(y)\psi(F(y))^2\le\rho(y),\qquad \|\rho-\widetilde\rho\|_1 =1-\int_\mathbb T\psi^2\,\mathrm da\le4\delta<\varepsilon.\] Its support is compact because that of \(\psi\) is compactly inside \((0,1)\). All the claims follow. ◻

Proof. Fix the old finite data and choose \(c\) as in Lemma 2, also satisfying its prescribed lower bound. Identify all but a small interval of the \(a\) circle with an interval in \(\mathbb R\). Choose an open arc \(I'\) whose complement is as small as requested, and \(0\le\psi\le1\) compactly supported there and equal to one except on a set of sufficiently small length. The boundedness of the finitely many old amplitudes allows us to make both their \(L^2\) cutoff errors and the removed squared masses arbitrarily small. Fix this cutoff before proceeding.

For a positive parameter \(L\), take a smooth periodic function \(J_L\) equal to \(La+\beta\) on \(I'\), extending it smoothly across the complementary interval. Such an extension exists by taking a smooth cutoff equal to one on the closure of \(I'\) in a slightly larger real coordinate interval. In particular \(J_L'=L\) on \(I'\). Put \(f_s(a)=W(s,a)\psi(a)^2\), extended by zero in this real coordinate. The periodization of its pushforward by \(sJ_L\) is \[ \sum_{k\in\mathbb Z}\frac1{|s|L} f_s\left(\frac{(y+kP)/s-\beta}{L}\right). \tag{24}\] This is \(1/P\) times a Riemann sum for \(\int f_s(a)\,\mathrm da\), of mesh \(P/(|s|L)\), with an arbitrary lattice offset and possibly reversed orientation. The error is bounded by \[ \frac{\|\partial_a f_s\|_{L^1(\mathbb R)}}{|s|L}. \tag{25}\] Indeed compare each lattice value times the mesh with the integral on its cell, using the fundamental theorem of calculus and summing the absolute derivative. Since \(s\) ranges over a fixed compact set away from zero, this bound tends to zero uniformly in \(s\) and \(y\). It remains valid for arbitrary fixed dependence of the original amplitude on \(a\).

Choose \(L\) large enough for these uniform errors to be as small as needed for the finite packet list. Now \(J_L\) and all its norms are fixed. Choose the support cutoffs in Lemma 2, and then take \(N\) sufficiently large. Its fine-density rule, together with (24), approximates each unfolded residue density by \(P^{-1}\int W(s,a)\psi(a)^2\,\mathrm da\) in \(L^1\). Folding modulo one gives \(p_r/P\), up to the small cutoff loss; the integer \(mc\) in \(mV\) has no effect on this folding. The estimate is uniform in the periodic phase, so the folding offsets \(Nl\) cause no difficulty. Summing the residue estimates gives the total density estimate, upon initially making each error smaller by the fixed factor \(P\) if necessary. Norm transport, exact support properties, and smooth proximity follow from the same passage lemma. All constraints on \(N\) are lower bounds imposed after the physical chart, the amplitudes, \(c\), the cutoff, \(L\), and the support neighborhoods have been fixed. ◻

Proof. A positive law. We first construct a positive law without using \(B\). We use the positive-exponent Fourier convention \[\widehat q(v)=\int_\mathbb Rq(R)\mathop{\mathrm{e}}(vR)\,\,\mathrm dR.\] Choose \(0<\alpha<b/2\), put \(a_0=(2\alpha)^{-1}\), and define \[ s_\alpha(R)=\alpha\left(\frac{\sin(\pi\alpha R)} {\pi\alpha R}\right)^2, \qquad q_0(R)=\frac1{2a_0}\int_{-a_0}^{a_0}s_\alpha(R-t)\,\,\mathrm dt, \tag{29}\] with the removable value at the origin understood. Fourier inversion for an interval, followed by convolution, gives \[\widehat{s_\alpha}(v)=(1-|v|/\alpha)_+.\] Consequently \(q_0\) is a smooth probability density with Fourier support in \([-\alpha,\alpha]\). It is strictly positive: the interval of translations in (29) cannot be contained in the discrete zero set of \(s_\alpha\).

Here a quantitative lower tail is useful. Since the averaging interval has length \(1/\alpha\), it contains exactly one period of \(\sin^2(\pi\alpha(R-t))\) as a function of \(t\). For \(|R|\ge2a_0+1\) this gives \[\frac1{2\pi^2\alpha(|R|+a_0)^2} \le q_0(R) \le\frac1{\pi^2\alpha(|R|-a_0)^2}.\] Positivity and continuity on the remaining compact interval therefore give constants \(c_0,C_0>0\) such that \[ \frac{c_0}{1+R^2}\le q_0(R)\le\frac{C_0}{1+R^2} \qquad(R\in\mathbb R). \tag{30}\]

Choose \(D>b\) such that \(\mathop{\mathrm{supp}}d\subset[-D,D]\), and then an integer \(L>\max(2,D/b)\). Let \(P\) be the product of all primes at most \(L\). The strict inequality in this choice will allow a small dilation of the Fourier arguments. Define \[g(R)=\int_\mathbb Rd(v)\mathop{\mathrm{e}}(-vR)\,\,\mathrm dv, \qquad C_P(h)=\sum_{k\in(\mathbb Z/P\mathbb Z)^\times}\mathop{\mathrm{e}}(-kh/P) \quad(0\le h<P).\] The function \(g\) is Schwartz by repeated integration by parts. The symmetry in (26) makes it real, and \(\int g=d(0)=0\). The number \(C_P(h)\) is real because the set of units is invariant under negation. Set \[ Q_h(R)=\frac1P\bigl[q_0(R)+\gamma C_P(h)g(R)\bigr]. \tag{31}\] For \(G=\sup_R(1+R^2)|g(R)|\) and \(C_* =\max_h|C_P(h)|\), choose \(\gamma_0>0\) so small that \[\gamma_0 C_*G\le c_0/2, \qquad 2\gamma_0\|d\|_\infty\le1/2;\] a zero factor imposes no restriction. Then, for every permitted \(\gamma\), \[ \frac{q_0(R)}{2P}\le Q_h(R)\le\frac{3q_0(R)}{2P}, \qquad \int_\mathbb RQ_h(R)\,\,\mathrm dR=\frac1P. \tag{32}\] Thus \(Q_h(R)\,\,\mathrm dR\) defines a probability law on \(\mathbb R\times\{0,\ldots,P-1\}\). Finite Fourier inversion gives, for every \(v\in\mathbb R\) and \(k\in\mathbb Z\), \[ \sum_{h=0}^{P-1}\int_\mathbb RQ_h(R)\mathop{\mathrm{e}}(vR+kh/P)\,\,\mathrm dR =\widehat q_0(v)\mathbf 1_{k\equiv0\ (P)} +\gamma d(v)\mathbf 1_{\gcd(k,P)=1}. \tag{33}\] Indeed the sum over \(h\) pairs \(C_P(h)\) with the character at \(k\), leaving precisely the indicator that \(k\) is a unit. All choices up to this point are independent of \(B\), the map, and its finite packet list.

Moments and the finite sieve. For fixed \(B\) and an integer \(c>0\), consider the random variable \[ J_*=(R/B+h/P)/c \tag{34}\] under this law. Its density is \[\rho_{B,c}(y)=cB\sum_{h=0}^{P-1}Q_h\bigl(B(cy-h/P)\bigr).\] It is a strictly positive smooth probability density. The bounds (32) imply quadratic upper and lower tails, with constants now allowed to depend on \(B,c\).

We calculate the periodized density of \(sJ_*\). Write \(x=m/B\) and \(s=mct\). For its positive integer Fourier moments, the exact formula is \[ M_\ell(t)=\frac1P\sum_{h=0}^{P-1}\mathop{\mathrm{e}}(\ell mt h/P) \bigl[\widehat q_0(\ell xt) +\gamma C_P(h)d(\ell xt)\bigr],\qquad \ell\ge1. \tag{35}\] At \(t=1\), (33) applies. Its baseline term vanishes since \(|\ell x|\ge b>\alpha\). If \(\ell\ge2\) and \(d(\ell x)\ne0\), then \(\ell\le D/b<L\). Some prime divisor of \(\ell\) therefore divides \(P\), so \(\gcd(\ell m,P)>1\). It follows that \[ M_1(1)=\gamma d(x)\mathbf 1_{\gcd(m,P)=1}, \qquad M_\ell(1)=0\quad(\ell\ge2). \tag{36}\]

For finite \(c\) the discrete phases in (35) need not be residue characters. We estimate their errors before using the sieve. Choose \(\tau\in(1/2,1)\) so that \(D/(\tau b)<L\), and take \(c\) large enough that \[t_N(u)=1+\frac{\phi(u)+\Delta_N}{c}\ge\tau, \qquad |\Delta_N|\le1/N,\] on the old supports, uniformly for \(N\ge1\). The baseline term in (35) then vanishes for every \(\ell\ge1\), since \(|\ell xt_N|\ge\tau b>\alpha\). The perturbation also vanishes identically when \(\ell>D/(\tau b)\). Hence nonintegral discrete phases cannot introduce an unbounded collection of harmonics. For the fixed remaining finite set, put \(C_1=P^{-1}\sum_h|C_P(h)|\). The elementary bound \(|\mathop{\mathrm{e}}(v)-\mathop{\mathrm{e}}(w)|\le2\pi|v-w|\) gives \[ |M_\ell(t)-M_\ell(1)| \le\gamma C_1\ell \bigl[D_0\|d'\|_\infty+2\pi|m|\|d\|_\infty\bigr]|t-1|. \tag{37}\] We may thus make the sum of all moment errors arbitrarily small by choosing \(c\) after the actual finite set of old indices. In particular \(c\) is allowed to grow with \(\max|m|\); \(\gamma\) does not need to decrease. The same choice can make the ranges \(m(c+\phi(\mathbb T))\) for distinct old indices disjoint and bounded away from zero.

To pass from moments to densities, let \[\mathcal P_s\rho(y) =\sum_{k\in\mathbb Z}\frac1{|s|}\rho((y+k)/s),\qquad y\in\mathbb T.\] Once \(c\) has been chosen, all the relevant \(s\) ranges are fixed compact sets away from zero. No bound on these sets uniform over future passages is needed. For \(s\) in a fixed compact subset of \(\mathbb R\setminus\{0\}\), the quadratic upper tail of \(\rho_{B,c}\) makes this a jointly continuous function of \((s,y)\), with uniformly summable tails. Its Fourier coefficients in the positive-exponent convention are the moments just computed. The compact bandwidth shows that its Fourier series is a finite sum on the relevant branches. Fourier uniqueness and (36)–(37) therefore give, uniformly on those branches and in \(y\), \[ \mathcal P_s\rho_{B,c}(y) =1+2\gamma\mathbf 1_{\gcd(m,P)=1}\mathop{\mathrm{Re}}\{d(m/B)\mathop{\mathrm{e}}(-y)\} +o_{c\to\infty}(1). \tag{38}\] The error here is for the fixed old list and scale \(B\). The negative exponent follows directly from our convention: \(\int\mathcal P_s\rho(y)\mathop{\mathrm{e}}(\ell y)\,\,\mathrm dy=M_\ell\).

Realization and support. We now realize this law by a passage. Lemma 4 provides a bounded smooth circle function \(J\), strictly increasing on a good arc, and a smooth cutoff \(\psi\) compactly supported in that arc, such that the subprobability law \[\widetilde\rho=J_*(\psi(a)^2\,\,\mathrm da)\] is arbitrarily close to \(\rho_{B,c}\) in total variation. Its missing mass is included in this error. The good arc has as small a complement as requested. Since \(|A_{r,m}|\) is independent of \(a\), its stationary weight is a function of \(s\) alone: \[w_{r,m}(s)=\frac{|A_{r,m}(u,a)|^2}{|m\phi'(u)|}, \qquad m(c+\phi(u))=s,\] extended by zero. For such a nonnegative compactly supported weight, Lemma 3 and contraction of total variation under pushforward and periodization give \[\begin{align*} &\lim_{N\to\infty}\int_\mathbb Rw_{r,m}(s) |\mathcal P_s\widetilde\rho(Ns) -\mathcal P_s\rho_{B,c}(Ns)|\,\,\mathrm ds \\ &\qquad=\int_\mathbb Rw_{r,m}(s)\int_\mathbb T |\mathcal P_s\widetilde\rho(y) -\mathcal P_s\rho_{B,c}(y)|\,\,\mathrm dy\,\,\mathrm ds \\ &\qquad\le\|w_{r,m}\|_1\|\widetilde\rho-\rho_{B,c}\|_{L^1(\mathbb R)}. \tag{39}\end{align*}\] Here densities of subprobabilities are used in the \(L^1\) norm. The realized law has a continuous compactly supported density; the ideal law has the uniform tail bound established above. These facts supply the continuity and domination required for fast averaging. The independence of the weight from \(a\) is essential in this step.

Lemma 2 identifies the output density, up to an arbitrarily small \(L^1\) error, as the folding to \(\mathbb T\) of \(w_{r,m}(s)\mathcal P_s\widetilde\rho(Ns)\). In using that lemma the rounding correction is removed at the coefficient-mass level: its replacement of \(w_{r,m}(s)\) by \(w_{r,m}(s-m\Delta_N)\) costs \(o(1)\) in the summed masses. It does not translate the fast argument \(Ns\). Alternatively, (37) also holds uniformly with \(t=t_N\). Since \(cm\) and \(N\) are integers, \[p_{r,m}(z)=\sum_{k\in\mathbb Z}w_{r,m}(z+k), \qquad \mathop{\mathrm{e}}(-N(z+k))=\mathop{\mathrm{e}}(-Nz).\] Now (38), (39), and the passage density formula prove (27). Choose \(c\) first to make its error sufficiently small, then the law approximation, and then \(N\). The finite packet list allows all errors to be summed.

For completeness we give the simultaneous support estimates, which will also apply to neutral passages. Put \(M=\max_{r,m}|m|\), with \(M=0\) for an empty packet list. By the support cutoff in Lemma 2, at each retained point of an output packet at index \(j\) there is a point \(u\) in its old support for which \[ j/N=m(c+\phi(u)+\Delta_N)+\eta, \qquad |\eta|\le\eta_0, \tag{40}\] where the image enlargement \(\eta_0\) can be as small as desired. If \(x'=j/(NcB)\) and \(z'=j/N+(j/N)J(a)/N\) is the new temporal frequency, then \[\begin{align*} |x'-m/B| &\le\frac{D_0(\|\phi\|_\infty+1)}c+\frac{\eta_0}{cB}, \tag{41}\\ \mathop{\mathrm{dist}}_\mathbb T(z',m\phi(u)) &\le\frac MN+\eta_0 +\frac{[M(c+\|\phi\|_\infty+1)+\eta_0]\|J\|_\infty}{N}. \tag{42}\end{align*}\] Choose \(c\) for the first bound and index separation, then \(\eta_0\) and the bounded \(J\), and finally \(N\) for the second bound. A large bound on \(J\) only increases the eventual requirement on \(N\). These choices also give physical smooth closeness of the passage and its inverse and the required \(L^2\) approximations, by Lemma 2.

Every new coefficient descending from old index \(m\) has the form \(\mathop{\mathrm{e}}(ma_{\rm new})\) times a scalar function of the new speed coordinate. Because different old indices have disjoint retained new index ranges, their spare-coordinate phases are never added at one new index. Thus the resulting packet moduli are independent of the new spare coordinate. All cutoffs can be chosen in conjugate pairs, so conjugation is retained as well. ◻

Proof. Use Lemma 5 with residue modulus \(P\). Denote the refreshed scale by \(\widetilde B\), its packet densities by \(\widetilde p_{r,m}\), and its color-residue densities by \[\widetilde p_{r,h}=\sum_{m\equiv h\ (P)}\widetilde p_{r,m}.\] That lemma makes \[ E_r=\sum_{h=0}^{P-1} \|\widetilde p_{r,h}-p_r/P\|_1 \tag{46}\] arbitrarily small. It also retains disjoint output angular-index sets and gives moduli independent of the new spare coordinate. To control its support, first choose its \(c\) as large as necessary for (41); next choose its cutoff and broad linear jitter for the residue equalization; finally choose its passage integer for (42). Thus equalizing residues does not prevent either displacement from being arbitrarily small.

For a continuous function \(f\) on its relevant compact domain, denote its modulus of continuity by \(\nu_f(t)=\sup_{\mathop{\mathrm{dist}}(v,w)\le t}|f(v)-f(w)|\). If the refresh changes normalized indices by at most \(\delta_x\) and temporal frequencies by at most \(\delta_z\), its retained color supports satisfy \[ |d(m/\widetilde B)-d_r(z)| \le\rho_r:=\omega+\nu_d(\delta_x)+\nu_{d_r}(\delta_z). \tag{47}\] This follows pointwise by linking an output support point to an old one and using (44). In particular the extra terms tend to zero. The compact margin in the normalized band lets us apply Lemma 7 with the same \(P,\gamma\) after this refresh, although its actual indices and scale may be very large.

By that lemma the transmission density, with an arbitrarily small summed error \(T_r\) for color \(r\), is \[\widetilde p_r(z)+2\gamma\mathop{\mathrm{Re}}\left\{\mathop{\mathrm{e}}(-Nz) \sum_{\gcd(m,P)=1}d(m/\widetilde B) \widetilde p_{r,m}(z)\right\}.\] Replacing the coefficient in this sum by \(d_r(z)\) costs at most \(2\gamma\rho_r\|\widetilde p_r\|_1\) in \(L^1\). Also, \[\|\widetilde p_r-p_r\|_1\le E_r, \qquad \left\|\sum_{h\in(\mathbb Z/P\mathbb Z)^\times}\widetilde p_{r,h} -\frac{|(\mathbb Z/P\mathbb Z)^\times|}{P}p_r\right\|_1\le E_r.\] Consequently the color’s total error against the target in (45) is bounded by \[ T_r+(1+2\gamma\|d_r\|_\infty)E_r +2\gamma\rho_r(\|p_r\|_1+E_r). \tag{48}\] Choose the refresh errors, displacements, and transmission errors small enough and sum this bound over the finite list. This proves (45), including the exact value of \(\lambda\). Only residue balancing after summing over a color was needed, because (47) holds pointwise before the sum.

The remaining assertions follow from the two passage lemmas and (41)–(42). Specifically, the triangle inequality gives \[\mathop{\mathrm{dist}}_\mathbb T(x',z')\le\mathop{\mathrm{dist}}_\mathbb T(x,z)+|x'-x|+\mathop{\mathrm{dist}}_\mathbb T(z',z).\] Positive separation distances and the fixed finite number of passages allow displacement budgets smaller than all band margins and color gaps. Constants such as \(\lambda\) are fixed before these error budgets are imposed. The required \(c\) and passage integers may then be chosen afresh after each actual finite packet list is known. ◻

Proof. Let \(D\) bound the absolute values of all the exponents in all the \(\mathcal E_j\). The integral of a nonconstant monomial on the product torus is zero. On the one-dimensional diagonal its contribution is a Fourier coefficient of \(c_{j,\nu}\) at \(-\sum_i\nu_iL_i\). The Fourier coefficients of an \(L^1\) function tend to zero: this follows by approximation in \(L^1\) by trigonometric polynomials, whose sufficiently high Fourier coefficients vanish. Since the coefficient list is finite, choose \(R\) so that the sum of the bounds for all these nonconstant contributions is less than \(\eta\) whenever their frequencies have absolute value greater than \(R\). Choose successively \[L_j>R+D\sum_{i<j}L_i,\] as well as the additional finite lower bounds. If \(j\) is the last nonzero position of \(\nu\), then \[\left|\sum_i\nu_iL_i\right| \ge L_j-D\sum_{i<j}L_i>R.\] This proves the assertion. No independence assertion about the distribution of the entire diagonal is made or needed. ◻

Proof of Proposition 9. Zero colors may be omitted. If all colors vanish, a neutral passage with zero output packets and \(\Psi=0\) proves the assertion, including the requested new good arc. Hence assume that a nonzero color is present. Dividing all tags by \(\max(1,\max_r|b_r|)\) and reducing the requested prediction tolerance reduces the proof to \(|b_r|\le1\). The transport and square-root tolerances remain independently adjustable. We work under this normalization until the end.

Proof. Choose a positive integer \(m\) and \(u_0<u_1\) in \(I\) such that \(m(\phi(u_1)-\phi(u_0))=1\). This is possible by first choosing a compact interval on which \(\phi\) increases and then making \(m\) large. Put \[A(u)=\mathbf 1_{[u_0,u_1]}(u) \sqrt{\frac{|m\phi'(u)|}{2}}, \qquad v(\theta,u,a)=A(u)\mathop{\mathrm{e}}(m\theta)+A(u)\mathop{\mathrm{e}}(-m\theta).\] The pushforward of \(|A(u)|^2\,\mathrm du\,\,\mathrm da\) by each of \(\pm m\phi(u)\) modulo one is \(\tfrac12\,\mathrm dz\). The two angular modes are orthogonal, so the spectral density of \(v\) is \(1\). The vector is real and centered. Its endpoint discontinuities are harmless: no smoothness is required of the vector. ◻

Proof. We first prepare smooth packets for \(f\) and \(h\), then put constant tags on disjoint packets, and finally use Proposition 9. Every accuracy appearing below may be made arbitrarily small after the finite data on which it depends have been fixed.

At a fixed map we use the density estimate (4), which also applies to the nonsmooth \(L^2\) coefficients of an admissible vector. Across a passage, density preservation instead comes from Lemma 5.

Proof of Theorem 1. Choose a smooth periodic speed \(\phi_0\) which agrees with an increasing affine function on a nonempty closed interval inside \((0,1)\), and let \(I_0\) be an open subinterval where \(\phi_0'>0\). Start with the twist \(S_0\) and its flat vector \(f_0\) from Lemma 11. Choose a countable family of real smooth centered functions, bounded by one, whose real linear span is dense in real \(L^2_0(\mathbb T^3)\); the physical torus sines and cosines will do. List them as \((h_n)_{n\ge1}\) so that each occurs infinitely often.

We record the continuity needed to retain approximations. If volume-preserving maps \(T_i\) and their inverses converge uniformly to \(T\) and \(T^{-1}\), then for every fixed \(k\in\mathbb Z\), \[ U_{T_i}^kg\longrightarrow U_T^kg\quad\hbox{in }L^2 \quad(g\in L^2). \tag{78}\] Uniform convergence of the corresponding fixed iterates follows by composition on the compact torus. For continuous \(g\) this proves uniform convergence of the compositions; for general \(g\), approximation by continuous functions and unitarity prove (78). Consequently, for every fixed Laurent polynomial \(P\) the map \[(T,g)\longmapsto P(U_T)g\] is continuous into physical \(L^2\) when maps and inverses have the \(C^\infty\) topology and vectors have the \(L^2\) topology. Explicitly, if \(P(w)=\sum_k c_kw^k\), its difference is bounded by \[\sum_k|c_k|\|g_i-g\|_2+ \sum_k|c_k|\|U_{T_i}^kg-U_T^kg\|_2.\]

Inductively apply Proposition 12 with \(h=h_n\) and \(\eta=2^{-n}\), obtaining \((S_n,f_n)\), a good arc \(I_n\), and a finite Laurent polynomial \(P_n\), such that \[ |\mathbb T\setminus I_n|<4^{-n},\qquad \|h_n-P_n(U_{S_n})f_n\|_2 <|\mathbb T\setminus I_{n-1}|^{1/2}+2^{-n}. \tag{79}\] Use the arbitrary closeness in that proposition also to impose \[\|f_n-f_{n-1}\|_2<2^{-n}, \qquad d_k(S_n,S_{n-1})+d_k(S_n^{-1},S_{n-1}^{-1})<2^{-n} \quad(0\le k\le n),\] where \(d_k\) are ambient \(C^k\) sup-norm distances induced by a fixed smooth embedding of the torus. At the same time impose the finitely many open conditions \[ \|P_j(U_{S_n})f_n-P_j(U_{S_{n-1}})f_{n-1}\|_2<2^{-n} \qquad(1\le j<n). \tag{80}\] Their compatibility follows from the continuity just proved. No bound uniform in \(j\) on the degrees or coefficients of \(P_j\) is required.

For each fixed \(k\), the map and inverse sequences have summable tails in \(C^k\). They therefore converge smoothly to maps \(T\) and \(R\). Uniform convergence, compactness, and \(S_nS_n^{-1}=S_n^{-1}S_n=\operatorname{id}\) give \(TR=RT=\operatorname{id}\), so \(T\) is a smooth diffeomorphism with smooth inverse. For every continuous \(g\), \[\int g\circ T=\lim_n\int g\circ S_n=\int g,\] so normalized volume is invariant. The vectors \(f_n\) converge in \(L^2\) to a real centered vector \(f\).

Every completed stage has density exactly one. Combining (78) with \(f_n\to f\), for all \(r,s\in\mathbb Z\) we obtain \[ \langle U_T^rf,U_T^sf\rangle =\lim_n\langle U_{S_n}^rf_n,U_{S_n}^sf_n\rangle =\delta_{rs}. \tag{81}\] The retained finite polynomials give, for every \(j\), \[\|h_j-P_j(U_T)f\|_2 \le |\mathbb T\setminus I_{j-1}|^{1/2}+2^{-j} +\sum_{n>j}2^{-n} =|\mathbb T\setminus I_{j-1}|^{1/2}+2^{1-j}.\] Along the infinitely many occurrences of any fixed test the right side tends to zero. Hence every test belongs to \(\mathcal C(U_T,f)\). Their complex linear span is dense in complex \(L^2_0(\mathbb T^3)\), whereas every orbit vector is centered. Thus \[\mathcal C(U_T,f)=L^2_0(\mathbb T^3).\] Together with (81), this says that the bilateral orbit of \(f\) is an orthonormal basis of the entire centered space. With \(w=\mathop{\mathrm{e}}(z)\), the assignment \(U_T^nf\mapsto w^n\) extends to a unitary map onto \(L^2(S^1,m)\) intertwining \(U_T\) with multiplication by \(w\).

Multiplication by \(w\) has no nonzero invariant vector, since \((w-1)g(w)=0\) forces \(g=0\) almost everywhere. Consequently every \(T\)-invariant \(L^2\) function is constant, which proves ergodicity. The manifold \(\mathbb T^3\) is compact, connected, smooth, and without boundary, and its normalized flat volume has strictly positive smooth density. These are all the properties required in Theorem 1. ◻

Proof. Let \(V:L^2_0(\mathbb T^3,\mu)\to L^2(S^1,m)\) be the unitary spectral model in Theorem 1. For \(a,b\in L^2_0(\mathbb T^3,\mu)\), \[\int_{\mathbb T^3}(a\circ T^n)\overline b\,\mathrm d\mu =\int_{S^1}w^n(Va)(w)\overline{(Vb)(w)}\,\mathrm dm(w) \longrightarrow0\qquad(|n|\to\infty).\] Indeed, \((Va)\overline{Vb}\in L^1(m)\) by Cauchy–Schwarz, so the limit is the Riemann–Lebesgue lemma. Taking centered indicators gives ordinary mixing. The map \(T\) is an invertible probability-preserving transformation with measurable inverse, so the multiple-mixing theorem (OpenAI 2026, Theorem 1.1) gives every order \(k\ge3\). Invariance shifts \(t_1\) to zero and identifies its positive time gaps with \(t_{i+1}-t_i\); the case \(k=2\) is ordinary mixing.

The full Koopman operator is the direct sum of the one-dimensional constants and the simple Lebesgue model on all of \(L^2_0\), and hence has finite spectral multiplicity. Normalized torus volume is a Lebesgue probability space, and \(T\) is a measure-preserving automorphism. Rokhlin’s finite-multiplicity entropy theorem (Rokhlin 1960, sec. 7, p. 13) therefore gives \(h_\mu(T)=0\).

The derivative and inverse derivative are bounded on the compact torus, so the invertible Oseledets integrability conditions hold. Write \(\lambda_1(x),\lambda_2(x),\lambda_3(x)\) for the exponents repeated according to multiplicity at regular points. Pesin’s entropy formula, as stated and reproved by Mañé (Mañé 1981, 95, equation (2)), applies to the \(C^\infty\) diffeomorphism and its absolutely continuous invariant probability \(\mu\). Thus \[0=h_\mu(T)=\int_{\mathbb T^3}\sum_{i=1}^3\max\{\lambda_i(x),0\}\,\mathrm d\mu(x).\] The nonnegative integrand vanishes almost everywhere, so every exponent is nonpositive there. Preservation of standard volume gives \(|\det DT_x|=1\), and at every Oseledets-regular point \[\sum_{i=1}^3\lambda_i(x) =\lim_{n\to\infty}\frac1n\log|\det D(T^n)_x|=0.\] The three nonpositive exponents must therefore all be zero at \(\mu\)-almost every point. ◻