Horizon joins and hinge images

An FPRD–Fold-and-Look research packet

Date: 2026-07-29
Research status: theorem packet, independently attacked and computationally regression-checked; not integrated into the public repository
Repository inspected: insight-forge/boxlab/fprd-lab, current main at 4f4fee67a42bb374e086f22b00788ce0c53fcfab
Source packet verified: SHA-256 d7a367915c6e0e89d0a107c3c39d4ff992f61bf2d9078c916112de3d4dcf3518

1. Executive determination

The passage from “works at every individual cut or exact horizon” to “one finite causal presentation works everywhere” is not governed by one compatibility condition. It has at least three independent seams:

1. horizon compatibility at one cut: the same complete cut snapshot and the same track layout must work for every possible future stopping time;
2. rooted stationarity across cuts: those common-horizon snapshots must arise from one initializer, legal one-symbol transitions, and one finite stationary role coloring; and
3. geometric realizability: if the presentation is physical origami, its relational configurations must additionally come from one continuous, nonduplicating signed-isometric sheet map with a valid global layer order.

The first seam already refutes the naïve uniform-lifting conjecture in a strong nonzero-track form.

Main causal result

There is an explicit family with $R$ independently addressable $d$-bit blocks such that every exact horizon has a one-track factorization with one snapshot, while a single layout that must serve all horizons has exact frontier

$$\boxed{\widehat\chi_k=2^{d(R-\min\{k,R\})}.}$$

For $R=d$, the one-track sewing tax is

$$\max_h\chi_1(h)=1, \qquad \widehat\chi_1=2^{d(d-1)}.$$

Embedding these stages into one total computable Boolean target gives a presentation-level lower bound. At cuts $m=4d^2$, every bare stationary online presentation with $s$ finite roles and $k$ trailing heads obeys

$$\boxed{\log_2 s+k\log_2(m+1)\ge d(d-k)} \qquad(k\le d).$$

Thus fixed $k$ forces $s=2^{\Omega(m)}$, while polynomially many roles force

$$k\ge \frac{\sqrt m}{2}-O(\log m).$$

Yet the same total target satisfies

$$\chi_2(F;c,h)\le2$$

at every exact cut and horizon. It is therefore a global counterexample: the entire marginal field stays uniformly bounded while no fixed finite root-accessible stationary role/head presentation exists.

This is not square length in disguise. Every selected marginal has $2^d$ distinct zero-track continuation rows, and one correctly placed track really collapses them to one snapshot. What fails is reuse of the same finite configuration over many incompatible future horizons.

Exact accessible image

For a pinned real-time trailing-head doctrine, a target has one accessible $(s,k)$-presentation exactly when its prefix tree admits a stationary rooted atlas: reachable head positions from the fixed root, legal hold/advance bonds on every edge, an $s$-coloring, and stationary transition/readout consistency. Every finite depth is a finite constraint problem, and solutions at all depths sew by König’s lemma.

This theorem is an exact doctrine-specific specialization of the existing accessible-certificate machinery. It also identifies why the semantic comparison field alone is insufficient: common layouts, root reachability, edge motions, and stationary colors are witnesses that marginal resolving cells do not determine.

Main origami results

For a pinned one-dimensional, one-fold finite-precision WFL fragment:

• at each fixed prepared-substrate instance, the exact coordinate-full terminal image is characterized by one global reflection center, interval closure, the hinge quotient metric, opposite orientations, one global over/under order per branch, and nonduplication;
• the Boolean semantic image is exactly the functions admitting a DNF of maximum term width at most two;
• a $T$-term reduced $2$-DNF has a one-hinge realization with $F=1$, depth $D=2$, global Look fan-in $T$, material $2T$, and $2T$ prepared sites in a uniform midpoint construction;
• NONPALINDROME on $2n$ bits has a one-fold construction, but exact DNF minimality forces $2n$ Look terms and $4n$ variable-literal sites;
• a family of individually legal terminal hinge records has one finite-role causal realization from a finite affine fold library exactly when it has an ultimately periodic admissible-template selector.

The power-of-two selector supplies the origami analogue of the sewing obstruction: fold count, depth, grid resolution, affine-library size, and branch degree stay constant at every horizon, but no finite-role stationary controller selects the required fold family.

The most important replacement

A natural proposed compatibility law—restriction of the terminal folded map from longer strips to shorter strips—is false even for the uniformly realizable midpoint fold. Appending material changes where old material lands after the terminal fold. The coherent objects are the reachable latent flat states plus one stationary terminal action schema, not a nested inverse system of already-folded terminal geometries.

Status of the claims

No novelty conclusion is drawn from the absence of an exact literature hit.

2. The pinned online doctrine

The following doctrine is narrow on purpose. It is the real-time leading-head/trailing-head fragment needed to make the cut geometry and accessibility obligations exact.

Let the input be $u=u_0\cdots u_{n-1}\in\Sigma^*$. A presentation has:

• one leading clock that consumes exactly one new symbol per round;
• $k$ labeled trailing heads on the already exposed prefix;
• a finite role set $Q=[s]$;
• all trailing heads placed at the first input cell when it appears;
• on each input round, every trailing head either holds or advances one cell;
• no late birth, leftward motion in absolute coordinates, teleportation, or random-access initial placement;
• observation of the symbols currently under the trailing heads, optionally augmented only by a fixed target- and prefix-independent local sensing map whose old-input support is included in the declared head cones;
• one stationary transition table depending on the current role, current observation, and the next input symbol; and
• one stationary readout table depending on the current role and current observation.

The controller is deterministic in this section. Relational physical outcomes are treated in Section 8.

At a cut after $m$ symbols, reflect an absolute old-input address $i$ by

$$\bar i=m-1-i.$$

If an absolute head holds or advances during the next $h$ rounds, the old symbols it can encounter map into

$$I_h(\bar i)=\{\max(0,\bar i-h),\ldots,\bar i\}.$$

Thus the doctrine is the reflected instance of the D30 cone geometry. It embeds into a standard one-way $(k+1)$-head automaton by treating the leading clock as a distinguished head forced to advance every round, while the $k$ charged trailing heads may hold or advance and may not outrun the exposed frontier. Only the trailing heads are charged in the displayed resource laws. In the classical definition, all heads begin at the first tape square and move one way under finite control; see Yao and Rivest, $k+1$ Heads Are Better Than $k$.

2.1 Stationary rooted atlases

For every nonempty prefix $u$, an $(s,k)$-atlas assigns

$$q(u)\in[s], \qquad p(u)\in\{0,\ldots,|u|-1\}^k.$$

Let

$$z(u)=\bigl(u[p_1(u)],\ldots,u[p_k(u)]\bigr)$$

with the declared finite sensing augmentation if one is present. The atlas laws are:

1. fixed root $$p(a)=(0,\ldots,0)\qquad(a\in\Sigma);$$
2. legal accessibility $$p(ua)-p(u)\in\{0,1\}^k;$$
3. stationary readout $$(q(u),z(u))=(q(v),z(v)) \Longrightarrow f(u)=f(v);$$
4. stationary transition for every $a\in\Sigma$, $$(q(u),z(u))=(q(v),z(v)) \Longrightarrow \bigl(q(ua),p(ua)-p(u)\bigr) = \bigl(q(va),p(va)-p(v)\bigr).$$

The empty prefix has one fixed left-marker observation and one initial role.

Theorem 2.1 — exact rooted image

A target $f:\Sigma^*\to B$ has a presentation in the pinned $(s,k)$-doctrine if and only if it has an $(s,k)$-stationary rooted atlas.

Proof

Run a presentation on every prefix and record its role and head positions. The initializer gives law 1, the permitted motions give law 2, and the two stationary tables give laws 3 and 4.

Conversely, on every attained tuple define

$$\omega(q,z)=f(u)$$

from any $u$ carrying $(q,z)$, and define

$$\delta(q,z,a)=\bigl(q(ua),p(ua)-p(u)\bigr).$$

Laws 3 and 4 make these definitions independent of the chosen witness prefix. Complete unattained transition entries by holding every head and choosing arbitrary roles/readouts. Hold is always legal in this doctrine, so this completion does not increase the budget. Induction from the fixed root recovers the atlas labels on every prefix, and the readout law proves correctness. $\square$

At a cut $m$, define fibers

$$X_{m,q,p} = \{u\in\Sigma^m:q(u)=q,\ p(u)=p\}.$$

After reflecting $p$ at the cut, every such fiber is simultaneously $A_h(\bar p)$-resolving for every continuation length $h$. Indeed, two prefixes in the same fiber that agree on the old symbols in the possible cones produce the same sequence of observations, moves, roles, and outputs under a common continuation.

This gives the exact target-level, witness-bearing image statement:

A target has a pinned $(s,k)$ lift exactly when it has a stationary rooted atlas. The atlas fibers are common-horizon resolving.

If a named, preassigned factorization family is required to be realized, add two explicit conditions: each atlas fiber must refine the named cell containing it, and its reflected head tuple must be one of that cell’s admitted layouts. Without a definition of these refinement/layout maps, “the family lifts” is underdetermined even though target realizability is exact.

The witness variables are not cosmetic. The block theorem below shows that one common layout cannot be reconstructed from the scalar marginal frontiers. Supplied-cut examples show that reachability cannot be reconstructed from a legal cut layout. Symmetric optimal partitions show that there need not be one canonical optimal layout to select.

2.2 Finite verification and sewing

For fixed $s,k,N$, the atlas laws on $\Sigma^{\le N}$ form a finite CSP. The variables are roles and legal positions at finitely many prefix nodes; all equality, motion, and output constraints are finite.

Let $\mathsf{Atlas}_f(N;s,k)$ be its solution set.

Theorem 2.2 — doctrine-specific accessible sewing

$$\boxed{ f\text{ has one global }(s,k)\text{-presentation} \iff \forall N\; \mathsf{Atlas}_f(N;s,k)\ne\varnothing. }$$

Proof

Restriction proves the forward implication. Normalize roles to $[s]$. Finite-depth atlas solutions form a finitely branching tree under restriction. If every level is nonempty, König’s lemma gives one compatible infinite branch. Its union satisfies every atlas law because each instance occurs at finite depth. Theorem 2.1 compiles the union. $\square$

This is compactness, but the content before compactness is doctrine-specific: the root, legal head bonds, observation equality, and stationary coloring are exactly the finite seams that must be witnessed.

3. Common-horizon causal factorization

Fix a cut $m$, a prefix set $U\subseteq\Sigma^m$, a maximum horizon $H$, and a Boolean target on pairs $uv$ with $u\in U$ and $|v|\le H$.

For a reflected $k$-track placement $p$, let

$$A_h(p)=\bigcup_{i=1}^k I_h(p_i)$$

be its old-input reach cone at exact horizon $h$.

Definition 3.1

A common-horizon $k$-cone factorization consists of:

• a finite complete-snapshot set $C$;
• a surjection $e:U\to C$;
• one placement $p(c)$ for each snapshot, independent of the future stopping time; and
• decoders $$d_{c,h}: \Sigma^{A_h(p(c))}\times\Sigma^h\to\{0,1\}, \qquad 0\le h\le H,$$

such that

$$f(uv) = d_{e(u),h} \bigl(u|_{A_h(p(e(u)))},v\bigr)$$

for all $u\in U$, $0\le h\le H$, and $v\in\Sigma^h$.

Write

$$\widehat\chi_k(f;U,H) = \min |C|.$$

Equivalently, every encoder cell must be $A_h(p)$-resolving for every $h\le H$ with one common $p$. This quantifier order is load-bearing:

$$\exists e\ \exists p\ \forall h\ \exists d_h,$$

not

$$\forall h\ \exists e_h\ \exists p_h\ \exists d_h.$$

Theorem 3.2 — finite common-horizon sewing

A common-horizon factorization with $C$ snapshots and $k$ tracks exists if and only if one cut-initialized deterministic strict-layered $k$-track continuation protocol, using those snapshots and layouts, computes the target at every depth $h\le H$.

The theorem preserves $C$ and $k$. As in the earlier fixed-cut converse, finite downstream layered state is not charged.

Proof

A protocol begins from one complete snapshot and layout before it knows when the continuation will terminate. Apply the fixed-cut cone induction at every depth to obtain the simultaneous resolving laws.

Conversely, initialize from $e(u)$ and $p(e(u))$. During the next $H$ rounds, sweep each track through its cone and cache the old symbols and continuation symbols read so far. At depth $h$, apply $d_{e(u),h}$. Because $H$ is finite, this cache and the family of decoders compile into a finite layered controller. $\square$

The theorem is deliberately cut-initialized. It does not assert that the encoder or layout is reachable from the online root. That is the separate atlas seam.

4. The exact block-switching obstruction

Let $d=2^r$ and $R\ge1$. Choose distinct horizons

$$h_t=d+t,\qquad 0\le t<R, \qquad H=d+R-1.$$

Choose a cut length $m$ and $R$ disjoint blocks $B_t\subseteq \{0,\ldots,m-1\}$ of $d$ old-input cells. Require the distance between the closest cells of any two blocks to exceed $H$. The theorem’s prefix domain is the fixed-padding subcube

$$U = \left\{ u\in\{0,1\}^m: u_i=0\text{ for }i\notin\bigcup_tB_t \right\}.$$

Thus exactly the $dR$ block positions vary independently. Fixing the padding is load-bearing in the row-count proof; visible padding may not be used as an uncharged side channel.

At horizon $h_t$, interpret the first $r$ continuation bits as $j\in\{0,\ldots,d-1\}$, and set

$$f(u,v)=u[B_t[j]].$$

At every other horizon through $H$, set the output to zero.

Proposition 4.1 — cheap exact marginals

For every special horizon $h_t$,

$$\chi_0(f;U,h_t)=2^d, \qquad \chi_1(f;U,h_t)=1.$$

The $2^d$ block assignments give distinct continuation rows when no old symbol can be read. One track rooted at the appropriate block endpoint has a cone containing the whole $d$-bit block because $h_t\ge d-1$.

Thus the track is substantive: removing it costs $2^d$ snapshots.

Theorem 4.2 — exact common-horizon Pareto frontier

For every $k\ge0$,

$$\boxed{ \widehat\chi_k(f;U,H) = 2^{d(R-\min\{k,R\})}. }$$

Equivalently, for $0\le k\le R$,

$$\widehat\chi_k=2^{d(R-k)},$$

and $R$ tracks reduce the snapshot count to one.

Lower bound

Fix one complete snapshot cell $X_c=e^{-1}(c)$ and its layout. Since the cones are nested in $h$,

$$\bigcup_{h\le H}A_h(p(c))=A_H(p(c)).$$

The strict spacing condition prevents one $H$-cone from meeting two data blocks. Hence $k$ cones reveal at most $d\min(k,R)$ of the $dR$ independent data bits.

The tuple of all special-horizon future rows determines all $dR$ bits: for each $t,j$, choose a suffix selecting $j$ at horizon $h_t$. If two prefixes in $X_c$ agree on every visible data bit, every decoder receives the same old-input restriction, so their joint row tuples agree. Therefore the prefixes must have the same full block assignment.

The visible projection is consequently injective on $X_c$, giving

$$|X_c|\le 2^{d\min(k,R)}.$$

There are $2^{dR}$ block assignments, so

$$2^{dR} = \sum_c|X_c| \le |C|\,2^{d\min(k,R)}.$$

This proves the lower bound even when different snapshot cells choose different layouts; the layout is part of the charged complete snapshot.

Upper bound

Choose $\min(k,R)$ blocks and permanently place one track at the endpoint of each. Index snapshots by the complete assignments to the untracked blocks. This uses

$$2^{d(R-\min(k,R))}$$

snapshots. At horizon $h_t$, retrieve the selected bit from the swept block if it is tracked and from the snapshot identity otherwise. $\square$

Corollary 4.3 — the sewing tax

For $R=d$,

$$\max_{h\le H}\chi_1(h)=1, \qquad \widehat\chi_1(H)=2^{d(d-1)}.$$

The exact Pareto boundary in snapshot/track coordinates is

$$\left\{ \left(2^{d(R-k)},k\right):0\le k\le R \right\}.$$

In logarithmic form the demand law is affine:

$$\log_2 |C|+dk\ge dR \qquad(0\le k\le R).$$

This is a genuine multi-horizon tradeoff, not the maximum of the individual frontiers.

Smallest incompatibility

Take $d=1$, $R=2$, horizons $1$ and $2$, and singleton blocks whose closest cells are more than two positions apart. Each marginal has one-track snapshot width one, while the common one-track width is two. Two horizons are necessary. This is minimal in the parameters $(d,R)$, not in maximum horizon: if depth-zero output after root sensing is admitted, horizons $0$ and $1$ give a shorter timing variant. If one track cone can meet both blocks, the example collapses.

What the theorem does not say

• If the layout is allowed to depend on the horizon, one track moves to the required block and the common lower bound vanishes.
• The theorem grants a separately supplied cut initializer; it does not construct a reachable upstream encoder.
• Finite layered state may cache an entire cone. Charging that state would add another coordinate rather than invalidate the snapshot/track law.
• Random-access placement after seeing the suffix selector is outside the doctrine and destroys the lower bound.

5. One total target and uniform resource escape

For every $r\ge1$, put

$$d_r=2^r,\qquad m_r=4d_r^2,\qquad H_r=2d_r-1,$$

and define

$$B_{r,t} = \{3d_rt,\ldots,3d_rt+d_r-1\}, \qquad 0\le t<d_r.$$

Adjacent blocks have closest-point distance $2d_r+1>H_r$, and every block lies below the cut $m_r$.

Let

$$U_r = \left\{ u\in\{0,1\}^{m_r}: u_i=0\text{ outside }\bigcup_{t<d_r}B_{r,t} \right\}$$

be the corresponding fixed-padding subcube.

Define one total computable Boolean word-value target $F$. If

$$|w|=m_r+d_r+t$$

for some $r\ge1$ and $0\le t<d_r$, let

$$j=\operatorname{value}_2\bigl(w[m_r:m_r+r]\bigr)$$

and set

$$F(w)=w[3d_rt+j].$$

At every other length, set $F(w)=0$. The stage windows

$$[m_r+d_r,m_r+2d_r-1]$$

are disjoint, so the definition is unambiguous.

At cut $m_r$, every horizon below $d_r$ is constant and every horizon from $d_r$ through $H_r$ queries one block. Consequently every exact horizon has $\chi_1=1$, but

$$\widehat\chi_k = 2^{d_r(d_r-\min(k,d_r))}.$$

For the lower bound, restrict any full-prefix factorization to $U_r$ and apply Theorem 4.2. For the matching upper bound, ignore all nonblock coordinates, track $k$ blocks, and encode the remaining blocks in the snapshot. Thus the displayed equality holds on the full prefix cube as well as on $U_r$.

Theorem 5.1 — presentation-level escape

Any bare root-accessible stationary online presentation of $F$ with $s$ non-position roles and $k$ labeled trailing heads satisfies

$$\boxed{ \log_2s+k\log_2(m_r+1) \ge d_r(d_r-k) }$$

whenever $k\le d_r$.

Proof

At cut $m_r$, there are at most

$$s(m_r+1)^k$$

role/position snapshots. The actual online run supplies one snapshot and layout before the future stopping time is known. Its fibers therefore form a common-horizon factorization. Restrict those fibers to the fixed-padding subcube $U_r$; Theorem 4.2 gives

$$s(m_r+1)^k\ge2^{d_r(d_r-k)}.$$

Taking logarithms proves the claim. $\square$

Because $m_r=4d_r^2$, fixed $k$ implies

$$\log_2s \ge \frac{m_r}{4}-O(\sqrt{m_r}),$$

and hence $s=2^{\Omega(m_r)}$.

If $s\le m_r^C$, rearranging gives

$$k \ge \frac{d_r^2-C\log_2m_r} {d_r+\log_2(m_r+1)} = d_r-O(\log m_r) = \frac{\sqrt{m_r}}2-O(\log m_r).$$

The lower bound grants arbitrary cut configurations, exact knowledge of the stage, and arbitrary downstream decoding. Root inaccessibility can only make the task harder, so it is not doing the work.

Theorem 5.2 — every exact cut remains uniformly cheap

For every exact cut $c$ and continuation horizon $h$, the target $F$ has a cut-initialized cone factorization with at most two snapshots and two tracks:

$$\boxed{\chi_2(F;c,h)\le2.}$$

Proof

Put $N=c+h$. If $N$ is outside every stage window, $F$ is constant and one snapshot with no live track suffices.

Suppose

$$N=M+d+t, \qquad M=m_r,\quad d=d_r,\quad 0\le t<d.$$

If $c\le M$, all selector bits are in the continuation. The portion of the queried block $B_{r,t}$ lying before the cut is one interval of at most $d$ cells. One track rooted at its rightmost old cell sweeps that interval; the continuation supplies the remaining data cells and the selector. Because $h=N-c\ge d$, the cone is long enough. One snapshot suffices.

If $M<c<M+r$, write $a=c-M$, so $1\le a<r$ selector bits are old. Use two snapshots indexed by the first old selector bit. That bit chooses one of the two $d/2$-cell halves of $B_{r,t}$; place the first track to sweep the chosen half. Place the second track at $c-1$ to sweep the $a$ old selector cells. The remaining selector bits arrive in the continuation. Here

$$h+1=d+t-a+1\ge d-r+2\ge d/2,$$

so the first cone covers its half; the second easily covers the old selector prefix. The case $r=1$ has no intermediate cut.

If $c\ge M+r$, the entire selector and queried data bit are old. Two zero-track snapshots encode the resulting output bit.

These cases exhaust all exact cuts and use at most two snapshots and two tracks. $\square$

Corollary 5.3 — bounded marginals, no bounded uniform presentation

The one total target $F$ has a uniformly bounded exact cut/horizon field: $(|C|,k)\le(2,2)$ everywhere. Nevertheless Theorem 5.1 excludes every fixed finite pair $(s,k)$ for one root-accessible stationary presentation.

At each staged cut $m_r$, every nonconstant marginal $h\in[d_r,2d_r-1]$ has the sharper values

$$\chi_0=2^{d_r}, \qquad \chi_1=1,$$

so the one track is genuinely load-bearing there. The global two-track upper bound does not weaken the common-horizon lower bound; it shows exactly how much horizon-indexed layout advice the marginal field is using.

Cone-local extension

Now let a richer doctrine satisfy the cone-local Markov law for the same $k$ unit-speed old-input cones: each complete cut type fixes the roots, future behavior depends on the old prefix only through those cones, and no uncharged old-input oracle exists. Let $V_\rho(m)$ bound the number of reachable charged non-role configuration types at the cut. Then the same argument yields

$$\boxed{ \log_2\!\bigl(sV_\rho(m_r)\bigr) \ge d_r(d_r-k). }$$

Consequently a fixed-$s,k$ core is excluded whenever

$$\log_2 V_\rho(m_r)<d_r(d_r-k)-\log_2s$$

infinitely often. In particular, cores with $\log V_\rho(m_r)=o(d_r^2)$ cannot realize $F$ at fixed $k$. A fixed fine core may still have exponentially many runtime configurations, so no unconditional “every compact core escapes” claim is made from this count alone. Converting the inequality into escape of precision, initializer magnitude, canonical height, or another named coordinate needs an additional theorem bounding $V_\rho$ by those coordinates.

This is the point at which the established FPRD machinery does real work:

• sewing identifies the common-horizon witness that a genuine run must provide;
• demand becomes the affine inequality $\log|C|+dk\ge dR$;
• Pareto analysis gives the exact frontier rather than one isolated count; and
• escape analysis upgrades the staged finite obstruction into a lower bound for one uniform bare presentation and, under an explicit volume bound, for cone-local charged cores.

No canonical-height conclusion is asserted without a pinned normalization showing how raw snapshot volume and the relevant gauge-invariant height compare.

6. What marginal semantic fields do and do not determine

The closest exact automata comparator is the Myhill–Nerode theorem with advice. For a language $L$, restrict the full residual equivalence to each grade:

$$x\equiv_{L,n}y \iff |x|=|y|=n \ \text{and}\ \forall z\in\Sigma^*\ (xz\in L\iff yz\in L).$$

Kruckman, Rubin, Sheridan, and Zax prove that $L$ is regular with advice if and only if these grade-$n$ quotients have uniformly bounded size. In their sufficiency proof, the transition map from grade $n$ to grade $n+1$ is encoded in the $n$-th advice symbol; see Theorem 4 and its construction.

This is the correct interpretation of a uniformly bounded exact same-grade all-suffix residual field: it can still contain grade-indexed transition advice. Arbitrary finite-horizon resolving fields require the additional horizon sewing of Section 3 before this comparator applies.

Let

$$C_n=\Sigma^n/{\equiv_{L,n}}, \qquad h_n([x],a)=[xa].$$

An ordinary stationary DFA exists exactly when there are a finite set $Q$, one fixed transition law $\delta$, one output map $o$, and grade maps $e_n:C_n\to Q$ such that

$$e_{n+1}(h_n(c,a))=\delta(e_n(c),a),$$

with output preserved. This is an elementary classical Myhill–Nerode corollary: it says that the infinite layered quotient graph admits one finite edge-respecting coloring.

The FPRD geometry adds another obligation. A semantic color must be realized by a reachable charged configuration, and its colored edges must be legal head or physical transitions. The atlas of Section 2 is exactly this resource-enriched refinement.

A two-cell warning control

Let $s_1s_2\ldots$ be a non-ultimately-periodic binary sequence and define

$$L_s = \left\{ x\in\{0,1\}^*: \bigoplus_{i=1}^{|x|}s_ix_i=1 \right\}.$$

At each grade the full residual quotient has at most two cells, and the grade transition is

$$h_n(c,b)=c\oplus s_{n+1}b.$$

Thus the entire all-suffix field is uniformly two-cell and advice-realizable. But $L_s$ is regular if and only if $s$ is ultimately periodic. For necessity,

$$s_{n+1}=1[0^n1\in L_s],$$

and a DFA’s orbit under $0$ is eventually periodic. Sufficiency uses an eventual phase and one parity bit.

This example isolates stationary transition advice but does not use charged heads. The block theorem is stronger in the direction needed here: every marginal uses a genuinely load-bearing track, yet a common configuration still explodes.

Limits on witness elimination

The atlas theorem is finite and exact once $s,k,N$ are fixed, but this does not imply a uniform effective intrinsic criterion for arbitrary multihead presentations. Known undecidability and non-semidecidability results for fixed multihead automata sharply limit such a promise; the relevant boundary is summarized in Holzer, Kutrib, and Malcher, Multi-Head Finite Automata: Characterizations, Concepts and Open Problems, Theorem 15. That theorem concerns decision problems for standard one- and two-way multihead automata. No reduction from those problems to the pinned real-time atlas doctrine is claimed here; it is an algorithmic warning against extrapolating the finite-core CSP into an unrestricted effective witness-elimination theorem.

The safe conclusion is narrower:

• a proposed bounded physical core has a finite checker;
• a finite-depth atlas is a finite CSP;
• failure at fixed $s,k$ has a finite unsatisfiable atlas instance; but
• the marginal semantic field does not canonically eliminate the layout, root, and transition witnesses.

7. Relational physical outcomes

Origami can have a causal controller action followed by more than one physical outcome. The quantifier order must remain

$$\forall\text{ next input}\; \exists\text{ controller action}\; \forall\text{ physical outcomes}.$$

For a fixed finite semantic transition system $C$, fixed finite fully observed physical core $P$, controller action set $M$, and physical successor relation $\operatorname{Post}(p,a,\mu)$, define

$$\mathcal G(X) = \left\{ (c,p): \operatorname{out}(c)=\operatorname{out}(p) \ \land\ \forall a\ \exists\mu\ \bigl[ \operatorname{Post}(p,a,\mu)\ne\varnothing \ \land\ \forall p'\in\operatorname{Post}(p,a,\mu),\ \bigl(h(c,a),p'\bigr)\in X \bigr] \right\}.$$

For branch-total safety semantics, a lift exists exactly when the root pair lies in the greatest fixed point

$$\nu X.\mathcal G(X).$$

Descending fixed-point pruning gives a finite checker and a memoryless controller when one exists. The displayed three-stage operator is obtained by expanding a round into input, controller-choice, and physical-outcome states, then specializing the alternating-simulation greatest fixed point to a deterministic semantic successor. This is standard alternating-refinement/safety-game machinery, not a new FPRD theorem; compare Alur, Henzinger, Kupferman, and Vardi, Alternating Refinement Relations. For partial observation, the selected action must additionally be uniform on each controller-observation class; the displayed full-information operator must not be used to grant access to hidden physical state.

Exact reachable-set semantics is stronger. It must preserve the complete successor set, not merely show that every physical branch remains safe. In the prefix certificate language, the load-bearing law is

$$K_{ua} = \bigcup_{(q,p)\in K_u} \{q'\}\times\operatorname{Post}(p,a,\mu(q,p,a))$$

with equality. Replacing equality by inclusion permits inconvenient physical branches to be silently deleted and turns exact or must semantics into an angelic may semantics.

The causal sewing theorem for a finite relational core then follows by the same finite-branching argument as the deterministic atlas theorem: one stationary policy and exact reachable sets at every finite depth sew to one global policy. Again, compactness is not the missing physics; the hard work is pinning a finite core and proving the physical successor relation correct.

8. A causal finite-precision Fold-and-Look fragment

Call the pinned fragment $\mathsf{THFL}(\lambda,\mathcal M)$, for terminal-hinge Fold-and-Look.

8.1 Substrate and encoding

• Inputs are $w\#$, with $w\in\Sigma^n$.
• Each ordinary symbol supplies and locally encodes a fixed block of $\lambda$ unit cells, appended to the right.
• Before the terminal marker the sheet is the flat interval $$I_n=[0,L_n],\qquad L_n=\lambda n.$$
• Material is charged and never duplicated.
• Potential sites lie at half-integer cell centers, at most one per cell.
• A site carries a control literal $x_i,\neg x_i,\mathsf{TRUE}$, or $\mathsf{FALSE}$.
• The punch radius is fixed with $0<R<1/2$.
• Writing or preparing a site is charged. We distinguish total site preparation $T_{\rm prep}$ from variable-dependent literal evaluations/writes $T_{\rm var}$.

The local append-only encoder is part of the causal doctrine. A prepared-substrate construction that scatters repeated occurrences of an input bit is not automatically a construction in this causal encoding subdoctrine.

8.2 Controller and fold library

The finite-role controller is a Moore transducer

$$(Q,q_0,\delta,\gamma), \qquad \delta:Q\times\Sigma\to Q, \qquad \gamma:Q\to\mathcal M.$$

On ordinary symbols it receives only its current role and the current input symbol; a fixed terminal-phase flag distinguishes #. It receives no substrate endpoint, layer map, $n$, $L_n$, coordinate register, or length predicate. At # it selects $\gamma(q)$ from a fixed finite library

$$\mathcal M = \{\mu=(a_\mu,b_\mu,d_\mu,\Omega_\mu)\}.$$

Here:

• $a_\mu,b_\mu\in\mathbb Q$ have a fixed bit bound;
• the crease schema is $$c_\mu(L)=a_\mu L+b_\mu;$$
• $d_\mu$ declares the moving side; and
• $\varnothing\ne\Omega_\mu\subseteq \{\mathrm{over},\mathrm{under}\}$ is the exact physical outcome set.

The schema is legal only when it gives an interior integer crease on the current strip. The physical transition relation may use the physical endpoint $L$ to apply the already selected affine law. The controller cannot choose a different law after seeing the physical outcome.

There is one all-layers fold, no cut, punch-after-fold, unfold, or material duplication, followed by one final Look.

The unit grid, integer crease, half-integer sites, and $0<R<1/2$ put each prepared terminal strip/fold state inside the repository’s WFL normal-form hypotheses. Material feed and online site encoding are a new causal wrapper, not operations in the original WFL doctrine. THFL itself is a custom pinned restriction/adaptation, not a published machine model. The published Look primitive globally asks whether the folded paper has a through-hole. In the separated-cell normal form adopted here, this specializes to AND over every complete layer at one candidate location and OR over candidate locations; compare An, Demaine, Demaine, and Ku, Computing 3SAT on a Fold-and-Cut Machine.

8.3 Charged resource vector

Use

$$\rho_{\rm orig} = (q,F,\Lambda,M,T_{\rm prep},T_{\rm var}, p_{\rm grid},b_{\rm coord},A,D,S,\beta,B),$$

where:

• $q$ is the role count;
• $F$ is the fold count;
• $\Lambda$ is material length;
• $M$ is the number of physical sites;
• $p_{\rm grid}$ is grid resolution/denominator precision;
• $b_{\rm coord}$ is the maximum bit magnitude needed for coordinates and affine fold coefficients;
• $A$ is the number of affine pieces;
• $D$ is maximum local layer/AND depth;
• $S$ is global Look OR fan-in;
• $\beta$ is physical branch degree; and
• $B$ is program/library description.

This explicitly charges the two coordinates most easily hidden by “one fold and one Look”: global Look fan-in and encoder/site-preparation work.

9. Exact coordinate-full image of one hinge

Let $I=[0,N]$, let

$$r_c(x)=2c-x,$$

and retain the complete material-address relation, not just marked coincidences. A normalized relational terminal record contains:

• every material address and its quotient location;
• the complete coincidence fibers;
• local orientation;
• layer order in every physical branch;
• the sites, literals, radius, grid resolution, and coordinate magnitude; and
• the exact physical outcome set.

Definition 9.1 — hinge certificate

A coordinate-full record is a hinge certificate when:

1. common center: there is one $\alpha=2c$ such that every non-singleton fiber is $$\{x,\alpha-x\};$$
2. interval closure: whenever $x,\alpha-x\in I$, those two addresses belong to the same fiber; all unmatched addresses are singleton tails;
3. hinge quotient metric: for the declared moving side, $$h_{c,d}(x)= \begin{cases} x,&x\text{ is stationary},\\ \alpha-x,&x\text{ moves}; \end{cases}$$
4. orientation: after normalizing the stationary side, it has orientation $+1$ and the moving side has orientation $-1$;
5. global layer order: in each branch, the entire moving overlap is above the stationary overlap or the entire moving overlap is below it;
6. nonduplication: the full fibers partition the connected source interval;
7. nontrivial WFL geometry: $N,c\in\mathbb Z$, $0<c<N$, at least one complete fiber is non-singleton, sites are half-integer cell centers, and $0<R<1/2$;
8. exact branching: the declared over/under outcome set is nonempty and retains every permitted physical branch; and
9. resource consistency: if resource labels are carried in the record, they equal or safely bound the represented objects—for example $T_{\rm prep}\ge M$ and $T_{\rm var}\ge M_{\rm var}$ under the one-site preparation convention, and the description budget stores the declared geometry and branch set.

Theorem 9.2 — fixed-instance one-hinge geometric image

For one fixed prepared strip, the forward abstraction from normalized one-fold WFL terminal states to coordinate-full relational records has exactly the hinge certificates as its essential image.

Proof

For necessity, one connected side is fixed and the other is carried by the single reflection $r_c$. Two distinct addresses coincide exactly when they are symmetric about $c$, so all nontrivial fibers have common sum $2c$. The entire connected moving interval is reflected, which pairs every available symmetric address and leaves only unmatched tails. Reflection has unit metric slope $-1$, is involutive, and reverses orientation. One whole moving layer is placed above or below the stationary layer, so the order cannot flip inside the overlap. The address map is a total function and therefore cannot duplicate source material.

For sufficiency, recover $c=\alpha/2$ and physically fold the declared side about $c$. The resulting address map is exactly $h_{c,d}$. Common center and interval closure make its fibers exactly the declared fibers, including the singleton tails. Include every branch in the declared nonempty over/under outcome set. The orientation, nonduplication, grid, radius, and resource-consistency laws then match the certificate. $\square$

Whenever the fold is nontrivial, the center is unique. After rigid normalization, abstraction followed by realization returns the same coordinate-full certificate. Realization followed by abstraction preserves the terminal map, complete incidences, orientations, branch set, sites, and Look result. Only the irrelevant continuous motion path is forgotten.

This is an exact terminal-state round trip, modulo rigid normalization and the forgotten continuous motion path. It does not characterize which families of terminal certificates arise from one append-only causal THFL history; that additional question needs the rooted controller/encoder conditions of Sections 8 and 12. The terminal result is an elementary specialization of reflection geometry, not a novelty claim.

9.3 Small independent obstructions

The unrestricted accessible-relational field is strictly larger.

Mixed center. On cell centers in $[0,5]$, take fibers

$$\left\{\tfrac12,\tfrac72\right\}, \qquad \left\{\tfrac32,\tfrac92\right\}.$$

Their address sums are $4$ and $6$. Each pair separately looks reflection-generated, and the pairs do not duplicate an address, but no single crease realizes both. Two nontrivial fibers are minimal for detecting this failure.

Closure gap. About $c=2$, pair $\{1/2,7/2\}$ but leave $3/2$ and $5/2$ as singletons. The record has a candidate common center and consistent order but cannot arise from moving a connected side.

Layer-order flip. Use the complete reflection pairs

$$\left\{\tfrac12,\tfrac72\right\}, \qquad \left\{\tfrac32,\tfrac52\right\}$$

about $c=2$, but declare the moving layer above in the first fiber and below in the second. No one-hinge folded state realizes this record.

Each example is a finite, branch-total accessible-relational certificate. The obstruction is geometric rather than causal.

Layer order remains nontrivial for larger crease patterns. For a supplied isometric flat folding, facewise order is governed by antisymmetry, transitivity, taco–taco, taco–tortilla, and tortilla–tortilla constraints; see Akitaya, Demaine, and Ku, Computing Flat-Folded States, Theorem 1.

9.4 General geometric invariants exposed by the round trip

The one-hinge laws have multi-fold analogues that an unrestricted FPRD relation does not see. Let a complete one-dimensional folded strip be represented by a continuous finite piecewise-linear map

$$\phi:[0,L]\to\mathbb R, \qquad \phi'\in\{-1,+1\}$$

off its crease points. At any regular value $y$—one that is not the image of a crease point or either substrate endpoint—list the complete fiber

$$\phi^{-1}(y)=\{x_1<\cdots<x_m\}.$$

Continuity forces the crossing orientations to alternate. Moreover,

$$\sum_{\phi(x)=y}\operatorname{sgn}\phi'(x) = 1_{\phi(L)>y}-1_{\phi(0)>y} \in\{-1,0,1\}.$$

The proof is the signed crossing count of the horizontal level $y$. In particular, a claimed complete two-layer fiber with equal orientations is impossible. The word complete is essential: a same-sign observed pair may be physically completed by an unobserved intervening layer.

Nonduplication also gives the classical one-dimensional area/Banach-indicatrix identity

$$\int_{\mathbb R}|\phi^{-1}(y)|\,dy=L.$$

Because every affine piece has unit slope, the integral of fiber multiplicity is exactly the original material length. A discrete unit-cell model has the corresponding sum rule. These laws become FPRD obstructions only after the abstraction retains or existentially completes the entire sheet and all fibers.

10. Exact Boolean image of one WFL hinge

The coordinate-full image is rigid. Its Boolean quotient is much larger because Look forgets coordinates, orientations, and order.

For a WFL presentation with address map $\phi$, let $\Phi(y)=\phi^{-1}(y)$ be the complete material fiber. The WFL incidence normal form gives

$$\operatorname{Look}(w) = \bigvee_{y\in Y_{\rm eff}} \bigwedge_{p\in\Phi(y)}\ell_p(w).$$

One hinge has fiber depth at most two. Therefore every surviving term has at most two distinct nonconstant literals after deleting any term that contains FALSE or a contradiction, removing TRUE literals, and collapsing repeated literals.

Theorem 10.1 — semantic essential image

The Boolean functions computed by one prepared-substrate WFL hinge are exactly the Boolean functions admitting a DNF of maximum term width at most two.

Here “$2$-DNF” means width $\le2$, not that every term has exactly two literals.

Forward proof

The incidence normal form is an OR over effective fibers and an AND over the sites in each fiber. A one-fold address has at most two preimages at every location, so the displayed formula has width at most two. $\square$

Reverse proof

Normalize an exact DNF by deleting false or contradictory terms, removing TRUE and repeated literals, and optionally deleting duplicate terms. Let it have $T\ge1$ terms and $L\le2T$ nonconstant literal occurrences.

Use a strip of $2T$ unit cells and fold at the integer midpoint $c=T$. For clause $j=0,\ldots,T-1$, use the symmetric cell centers

$$p_j=j+\tfrac12, \qquad q_j=2T-p_j.$$

For a binary term $\ell_1\wedge\ell_2$, label the two sites $\ell_1,\ell_2$. For a unary term $\ell$, label them $\ell,\mathsf{TRUE}$. For an empty true term, label both $\mathsf{TRUE}$. Choose $R=1/4$.

After folding, $\{p_j,q_j\}$ is one complete depth-two fiber and opens exactly when clause $j$ is true. The final Look is the desired disjunction. The integer/half-integer grid ensures clean alignment, no crease through a punch, uniform overlay, and interiority. $\square$

For the constant-false function, use a solid two-cell strip, crease $1$, and no sites. Constant true uses one pair of TRUE sites.

10.2 Resource distortion

For $T\ge1$, the midpoint reverse construction has:

An off-center construction can save filler sites: pair the $b$ binary terms across a crease at $b$, and put the $u$ unary terms on a singleton tail. For $b>0$, this uses

$$\Lambda=2b+u,\qquad M=2b+u=L.$$

If $b=0$, two blank hinge cells plus the unary tail use $\Lambda=u+2$. The supplied checker exercises this optimized version.

This reverse theorem is a prepared-substrate result. Repeated literal occurrences are distinct physical sites controlled by the same input bit. That duplicates features, not paper, and it incurs one preparation cost per occurrence. It does not by itself prove that an append-only local causal encoder can place every arbitrary formula family.

A smallest semantic obstruction

The three-variable function

$$x_1\wedge x_2\wedge x_3$$

has no width-$\le2$ DNF and therefore no one-hinge WFL realization. It is not enough that three marked incidences can be drawn: one final location can contain at most two layers after one hinge.

This obstruction is semantic. The mixed-center, closure-gap, and order-flip examples are geometric obstructions invisible after taking the Boolean quotient.

11. The palindrome control becomes an exact resource theorem

For a word $w_1\cdots w_m$, use two unit cells per bit, with sites

$$p_i^+=2i-\tfrac32 \quad\text{labeled }w_i, \qquad p_i^-=2i-\tfrac12 \quad\text{labeled }\neg w_i.$$

Fold the length-$2m$ strip at $c=m$. If $j=m+1-i$, reflection pairs

$$p_i^+\longleftrightarrow p_j^-, \qquad p_i^-\longleftrightarrow p_j^+.$$

The corresponding terms are

$$w_i\wedge\neg w_j, \qquad \neg w_i\wedge w_j.$$

Look is therefore exactly NONPALINDROME. For odd $m$, the central pair gives the contradictory term $w_i\wedge\neg w_i$. On the all-zero word, every mismatch term is false, so the construction returns zero without candidate-lineage inflation.

For even $m=2n$, the exact construction resources are

$$F=1,\quad A=2,\quad D=2,\quad S=2n,\quad M=T_{\rm prep}=T_{\rm var}=4n,\quad \Lambda=4n.$$

Theorem 11.1 — exact one-fold NONPAL resources

Every exact DNF for NONPALINDROME on $2n$ bits has at least $2n$ terms and at least $4n$ literal occurrences. Therefore every WFL realization satisfies

$$S\ge2n, \qquad M\ge4n, \qquad T_{\rm var}\ge4n$$

under one-operation-per-variable-site encoding. The construction attains all three bounds.

Proof

For each mirrored pair $i,j$, the two mismatch cubes

$$w_i\wedge\neg w_j, \qquad \neg w_i\wedge w_j$$

are prime implicants. Any implicant of NONPALINDROME must force at least one mirrored pair to mismatch; otherwise it has a palindromic completion. Removing extra literals reduces it to one of the $2n$ displayed prime implicants.

Each displayed implicant is essential: choose an assignment where exactly that mirrored pair mismatches in the named orientation and all other mirrored pairs agree. No other prime mismatch implicant is true. Hence all $2n$ terms are required, each with two literals, for a total of $4n$ literal occurrences.

The WFL incidence normal form has at most one term per effective Look fiber and uses every physical site in at most one complete fiber. Therefore $S$ and $M$ dominate the minimum term and literal-occurrence counts. The encoding convention turns the site lower bound into the stated work lower bound. $\square$

This is the FPRD-to-origami transfer in its cleanest form: demand and Pareto lower bounds for formula terms and literal occurrences become physical lower bounds on Look fan-in and encoder work, even though the fold count is one.

12. Uniform causal lifting of terminal folds

For $n\ge1$, let $C_n$ be a desired normalized structural and control-labeled terminal hinge record on the length $L_n=\lambda n$ strip. It retains site locations and their controlling literals, not the valuation-specific subset of sites that happens to be open on one word. Geometry is input-independent, although site openings depend on the input bits.

Fix one append-only encoder and write $U_n$ for its flat structural/control-labeled substrate after $n$ symbols. The theorem below characterizes the remaining finite-controller selection problem; it does not decide whether an arbitrary prescribed family $U_n$ is locally encodable.

For the fixed finite fold library $\mathcal M$, define

$$A_n = \{ \mu\in\mathcal M: \mu\text{ is legal at }L_n \text{ and }\operatorname{Terminal}(U_n,\mu)=C_n \}.$$

The comparison includes crease, direction, full incidence, orientation, layer order, and the exact physical branch set. Assume $A_n\ne\varnothing$ for every $n$, so each horizon is individually realizable.

Theorem 12.1 — periodic-selector criterion

The family $(C_n)$ has one finite-role stationary causal lift in $\mathsf{THFL}(\lambda,\mathcal M)$ if and only if the sets $A_n$ admit an ultimately periodic selector:

$$\exists s_n\in A_n,\ t\ge0,\ p\ge1 \quad s_{n+p}=s_n\quad(n\ge\max\{1,t\}).$$

Equivalently, after a finite prefix, every residue class modulo $p$ has one template belonging to all of its admissible sets.

For exact role counting, the unused pre-input role must also be charged. The minimum role count is the minimum orbit size of an ultimately periodic extension

$$s_0,s_1,s_2,\ldots, \qquad s_n\in A_n\quad(n\ge1),$$

where $s_0$ is arbitrary because no fold is performed on the empty substrate. Equivalently, minimize transient-plus-cycle size over all choices of $s_0$ and all admissible selectors. A selector described only from $n=1$ with preperiod $t$ and period $p$ always gives the safe upper bound $q\le t+p+1$, and can sometimes merge the unused initial role.

Proof

Run any realizing finite-role controller on the unary input ray $0^n$. The ordinary-symbol transition is one function on a finite role set. Its orbit is a finite transient chain followed by a cycle. The template selected at # is therefore ultimately periodic in $n$, and it must lie in $A_n$.

Conversely, extend the admissible selector by one arbitrary value at $n=0$, and realize that ultimately periodic sequence by the transient chain and cycle of a Moore controller. Ignore ordinary input values, advance one role per symbol, and output the role’s template at #. The exact physical relation then produces $C_n$. Every one-generator finite orbit has precisely this chain-cycle form, giving the orbit-size characterization. $\square$

This is classical unary finite-state eventual periodicity instantiated on a finite physical fold library. It is also the origami specialization of the stationary atlas seam.

Corollary 12.2 — power-of-two incompatible family

Take $\lambda=4$, so $L_n=4n$, and two fixed affine templates:

$$\mu_0:c=L/2=2n, \qquad \mu_1:c=L/2+1=2n+1.$$

Both use the same moving side and exact outcome set. Require the $\mu_1$-record exactly when $n$ is a power of two and the $\mu_0$-record otherwise.

Every horizon has constant fold count, depth, grid resolution, affine template description, branch degree, and library size; material and absolute crease magnitude grow with $n$. But the unique selector is not ultimately periodic. If period $p$ held beyond $t$, choose $2^k>\max(t,p)$. Periodicity would also select $\mu_1$ at $2^k+p$, while

$$2^k<2^k+p<2^{k+1},$$

so this length is not a power of two. Contradiction.

Thus finite roles must escape, unless the operation repertoire is enriched with a non-affine power-of-two length test. Adding that test merely moves the missing resource into the physical action semantics.

Why terminal-map restriction is the wrong compatibility

Consider the uniformly realizable midpoint fold

$$I_n=[0,2n], \qquad h_n(x)= \begin{cases} x,&x\le n,\\ 2n-x,&x>n. \end{cases}$$

One controller role always selects the fixed schema “fold at the current midpoint.” Nevertheless the terminal maps do not restrict under $I_n\hookrightarrow I_{n+1}$. For example,

$$h_n\!\left(n+\tfrac12\right)=n-\tfrac12, \qquad h_{n+1}\!\left(n+\tfrac12\right)=n+\tfrac12.$$

Even the final fiber of old material changes when another input block is appended.

The coherent diagram is instead

$$C_n=\tau(U_n,\mu_{\rm midpoint}),$$

where the latent flat states $U_n$ extend by material append and one stationary terminal action $\tau(-,\mu_{\rm midpoint})$ is applied only after #. “Append after terminal folding” need not equal “terminal fold after append.” This geometric counterexample rules out a tempting but incorrect inverse-system formulation of causal sewing.

13. Branch semantics under a layer-sensitive observational extension

Let a midpoint fold permit both over and under outcomes. Put distinct permanent face tags on the stationary and moving layers at one overlap probe. The tag-reading observation in this section is an explicit extension of standard WFL/THFL; ordinary through-hole Look does not report the top layer.

• In the over branch, a layer-sensitive Look reports top tag $M$.
• In the under branch, it reports top tag $S$.

The exact reachable observation set contains both:

$$\{(\mathrm{hole},M),(\mathrm{hole},S)\}.$$

Therefore:

• may-$M$ succeeds;
• must-$M$ fails;
• exact singleton-$M$ fails; and
• exact $\{M,S\}$ succeeds.