Background and the two mechanisms

Turing proved the undecidability of whether a described machine ever prints a designated symbol (Turing 1936--1937, sec. 8). Replacing that printing event by a halt gives the decision problem used here. Moore’s generalized shifts encode tapes in positional real coordinates and represent local instructions by piecewise affine maps (Moore 1990, 1991). Smooth flows require injective finite-time maps, whereas a machine may erase a symbol. Landauer (Landauer 1961, sec. 3) discusses retaining information that an irreversible operation discards; Bennett (Bennett 1973) gives a reversible simulation that records the instruction history. The particular one-head compiler and planar realization used here are supplied by (OpenAI 2026c, Lemma 3.1 and Theorem 4.1); the diffusion estimates below are proved here.

Material-particle computation was realized by Cardona, Miranda, Peralta-Salas and Presas in steady Euler flow on a three-sphere with a chosen Riemannian metric (Cardona et al. 2021, Theorem 6.1). Dyhr, González-Prieto, Miranda and Peralta-Salas construct steady unforced Navier–Stokes flows on suitable Riemannian three-manifolds, using harmonic fields for the Hodge Laplacian and an adapted metric (Dyhr et al. 2026, Theorem A). These results concern trajectories of a stationary velocity, whereas the observations here test the evolving velocity itself.

Velocity-space observation has a direct predecessor in the work of Cardona, Miranda and Peralta-Salas (Cardona et al. 2022). Their construction uses the unforced incompressible Euler evolution on a constructed high-dimensional compact Riemannian manifold; computation is observed by entry into an open set of divergence-free velocity fields. Here the geometry is flat and three-dimensional, the viscosity is positive, and the initial velocity is zero. The machine and input enter a prescribed external force, and the observations are the pointwise and integral tests above. These differences distinguish the present problem from both stationary-particle and Euler velocity-space computation.

The common analytic device is particularly simple. For a planar divergence-free field \(a(t,Y)\) and a scalar \(w(t,Y)\), the three-dimensional velocity \((a,w)\) is divergence free when it is independent of \(z\). Its third equation is an advection–diffusion equation. We prescribe its source and the horizontal force, and then solve for \(w\). Thus the unknown velocity is not concealed in a force formula.

In the torus construction, each block injects a very small bump of height one at the initial machine code and runs a longer prefix of the planar processor. A short physical duration keeps diffusion close to pure transport, while a long heat-only interval makes older bumps small. A halting run carries a fresh bump to the detector. In a nonhalting run, the entire transported bump remains separated from it, so even its diffusive tail stays below the threshold.

For bounded forcing, an initial impulse creates scalar mass one. The horizontal field translates large squares between integer addresses for all possible configurations. Longer and larger gates keep every force derivative bounded. A cutoff moving with the chosen address cancels transport exactly; its Laplacian measures the diffusion loss. Choosing the gate radii fast enough makes the sum of those losses less than \(1/24\).

Section 2 establishes the scalar reduction and the solution convention. Section 3 proves a parameterized injection, stirring and waiting theorem, including two quantitative specializations. Section 4 applies it to the fixed torus detector. The expanding-address construction and its mass estimate appear in Section 5. The classical background for the maximum principle and parabolic kernels is described, for example, by Aronson (Aronson 1968); all constants and comparison arguments needed for detection are included below.

An injection and a finite computation in each block

Use the effective nondecreasing smooth step \[b(r)=\begin{cases}e^{-1/r},&r>0,\\0,&r\le0,\end{cases} \qquad \theta(r)=\frac{b(r)}{b(r)+b(1-r)}.\] It is zero for \(r\le0\) and one for \(r\ge1\), with all derivatives flat at the endpoints. Define a fixed bump on \(\mathbb R^2\) by \[g(Z)=\prod_{j=1}^2\theta(2(Z_j+1))\theta(2(1-Z_j)).\] It lies in \([0,1]\), equals one near zero, and is supported in \([-1,1]^2\). Let an effective integer \(C\ge1\) bound its first and second derivative norms. For \(n\ge1\), put \[ \begin{aligned} b_n&=10^{-6}(2R)^{-n},&g_n(Y)&=g((Y-p)/b_n),\\ D_n&=2Cb_n^{-2}(1+nL)R^{3n},& \delta_n&=\frac1{100(1+D_n)},\qquad t_n=2n. \end{aligned} \tag{10}\] The width \(b_n\) makes the bump’s mass small and its transported support narrow. The quantity \(D_n\) is chosen to bound the Laplacian of that transported bump; Section 3.3 verifies the bound. The duration \(\delta_n\) then makes the accumulated diffusion error at most \(2\delta_nD_n<1/16\). These are separate roles: spatial width controls the mass and separation, while physical duration controls the error in height. Here \(g_n\) is periodized from the small square about a lift of \(p\); the copies have disjoint supports. In particular \[ \|\nabla g_n\|_\infty\le Cb_n^{-1},\qquad \|D^2g_n\|_\infty\le Cb_n^{-2},\qquad \int_{\mathbb T^2}g_n\le4b_n^2. \tag{11}\]

Block \(n\) injects this bump over time \(\delta_n\), runs \(n\) complete periods of \(V\) over another time \(\delta_n\), and then waits for the next block. In particular, its heat-only wait has length \(2-2\delta_n>1\). The time diagram in Figure 1 shows the roles of these intervals; their plotted lengths are schematic.

Let \(V_0\) be the single period of \(V\) extended by zero outside \([0,1]\). The phase collars make this extension smooth. Prescribe \[ \begin{split} s_n(t)&=n(t-t_n-\delta_n)/\delta_n,\\ a(t,Y)&=\sum_{n\ge1}\frac n{\delta_n} \sum_{j=0}^{n-1}V_0(s_n(t)-j,Y),\\ h(t,Y)&=\sum_{n\ge1}\delta_n^{-1} \theta'((t-t_n)/\delta_n)g_n(Y),\\ F(t,Y)&=\partial_ta+(a\cdot\nabla_Y)a-\Delta_Ya, \qquad f(t,Y,z)=(F(t,Y),h(t,Y)). \end{split} \tag{12}\] Each summand is supported in its indicated block. The sums are locally finite in physical time and join smoothly at all endpoints. Moreover \(\mathop{\mathrm{div}}_Ya=0\), \(h\ge0\), and \(a=h=0\) near time zero. This prescription uses \(V\) and the injection profiles, without using the scalar that it will produce.

Constructing the scalar and controlling old mass

Apply the lemma to (12) with \(w(0)=0\). Solutions on finite intervals agree by uniqueness, giving a global smooth nonnegative scalar. The velocity \((a,w)\) with pressure zero is the unique three-dimensional solution by Section 2. Integrating the scalar equation and using (11) gives \[ 0\le\int_{\mathbb T^2}w(t,Y)\,\mathrm dY \le M_0:=\sum_{n\ge1}4b_n^2 =\frac{4\cdot10^{-12}}{4R^2-1}<10^{-11}. \tag{13}\] Thus infinitely many height-one injections still have very small total mass. The heat kernel converts this mass bound into control of their older contributions.

Before block \(n>1\), there has been more than one unit of heat-only evolution. Therefore \[ 0\le w(t_n,\cdot)\le HM_0; \tag{15}\] for \(n=1\) the scalar is zero. From \(t_n\) to \(t_{n+1}\), linearity splits \(w=w^{\mathrm{old}}+w^{\mathrm{new}}\). The old part has initial value \(w(t_n)\) and no source, so remains between zero and \(HM_0\). The new part starts at zero and receives just the current injection. It remains to estimate this new part near its transported bump and away from it.

A controlled diffusion error during the burst

The variational equations for \(\Psi_s\) give, for \(0\le s\le n\), \[ \|D\Psi_s^{\pm1}\|_\infty\le e^{Ln}\le R^n, \qquad \|D^2\Psi_s^{\pm1}\|_\infty \le nLe^{3Ln}\le nLR^{3n}. \tag{16}\] For the first bound, differentiate the flow once and use Gronwall. The second variation has zero initial data, linear coefficient at most \(L\), and source at most \(L\) times the square of the first variation. Variation of constants bounds it by \(\int_0^s e^{L(s-r)}Le^{2Lr}\,dr\), and hence by \(nLe^{3Ln}\). The inverse is the flow of \(-V(s-r,\cdot)\) over the reversed interval, which obeys the same derivative bounds. These estimates are uniform over all initial points; they do not require following the selected machine execution.

During the two active intervals of block \(n\), define the scalar that would evolve without diffusion: \[ w^0(t,Y)= \begin{cases} \theta((t-t_n)/\delta_n)g_n(Y), &t_n\le t\le t_n+\delta_n,\\ g_n(\Psi_{s_n(t)}^{-1}(Y)), &t_n+\delta_n\le t\le t_n+2\delta_n. \end{cases} \tag{17}\] The endpoint flatness and phase collars make these expressions agree smoothly. They satisfy \(w^0_t+a\cdot\nabla w^0=h\). The chain rule and (16) bound the Hessian during stirring by \[Cb_n^{-2}R^{2n}+Cb_n^{-1}nLR^{3n}.\] Taking the trace costs at most two. Since \(b_n<1\) and \(R\ge1\), the result is at most \(D_n\) from (10); the injection bound is smaller. Subtracting the inviscid equation from the diffusive one shows that \(w^{\mathrm{new}}-w^0\) has zero initial data and source \(\Delta w^0\). Lemma 3 gives \[ \|w^{\mathrm{new}}-w^0\|_\infty \le2\delta_nD_n<\frac1{50}<\frac1{16} \tag{18}\] throughout the injection and stirring intervals.

Suppose first that \(\Psi_k(p)\in E\) for an integer \(k\ge0\). Choose \(n\ge\max\{1,k\}\). At the stirring time with \(s_n(t)=k\), the reference at \(Y=\Psi_k(p)\) equals \(g_n(p)=1\). The old part is nonnegative, so (18) gives \(w(t,Y)>15/16>1/2\). This proves the first implication of Theorem 2.

Now suppose the orbit stays at distance at least \(2d\) from \(E\). Every point of the reference support lies within distance \[\sqrt2\,b_nR^n=\sqrt2\,10^{-6}2^{-n}<1/32\le d\] of \(\Psi_{s_n(t)}(p)\) during stirring. The same bound follows directly during injection. The reference support is therefore at distance at least \(d\) from \(E\) at every active time. In particular, \(w^0=0\) on \(E\), and \[ w(t,Y)\le HM_0+1/16\qquad(Y\in E) \tag{19}\] during injection and stirring.

At the end of stirring, the reference still has this separation and, by area preservation, has mass \(\int g_n\). During the wait, the new part is the heat evolution of that reference plus the heat evolution of an error with supremum norm below \(1/16\). The latter bound is preserved; Lemma 4 bounds the former contribution on \(E\) by \(H\int g_n\). Including the old part gives \[ w(t,Y)\le2HM_0+1/16<1/2\qquad(Y\in E). \tag{20}\] The last inequality holds for both pairs in (9). Before the first block \(w=0\). Equations (19) and (20) cover all later times and every endpoint, proving the second implication.

Effective prescription and physical viscosity

For a computably supplied \(t\ge0\), choose rational \(q\) with \(|q-t|\le1\) and put \(N_t=\max\{1,\lceil(q+1)/2\rceil\}\). Every summand of (12) with \(n>N_t\) starts after \(t\) and vanishes there with all derivatives. The remaining sum is finite, and its \(n\)th term contains only \(n\) translates of \(V_0\). Effective derivative bounds give \(L,C\), and hence all scales. No comparison of an exact real with a block boundary is needed: near zero, derivatives of \(b\) are polynomial factors times \(e^{-1/r}\), and \(s^m e^{-s}\le(m+j)!s^{-j}\) bounds their tails effectively. Also \(b(r)+b(1-r)\ge e^{-2}\). These estimates give effective evaluation of the smooth profiles even at undecidable endpoint equalities. Periodic charts are evaluated by finite supersets of their possibly active translates.

The force formula repeatedly uses the same whole-period processor, which contains all its local branches. Increasing \(n\) asks the flow to perform more periods, not the force evaluator to determine the visited configurations. In particular evaluation of \(f\) never uses the unknown scalar \(w\).

For a positive computable physical viscosity \(\nu\), set \[ U_\nu(t,Y,z)=(\nu a(\nu t,Y),w(\nu t,Y)),\qquad f_\nu(t,Y,z)=(\nu^2F(\nu t,Y),\nu h(\nu t,Y)). \tag{21}\] Every horizontal equation acquires the factor \(\nu^2\), and every vertical term the factor \(\nu\). Thus direct substitution gives viscosity \(\nu\) with pressure zero. The vertical threshold and mass are unchanged apart from the time parameter. The time intervals \(\delta_n\) and the waits above are measured in unit-viscosity time; their physical lengths are divided by \(\nu\). Smoothness, effective evaluation and finite-time uniqueness persist. For an arbitrary positive real \(\nu\), the same formulas have the corresponding oracle-relative interpretation. This completes the proof of Theorem 2.

Finite addresses and their transitions

The only computational input is the injective local compiler of (OpenAI 2026c, Lemma 3.1). Its output is a finite partial one-head table with states \(i=1,\ldots,N\), tape alphabet \(\Sigma\), and an initial tape differing from the full blank symbol at finitely many cells. It has one terminal state, with no outgoing rule. Its initialized run reaches that state exactly when the original machine halts; otherwise every successive transition is defined. Every target state fixes the displacement of its incoming rules, and the target state together with the full written symbol identifies the incoming rule. The compiler first adds a nonterminal initial step, so these assertions include an input on which the original machine is already halted. Its history mechanism is the reversible recording method used by Bennett (Bennett 1973); here we use the stated finite table, including its full-domain incoming-rule property.

Let \(b=|\Sigma|\ge2\) and assign its symbols distinct digits \(d_\alpha\in\{0,\ldots,b-1\}\), with the full blank symbol assigned zero. The integer \(\sum_{j\ge0}d_{\alpha_j}b^j\) encodes an eventually blank stack whose top is \(\alpha_0\). Zero tails are interpreted as infinitely many blanks. For a configuration, let \(x\) encode the tape strictly left of the head, nearest cell first, and let \(y\) encode the head cell and the tape to its right. If the rule reads \(\alpha\), writes \(\beta\) and enters state \(i'\), then \(d_\alpha=y\bmod b\), and the new stacks are \[ \begin{array}{c|cc} \text{move}&x'&y'\\ \hline \mathsf R&bx+d_\beta&\lfloor y/b\rfloor\\ \mathsf N&x&y-d_\alpha+d_\beta\\ \mathsf L&\lfloor x/b\rfloor& d_\gamma+bd_\beta+b^2\lfloor y/b\rfloor, \quad d_\gamma=x\bmod b. \end{array} \tag{22}\] These formulas push and remove exactly the indicated tape symbols.

They also give an injective map on all configurations where a rule is defined. The target state determines which row applies. In the right row the lowest digit of \(x'\) recovers \(d_\beta\); in the stationary row the lowest digit of \(y'\) does so; in the left row the second lowest digit of \(y'\) does so, while its lowest digit recovers \(d_\gamma\). The compiler then identifies the old state and read symbol. The inverses are, respectively, \[(x,y)=\bigl((x'-d_\beta)/b,\,by'+d_\alpha\bigr),\qquad (x,y)=\bigl(x',\,y'-d_\beta+d_\alpha\bigr),\] and \[(x,y)=\bigl(bx'+d_\gamma,\, b\lfloor y'/b^2\rfloor+d_\alpha\bigr).\] This proves injectivity without restricting to the initialized run.

Choose \(m\ge1\) effectively so that the two initial stacks are smaller than \(b^m\), and define \[ B_n=b^{m+n},\qquad K_n=NB_n^2=K_0D^n,\qquad D=b^2. \tag{23}\] Every update with \(0\le x,y<B_n\) has \(0\le x',y'<B_{n+1}\). For the largest expression, in the left row, \[y'\le(b-1)+b(b-1)+b^2(B_n/b-1)=bB_n-1.\] All other bounds follow directly from (22). At level \(n\), give the triple \((i,x,y)\) the address \[ k=1+x+B_ny+B_n^2(i-1),\qquad 1\le k\le K_n. \tag{24}\] Integer division recovers the triple. Thus every address with a defined instruction determines one target address \(k'\) at level \(n+1\), and distinct defined addresses have distinct targets. Only this one-step lookup is used to prescribe the force.

Disjoint translation gates at increasing scales

We now assign each address a square and move all defined source squares to their targets simultaneously. Three successive motions keep the squares separated: descent along source columns, horizontal motion along private rows, and vertical motion along target columns.

Use the smooth step \(\theta\) from Section 3.1, and put \[\Theta(r)=\theta(2r-1/2).\] The function \(\Theta\) makes its transition between \(1/4\) and \(3/4\). Introduce the scales \[ \begin{aligned} C_*&=3000,&\Lambda&=4D, &R_0&=1024C_*(1+\nu)\Lambda(K_0D+2),\\ R_n&=R_0\Lambda^n,&S_n&=16R_n, &T_n&=S_{n+1}(K_{n+1}+2),\\ t_0&=1,&t_{n+1}&=t_n+T_n. \end{aligned} \tag{25}\] The constant \(C_*\) will bound the Laplacian of a cutoff at radius \(R\) by \(C_*/R^2\). For scalar mass at most one, a stage of duration \(T_n\) will therefore lose at most \(\nu C_*T_n/R_n^2\) from that cutoff. The radii grow faster than the number of addresses so that these losses form a summable series. The durations \(T_n\) are long enough for the most distant gate to move at bounded speed. In particular \(R_n,T_n\ge1\), and \(t_n\ge n+1\).

Let \(s_i=1\) for the terminal state and \(s_i=-1\) for every other state. An address \(k\) of state \(i\) has center \[p_{n,k}=(S_nk,s_iS_n).\] The known initial configuration determines a center \(p_*\) at level zero. It lies on the negative row. For a defined source address \(k\) at level \(n\), with target \(k'\) in state \(i'\), put \[Y_k=-(k+2)S_n,\qquad \theta_j(t)=\Theta\bigl(3(t-t_n)/T_n-(j-1)\bigr), \quad j=1,2,3.\] Its center path on \([t_n,t_{n+1}]\) is \[ \begin{aligned} c_{n,k,1}(t)&=(1-\theta_2(t))S_nk +\theta_2(t)S_{n+1}k',\\ c_{n,k,2}(t)&=-(1-\theta_1(t))S_n +(\theta_1(t)-\theta_3(t))Y_k +\theta_3(t)s_{i'}S_{n+1}. \end{aligned} \tag{26}\] It starts at \(p_{n,k}\) because every defined source is nonterminal, and it ends at \(p_{n+1,k'}\). The supports of \(\dot\theta_1,\dot\theta_2,\dot\theta_3\) occur in successive disjoint time slots, with stationary collars between them.

During the first slot, distinct centers have first coordinates separated by at least \(S_n\). During the second, their private heights \(Y_k\) are separated by at least \(S_n\). During the third, their target first coordinates are separated by at least \(S_{n+1}\), by the injectivity just proved. The same separations hold through the intervening collars. If the target is nonterminal, every vertical segment stays at or below the source row: \[ c_{n,k,2}(t)\le-S_n\qquad(t_n\le t\le t_{n+1}). \tag{27}\]

To realize one center path, set \[\chi(a)=\prod_{\ell=1}^2\theta(a_\ell+3)\theta(3-a_\ell), \qquad \nabla^\perp=(\partial_{X_2},-\partial_{X_1}).\] For \(c=c_{n,k}\), \(G=\dot c\) and \(\xi=X-c(t)\), define \[ W_{n,k}(t,X)=\nabla^\perp \left[\chi(\xi/R_n)(G_1\xi_2-G_2\xi_1)\right]. \tag{28}\] This field is divergence free, equals \(G\) when \(|\xi|_\infty\le2R_n\), and vanishes when \(|\xi|_\infty\ge3R_n\). The gate supports are disjoint at each time, since their centers are separated in one coordinate by at least \(16R_n>6R_n\). Figure 2 displays the separation coordinate in each time slot.

Let \(W=0\) on \([0,1]\), and on \([t_n,t_{n+1}]\) sum (28) over every level-\(n\) address with a defined rule. The sum is finite. Every gate is zero near the time-slot endpoints, so these formulas join smoothly, including at \(t=1\). The resulting field has compact horizontal support on each finite time interval and has the exact plateau property \[ W(t,X)=\dot c_{n,k}(t) \quad\hbox{whenever }|X-c_{n,k}(t)|_\infty\le R_n. \tag{29}\]

Here is why the growing array costs no derivative bound. Every coordinate used in (26) is bounded by a fixed multiple of \(T_n\): for example, \(|Y_k|\le S_n(K_n+2)\le T_n\) and \(S_{n+1}k'\le T_n\). Therefore, for every \(j\ge1\), \[ |c_{n,k}^{(j)}(t)|\le C_j T_n^{1-j}, \tag{30}\] with \(C_j\) independent of \(n\) and \(k\). Differentiating the potential in (28) writes the gate as \(A((X-c)/R_n)\dot c\) for a fixed smooth compactly supported matrix \(A\). A spatial derivative contributes \(R_n^{-1}\); time derivatives contribute derivatives of \(c\) divided by \(R_n\), together with higher derivatives of \(c\). Since \(R_n,T_n\ge1\), the product and chain rules and (30) bound each fixed mixed derivative uniformly in \(n,k\). At most one gate contributes at a point. The same global bounds thus hold for \(W\).

A prescribed impulse and its scalar evolution

The horizontal field is now fixed. The remaining force is a unit vertical impulse near \(p_*\). Prescribe \[ \begin{aligned} f_h&=\partial_tW+(W\cdot\nabla_X)W-\nu\Delta_XW,\\ g(t,X)&=\theta'(t)\prod_{\ell=1}^2\theta'\bigl(X_\ell-(p_*)_\ell+1/2\bigr),\\ f(t,X,z)&=(f_h(t,X),g(t,X)). \end{aligned} \tag{31}\] All terms are prescribed independently of the unknown vertical velocity. The bounds for \(W\) prove global boundedness of all mixed derivatives of \(f\), and its horizontal support is compact on each finite time interval. Moreover, \(g\ge0\), its support lies in \(0\le t\le1\) and \(|X-p_*|_\infty\le1/2\), and \[ \int_{\mathbb R^2}g(t,X)\,dX=\theta'(t). \tag{32}\]

Define \(\rho\) by solving \[ \partial_t\rho+W\cdot\nabla_X\rho =\nu\Delta_X\rho+g,\qquad \rho(0,X)=0. \tag{33}\] We need integrability as well as smoothness, because both the mass test and the noncompact uniqueness argument use it.

We can now verify the fluid equation and its solution class. Set \[ u(t,X,z)=(W(t,X),\rho(t,X)),\qquad p(t,X,z)=0. \tag{36}\] The divergence vanishes and \(u\cdot\nabla=W\cdot\nabla_X\) on these \(z\)-independent components. Thus the horizontal equation is the definition of \(f_h\) in (31), and the vertical equation is (33). Both components start from zero.

On \([0,T]\), the horizontal field and its derivatives have support in one compact set. Every derivative of \(\rho\) is continuous in both \(L^1\) and \(L^\infty\) by Lemma 5. The inequality \[\|v\|_2^2\le\|v\|_1\|v\|_\infty\] applied also to differences at two times gives continuity into \(L^2\). Applying it to the spatial derivatives through order two and to \(\partial_t\rho\) proves exactly the velocity regularity in (4); the same lemma gives the required bounded velocity and gradient. Pressure zero belongs to the stated pressure class.

The comparison argument of Section 2 now applies in exactly the established class. In particular, its pressure cutoff error uses \(L^2\) pressure, and its velocity errors use the scalar’s integrable tails, not compact support of the scalar. It gives uniqueness among all three-dimensional competitors in (4). The half-plane integral is well defined by Lemma 5.

A moving cutoff and the detection margin

We have constructed the force and its unique solution without following the selected computation. Only to prove the halting equivalence do we now follow that run. During loading put \(c(t)=p_*\). Thereafter concatenate the paths (26) corresponding to its successive configurations, up to the first terminal endpoint if one occurs. Each required gate is present because of the integer-stack bounds and the compiler’s continuation property. The paths agree at endpoints and are stationary in endpoint collars.

An explicit test function will quantify the mass retained near \(c(t)\). On \([0,1]\) put \(P(s)=10s^3-15s^4+6s^5\), extending it by zero to the left and one to the right. This extension is \(C^2\), takes values in \([0,1]\), and satisfies \(|P''|\le360\). Define \[A_*(s)=P(2(1+s))P(2(1-s)),\qquad \varphi_R(a)=A_*(a_1/R)A_*(a_2/R).\] Then \(0\le\varphi_R\le1\), it equals one on \(|a|_\infty\le R/2\), and it vanishes for \(|a|_\infty\ge R\). The transition intervals of the two factors in \(A_*\) are disjoint; on each, the other factor equals one. Consequently \(|A_*''|\le1440\), and \[ \|\Delta\varphi_R\|_\infty\le2880R^{-2}\le C_*R^{-2}. \tag{37}\]

Take \(R=R_0\) during loading and \(R=R_n\) during the \(n\)th transition, and within each such interval put \[\mathcal I(t)=\int_{\mathbb R^2} \varphi_R(X-c(t))\rho(t,X)\,dX.\] Differentiating under the integral and integrating by parts against the compactly supported \(C^2\) cutoff gives \[ \begin{aligned} \mathcal I'(t) ={}&\int_{\mathbb R^2} \bigl((W-\dot c)\cdot\nabla\varphi_R(X-c) +\nu\Delta\varphi_R(X-c)\bigr)\rho\,dX\\ &+\int_{\mathbb R^2}\varphi_R(X-c)g\,dX. \end{aligned} \tag{38}\] The transport term vanishes exactly by (29); during loading both \(W\) and \(\dot c\) are zero. Positivity, \(\mathcal M\le1\), and (37) bound the diffusion term below by \(-\nu C_*R^{-2}\). The loading source is supported where \(\varphi_{R_0}=1\), because \(R_0\ge1\); it contributes \(\mathcal M'(t)\). After loading the source vanishes.

There is no loss at a radius change. Indeed \(R_{n+1}\ge2R_n\), so the new cutoff equals one on the entire support of the previous one, with the same center at the joint. Nonnegativity then implies that \(\mathcal I\) cannot decrease at that change. Starting with zero mass and integrating (38) yields, at every time through the first terminal endpoint, \[ \mathcal I(t)\ge\mathcal M(t)-\delta,\qquad \delta=\nu C_*\left(R_0^{-2} +\sum_{n=0}^{\infty}\frac{T_n}{R_n^2}\right) <\frac1{24}. \tag{39}\] The last inequality follows directly from the chosen scales: \[\frac{T_n}{R_n^2}= \frac{16\Lambda}{R_0} \left(K_0D(D/\Lambda)^n+2\Lambda^{-n}\right).\] Here \(D/\Lambda=1/4\) and \(\Lambda\ge4\), so each geometric series has sum at most \(4/3\). Therefore \[\nu C_*\sum_{n\ge0}\frac{T_n}{R_n^2} \le\frac{\nu}{1+\nu}\frac{64}{3\cdot1024}<\frac1{48}.\] Also \(R_0\ge1024C_*(1+\nu)\) gives \[\frac{\nu C_*}{R_0^2} \le\frac{\nu}{1024^2C_*(1+\nu)^2}<\frac1{48}.\] This proves the strict bound in (39).

At every completed level before halting, more than \(3/4\) of the unit mass lies in the address square of radius \(R_n\) about the correct center. At a terminal endpoint the preceding cutoff radius already gives that conclusion, and its square lies in the larger level-\(n\) square. Other level-\(n\) address squares are disjoint from this one, so none can contain mass \(1/2\). Thus the velocity field records the intermediate configurations as well as their eventual outcome.

If the run reaches a terminal center at level \(n\), its height is \(S_n=16R_n\). The cutoff at arrival lies wholly in \(X_2>0\) and captures at least \(1-\delta>3/4\) of the mass. Its arrival time is finite, so (3) holds. This also handles an initially halted original input: the compiler starts nonterminal and reaches its terminal state after the finite preliminary simulation. No assertion about the captured mass after the first terminal endpoint is needed for this existential event.

If the machine never halts, the loading center is on the negative row and every transition has a nonterminal target. By (27), the moving cutoff remains entirely in \(X_2<0\) at every physical time. Hence \[\int_{\{X_2>0\}}\rho(t,X)\,dX \le\mathcal M(t)-\mathcal I(t) \le\delta<\frac1{24}<\frac14.\] The vertical period has length one, so this integral equals the one in (3). The threshold \(1/2\) therefore separates the two cases with a strict margin.

Effective finite prescriptions

Every integer, scale, address lookup and single-rule update above is computed directly from the finite machine table and input. For any finite interval \([0,T]\), choose an integer \(Q>T\). Since \(t_n\ge n+1\), only the levels \(0\le n<Q\) can contribute there. Extend each gate by zero through its time collars. Evaluation on \([0,T]\) then uses a finite sum over those levels and their finite address sets; it requires no decision about which side of a computable real switching time contains the argument.

The smooth profiles and every requested derivative are effectively evaluable. Near a flat endpoint, derivatives of \(e^{-1/r}\) are \(e^{-1/r}\) times explicit polynomials in \(1/r\). The estimate \(s^d e^{-s}\le(d+k)!s^{-k}\) supplies an effective error bound there, and the denominator in \(\theta\) is bounded below by \(e^{-2}\). Finite differentiation of the gate formulas gives effective bounds of every specified derivative order. Formula (31) is consequently a finite program for the force and its derivatives to any prescribed accuracy.

The force installs every defined one-step transition. It never follows the distinguished computation to decide which gate to install. The initial impulse selects its configuration; the subsequent solution carries the mass through the prescribed array. This completes the proof of the second part of Theorem 1.

Effective evaluation of the scalar

The scalar can also be evaluated effectively on each finite time interval, including its derivatives and integrable tails. This is a stronger computability property, not an input to the force prescription. We give the error control because pointwise computation alone would not justify restarting the scalar solver on this unbounded domain. Fix a derivative order \(a\) and a time bound \(T\). Let \(M_a\) be an effective bound for multiplication by each \(W_\ell(t)\) on \(E_a\), and let \(C_\nu\) be a constant large enough to include the sum of both derivative-kernel bounds. On a step of length \(h\) choose \[q=2C_\nu M_a\sqrt h\le\tfrac12.\] Then the Volterra operator \(B\) in the second term of (35), with the lower time limit replaced by the step’s initial time, has norm at most \(q\). Write \(y\) for the sum of heat evolution of the initial datum and the source integral on that step. The truncation \(\sum_{j=0}^N B^j y\) has error at most \[\frac{q^{N+1}}{1-q}\,\|y\|_{C_tE_a}.\] An initial-data error \(\epsilon_0\) and an additional error \(\epsilon_1\) in the integral equation, measured in \(C_tE_a\), change its solution by at most \((\epsilon_0+\epsilon_1)/(1-q)\). All the norms used here have computable upper bounds from the finite coefficient formulas and the preceding step. Thus choosing \(N\) and the quadrature accuracies gives a prescribed step error. Finitely many steps cover \([0,T]\), so their error budgets can be chosen backwards to achieve any specified final accuracy.

For completeness, the approximants carry a computable spatial-tail bound for every derivative through order \(a\), in both supremum and \(L^1\) norm. The source \(g\) and each product \(W_\ell v\) have known compact support on \([0,T]\), even when the approximant \(v\) does not. In a derivative-kernel integral, split the time lag at \(\eta>0\). If \(\|v\|_{C_tE_a}\le A\), the part with lag less than \(\eta\) has \(E_a\) norm at most \(2C_\nu M_a A\sqrt\eta\). For lags in \([\eta,T]\), derivatives of the explicit Gaussian have computable supremum and \(L^1\) tails, uniform in that interval. Convolution with data of known compact support and known \(E_a\) norm therefore has both tail bounds outside the support radius plus a computably chosen radius. The source integral is treated in the same way, with its bounded heat-kernel operator in place of the integrable derivative-kernel singularity.

At a restart, the exact previous scalar is not assumed compactly supported. Carry forward its certified \(E_a\) approximant and error. Multiply that approximant by a smooth cutoff equal to one on a large ball. Its certified derivative tails bound the cutoff error in \(E_a\) by the product rule. Heat convolution contracts that error, while the compactly supported truncation has explicit Gaussian tails, uniformly from lag zero to \(h\). This supplies the homogeneous term at the next step with both its \(E_a\) error and its tail bounds. Starting from zero, induction proves that every finite Picard approximation and every endpoint datum has the required effective representation. On the remaining compact sets, the computably smooth integrands can be integrated with derivative error bounds. A uniform error for each derivative on the retained spatial ball bounds its supremum error there and its \(L^1\) error by the ball’s area times that error; the certified tails control the complement. Time derivatives are recovered from (33).