Main results

Write \(\mathbb T_L^3=(\mathbb R/L\mathbb Z)^3\) for the flat torus. At a fixed viscosity \(\nu>0\) we use \[ u_t+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f, \qquad \operatorname{div}u=0,\qquad u(0,\cdot)=0. \tag{1}\] Pressure is spatially periodic and normalized to have mean zero. A classical comparison solution has continuous \(u,u_t\), spatial derivatives of \(u\) through order two, and \(p,\nabla p\) on every finite closed time cylinder. The material flow is defined by \(\partial_tX(t,a)=u(t,X(t,a))\), \(X(0,a)=a\). An effective force prescription evaluates every requested mixed derivative at computably supplied space and time arguments to any positive rational error. The construction also supplies effective global bounds when these are asserted. Its finite formula describes all instruction branches; it is not a recording of an executed computation.

Section 4 proves this theorem after the common toolkit in Section 3. It gives two initialization mechanisms: an input-dependent chart and a reserved loading branch in the repeated table. The latter gives the one-component option. Both incorporate initialization into a schedule that repeats from time zero. Alternative recorder guards and observers are developed separately in Section 5.

The second result allows an instruction to change planar area. Its normal action then changes as well, as volume preservation requires.

The sequential storage construction in Section 8.1 proves this theorem for arbitrary such rectangle data; Section 8.2 applies it to a complete tape computation. Section 8.4 gives a simultaneous construction for the narrow-strip rectangles of a different recorder, using separate heights: contraction, translation, and expansion take fifteen phases. Both proofs exhibit the compensating normal action instead of treating an area-changing planar map as area preserving. If the required action is on a whole box of positive thickness, Section 7.1 provides a separate reciprocal box-transfer theorem with an all-time sign guard.

How the constructions fit together

A finite local recorder stores enough information to reverse each rule on its entire partial domain. Two positional coordinates encode the tape halves. Fixing the finitely many letters read by a rule gives a closed rectangle, and the inverse ensures that the image rectangles are separated. The toolkit in Section 3 makes this passage explicit.

Each reciprocal rectangle map is then performed at a private height. A vertical pulse lifts the source rectangle; four centered planar shears produce the diagonal change of shape; a translation moves it to its target center; a final vertical pulse returns it to the coding height. Separating these stages in time makes each active field transverse, so convection vanishes. Spatial selectors have zero mean and are constant on the tracked paths. Thus the direct force formula gives the fluid equation, while the partial shear paths give the observation bounds. Sections 6 and 7 adapt this mechanism to a one-time loader, other clocks and charts, and whole boxes.

For arbitrary planar area change, the normal direction must also move. A vertical shear records a horizontal offset in height, a horizontal shear changes that offset, and a second vertical shear restores the sheet while rescaling nearby normal offsets. To allow a target to intersect any source, the general construction first evacuates all source sheets into a separate storage region. It then delivers them one at a time. This geometry proves the second theorem before any symbolic application is chosen; its low-height bound supplies the application’s fixed detector.

Ideas and relation to earlier work

An inverse cannot discard the information that distinguishes predecessors. Landauer discusses this retention principle and its storage implications [5]. Reversible Turing machines already appear in Lecerf’s work on undecidability for code isomorphisms [6]. Bennett’s reversible simulation records each instruction and verifies that the local rules have disjoint domains and disjoint ranges [1]. Our finite local recorders use these principles; their particular guards are checked on the full partial domains, including tapes outside the initialized run.

Moore’s generalized shifts connect positional tape coding to dynamical systems [8]. They admit finite affine Cantor-block representations, and invertible generalized shifts have smooth realizations [9]. Our geometric constructions implement the resulting closed-rectangle maps with explicit transverse pulses, mean-zero forcing, and observation bounds throughout each operation. Separation of sources and of images is the interface between the symbolic inverse and these pulse formulas.

The fluid setting also has substantial predecessors. Cardona, Miranda, Peralta-Salas and Presas realize computation by stationary Euler flows on an adapted Riemannian three-sphere [3]. Their Proposition 5.1 constructs an area-preserving realization by contracting, transporting, and expanding neighborhoods with a relative area correction. That geometric organization is related to the sheet construction below. For a fixed Euclidean metric, Cardona, Miranda and Peralta-Salas construct Turing-complete stationary Euler fields on \(\mathbb R^3\); their simulation uses a noncompact set and the fields have infinite energy [2]. On the standard flat three-torus they robustly simulate tape-bounded machines, with reachability tested before a trajectory leaves a prescribed compact region. Their construction also gives time-dependent Navier–Stokes realizations of these bounded computations [2].

Dyhr, González-Prieto, Miranda and Peralta-Salas obtain stationary unforced viscous computation after a metric deformation, using the Hodge Laplacian on one-forms as the viscous operator [4]. Here the flat metric, the usual flat Laplacian, positive viscosity, and zero initial velocity are fixed; the finite machine and input specify the external force. The local shear formulas supply the required geometry directly.

The companion Finite Instructions and Solenoidal Shear Flows [10] gives one complete initialized construction and the common reciprocal-rectangle theorem. We restate its geometric realization (Theorem 3.2) in Section 2, and its recorder and extensions (Lemma 2.1 and Section 5.1) in Section 6.3. The present proofs address changed requirements: periodicity from zero, alternative full-domain guards, routing through a common center, positive thickness, and planar area change. A symbolic inverse is used to prove separation; it does not replace the geometric or intermediate-time argument.

At each instant our active velocity moves only in directions on which its coefficients do not depend. Consequently convection vanishes and \(f=U_t-\nu\Delta U\) gives solenoidal forcing with pressure zero. Section 2.1 proves the relevant uniqueness statement by the classical difference-energy method associated with Leray [7]. Slower onto clocks preserve the complete trajectory, including initialization; their exact derivative estimates are proved in Section 6.1. A clock with finite accumulated time would not suffice.

Finally Section 9 writes the same equation in material labels, and Appendix 10 constructs a fixed universal table in the stated tape model. Thus the force compiler can either take an arbitrary finite machine table or keep one interpreter fixed and vary only its input. Composing Theorem 1 with any decision procedure for its fixed particle event would decide halting. This includes Turing’s symbol-printing question: stop when the designated symbol is printed, and otherwise continue forever, including when the original machine stops without printing it [11]. The constructions use exact real codes and assert no uniform finite-precision tolerance.

Transverse pulses and classical uniqueness

A transverse field has either one nonzero component independent of its direction of motion, or the form \((a(z),b(z),0)\). Both its divergence and its self-advection vanish. A finite sum of such fields need not have zero self-advection if several act simultaneously; we instead give them disjoint time supports. This scheduling restriction is part of every construction below.

The reciprocal geometric input

Theorem 3.2 of [10] has the following precise interface. Fix rational \(L,h>0\) and rational coordinate charts on \(\mathbb T_L^3\). Take finite families of closed axis-parallel rational rectangles \(P_i,Q_i\) with positive side lengths and centers \(p_i,q_i\), each family pairwise positively separated, and maps \(F_i(y)=q_i+\operatorname{diag}(r_i,r_i^{-1})(y-p_i)\) with \(F_i(P_i)=Q_i\) and \(r_i>0\) rational. Suppose every source and target half-width is at most \(h\), their centers lie in a rational rectangle \(K\), and \(K+[-2h,2h]^2\) is compactly contained in the planar chart. For any rational coding height \(z_0\) inside its chart, that theorem supplies seven effective mean-zero transverse pulses, periodic with zero collars, whose period map is \((y,z)\mapsto(F_i(y),z)\) on a positive effectively specified neighborhood of each \(P_i\times\{z_0\}\). For a point starting on \(P_i\) at height \(z_0\), the four deformation stages remain within sup distance \(2h\) of the source center; the following translation interpolates between centers with a fixed offset of sup norm at most \(h\). Lifting and lowering preserve the planar coordinates. Source–target intersections are allowed. The theorem also proves the empty-list case and the force conclusions of Lemma 3.

This seven-stage construction supplies the reciprocal template used below. Alternative shear lists and translation routes are checked locally, including their cutoff plateaus and intermediate paths. The constructions on boxes of prescribed thickness likewise prove their own bounds on every point of those boxes.

Profiles that preserve the exact shear paths

Write \(X(a)(x,y)=(x+ay,y)\) and \(Y(a)(x,y)=(x,y+ax)\). Besides the four shears in Theorem 3.2 of [10], we will use these chronological lists about a branch center: \[\begin{align*} &Y(-r),\ X(r^{-1}-1),\ Y(1),\ X(r-1),\tag{2}\\ &Y(-r(r-1)),\ X(-r^{-1}),\ Y(r-1),\ X(1),\tag{3}\\ &X(r^{-1}-1),\ Y(1),\ X(r-1),\ Y(-r^{-1}). \tag{4}\end{align*}\] All have product \(\operatorname{diag}(r,r^{-1})\), with products taken in reverse chronological order. For example, the leftmost three factors of the full product in (3) multiply to \(\left(\begin{smallmatrix}r&0\\r-1&r^{-1}\end{smallmatrix}\right)\); right multiplication by \(Y(-r(r-1))\) finishes the identity. For (4) the rightmost three factors are \(\left(\begin{smallmatrix}r&0\\1&r^{-1}\end{smallmatrix}\right)\), which the final shear makes diagonal.

If \(r,r^{-1}\le B\) and initial half-sides are at most \(\ell/2\), the crude bounds for every partial shear are \((1+B)^4\ell/2\) for (2),(4), and \((1+B^2)^4\ell/2\) for (3). Each follows by four applications of \(\|X(a)\|_\infty,\|Y(a)\|_\infty\le1+|a|\); partial pulse coefficients have smaller modulus. For (2), the stronger endpoint-size estimate of Theorem 3.2 of [10] bounds offsets by \(3\ell/2\) whenever both the source and target sides are at most \(\ell\), independently of \(r\). These estimates are useful for different chart choices.

Here are three interchangeable ways to impose zero mean while retaining the specified paths. A box cutoff always equals one on a neighborhood of the indicated closed box, and is supported in a rational collar. For an explicit convention set \(E(s)=e^{-1/s}\) for \(s>0\) and zero otherwise, and \(H(s)=E(s)/(E(s)+E(1-s))\). For \(I=[a,b]\) use \[\chi_{I,\delta}(s)= H\bigl(2(s-a+\delta)/\delta\bigr) H\bigl(2(b+\delta-s)/\delta\bigr).\] It is one on \([a-\delta/2,b+\delta/2]\), supported in \([a-\delta,b+\delta]\), and symmetric about the interval center. Products give box cutoffs. Positive-side derivatives of \(E\) are polynomials in \(1/s\) times \(e^{-1/s}\), and the bound \(e^{-1/s}\le n!s^n\) gives effective flatness at zero to every order. The denominator of \(H\) is at least \(e^{-2}\). Thus these profiles and all their derivatives can be evaluated without deciding equality with a cutoff endpoint.

1. A rectangle selector may be \(\partial_x((x-c_x)\chi_P(x,y))\), and a height selector may be \(\partial_z((z-z_i)\zeta_i(z))\). They equal one on their plateaus and have integral zero by periodic differentiation.

2. A selector may be \(\chi_P(x,y)-\chi_P(x,y-\delta)\), or \(\zeta_i(z)-\zeta_i(z-\delta)\), when the translated support misses all tracked rectangles or heights. Equal translated integrals cancel. A mean-zero height selector makes every supported horizontal linear profile have zero spatial mean, even if that linear profile does not itself have zero mean.

3. With an even local cutoff, the periodic local function \(\Xi_c(s)=(s-c)H(2-2(s-c)^2/D^2)\) is odd about \(c\), hence has zero mean. Its derivative also has zero mean. On \(|s-c|\le D/2\) they equal \(s-c\) and \(1\), respectively. Alternatively, \(\partial_s(\tfrac12(s-c)^2\chi(s))\) and \(\partial_s(s\chi(s))\) give the same linear and constant plateaus.

All profiles are extended periodically only across a region where they vanish. The first recipe requires a neighborhood plateau because a derivative is taken; the second only requires the explicit separation from translated supports.

The seven-stage construction and its pulse convention

The reciprocal theorem in Section 2 uses a source selector to assign each rectangle its own height, performs its planar map there, and uses a target selector to return it to the coding height. We write the fields from its proof [10] in a form allowing separate shear and translation selectors, since later constructions change individual profiles or stages.

Write the centers of \(P_i,Q_i\) as \(c_i^-,c_i^+\) and the prescribed linear part as \(\operatorname{diag}(r_i,r_i^{-1})\). Let \(z_0\) be the coding height. Choose smooth periodic selectors \(A_i^-,A_i^+\) with zero planar mean, equal to one near their own source or target rectangle and zero near all other rectangles of the same family. Choose distinct private heights \(z_i\). Two height selectors \(\sigma_i,\tau_i\) equal one near \(z_i\) and zero near every other private height. Require \(\tau_i\) to have zero mean. For \(j=1,2\), choose a periodic profile \(g_{ij}\) equal to \(s-c_{ij}^-\) on a neighborhood of the coordinate range of the four centered shear paths, and require \[\left(\int\sigma_i\right)\left(\int g_{ij}\right)=0.\] Thus the shear selector \(\sigma_i\) may have nonzero mean when both linear profiles have zero mean. Usually one can take \(\sigma_i=\tau_i\). The derivative, odd-profile, and translated-copy recipes above provide these choices; their compensating parts must miss the other tracked rectangles or private-height plateaus as specified there.

For the chronological shear list (2), the seven spatial fields are \[\begin{align*} W_1&=e_z\sum_i(z_i-z_0)A_i^-(x,y),\\ W_2&=e_y\sum_i(-r_i)\sigma_i(z)g_{i1}(x),& W_3&=e_x\sum_i(r_i^{-1}-1)\sigma_i(z)g_{i2}(y),\\ W_4&=e_y\sum_i\sigma_i(z)g_{i1}(x),& W_5&=e_x\sum_i(r_i-1)\sigma_i(z)g_{i2}(y),\\ W_6&=\sum_i\tau_i(z)(c_i^+-c_i^-,0),\\ W_7&=e_z\sum_i(z_0-z_i)A_i^+(x,y). \tag{5}\end{align*}\] At each stage all branch contributions have the same direction of motion, except in \(W_6\), whose two horizontal components depend only on height. Consequently each entire stage is transverse, even where profile supports overlap. The stated integral conditions give zero spatial mean to every summand and hence to each \(W_j\).

To turn the stages into smooth motion, choose successive closed rational intervals \([a_j,b_j]\) with \(0<a_j<b_j<1\) and positive gaps, and set \[ \beta_j(t)=\frac1{b_j-a_j} H'\!\left(\frac{t-a_j}{b_j-a_j}\right),\qquad U(t,x)=\sum_j\beta_j(t)W_j(x)\quad(0\le t\le1). \tag{6}\] The step function \(H\) defined above is nondecreasing, so each pulse is nonnegative, has integral one, and vanishes to every order at its endpoints. The finite list repeats smoothly with period one and zero collars at integer times. This convention applies to any finite number of repeated stages below. The same pulse formula is used on a separately specified finite loading interval. Its derivatives and their bounds are effective by the same flat-profile estimates. Distinct stages have disjoint temporal supports; branch contributions within one stage act simultaneously. Thus the summed velocity still has zero self-advection, and Lemma 3 applies.

Here is the exact path represented by these fields. Starting from \((y,z_0)\) with \(y\in P_i\), \(W_1\) lifts the point to \((y,z_i)\). Only the \(i\)th height selectors act during the next five stages. The four shears send the offset \(y-c_i^-\) to \(\operatorname{diag}(r_i,r_i^{-1})(y-c_i^-)\), still based at \(c_i^-\). The sixth stage translates that center to \(c_i^+\), and \(W_7\) lowers the resulting point of \(Q_i\) back to \(z_0\). Nonnegative pulse masses parameterize each partial shear or translation by a number in \([0,1]\). The excursion estimates above therefore verify the linear plateaus on the full closed source rectangle before ODE uniqueness identifies these formulas with the actual trajectories. Positive plateau margins also permit a sufficiently small neighborhood of that rectangle and coding height; its size must be chosen from the margins and the affine partial maps. The bounds for points on the original rectangle are not automatically bounds for that neighborhood.

The other four-shear lists replace \(W_2,\ldots,W_5\) by their listed coefficients and directions, using their own excursion estimates. Splitting \(W_6\) into an \(x\)-translation and a \(y\)-translation gives the eight-stage version. If extra transverse controls are used, the first is one on the current \(y\)-interval and the second on the target \(x\)-interval already reached; both retain the private-height selector. Their scalar coefficients must have zero integral, supplied by either factor or by a correction supported off every tracked path. Remote centers and routes with an extra translation are specified separately below: their endpoint maps use the same principles, while their intermediate paths require the bounds proved for those routes.

A digit interface for full rule domains

A relative tape is a bi-infinite sequence \(T=(T_j)_{j\in\mathbb Z}\) whose index zero is the current head position. If an edit produces \(\widetilde T\), displacement \(d\) gives \(T'_j=\widetilde T_{j+d}\). A full cylinder fixes a mode and finitely many tape letters, with every other letter arbitrary. This convention specifies the domains on which we will prove local inverses.

Let an alphabet of size \(m\) have distinct digits separated by at least two in base \(B\), with all digits in \(\{0,\ldots,B-2\}\). Write \[C(a_1a_2\cdots)=\sum_{j\ge1}d(a_j)B^{-j},\qquad I(v)=\left[\sum_{j=1}^{|v|}d(v_j)B^{-j}, \sum_{j=1}^{|v|}d(v_j)B^{-j}+B^{-|v|}\right].\] The two coordinates of a relative tape are \(C(T_{-1}T_{-2}\cdots)\) and \(C(T_0T_1\cdots)\). Place the modes in pairwise positively separated translated squares of common side length \(\ell\), and use these two codes as the coordinates within each square. This is the positional-coding principle of Moore [9]; the following lemma records the full-rectangle and separation properties used here.

For a single-cell table there is a shorter sufficient inverse test: the target mode determines the incoming displacement, and the target mode together with the written letter determines the old mode and old letter. One first reverses the displacement, then the unique write. Let \(T_a(v)=(d(a)+v)/B\) and \(I_a=T_a([0,1])\). Reading \(a\), writing \(b\), and moving by \(d\) gives the following complete rectangle formulas: \[ \begin{array}{c|c|c|c} d&\text{source}&(L',R')&\text{target}\\\hline 1 &[0,1]\times I_a &(T_b(L),T_a^{-1}(R))&I_b\times[0,1]\\ 0 &[0,1]\times I_a &(L,T_bT_a^{-1}(R))&[0,1]\times I_b\\ -1&I_c\times I_a &(T_c^{-1}(L),T_cT_bT_a^{-1}(R)) &[0,1]\times T_c(I_b). \end{array} \tag{7}\] Include the last row for every \(c\). Determinism separates sources. The inverse test separates target written letters; in the last row the pair \((c,b)\) separates the two-digit target intervals. If mode-square gaps are at least \(\ell\), all within-family gaps are at least \(\ell/B^2\) in one coordinate.

Recording before the simulated move

This recorder marks the old head, logs the instruction, and only then makes the simulated move. Let \(e\in\{-1,0,1\}\) remember the preceding move, let \(r=(q,e,a)\), and give its instruction the data \((q_r,b_r,d_r)\). Letters are \((a,h,m)\), with \(h\in\{E,F\}\sqcup\{r\}\) and \(m\in\{0,1\}\). The controls are \(M_{q,e},R_r,P_{q,d},L_{q,d}\). The following table uses \(h\ne F\): \[ \begin{array}{c|c|c|c|c} \text{source}&\text{read}&\text{written}&\text{move}&\text{target}\\\hline M_{q,e}&(a,h,0)&(b_r,h,1)&1&R_r\\ R_r&(c,h,0)&(c,h,0)&1&R_r\\ R_r&(c,F,0)&(c,r,0)&1&P_{q_r,d_r}\\ P_{q,d}&(c,E,0)&(c,F,0)&-1&L_{q,d}\\ L_{q,d}&(c,h,0)&(c,h,0)&-1&L_{q,d}\\ L_{q,d}&(c,h,1)&(c,h,0)&d&M_{q,d}. \end{array} \tag{8}\] Only nonhalting \(q\) have a first row. Initialize the frontier at absolute cell \(c_0\), where either \(c_0=1\) or \(c_0=2\), and start in \(M_{q_0,0}\). At the \(n\)th main visit the head is the machine head \(h_n\), the records occupy \(c_0,\ldots,c_0+n-1\), and the frontier is \(J_n=c_0+n\). Since \(h_n\le n<J_n\), the first row is defined. It marks \(h_n\), the right scan logs at \(J_n\), the placement row moves the frontier to \(J_n+1\), and the left scan returns to the unique mark. The last row makes the original move. This is exactly \(2(J_n-h_n)+3\) local steps, a finite positive number; for \(c_0=2\) it lies between \(7\) and \(4n+7\). Thus main halts and original halts agree, with no intervening main visit.

This table also passes the inverse test on arbitrary tapes. Into \(R_r\), the written mark distinguishes entry from its loop and \(r\) recovers the old work symbol and control. Into \(P\), the written record identifies the predecessor. Into \(L\), a written frontier distinguishes placement from scanning. Into \(M_{q,d}\), the sole predecessor restores a mark and has known displacement \(d\). All other tracks are retained. Incoming displacements are respectively \(1,1,-1,d\), proving the test.

The choices \(c_0=1\) and \(c_0=2\) have the same entire rule table but different initialized histories. A permanent origin bit may be added and copied by every rule. When \(c_0=1\) without such a bit, the leftmost nonempty history cell remains absolute cell one and also determines absolute indexing. These are precise ways of recovering the head position, rather than additional assumptions on arbitrary tapes.

Writing the input inside the repeated rule table

A one-time loader gives eventual periodicity. To keep periodicity from time zero, the loading operation can instead be a distinguished local branch. Here is a version starting from an entirely blank tape. Let \(r=(q,e)\), with \(e\in\{-1,0,1,\star\}\); only initialization uses \(\star\). For a nonhalt instruction put \(k=(r,a)\), \(s(k)=(q',d)\). Letters are \((a,m,g)\) with \(g\in\{E,P\}\sqcup\{k\}\); modes are \(I,C_r,S_k,B_s\).

1. In \(I\) require cells \(0,\ldots,R_0-1\) fully blank, where \(R_0=\max(2,|w|)\). Write the input and a history pointer at cell one, and enter \(C_{(q_0,\star)}\) without moving.

2. In nonhalt \(C_r\) require \(m_0=0,g_0\ne P\); write the instruction symbol, mark cell zero, move right into \(S_k\).

3. In \(S_k\) require \(m_0=0\). Scan right if \(g_0\ne P\); if \(g_0=P,g_1=E\), write \(k,P\), move left into \(B_{s(k)}\).

4. In \(B_s\) require \(g_1\ne P\). At \(m_0=0\) scan left; at \(m_0=1\) erase the mark, move by the displacement in \(s\), and enter \(C_s\).

The star tag distinguishes initialization from every other arrival in a primary mode; it restores exactly the required blank block. At an \(S_k\) output, the mark at \(-1\) distinguishes entry and scanning. At a \(B_s\) output, the pointer at two distinguishes turning and scanning; the record at one identifies \(k\). An ordinary primary output has a specified return state. These are full-domain inverse tests. At the main visit after \(n\) steps, the frontier is \(n+1>h_n\) and the records occupy \(1,\ldots,n\). The old-head shuttle therefore takes \(2(n-h_n)+3\) rules, including the final erase-and-move rule. An initially halting machine is reached immediately after the initializer. The ordinary rules are resolved on \([-1,1]\), the initializer on \([0,R_0-1]\); Lemma 4 applies to both.

Neighbor guards and a persistent head

Here the history append writes two adjacent cells at once. This saves a local step, but the inverse now needs a neighbor guard.

A remote center and positive thickness

The persistent-head recorder of Section 5.2 has an initial mode with no incoming rule. Place its mode squares first in unscaled coordinates: the first origin is \(2j\) for a main halt and \(-2j\) for every other mode, with distinct positive indices \(j\); the second origin and coding height are zero. Use odd digits in base \(B\) and full triples. The rectangle gaps are at least \(B^{-3}\), and half-widths are at most \(1/2\). Let \(N\) be the number of ordinary branches plus one, let \(O_0\) be the largest absolute mode origin, and set \[K=(1+B^2)^4,\qquad R=10N+O_0+K+10,\qquad L=64R.\] The loading branch is the translation of the square of half-width \(1/(4B)\) centered at \((-4R,0)\) to the congruent square centered at the rational initial code. Odd-digit codes have coordinate margins at least \(1/(B-1)\), so its target fits inside the mode square. That square has no incoming branch. Both augmented rectangle families are consequently separated.

Choose heights \(Z_i=10i\), source and target collars \(1/(4B^3)\), and height cutoffs \(g_i\) equal to one on \([Z_i-1,Z_i+1]\), supported within a further unit collar. The common center is \(c^\circ=(-2R,0)\). Use eight motions: lift by \(Z_i\), translate the center to \(c^\circ\), perform (3) there, translate to the target center, and lower by \(Z_i\). The central linear profiles are \[(X+2R)\chi_{[-K,K],1}(X+2R)g_i(Z),\qquad Y\chi_{[-K,K],1}(Y)g_i(Z).\] All scalar profiles have compact support in \((-L/8,L/8)\) in their dependent coordinates. For a scalar profile \(a\) depending on \(d\) coordinates define \[ a^\#(v)=\sum_{m\in\mathbb Z^d} \left(a(v-Lm)-a(v-Lm-(L/2)e_1)\right). \tag{9}\] This is \(L\)-periodic and has zero mean. In the central cube it agrees with \(a\), since every noncentral or negatively translated copy is disjoint there. Replace every scalar profile by this correction.

For every starting height \(z\in[-1,1]\), lifting puts the point in \([Z_i-1,Z_i+1]\). The four-shear offset bound is \(K/2\), with partial coefficients included, and the two translations keep offsets at most \(1/2\). All paths lie in \((-8R,8R)^3\); every controlling profile remains on its intended plateau and all correction terms vanish. The time-one map is therefore the specified reciprocal affine map on the entire \(P_i\times[-1,1]\), with the height unchanged. This is a neighborhood statement, not merely an assertion on the coding sheet.

Let \(W\) denote the resulting \(L\)-periodic spatial field and \(a=(1/2,1/2,1/2)\). At the prescribed unit-torus viscosity \(\nu\), define \[U(t,x)=L^{-1}W(t,L(x-a)),\qquad f(t,x)=\bigl(L^{-1}W_t-\nu L\Delta W\bigr)(t,L(x-a)).\] The factor of the Laplacian is \(L\). These formulas directly give zero pressure, mean-zero solenoidal forcing, and the same time period. The loading center becomes the fixed label \((7/16,1/2,1/2)\). During a nonhalting transition the endpoint centers are at most \(-1\) in first coordinate. Translation paths have \(X\le-1+1/2<0\); central shear paths have \(X\le-2R+K<0\). Loading obeys the same bound, and no path crosses the chart seam. A halt endpoint has positive first coordinate. Thus the fixed event on the unit torus is \(1/2<x<1\), and the loader is part of the period repeated from zero. The locally finite sum (9) is effectively evaluated from known supports and a bounded coordinate enclosure.

Moving the head and retaining the origin

The persistent map admits a distinct five-rule form. Normalize to one halt state and a different, fresh nonhalting start. Use a return control \(B_q\) for every \(q\), including that start; this version does not impose the preceding no-incoming-start restriction. Add an origin bit, fixed at absolute zero and never edited, and combine the first two motions: at \(M_q\), perform the same guarded write and mark move but shift by \(d_i+1\) into \(O_i\). The \(O_i\) scan and append and the \(B_q\) return and exit are the preceding \(R_i,L_q\) rules. An \(O_i\) output from the main rule has mark one at index \(-1\); a scan output has mark zero there. The append/return distinction is still the pointer at two. This proves the full inverse, including the possible shifts zero and two. At a main checkpoint the history end \(J\) satisfies \(J\ge h+2\). After the work move to \(h'=h+d_i\), the working cursor is \(h'+1\le J\). There are \(J-1-h'\) outward and the same number of backward scan steps, so one machine transition uses \(2(J-h')+1\) rules. The origin flag recovers absolute indexing independently of the history. Resolving on \([-2,1]\) gives prefix lengths \(2,2\) and target lengths \(2+d,2-d\) for each actual working shift \(d\in\{-1,0,1,2\}\); an empty target prefix when \(d=2\) is allowed in Lemma 4.

The preceding three guarded recorders use complete triples on \([-1,1]\), with target prefix lengths \(1+d,2-d\). Thus all four maps provide the full closed rectangle interface. The initialized marker invariant proves halting equivalence; the separate inverse proofs prove geometric separation.

Onto clocks and the exact derivative estimates

Slow forcing must still complete every finite logical step. The relevant condition is that the new clock maps the nonnegative half-line onto itself. The following elementary calculation applies to our transverse pulse templates, including their finite loading intervals.

The force obtained after slowing is a different prescription from the unslowed periodic force. In particular these conclusions do not combine decay with physical period one. They also do not imply decay faster than every power of time for an arbitrary repeated processor: a speed bounded by \(C(1+t)^{-2}\) has only finite accumulated logical time. The same obstruction appears in the viscous example of Cardona–Miranda–Peralta-Salas–Presas [3]: their exponentially damped velocity traverses only a finite interval of the underlying stationary trajectory. The onto condition above ensures that slowing preserves every finite stage of the computation.

A separate loader and changes of clock

The following constructions keep the repeated processor independent of the input. A horizontal transverse pulse depending only on height first sends a fixed particle to the rational initial code. Its height profile has mean zero and value one at the coding height. The loader occupies \([0,1]\) and the processor starts afterward.

For the delayed-move table (8) with frontier at two, let \(\ell=1/(32(N+1)(B+2)^4)\) and place mode \(s\) at \((\xi_s,1/4+2\ell i(s))\), where \(1\le i(s)\le N\) and \(\xi_s=1/4\) or \(5/8\) according to halting status. Use derivative rectangle masks, height selectors \(p_i=\zeta_i\) and \(q_i=\partial_z((z-z_i)\zeta_i)\), and mean-zero linear profiles \(\partial_s(\tfrac12(s-c)^2\chi(s))\), with \(z_i=1/2+i/(4(J+1))\). Here \(p_i\) need not have zero mean: the linear profiles provide it in the four shear phases. The seven phases (4) have excursion at most \(1/(64(N+1))\). Load from \((1/4,1/4,1/4)\), using the same derivative height recipe at height \(1/4\). In a nonhalting run both the loader and every processor path have \(x<1/2\); terminal codes lie near \(5/8\). The observer is \(1/2<x<7/8\). The fourth-root clock gives \[\|\partial_t^kD_x^\alpha u(t)\|_\infty+ \|\partial_t^kD_x^\alpha f(t)\|_\infty =O_{k,\alpha}((1+t)^{-3/4}),\] for every fixed \(k,\alpha\), by Proposition 6. The codes occur at physical times \((2+j)^4-1\).

The move-first recording input

For an instruction \(r=(q,a)\) with data \((q_r,b_r,d_r)\), the recorder of [10] is written here with its frontier symbol \(P\) renamed \(F\) and its placement controls \(F_q\) renamed \(P_q\); all other controls and guards are unchanged. It uses letters \((a,h,m)\), \(h\in\{E,F\}\sqcup\{[r]\}\) and \(m\in\{0,1\}\), with modes \(S_q,A_r,R_r,P_q,L_q\). Here is its complete table; every occurrence of \(h\) requires \(h\ne F\): \[\begin{array}{c|c|c|c|c} S_q&(a,h,0)&(b_r,h,0)&d_r&A_r\\ A_r&(c,h,0)&(c,h,1)&1&R_r\\ R_r&(c,h,0)&(c,h,0)&1&R_r\\ R_r&(c,F,0)&(c,[r],0)&1&P_{q_r}\\ P_q&(c,E,0)&(c,F,0)&-1&L_q\\ L_q&(c,h,0)&(c,h,0)&-1&L_q\\ L_q&(c,h,1)&(c,h,0)&0&S_q. \end{array}\] There are no first-row instructions for halting \(q\). The cited lemma proves that the target mode determines the incoming move, and the target mode with the written complete letter determines the unique predecessor. Initialize at the original machine head with frontier at any fixed \(r_0\ge2\), empty history elsewhere, and zero marks everywhere. After \(n\) machine steps the frontier is \(r_0+n\); the work state, tape and head are recovered at the \(S\)-visits. If the next machine head is \(h'\), its next checkpoint takes exactly \(2(r_0+n-h')+4\) local transitions. In particular these counts are finite and positive. Halting and indefinitely defined nonhalting runs are preserved. We use \(r_0=2\) below.

Section 5.1 of [10] proves two extensions on the whole partial domain: toggle the current mark only in ready states to obtain the marker-preserving version, or copy an arbitrary passive track at every write. The latter permits a permanent origin flag. We specify the two changed rows again when the whole-box application uses the former. Every other recorder in this paper has its own complete inverse proof.

Eight phases about the center of the chart

Use the seven-row move-first recorder of Section 6.3 with one halt label. For its odd-digit alphabet let the base be \(D=2m+1\). If there are \(n\) nonterminal modes, set \(\lambda=1/(16(n+1))\), use common second origin \(3/8\), nonterminal first origins \(1/4+2j\lambda\) (\(0\le j<n\)), and halt origin \(5/8\). Refine every single-cell rule by its left letter, including right and stay moves. Its source then has side lengths \(\lambda/D\) in both directions. Formula (7) gives target rectangles; the right-move target is \(T_b(I_c)\times[0,1]\), the stay target is \(I_c\times I_b\), and the left-move target is \([0,1]\times T_c(I_b)\). The target separation follows from the incoming rule and then the extra left letter. Both families have gaps at least \(\lambda/D^2\).

Let branch \(i\) have centers \(c_i,c_i'\) and reciprocal factor \(\mu_i\). Number the \(J\) branches by \(j_i=0,\ldots,J-1\), put \(B_0=\max(2,J+1)\), and use coding height zero. Source and target cutoffs with collars \(\lambda/(8D^2)\) have second-coordinate supports in \((1/4,1/2)\). Therefore \[w^\pm(x,y)=\sum_i j_i\bigl(\chi_i^\pm(x,y) -\chi_i^\pm(x,y-1/2)\bigr)\] have zero mean and value \(j_i\) on the appropriate rectangle. For every \(j<B_0\), choose a periodic cutoff \(P_j\) equal to one within radius \(1/(8B_0)\) of \(j/B_0\), supported within radius \(1/(4B_0)\), and set \(G_j(z)=P_j(z)-P_j(z-1/(2B_0))\). Then \(G_j\) has zero mean and \(G_j(k/B_0)=\delta_{jk}\), including the height at the coordinate seam.

Put \(c_*=(1/2,1/2)\) and let \(h(s)=(s-1/2)\chi(s)\) be a periodic profile with \(\chi=1\) on \([3/8,5/8]\) and support in \([1/4,3/4]\). At height \(j_i/B_0\), perform these eight motions: lift by \(j_i/B_0\), translate by \(c_*-c_i\), apply \[X(1),\quad Y(\mu_i^{-1}-1),\quad X(-\mu_i),\quad Y(-(\mu_i^{-1}-1)/\mu_i),\] translate by \(c_i'-c_*\), and lower by \(j_i/B_0\). The lift and lower use \(\pm B_0^{-1}w^\pm e_z\); translations use \(G_{j_i}(z)\); each central shear uses the same selector and the appropriate profile \(h\). Thus all fields have zero mean and zero self-advection.

To verify the plateaus, let the initial centered offsets be \((\xi,\eta)\), with \(|\xi|,|\eta|\le\lambda/(2D)\). The successive endpoints are \[(\xi+\eta,\eta),\quad (\xi+\eta,(\mu^{-1}-1)\xi+\mu^{-1}\eta),\quad (\mu\xi,(\mu^{-1}-1)\xi+\mu^{-1}\eta),\quad (\mu\xi,\mu^{-1}\eta).\] Since \(\mu,\mu^{-1}\le D\), all second deviations are at most \(\lambda\) and all first deviations at most \(\lambda/2\); partial shears interpolate between these endpoints. The profiles remain linear, and the complete endpoint is the prescribed target point at height zero.

Load the fixed particle \((1/2,1/2,0)\) to the rational initial code by a horizontal pulse with height selector \(G_0\). In a nonhalting run all endpoint centers have first coordinate in \([1/4,3/8]\). During the centered shears \(|x-1/2|\le\lambda/2\); translations and loading have the same upper bound. Hence \(x\le1/2+\lambda/2<9/16\) throughout. A halt code lies in \([5/8,5/8+\lambda]\subset(9/16,3/4)\). The fixed event is therefore \(9/16<x<3/4\). The cube-root clock \(g(t)=(1+t)^{1/3}-1\) gives the samples \((2+j)^3-1\) and, by Proposition 6, for every \(k,\alpha\), \[\|\partial_t^kD_x^\alpha u(t)\|_\infty+ \|\partial_t^kD_x^\alpha f(t)\|_\infty =O_{k,\alpha}((1+t)^{-2/3}).\] Every such derivative is square integrable in time in supremum norm.

Keeping the physical scale of a side-ten construction

The seven-row move-first table also admits a useful larger chart. Normalize the original machine to one halt label before constructing this table, and let \(\Gamma\) be its augmented tape alphabet. Add a copied origin bit at absolute cell zero, put the frontier at two, and retain the table’s guards on every other track. The incoming displacement and written-letter inverse are unchanged. If there are \(m\) modes, set \(r=1/(4m)\), put nonhalt origins at \((2+2rj,2)\) and the halt origin at \((6,2)\), and use coding height two in \(\mathbb T_{10}^3\). Odd digits in base \(K=2|\Gamma|+1\) and (7) give gaps at least \(r/K^2\). Use rectangle collars \(d=r/(4K^2)\) and subtract their copies shifted by four in \(y\). Use processing heights \(Z_i=4+i/(J+1)\) with collar \(1/(4(J+1))\), subtracting copies shifted by three in \(z\). The linear profile around a center \(c\) is \((s-c)\chi_{[c-1/2,c+1/2],1/2}(s)\).

Apply (2) between lift, translation, and lower. The source half-widths \(\alpha,\beta\) satisfy \[\alpha,\beta,a\alpha,\beta/a\le r/2.\] Its second intermediate horizontal coordinate is \(a\xi+(a^{-1}-1)\eta\); its modulus is at most \(r\), since \(|a^{-1}-1|\beta=|\beta/a-\beta|\le r/2\). All other intermediate deviations are at most \(r\) as well. Hence every profile is linear on the required paths. A mean-zero height pulse at height two loads \((2,2,2)\) to the rational initial code. A nonhalting path stays in \(0<x<5\), while halt codes have \(6\le x\le6+r\). The fixed event is \(5<x<8\). This is a side-ten statement at the given physical viscosity \(\nu\).

For completeness, a general scale change from side ten with viscosity \(\nu_{10}\) is \[ \begin{aligned} v(s,y)&=\frac T{10}U(Ts,10y),& p(s,y)&=\frac{T^2}{100}P(Ts,10y),\\ f(s,y)&=\frac{T^2}{10}F(Ts,10y),& \nu_1&=\frac T{100}\nu_{10}. \end{aligned} \tag{10}\] Every term of the equation has factor \(T^2/10\). With \(T=1\) the label becomes \((1/5,1/5,1/5)\) and the event \(1/2<x<4/5\); to obtain a prescribed unit-torus viscosity \(\nu\) one must start with \(\nu_{10}=100\nu\). Keeping the viscosity unchanged instead requires \(T=100\) and changes the time period. These are coordinate transformations with their stated physical parameters.

Sequential parking of whole boxes

We next protect every point in a box of fixed positive thickness. Let \(J_0=[-1/2,1/2]\). Suppose closed rational rectangles \(P_i,Q_i\) are separately pairwise disjoint and have positive side lengths at most one. Their prescribed center-to-center affine maps \(F_i\) satisfy \(F_i(P_i)=Q_i\) and have reciprocal factors \(\operatorname{diag}(a_i,a_i^{-1})\), \(a_i>0\) rational. The box map fixes the third coordinate on \(P_i\times J_0\). There is no requirement that a source miss another target.

Normalize the original machine to one halt label before constructing its recorder. To apply the lemma, take the marker-preserving version of that seven-row recorder. Explicitly, change its first ready rule to read mark one and write mark zero, and its last return rule to write mark one; initialize one mark at the head. All intermediate rows are unchanged. The written symbol still distinguishes every incoming case; at a ready checkpoint the unique mark is retained rather than erased. The first row removes it, the next marks the new head, and the return retains it. This proves the same full-domain inverse and initialized simulation directly. Use odd digits and (7), and place its unit mode squares at \((\alpha_s,0)\), where \(\alpha_h=1\) and other \(\alpha_s=-2j\), \(j\ge1\). Thicken every branch by \(J_0\). The initial code \(y_{\rm in}\) is rational and has height zero. Nonhalting boxes lie wholly in \(x<0\); halting codes lie in \(1<x<2\).

Apply Lemma 7 to the stepping list, and separately to the single loading translation from the cube \([-1/2,1/2]^3\) to the congruent cube centered at \(y_{\rm in}\). If there are \(N\) modes and \(l\) stepping branches, take \(R_*=100+2N+4\max(1,l)\). All paths and bounded transverse supports lie strictly in \((-R_*,R_*)\) in each relevant coordinate: the original rectangles lie in \([-2N-1,3]\times[-1,2]\); parking adds at most \(5+3\max(1,l)\) to the negative first extent; shearing adds at most two and a collar at most \(1/4\); heights are below \(4+1/2+1/4\). This also covers the loading cube.

Scale by \(\lambda=(8R_*)^{-1}\) into the unit torus centered at zero. For each pulse with transverse index set \(I\), form its scaled periodic field \(F(x)=\lambda v(x_I/\lambda)\) and replace it by \(F(x)-F(x+e_j/2)\), where \(j=\min I\). Its transverse support is inside \((-1/8,1/8)^I\); the translated term vanishes on every tracked path. The corrected pulse has zero mean and still has zero divergence and self-advection. Concatenate the loading interval and the repeated stepping interval and prescribe \(f=U_t-\nu\Delta U\). Lemma 3 gives the resulting smooth solution, pressure zero, and uniqueness in the stated classical class.

Write \(\kappa\) for this configuration code, \(\mathcal T\) for the recorder transition, and \(c_0\) for its initialized configuration. The fixed label is the torus origin, and the event is the positive half-circle \(0<x<1/2\). At time \(1+j\) its code is \([\lambda\kappa(\mathcal T^jc_0)]\). A halt gives a point in that event, including a terminal initial configuration. For a nonhalting computation the sign part of the lemma excludes the event during stepping. During loading, the origin is the center of its cube: it first has coordinate zero, then moves to a negative parking center, is fixed relative to that center by every shear, and translates to the negative initial code. Its first coordinate is nonpositive throughout. No scaled path leaves \((-1/8,1/8)^3\), so reduction modulo one introduces no spurious visit to the positive half-circle.

The unslowed force is periodic after loading, with its repeating part depending only on the machine table. A separate logarithmic choice uses Lemma 5; every mixed derivative of force is bounded and square integrable on space–time. It reaches code \(j\) at \(t=\exp(1+j)-1\). These claims apply to the displayed finite pulse prescription, whose construction never executes the simulated machine.

Sequential realization of arbitrary sheet maps

Take the data of Theorem 2: separately separated closed rational rectangles \(P_i,Q_i\subset[2,3]^2\), with positive side lengths, and their prescribed positive diagonal affine maps \(F_i:P_i\to Q_i\). We realize every map on \(P_i\times\{2\}\) in one period. For an empty family use the zero field; henceforth let \(1\le i\le N\) with \(N>0\).

Write \(v_i,v_i'\) for the centers of \(P_i,Q_i\) and \(h_i,h_i'\) for their half-width vectors. All these half-widths are at most \(1/2\). Take one tenth of the minimum of one and the within-source and within-target sup-norm gaps as \(d>0\), omitting an empty list of gaps. Thus \(d\le1/10\). With an even cutoff \(\chi_h=1\) on \([-h-d/2,h+d/2]\) and zero outside \([-h-d,h+d]\), define side-ten periodic profiles \[C_{h,c}(s)=\sum_{k\in\mathbb Z}\chi_h(s-c-10k),\qquad D_{h,c}(s)=\sum_{k\in\mathbb Z}(s-c-10k)\chi_h(s-c-10k).\] The \(D\) profile has mean zero by oddness. Vertical transport over a footprint centered at \(v\) with half-widths \(h\) uses \[(z^+-z^-)e_z C_{h_1,v_1}(x) \bigl(C_{h_2,v_2}(y)-C_{h_2,v_2+4}(y)\bigr).\] Horizontal transport from center \(v^-\) to center \(v^+\) at height \(z\) uses \[((v^+-v^-),0)\bigl(C_{0,z}(x_3)-C_{0,z+3}(x_3)\bigr).\] These have zero mean and the indicated exact translations on plateaus.

A separator-stack application and its observer

We now supply rectangle data to the sequential construction. The recorder separates a finite left working word from its retained history. A prefix rule \((s;\alpha,\beta)\to(s';\bar\alpha,\bar\beta)\) replaces the two indicated finite stack prefixes and retains both arbitrary infinite tails. Normalize the original machine to one halt label, and let \(A\) be its tape alphabet, with blank \(b_0\). Use the stack alphabet \(A\sqcup\{\bot,\diamond\}\sqcup\{\eta_i\}\) and controls \(R_q,T_i,S_p\). At a ready control \(R_q\), the left stack has the form \(L=\ell\bot H\), with \(\ell\in A^*\) listing the working cells to the left of the head, nearest first, and implicit blanks beyond them. The separator \(\bot\) keeps these cells above the retained history \(H\); the right stack begins with the scanned cell. A temporary symbol \(\diamond\) marks the beginning of the updated right working word while \(T_i\) transfers the finite left word onto that stack above the marker. The control \(S_p\) then restores that word until \(\diamond\) is exposed and removed. A new symbol \(\eta_i\) retained below \(\bot\) records the instruction case. In the transfer and restore rules, \(c\) ranges over \(A\). Give each applicable case of an original instruction \((q,a)\mapsto(p,b,d)\) its own index \(i\), with \(p(i)=p\), and use \[ \begin{array}{c|l} d=0 &(R_q;\varnothing,a)\to(T_i;\varnothing,\diamond b),\\ d=1 &(R_q;\varnothing,a)\to(T_i;b,\diamond),\\ d=-1, c\in A &(R_q;c,a)\to(T_i;\varnothing,\diamond cb),\\ d=-1,\ \text{empty left}&(R_q;\bot,a)\to(T_i;\bot,\diamond b_0b). \end{array} \tag{13}\] Complete the table with \[ \begin{aligned} (T_i;c,\varnothing)&\to(T_i;\varnothing,c),& (T_i;\bot,\varnothing)&\to(S_{p(i)};\bot\eta_i,\varnothing),\\ (S_p;\varnothing,c)&\to(S_p;c,\varnothing),& (S_p;\varnothing,\diamond)&\to(R_p;\varnothing,\varnothing). \end{aligned} \tag{14}\] Initialize with \((R_{q_0},\bot b_0^\infty,wb_0^\infty)\). At a ready state with \(L=\ell\bot H\), the primary rule yields \((T_i,\ell'\bot H,\diamond Y')\) with the correct updated tape. The transfer moves \(\ell'\) right in reverse order, inserts \(\eta_i\) below \(\bot\), and the restore returns precisely those work symbols until \(\diamond\) is exposed and removed. The new ready configuration is \((R_{p(i)},\ell'\bot\eta_iH,Y')\), after \(2|\ell'|+3\) rules. This is finite and positive, including an empty left word. Its special left-move case exposes an implicit blank correctly. The retained instruction-case records also determine the displacement history if absolute indexing is desired.

The source families are disjoint by tested control and top symbol. Into \(T_i\), the unique primary arrival has right top \(\diamond\); loop arrivals have distinct real symbols. Into \(S_p\), transfer arrivals have left prefix \(\bot\eta_i\), while loops have real left top; record indices and symbols distinguish their respective cases. There is one incoming rule into each \(R_p\). This proves disjoint image cylinders on arbitrary tails and gives their inverses.

Use a common digit dictionary for controls and stack symbols. If there are \(m\) entries other than the halt control, let \(K=8(m+1)\), give them distinct digits \(2,4,\ldots,2m\), and give the halt control digit \(3K/4\). Set \(e(v)=\sum_jd(v_j)K^{-j}\) and \(I(v)=[e(v),e(v)+K^{-|v|}]\) for finite prefixes. The code is \[(s,L,Y)\longmapsto (2+e(sL),2+e(Y),2)\in\mathbb T_{10}^3.\] For a rule \((s;\alpha,\beta)\to(s';\bar\alpha,\bar\beta)\), its rectangles are \[P_i=(2+I(s\alpha))\times(2+I(\beta)),\qquad Q_i=(2+I(s'\bar\alpha))\times(2+I(\bar\beta)).\] Both families are separated in \([2,3]^2\). Their affine factors are \(K^{|\alpha|-|\bar\alpha|}\) and \(K^{|\beta|-|\bar\beta|}\); the control digit cancels in the first ratio. Nonhalt first coordinates are at most \(2+(2m+1)/K<9/4\), while halt codes are between \(11/4\) and \(3\). The initial code is rational by summing the blank tails.

These rectangles and maps satisfy every hypothesis of the sequential construction in Section 8.1. Load \((2,2,2)\) horizontally to the rational initial code during \([0,1]\), then repeat its schedule. The fixed event is \[5/2<x<7/2,\qquad 3/2<z<5/2.\] A halt code is in this set, including an initially halted input after loading. In a nonhalting run the loader has \(x<9/4\). During every later period both endpoint coordinates lie in \([2,9/4)\); (12) makes the low-height first coordinate either less than \(9/4\) or at least six. At other heights the event is excluded by its height condition. Thus there is no false intermediate hit even though an upper horizontal route may cross the observer’s projection. Idle collars make each integer-time halt witness persist on an interval.

Direct transverse forcing has zero pressure and is periodic after loading at the specified side-ten viscosity. The logarithmic choice of Lemma 5 instead has \(\|f(t)\|_\infty=O((1+t)^{-1})\), bounded mixed derivatives, and \(f\in L^2([0,\infty)\times\mathbb T_{10}^3)\) with the same event. This is a separate choice of forcing. Under the time-preserving scale change (10), the label is \((1/5,1/5,1/5)\) and the event is \(1/4<x<7/20\), \(3/20<z<1/4\), with the viscosity transformation stated there.

Prefix rules with retained instruction tags

The next recorder retains history as tags between the work symbols. Its rectangles will be arranged in a narrow strip so that a simultaneous construction can contract, translate, and expand all branches together. Let \(\Sigma\) be the machine alphabet, with blank \(\beta\). Replace a real tape symbol by \((a,o)\in\Sigma\times\{0,1\}\), where \(o\) marks the original cell zero and is preserved by every write. Write the enlarged transition as \((q,a)\mapsto(q',b,d)\), with \(d\in\{+,0,-\}\). Use controls \(M(q,e)\) for \(e\in\{\star,+,0,-\}\) and \(P_+(q),P_-(q)\). For every nonhalting main control \(s=M(q,e)\) and real symbol \(a\) introduce a distinct tag \(\gamma_{s,a}\). The stack alphabet \(\mathcal A\) is the disjoint union of these tags and the real symbols. At a main control \(M(q,e)\), erasing all tags gives the two tape halves: the left stack lists cells to the left of the head, nearest first, and the right stack begins with the scanned cell. The control index \(e\) records the preceding move, with \(\star\) reserved for initialization. The transfer controls \(P_+(q)\) and \(P_-(q)\) finish a right or left move by passing intervening history tags until the next real work symbol is available. Thus history occupies space in the stacks without being mistaken for a work cell.

A prefix rule \([s;\ell,r]\to[s';\ell',r']\) replaces the two displayed prefixes, retaining arbitrary infinite tails. For each main instruction use \[ [s;\varnothing,a]\longrightarrow \begin{cases} [P_+(q');b\gamma_{s,a},\varnothing],&d=+,\\ [P_-(q');\varnothing,b\gamma_{s,a}],&d=-,\\ [M(q',0);\varnothing,b\gamma_{s,a}],&d=0. \end{cases} \tag{15}\] For every tag \(g\) and real symbol \(c\) include \[ \begin{aligned} {}[P_+(q);\varnothing,g]&\to[P_+(q);g,\varnothing],& [P_+(q);\varnothing,c]&\to[M(q,+);\varnothing,c],\\ [P_-(q);g,\varnothing]&\to[P_-(q);\varnothing,g],& [P_-(q);c,\varnothing]&\to[M(q,-);\varnothing,c]. \end{aligned} \tag{16}\] Initialize in \(M(q_0,\star)\) with an unmarked blank left stack and the input followed by blanks on the right, marking the right top as the origin (a marked blank for empty input).

For a right move, the primary rule pushes the written symbol left and the transfer removes any tags above the next real right symbol. For a left move, it writes the new right top, removes tags from the left, and moves its next real symbol onto the right. A stay just writes and inserts its tag. Each original instruction adds exactly one tag; at every finite stage there are only finitely many tags to transfer. After \(j\) prior machine steps, the next step takes one rule for a stay or two plus the number of transferred tags for a nonzero move, hence at most \(j+3\) rules. If \(n_k\) is the rule count after \(k\) machine steps, then \[k\le n_k\le k(k-1)/2+3k.\] No intermediate control is terminal. The origin flag recovers the absolute head position: it is \(k\) when the mark is the \(k\)th real symbol of the left stack and \(1-k\) when it is the \(k\)th of the right.

Both rule-cylinder families are disjoint on arbitrary tails. Source controls and tested top symbols distinguish rules. Into \(P_+(q)\), primary arrivals have left prefix \(b\gamma_{s,a}\) beginning with a real symbol, while loops have a tag there. Distinct tags distinguish the primary arrivals. The analogous argument on the right applies to \(P_-(q)\). Exits into \(M(q,+)\) or \(M(q,-)\) have distinct real right top symbols, while arrivals into \(M(q,0)\) have distinct two-symbol prefixes \(b\gamma_{s,a}\). No rule enters a star-tagged main control. These tests also invert each rule; they do not assume that the tails encode an initialized computation.

Let \(K\) be the number of controls, set \(\lambda=1/(16(K+1))\), and enumerate nonhalting and halting controls separately from zero. Place their origins at \[(A_s,1/4),\qquad A_s=\xi_s+(2j_s+1)\lambda, \qquad \xi_s=\begin{cases}1/8,&s\text{ nonterminal},\\ 11/16,&s\text{ terminal}.\end{cases}\] Use odd digits in base \(B=2|\mathcal A|+1\) for the stack alphabet. A prefix \(w\) acts on its free tail parameter by \(F_w(t)=\sum_jd(w_j)B^{-j}+B^{-|w|}t\), with interval \(I(w)=F_w([0,1])\). Hence each prefix rule gives a positive diagonal map from \[(A_s+\lambda I(\ell))\times(1/4+\lambda I(r)) \quad\hbox{to}\quad (A_{s'}+\lambda I(\ell'))\times(1/4+\lambda I(r')).\] The factors are \(B^{|\ell|-|\ell'|}\) and \(B^{|r|-|r'|}\). At the first conflicting digit the closed intervals have a positive gap; the full-cylinder argument therefore gives separately disjoint closed rectangle families. It imposes no condition on a source intersecting a target. Constant-tail sums give the rational initial code.

Fifteen simultaneous phases

We use the rectangles just constructed in Section 8.3, on the unit torus. Both families are separately separated, every half-width is at most \(\lambda/2<1/2\), and all second coordinates lie in \([1/4,1/4+\lambda]\). These are the geometric properties needed for the simultaneous construction; source–target intersections remain allowed. Write the source and target rectangles as \(D_i=p_i+[-s_{i1},s_{i1}]\times[-s_{i2},s_{i2}]\) and \(T_i=q_i+[-t_{i1},t_{i1}]\times[-t_{i2},t_{i2}]\). Choose symmetric product cutoffs \(\chi_i^D,\chi_i^T\) on neighborhoods, separately disjoint, with collars smaller than \(1/32\). The original supports lie in the common strip \(1/4-1/32<y<1/4+\lambda+1/32\); their translates by \(\omega=(0,1/2)\) therefore miss all tracked rectangles and all untranslated supports. For \(N>0\) branches put \[z_\circ=1/2,\quad z_i=1/4+\frac{i}{2(N+1)},\quad d=\kappa=\frac1{16(N+1)},\quad \epsilon=\frac12\min_{i,j}\{s_{ij},t_{ij}\}.\] Let \(\zeta_i\) be symmetric about \(z_i\), one near \([z_i-d,z_i+d]\), with support in \((z_i-2d,z_i+2d)\). The closures of these supports are disjoint. Define \(G_i(z)=(z-z_i)\zeta_i(z)\). Both \(G_i\) and \(G_i'\) have zero mean, by oddness and differentiation, and are respectively \(z-z_i\) and one near the required height interval.

The essential operation changes one coordinate of one sheet. For centers \(o_i\), symmetric rectangle cutoffs \(\chi_i\), and factors \(\mu_i>0\), use three successive profiles \[ e_z\sum_i\kappa(y_j-o_{ij})\chi_i(y),\qquad e_j\sum_i\frac{\mu_i-1}{\kappa}G_i(z),\qquad -e_z\sum_i\frac{\kappa}{\mu_i}(y_j-o_{ij})\chi_i(y). \tag{17}\] Starting at height \(z_i\) with offset \(\xi\) in coordinate \(j\), these stages give height \(z_i+\kappa\xi\), horizontal offset \(\mu_i\xi\), and height \(z_i\). If \(|\xi|\le1/2\) and the horizontal segment remains in the cutoff rectangle, the height lies in the linear plateau and no other height profile acts. Every vertical profile has zero mean because its linear factor is odd under reflection about the rectangle center. Every profile is a transverse shear.

Now lift \(D_i\) from \(z_\circ\) to \(z_i\) using the mean-zero selector \(\chi_i^D(y)-\chi_i^D(y-\omega)\). Use (17) first in coordinate one and then coordinate two, with \(o_i=p_i\) and \(\mu_i=\epsilon/s_{ij}\). This takes six stages and contracts every source to \(p_i+[-\epsilon,\epsilon]^2\). Translate by \((q_i-p_i)G_i'(z)\). Next apply the same two triples with \(o_i=q_i\), \(\chi_i=\chi_i^T\), and \(\mu_i=t_{ij}/\epsilon\). Finally lower using the target selector difference. There are \(1+6+1+6+1=15\) stages. Assign them disjoint pulse supports inside \([1/4,3/4]\).

During contraction the entire projection stays in its source rectangle; during expansion it stays in its target rectangle. Thus all rectangle plateaus used in (17) remain valid, including when the other coordinate has already changed. The translation at height \(z_i\) is exactly by \(q_i-p_i\). Consequently the period map is \[ (y,z_\circ)\longmapsto \left(q_i+\operatorname{diag}(t_{i1}/s_{i1},t_{i2}/s_{i2}) (y-p_i),z_\circ\right),\qquad y\in D_i. \tag{18}\] For an empty rule list use zero. As in the sequential construction, the normal compensation follows directly from the three shears: in the linear core a triple acts on a small initial height deviation \(\zeta\) by \[(\xi,\zeta)\longmapsto \left(\mu_i\xi+\frac{\mu_i-1}{\kappa}\zeta, \frac{\zeta}{\mu_i}\right).\] Its determinant is one, while its restriction to \(\zeta=0\) multiplies the horizontal coordinate by \(\mu_i\). Figure 1 shows this three-shear mechanism on a one-dimensional slice. The sheet statement (18) therefore permits planar area change without claiming that a thick box has that same diagonal action.

Load the fixed label \((1/2,1/2,1/2)\) to the rational initial code at height \(z_\circ\). A profile \(G_\circ'(z)\) supplies a mean-zero horizontal loader if \(G_\circ\) equals \(z-1/2\) near \(1/2\) and vanishes near the coordinate endpoints. Run it once during \([0,1]\), then repeat the fifteen-stage processor. At \(1+n\) the particle represents rule \(n\) by (18). Nonhalt source and target rectangles have first coordinates in \((1/8,1/4)\); contractions, expansions and the middle translation preserve that band. Loading follows a segment from \(1/2\) into that band. Thus a nonhalting run never enters \(5/8<x<7/8\). Terminal codes are in \((11/16,13/16)\) and do enter it, including after loading for an initial halt. The finite positive rule counts proved above show that no machine step is left indefinitely unfinished. Direct transverse forcing gives the smooth mean-zero solenoidal force, pressure zero, and periodicity after loading. The repeated processor depends only on the transition table; the input affects only the rational loading code.