History of the ideas and the role of the forcing

Turing’s undecidable symbol-printing question supplies the logical obstruction behind the final reachability corollary (Turing 1936--1937, sec. 8). A machine can be made to halt when it prints the specified symbol, while each ordinary stop without that printing is redirected into an infinite dummy loop. This connects that formulation to the halting convention used here. Landauer discussed retaining intermediate information to avoid logical irreversibility (Landauer 1961, sec. 3); Bennett implemented reversible computation by recording a history of transitions (Bennett 1973). Our explicit local recorders use that information-retention principle. Their exact-real tape coordinates have no finite-storage bound, and we do not import thermodynamic or computational-complexity conclusions.

Moore’s generalized shifts encode tapes by positional expansions and turn local symbolic rules into piecewise affine maps (Moore 1990, 1991). His smooth realizations and suspensions explain the symbolic-to-dynamical framework used here. Cardona, Miranda, Peralta-Salas and Presas realize bijective generalized shifts by area-preserving disk diffeomorphisms and suspend them in a geometric construction of stationary Euler flows (Cardona et al. 2021, Proposition 5.1 and Theorem 3.1). Their Euler realization uses an adapted Riemannian metric. Cardona, Miranda and Peralta-Salas later constructed a Turing-complete stationary Euler field for the Euclidean metric on \(\mathbb R^3\); its computational invariant set is noncompact and the field has infinite energy (Cardona et al. 2023, arXiv v3, Theorem 1 and the following discussion). We work in fixed flat coordinates and explicitly prescribe a viscous force from zero initial velocity. The local proofs below supply their own closed-rectangle action, support, quantitative bounds and intermediate observation paths.

Dyhr, González-Prieto, Miranda and Peralta-Salas construct unforced stationary Turing-complete Navier–Stokes fields with Hodge viscosity on compact three-manifolds admitting a nowhere-zero harmonic one-form, after a generally non-small metric deformation (Dyhr et al. 2026, Theorem A). That geometric result supplies viscous context; the fixed-flat-metric, zero-initial-data force prescriptions here are proved separately.

Cardona, Miranda and Peralta-Salas also realize computation through the unforced Euler evolution on a constructed high-dimensional compact Riemannian manifold with a specially chosen metric: the computational input is encoded in the initial velocity, and the observation is entry into an open set of smooth divergence-free velocity fields (Cardona et al. 2022, arXiv v1, Theorem 1.1, Remark 1.2 and Definition 5.1). This supplies a velocity-field antecedent to the observation used in our diffusion application.

For a prescribed divergence-free velocity \(U\), the identity \(f=U_t+(U\cdot\nabla)U-\nu\Delta U\) always makes \((U,0)\) a solution. It does not by itself construct a finite computational mechanism. The work lies in defining \(U\) from all local instruction branches, proving its effectivity, and controlling the complete trajectories and force derivatives. The analytic uniqueness step is the classical energy comparison associated with Leray (Leray 1934, sec. 18), proved locally in Section 2 for the classes we use.

Proof organization

Section 2 fixes the analytic comparison classes and proves the residual realization lemma. Section 3 proves the onto-clock identity and logarithmic estimate, emphasizing that accumulated computational time must remain unbounded. Section 4 proves the compact-support theorem by a complete compiler, coding and scalar-potential argument. Section 5 then gives the optional root clocks and spatial startup profiles. Section 6 replaces reciprocal planar maps by general prefix maps and compensates their area change. Section 7 keeps the realization planar and derives the suspension and diffusion applications. Section 8 states the decision consequence and the scope of the exact detector.

Recording an instruction before moving the head

We first produce a finite local table whose images can be distinguished. Retaining erased information is the history method of reversible computation (Bennett 1973). Here the ordinary head displacement is delayed until the history has been written and the recorder has returned.

Let \(A\) be the work alphabet, \(Q\) the state set, and \(q_0\) the initial state. Identify all designated halt states with one state \(h\), redirect transitions into them to \(h\), and replace an initially terminal state by \(h\). Replace each missing instruction at \((q,a)\) by the stationary symbol-preserving instruction \((q,a)\mapsto(h,a,0)\). A transition into \(h\) completes its prescribed write and move before halting. For a one-sided tape, include an immutable end-marker track at cell zero in \(A\) and retain its prescribed boundary action in the finite table; every write preserves that track. Write \(\delta(q,a)=(q^+,b,d^+)\) with \(d^+\in\{-1,0,1\}\). Let \[I=\{(q,d,a):q\ne h,\ d\in\{-1,0,1\},\ a\in A\},\qquad \mathcal H=\{\bot,P\}\sqcup\{[i]:i\in I\}.\] A tape cell has a work letter, a history letter in \(\mathcal H\), and a bit. The auxiliary states are \(R_{q,d},B_{q,d},L_{q,d}\) and \(F_i\). At a ready state \(R_{q,d}\) after \(n\) ordinary steps, the intended invariant has its frontier at cell \(2+n\), at least two cells to the right of the work head; records occupy cells \(2,\ldots,1+n\), all remaining history cells except the frontier are blank, and all bits are zero. The recorder writes the work letter and marks the old head, scans right to append a history record, and returns to erase that mark. Only then does it make the ordinary head displacement, restoring a ready state; the proof below verifies the invariant and finite scan lengths.

The following rules, expanded over the indicated letters, are the entire table.

1. In \(R_{q,d_0}\) with \(q\ne h\), on \((a,\gamma,0)\) with \(\gamma\ne P\), write \((b,\gamma,1)\), move right, and enter \(F_i\), where \(i=(q,d_0,a)\).

2. In \(F_i\), move right over unmarked cells with history different from \(P\). On \((v,P,0)\), write \((v,[i],0)\), move right, and enter \(B_{q^+,d^+}\).

3. In \(B_{q,d}\), on \((v,\bot,0)\), write \((v,P,0)\), move left, and enter \(L_{q,d}\).

4. In \(L_{q,d}\), move left over unmarked cells whose history is not \(P\). On \((v,\gamma,1)\) with \(\gamma\ne P\), erase the bit, move by \(d\), and enter \(R_{q,d}\).

Initialize in \(R_{q_0,0}\) with head at zero and the given finite work input, blank work elsewhere. The history track has \(P\) at cell \(2\) and \(\bot\) at every other cell; every marker bit is zero.

From the local table to full rectangles

The positional encoding follows the generalized-shift viewpoint of Moore (Moore 1990, 1991); its explicit gap and endpoint estimates are proved here. Order the alphabet \(\Gamma=A\times\mathcal H\times\{0,1\}\) with its all-blank symbol first. Give its letters the digits \(c(a)=2\operatorname{index}(a)\), starting at index zero, in base \(G=2|\Gamma|\). Put \[E(a_1a_2\cdots)=\sum_{n\ge1}c(a_n)G^{-n},\qquad I_a=[c(a)/G,(c(a)+1)/G].\] The digits range from \(0\) to \(G-2\). Distinct first-letter intervals have a gap at least \(G^{-1}\), and \(GE(a\omega)-c(a)=E(\omega)\) uniquely decodes the tails, including the zero blank tail. Give the states distinct positive indices. Set \(o_s\) equal to three times its index for nonterminal states, and minus three times its index for \(R_{h,d}\). A relative tape with head at \(j\) is encoded by \[(x,y)=\bigl(o_s+E(w_{j-1}w_{j-2}\cdots),E(w_jw_{j+1}\cdots)\bigr).\] At an initialized ready state the consecutive records immediately left of the frontier give their number \(n\), so the frontier has absolute index \(2+n\). Its relative distance from the head recovers the absolute head index as well.

Split each local rule \((s,a)\mapsto(s',b,d)\) by the letter \(l\) immediately to the left, and index the resulting geometric branches by \(i\). This index labels the split branches, rather than the recorder triples used above. The source of branch \(i\) is \(C_i=(o_s+I_l)\times I_a\), and the affine instruction is \[ \Theta_i(x,y)=\begin{cases} (o_{s'}+(c(b)+x-o_s)/G,\ Gy-c(a)),&d=1,\\ (o_{s'}+G(x-o_s)-c(l),\ (c(l)+y+(c(b)-c(a))/G)/G),&d=-1,\\ (o_{s'}+x-o_s,\ y+(c(b)-c(a))/G),&d=0. \end{cases} \tag{7}\] Removing and prefixing digits proves the update on every applicable code. Its linear part is \(\operatorname{diag}(\lambda_i,\lambda_i^{-1})\) with positive \(\lambda_i\). Its full image rectangle is, respectively, \[(o_{s'}+(c(b)+I_l)/G)\times[0,1],\quad (o_{s'}+[0,1])\times(c(l)+I_b)/G,\quad (o_{s'}+I_l)\times I_b.\] The source family is separated. The image family is separated too: within a destination the displacement is fixed by Lemma 4, and the pair \((l,b)\) identifies the split rule. Distinct such pairs have distinct two-digit intervals in a moving case, or separated coordinate products in the stationary case. Every gap is at least \(G^{-2}\). This verifies the needed geometric property on full rectangles, not just on encoded points.

Lifting, interpolation, and lowering

For a closed interval \([a,b]\) and \(\varepsilon>0\) use \[\eta_{[a,b]}^\varepsilon(v)= \sigma(2(v-a+\varepsilon)/\varepsilon) \sigma(2(b+\varepsilon-v)/\varepsilon),\] where \(\sigma\) is the smooth step in (3). It is one on a neighborhood of the closed interval and vanishes outside its \(\varepsilon\) enlargement. Products give rectangle cutoffs. With \(\varepsilon=1/(4G^2)\), choose \(g_i\) for \(C_i\) and \(\bar g_i\) for \(\Theta_i(C_i)\); within each family their supports are disjoint.

Write \(\Theta_i(x,y)=(\lambda_i x+\alpha_i,\lambda_i^{-1}y+\beta_i)\) and let \(N\) be the number of branches. Put \(h_i=3i\). Choose \(\eta_i(z)\) equal to one near \(h_i\) and supported within distance one of it. Choose \(\vartheta(z)\) equal to one on a neighborhood of \([0,3N]\) and compactly supported. Let \(O=1+\max_s|o_s|\), and choose an integer \(A\) greater than \(O+1\) and every number \[\max(1,\lambda_i)O+|\alpha_i|,\qquad \max(1,\lambda_i^{-1})+|\beta_i|.\] Choose a compactly supported planar cutoff \(\Xi\) equal to one near \([-A,A]^2\). Define the two divergence-free differential operators \[D_yP=(-\partial_zP,0,\partial_xP),\qquad D_zP=(\partial_yP,-\partial_xP,0).\] They act on scalar potentials; their divergence vanishes because mixed partial derivatives commute.

Figure 1 shows why separate source and image families suffice. A source is selected before lifting; its image is selected only after the planar map.

For \(m=1,2,3\) let \(r_m(\tau)=\sigma(8\tau-(2m-1))\). The three derivative supports are disjoint and contained in \((0,1)\). Put \[a_i(r)=1+(\lambda_i-1)r,\qquad b_i(r)=(\lambda_i-1)/a_i(r).\] For \((x_0,y_0)\in C_i\), the desired planar path is the reciprocal interpolation \[ (x,y)=(a_i(r)x_0+r\alpha_i,\ a_i(r)^{-1}y_0+r\beta_i), \qquad 0\le r\le1. \tag{8}\] It starts at \((x_0,y_0)\) and ends at \(\Theta_i(x_0,y_0)\). The scalar potential generating this path is \[K_i(x,y,r)=b_i(r)xy+(1-rb_i(r))\alpha_i y -(1+rb_i(r))\beta_i x.\] All denominators have a positive lower bound. Use the potentials \[\begin{align*} P_1&=r_1' x\vartheta(z)\sum_i h_i g_i(x,y),\\ P_2&=r_2'\Xi(x,y)\sum_i\eta_i(z)K_i(x,y,r_2),\\ P_3&=-r_3' x\vartheta(z)\sum_i h_i\bar g_i(x,y),\qquad W=D_yP_1+D_zP_2+D_yP_3. \end{align*}\] The resulting field is divergence free, smooth, supported in one compact spatial set, and zero near the temporal endpoints.

For \((x_0,y_0)\in C_i\) at height zero, the first phase is exactly \((x_0,y_0,h_i r_1)\). On this path \(g_i=1\), all its derivatives vanish, and \(\vartheta=1\), so the curl correction has no horizontal component. At height \(h_i\) the second phase follows (8) with \(r=r_2(\tau)\). Indeed its derivatives with respect to \(r\) are \(b_i x+(1-rb_i)\alpha_i\) and \(-b_i y+(1+rb_i)\beta_i\), which are \(\partial_yK_i\) and \(-\partial_xK_i\). The choice of \(A\) keeps the path in the plateau of \(\Xi\); exactly one height cutoff is active. At its endpoint \(\bar g_i=1\), so the third phase lowers it as \(z=h_i(1-r_3)\). ODE uniqueness proves that these are the actual trajectories. One period therefore sends every point \((x,y,0)\) above \(C_i\) to \((\Theta_i(x,y),0)\).

The first coordinate during (8) is \((1-r)x_0+r\Theta_{i,1}(x_0,y_0)\). The other phases keep it fixed. Thus a branch with nonterminal endpoints cannot cross the negative detection half-space during its interpolation.

Loading and an unbounded slow clock

The initial code \((x_*,y_*)\) is rational because it has only finitely many nonblank digits. Put \(r_*(t)=\sigma(2t-1/2)\) and, on \(0\le t\le1\), use \[P_*(t,x,y,z)=r_*'(t)\Xi(x,y)\vartheta(z)(x_*y-y_*x),\qquad U=D_zP_*.\] The trajectory of the origin is \((r_*x_*,r_*y_*,0)\). The chosen plateau contains this segment. Extend \(W\) with period one in its time variable, writing the result as \(V\). The loader occupies the first unit of time; after it, the onto processor clock \(s=\log_2t\) starts at zero when \(t=1\). For \(t>1\) define \[U(t,x)=\frac1{t\log2}V(\log_2t,x).\] The zero time collars glue these definitions smoothly at \(t=1\). At \(t=2^m\), the particle has the code after \(m\) auxiliary steps, up to its first halt. This follows from the onto clock \(\log_2t\) and the exact full-rectangle map. Each finite ordinary computation thus finishes in finite physical time.

For a nonhalting computation every code has \(x\ge3\), and the loader travels from zero to such a code. Convexity of the first coordinate proves avoidance of \(x<-1\) at every real time. A terminal code has \(x\le-2\), including an initial halt immediately after loading. This proves the event equivalence.

Finally, \[\partial_t[t^{-a}Y(\log_2t,x)] =t^{-a-1}[-aY+(\log2)^{-1}\partial_sY](\log_2t,x).\] The proof of Lemma 3 therefore gives the stated \(t^{-1-j}\) bounds on \(t>1\); the finite loading interval adds bounded quantities. Define \(f=U_t+(U\cdot\nabla)U-\nu\Delta U\). Fixed compact support gives the Sobolev comparison conditions and every asserted norm. Lemma 2 proves uniqueness. All rules, cutoffs and clock formulas are finite effective data. Periodic evaluation uses a finite superset of possibly active translates, so it does not test equality at a seam or follow the machine’s run. This proves Theorem 1.

A record between ordinary tape symbols

Let \(Q\) be the finite state set, \(q_{\mathrm{in}}\) its initial state, \(Q_h\subseteq Q\) its terminal states, and \(\Gamma\) the work alphabet with blank \(\square\). Missing instructions are replaced by stationary transitions to an added terminal state, preserving the read letter. A transition into a terminal state is completed; an initially terminal configuration also counts as halting. Add a read-only origin bit to the work alphabet, one only at cell zero and preserved by every write. For empty input, the initial word is a marked blank. This allows recovery of the absolute head position after decoding.

The record symbols retain erased information in the sense of reversible history computation (Bennett 1973). For each nonterminal \(q\), previous displacement \(d\in\{-1,0,1\}\), and read letter \(a\), create a record \(g(q,d,a)\), and let \(G\) be the set of these records. Two infinite stacks \(L,R\) use the alphabet \(\mathcal A=\Gamma\sqcup G\). Labels are \(C(q,d)\) and \(S_+(q),S_-(q)\). At a checkpoint \(C(q,d)\), deleting the records from \(L\) gives the tape left of the head in reverse order; deleting them from \(R\) gives the tape from the head rightward. The top of \(R\) is a work letter. The scan labels transfer leading records between the stacks until the next work letter appears.

A prefix rule \((\ell;\alpha,\beta)\to(\ell';\alpha',\beta')\) replaces the indicated prefixes of \(L,R\) and retains their infinite tails. For \(\delta(q,a)=(q',b,m)\) and \(g=g(q,d,a)\), use \[ \begin{array}{lll} (C(q,d);\epsilon,a)\to(S_+(q');bg,\epsilon),& &m=1,\\ (C(q,d);\epsilon,a)\to(S_-(q');\epsilon,bg),&&m=-1,\\ (C(q,d);\epsilon,a)\to(C(q',0);g,b),&&m=0. \end{array} \tag{10}\] Here \(\epsilon\) is the empty word. For every \(q\), every \(g\in G\) and every \(c\in\Gamma\), also use \[ \begin{array}{ll} (S_+(q);\epsilon,g)\to(S_+(q);g,\epsilon),& (S_+(q);\epsilon,c)\to(C(q,1);\epsilon,c),\\ (S_-(q);g,\epsilon)\to(S_-(q);\epsilon,g),& (S_-(q);c,\epsilon)\to(C(q,-1);\epsilon,c). \end{array} \tag{11}\] The terminal labels are exactly the checkpoints \(C(q,d)\) with \(q\in Q_h\); there are no rules out of them. The scan labels \(S_\pm(q)\) remain nonterminal even for \(q\in Q_h\), so a pending move finishes before its terminal checkpoint is reached. Initialize at \(C(q_{\mathrm{in}},0)\) with left stack \(\square^\infty\) and right stack \(\omega\square^\infty\), where \(\omega\) includes the origin bit.

A planar motion for arbitrary diagonal rectangle maps

As in Moore’s positional representation of symbolic shifts (Moore 1991), assign the ordered letters of \(\mathcal A\) the digits \(1,3,\ldots,2M-1\) in base \(B=2M+1\), where \(M=|\mathcal A|\). Set \[\gamma(w)=\sum_{j\ge1}D(w_j)B^{-j},\quad c(\alpha)=\sum_{j=1}^{|\alpha|}D(\alpha_j)B^{-j},\quad I(\alpha)=[c(\alpha),c(\alpha)+B^{-|\alpha|}].\] The identity \(\gamma(\alpha w)=c(\alpha)+B^{-|\alpha|}\gamma(w)\) and the gaps between digits prove unique decoding. Child intervals lie strictly inside the parent, so incompatible prefix cylinders have separated intervals even as closed sets.

Put \(\Omega=(0,1)^2\) and \(\Omega_-=(0,1/2)\times(0,1)\). If the labels number \(N\), give label \(\ell_i\) a square \(K_i\) of half-side \(1/[16(N+1)]\), centered at height \(i/(N+1)\) and first coordinate \(3/4\) for terminal labels, \(1/4\) otherwise. Let \(\Phi_i\) be the positive axis-preserving affine map of \([0,1]^2\) onto that square. The encoded state is \(\Phi_i(\gamma(L),\gamma(R))\). A rule with prefixes \(\alpha,\beta\) and \(\alpha',\beta'\) has rectangles \[D_i=\Phi_\ell(I(\alpha)\times I(\beta)),\qquad E_i=\Phi_{\ell'}(I(\alpha')\times I(\beta')).\] Both families are separately disjoint by Lemma 8. Each source lies in \(\Omega_-\), and each target lies entirely in one open half of \(\Omega\). The positive diagonal affine map \(D_i\to E_i\) implements the prefix rule exactly. Its determinant is allowed to differ from one.

The shrink–move–expand interpolation also appears in the proof of (Cardona et al. 2021, Proposition 5.1); here we specify rational routes that preserve the required observation half-plane.

Compensating the planar divergence

The rational initial code is denoted by \(Z_{\mathrm{in}}\). Its rationality follows from the finite input and the blank tail \(D(\square)/(B-1)\). Let \(p_*=(1/4,1/2)\) and load it along \(c(s)=p_*+\zeta(s)(Z_{\mathrm{in}}-p_*)\). A translating bump supported in a sufficiently thin tube around this segment gives a field \(v_{\mathrm{in}}\) with precisely that trajectory. For instance, take the half-side of the moving square to be one quarter of the smaller endpoint distance from \(\partial\Omega\), and take the same enlargement margin. Convexity keeps the support in \(\Omega\). A nonterminal loader stays in \(\Omega_-\); a terminal loader reaches the right half.

Extend \(v_*,v_{\mathrm{in}}\) by zero in time and periodically in space, and define \[v(s,Y)=v_{\mathrm{in}}(s,Y)+\sum_{n\ge1}v_*(s-n,Y).\] All mixed derivatives are bounded because only one repeated phase template and one loader occur. At logical time \(1+n\), the trajectory from \(p_*\) is the code after \(n\) prefix steps, up to a terminal visit. The rectangle lemma proves absence of a right-half visit during every nonterminal transition as well as at its endpoints. Together with the finite-scan invariant, this gives the all-time halting equivalence.

Choose a periodic smooth function \(\eta(z)\) supported near \(1/2\) which equals \(z-1/2\) on a neighborhood of \(1/2\). Define \[ W(s,Y,z)=\bigl(\eta'(z)v(s,Y),-\eta(z)\operatorname{div}_Yv(s,Y)\bigr). \tag{12}\] The horizontal divergence and vertical derivative cancel. The spatial mean is zero: \(\int\eta'=0\), and the integral of a periodic planar divergence is zero. At \(z=1/2\), the vertical component vanishes and the horizontal component is exactly \(v\). Thus this plane carries the entire prescribed computation despite the planar area changes. The lift has bounded mixed derivatives and vanishes at logical time zero.

Apply Lemma 3 with \(g(t)=\log(1+t)\). The particle from \((p_*,1/2)\) is \((Y(g(t)),1/2)\), so its prefix-step observations occur at \(t=e^{1+n}-1\). Every finite logical time occurs at finite physical time, and the event equivalence is preserved. The lemma gives the exact derivative decay for the velocity and its residual, which has zero mean and pressure zero. Lemma 2 gives uniqueness on the torus.

For clarity, the optional solenoidal force changes pressure. Given the residual \(f\), let \(\phi\) be the zero-mean solution of \(\Delta\phi=\operatorname{div}f\) and replace \((f,p)\) by \((f-\nabla\phi,-\phi)\). This leaves \(-\nabla p+f\) and therefore the velocity unchanged. In Fourier variables, \[\phi(t,x)=-\sum_{k\ne0}\frac{i\,k\cdot\widehat f_k(t)}{2\pi|k|^2}e^{2\pi i k\cdot x}.\] Integration by parts in a coordinate of maximal \(|k_l|\) gives, for every \(m\), \(|\partial_t^j\widehat f_k(t)|\le C_{j,m}(1+t)^{-1-j}|k|_\infty^{-m}\). There are \(24r^2+2\) lattice points with \(|k|_\infty=r\). Choosing \(m\) large proves uniform convergence after any fixed derivatives, retains the decay, and gives effective tails. Finite Fourier coefficients are effectively integrated using derivative bounds. Termwise differentiation gives the Poisson equation, with zero mode zero; uniqueness of Fourier coefficients on continuous periodic functions identifies it pointwise. Thus the projected force is effective, mean zero and divergence free. This is the classical Helmholtz–Leray projection, with its pressure correction explicit.

Finally all infinite sums used to prescribe the force are evaluated from finite supersets of possibly active translates, obtained from a coarse enclosure of the time and position. The elementary flat-step derivatives have effective bounds near their seams. Neither construction nor evaluation uses a configuration reached after initialization. This completes Theorem 7.

A partial table with a recognizable inverse

This construction retains an ordinary partial instruction table: the machine halts when its state and scanned letter have no instruction. The optional one-sided-tape convention of Section 4 is retained by including its immutable end-marker in the work alphabet and keeping the prescribed boundary actions; the recorder preserves that track. Let \(Q\) be the finite state set, \(A\) the work alphabet, \(q_0\) the initial state, and \(E\subset Q\times A\) the set of defined instructions, and write \(e=(q,a)\mapsto(q'_e,b_e,d_e)\). Let \[H_0=\{\mathtt u,\mathtt P\}\sqcup\{\mathtt r_e:e\in E\}, \quad H_1=H_0\setminus\{\mathtt P\},\quad \Gamma=A\times H_0\times\{0,1\}.\] The extra tracks record instructions and mark the place to which the recorder must return, following the history principle of Bennett (Bennett 1973). States are \({\sf Run}_q,{\sf Turn}_q,{\sf Back}_q,{\sf Stop}_q\) and \({\sf Mark}_e,{\sf Seek}_e\). Put \[(\tau_{q,a},\beta_{q,a},\mu_{q,a})= \begin{cases}({\sf Mark}_{(q,a)},b_{(q,a)},d_{(q,a)}),&(q,a)\in E,\\ ({\sf Stop}_q,a,0),&(q,a)\notin E. \end{cases}\] At a checkpoint \({\sf Run}_q\) after \(k\) ordinary steps, the intended invariant has frontier \(F=2+k\) at least two cells to the right of the head, records in cells \(2,\ldots,F-1\), blank history elsewhere except at the frontier, and all bits zero. Here a defined work instruction moves the head first; the next row marks that new position, scans right to append the record and advance the frontier, then returns to erase the mark. This order restores the updated head position at the next checkpoint, as the proof below shows.

Use exactly the following seven rule families: \[ \begin{array}{lll} ({\sf Run}_q,(a,h,0))\mapsto(\tau_{q,a},(\beta_{q,a},h,0),\mu_{q,a}),&h\in H_1,\\ ({\sf Mark}_e,(a,h,0))\mapsto({\sf Seek}_e,(a,h,1),1),&h\in H_1,\\ ({\sf Seek}_e,(a,h,0))\mapsto({\sf Seek}_e,(a,h,0),1),&h\in H_1,\\ ({\sf Seek}_e,(a,\mathtt P,0))\mapsto({\sf Turn}_{q'_e},(a,\mathtt r_e,0),1),\\ ({\sf Turn}_q,(a,\mathtt u,0))\mapsto({\sf Back}_q,(a,\mathtt P,0),-1),\\ ({\sf Back}_q,(a,h,0))\mapsto({\sf Back}_q,(a,h,0),-1),&h\in H_1,\\ ({\sf Back}_q,(a,h,1))\mapsto({\sf Run}_q,(a,h,0),0),&h\in H_1. \end{array} \tag{13}\] There are no rules out of a stop state. Initially the work tape is the finite input with blanks elsewhere, the head is at zero, every bit is zero, and the history has \(\mathtt P\) at cell \(2\) and \(\mathtt u\) everywhere else.

Coding a fixed starting point

Give the \(m=|\Gamma|\) letters odd digits \(D(s)=1,3,\ldots,2m-1\) in base \(B=2m+1\). Let \[C(s_1s_2\cdots)=\sum_{j\ge1}D(s_j)B^{-j},\qquad I_s=[D(s)/B,(D(s)+1)/B].\] A code lies strictly inside its first-letter interval, and successive stripping uniquely decodes it. The left and right tape lists have coordinates \(\xi=C(s_{-1}s_{-2}\cdots)\) and \(\eta=C(s_0s_1\cdots)\). For a rule \((\varsigma,s)\mapsto(\tau,s',d)\) write \(\beta=(D(s')-D(s))/B\). Its normalized rectangles and maps are \[ \begin{array}{ccl} d&\text{source}&(\xi,\eta)\longmapsto\\ 0&[0,1]\times I_s&(\xi,\eta+\beta),\\ 1&[0,1]\times I_s&((D(s')+\xi)/B,B\eta-D(s)),\\ -1&I_l\times I_s&(B\xi-D(l),(D(l)+\eta+\beta)/B),\quad l\in\Gamma. \end{array} \tag{14}\] The image rectangles are \([0,1]\times I_{s'}\), \(I_{s'}\times[0,1]\), and \([0,1]\times(D(l)/B+B^{-1}I_{s'})\), respectively. The digit identities prove that these are the correct tape maps, with determinant one and positive diagonal part \(\operatorname{diag}(B^{-d},B^d)\).

Let \(K\) be the number of states and \(N\) the number of split branches. They are positive, even if \(E\) is empty, because the run-to-stop rules exist. Set \[r=[10^6(K+1)(N+1)(B+1)^2]^{-1}.\] Each state has a square \(o_\varsigma+r[0,1]^2\), based near \(c_R\) for stop states and near \(c_L\) otherwise. If \((\xi_*,\eta_*)\) is the rational initial tape code, set \(o_{{\sf Run}_{q_0}}=c_L-r(\xi_*,\eta_*)\). Enumerate the other states by \(j=1,\ldots,K-1\) and set their origins to their cluster base plus \((0,3jr)\). These squares are separated by at least \(r\) and lie within sup distance \((3K+1)r\) of their bases. The initial physical code is exactly \(c_L\).

Write the resulting physical affine branches \(T_i:R_i\to R_i'\) with centers \(p_i,p_i'\) and linear parts \(\operatorname{diag}(\lambda_i,\lambda_i^{-1})\). Distinct sources are separated by at least \(r/B^2\). Images have the same separation: in a target state the displacement is fixed and written symbols distinguish incoming rules, by Lemma 10. For a left move, the two-digit target interval distinguishes \((l,s')\); its gaps are at least \(B^{-2}\). This is a full-rectangle separation statement. At checkpoints, the record block length also recovers \(k\), hence the frontier’s absolute index \(2+k\) and the absolute head position.

Routing full rectangles inside the plane

Area-preserving realization of symbolic rectangle maps is central to the generalized-shift construction of Cardona, Miranda, Peralta-Salas and Presas (Cardona et al. 2021, Proposition 5.1). We give the full local construction needed here. Its quantitative collars and complete intermediate paths supply the later suspension and diffusion tests.

The same construction supplies the estimates needed for diffusion. Choose an effective integer \(L\ge1\) bounding the first and second spatial derivative operator norms of \(V\), and put \(R=4^L\). If \(\Psi_s\) is its flow from phase zero, then for \(0\le s\le n\), \[ \|D\Psi_s^{\pm1}\|_\infty\le e^{Ln}\le R^n, \qquad \|D^2\Psi_s^{\pm1}\|_\infty\le nLe^{3Ln}\le nLR^{3n}. \tag{15}\] Indeed the first variation equation has coefficient norm at most \(L\). The second has the same linear coefficient, zero initial data, and forcing at most \(L\) times the square of the first variation; variation of constants gives the bound. The reversed-time field has the same spatial derivative bounds, proving the inverse estimates as well.

Periodic and eventually stationary forcing

A velocity threshold for the same strip

The injection, stirring and waiting theorem in (OpenAI 2026, Theorem 3.1) applies to this planar field. Each block injects a fresh narrow scalar bump near \(c_L\), carries it through \(n\) processor periods in a short physical interval, and then lets heat diffuse during a long wait. The short stirring time keeps the new bump close in height to its transported profile; its small mass and the waits keep old injections below the detection threshold. The scalar becomes the third velocity component, so a visit of a transported bump to the strip yields a velocity-field observation. We now state the exact substitution and the bounds supplied by that theorem.

Take its initial point to be \(c_L\), its observed planar set to be \(E=\{2/3<Y_1<5/6\}\), and its constants to be \(d=1/16\) and \(C_H=10^8\). The field \(V\) is Hamiltonian, smooth, period one, has zero phase collars and a fixed compact chart support. Lemma 11 gives a terminal integer-phase visit, which is precisely the companion theorem’s halting hypothesis; the constant \(d\) controls the nonhalting separation below. In a nonhalting run the trajectory lies in \(1/8<Y_1<3/8\), whose infimum circle distance from the observed arc is \[\min\{2/3-3/8,\ 1+1/8-5/6\}=7/24>2d.\] Thus the whole nonhalting orbit has distance at least \(7/24>2d\) from \(E\). Equation (15) supplies the effective forward and inverse first and second derivative bounds. These verify every geometric hypothesis of the parameterized compact detector theorem.

For specificity, its unit-viscosity construction uses \(R=4^L\), a smooth bump \(g\) supported in \([-1,1]^2\), equal to one near zero and between zero and one, and a bound \(C\ge1\) on its first two derivative operator norms. All derivative norms here are Euclidean operator norms, as in the companion theorem. Its parameters here are \[\begin{align*} b_n&=10^{-6}(2R)^{-n},&g_n(Y)&=g((Y-c_L)/b_n),\\ D_n&=2Cb_n^{-2}(1+nL)R^{3n},&\delta_n&=[100(1+D_n)]^{-1},\qquad t_n=2n. \end{align*}\] Here \(D_n\) bounds the Laplacian of the bump after transport through at most \(n\) periods; the choice of \(\delta_n\) controls the accumulated diffusion error. If \(V_0\) is one period extended by zero, the prescribed horizontal field and scalar source are \[a(t,Y)=\sum_{n\ge1}\frac n{\delta_n}\sum_{j=0}^{n-1} V_0\left(\frac{n(t-t_n-\delta_n)}{\delta_n}-j,Y\right),\qquad h(t,Y)=\sum_{n\ge1}\delta_n^{-1}\sigma'((t-t_n)/\delta_n)g_n(Y).\] Let \(F=a_t+(a\cdot\nabla)a-\Delta a\). The force is the precomputed pair \((F,h)\); the unknown transported scalar is absent from it. The theorem supplies the scalar \(w\) solving \(w_t+a\cdot\nabla w=\Delta w+h\), \(w(0)=0\), and the velocity \((a,w)\) with pressure zero. In this specialization, the total injected mass is bounded by \(M_0\), with \[M_0=\sum_{n\ge1}4b_n^2<10^{-11},\qquad 2\delta_nD_n<1/50<1/16,\qquad 2-2\delta_n>1.\] The companion theorem uses these inequalities with the off-diagonal heat bound \(C_H=10^8\) at separation \(1/16\). It proves an all-time velocity threshold, including the waits between bursts, rather than only a sampled-time statement.