Relational admission and causal sewing

Status. Working theorem draft. No novelty claim. This extends the accessible-field sewing note to nonfunctional doctrines; it is intended to make the origami/virtual-crease uncertainty mathematical rather than to choose a crease algorithm prematurely.

1. The distinction

There are three increasingly strong requirements.

1. A presentation-demand field needs a set of finite presentations, a well-defined (possibly set-valued) semantic map, and pinned resources.
2. A relational operational doctrine additionally needs a specified relation of permitted physical successor configurations.
3. A causal controller chooses one operation as a function of its current finite observation; the physical relation may still return several outcomes.

The controller policy need not be known before admission: it can be quantified over as part of the presentation field. But one selected presentation must use one causal policy on every continuation. What cannot be omitted is the relation at level 2: without it, accessibility from the preceding cut is undefined.

2. A finite relational storage doctrine

Fix finite alphabets:

$$\Sigma\quad\text{(input)},\qquad Y\quad\text{(observable outcomes)},\qquad O\quad\text{(finite local observations)},\qquad M\quad\text{(declared operation names)}.$$

For each prefix $u$, let $P(u)$ be a finite set of admissible physical configurations, with

$$\operatorname{obs}_u:P(u)\to O .$$

For every symbol $a$, operation $\mu$, and configuration $p$, declare a set of legal successors

$$\tau_{u,a}(p,\mu)\subseteq P(ua). \tag{1}$$

This relation may retain several folds, several layer orders, or several virtual-crease resolutions. It may also be empty. It is target-independent: the target does not get to invent new physical transitions.

A normalized $s$-role controller over relational physics consists of

$$Q_s=[s],$$

a finite stationary causal policy

$$\delta:Q_s\times\Sigma\times O\to Q_s\times M, \tag{2}$$

a finite readout

$$\omega:Q_s\times O\to 2^Y, \tag{3}$$

and a nonempty finite set of permitted root configurations

$$K_\varepsilon\subseteq Q_s\times P(\varepsilon). \tag{4}$$

The reachable set after $u$ is defined, not selected independently:

$$\begin{aligned} K_{ua}=\{(q',p'):\;&(q,p)\in K_u,\ o=\operatorname{obs}_u(p),\\ &\delta(q,a,o)=(q',\mu),\ p'\in\tau_{u,a}(p,\mu)\}. \end{aligned} \tag{5}$$

Its exact set-valued observation is

$$\operatorname{Sem}(u) = \bigcup_{(q,p)\in K_u} \omega(q,\operatorname{obs}_u(p)). \tag{6}$$

For a set-valued target $F:\Sigma^*\to2^Y$, exact correctness means

$$\operatorname{Sem}(u)=F(u). \tag{7}$$

Other predicates can be pinned without changing the dynamics:

• feasibility (may): $\operatorname{Sem}(u)\cap F(u)\ne\varnothing$;
• safety (must): $\operatorname{Sem}(u)\subseteq F(u)$;
• nonblocking refinement: $\varnothing\ne\operatorname{Sem}(u)\subseteq F(u)$.

Only exact equality is called target realization here. The other predicates are useful specifications but can be vacuous without their stated nonblocking conditions. If silent failure matters, add an explicit $\bot$-outcome, or require every reachable branch to have a nonempty readout and a legal continuation.

3. Relational finite-horizon certificate

A depth-$H$ certificate consists of one stationary $(\delta,\omega,K_\varepsilon)$ and sets

$$K_u\subseteq Q_s\times P(u) \qquad (|u|\le H)$$

such that:

1. $K_\varepsilon$ is the declared root set;
2. equation (5) holds with equality on every edge $u\to ua$ through depth $H$;
3. the selected correctness policy holds at every $u$ through depth $H$.

Equality in (5) is load-bearing. Mere inclusion would permit a certificate to discard inconvenient fold outcomes at each new cut, silently replacing a must or exact semantics by existential advice.

4. Relational sewing theorem

Theorem

Fix $s$ and a finite relational storage doctrine. Under exact semantics, or under any one of the three explicitly named local specifications above, the following are equivalent:

1. one global stationary $s$-role relational controller satisfies the selected exact/specification policy;
2. a depth-$H$ relational certificate exists for every $H$.

Proof

Restriction of a global controller and its reachable sets proves $1\Rightarrow2$.

For the converse, normalize roles to $[s]$. At each depth there are only finitely many certificates:

• finitely many stationary causal policies $\delta$;
• finitely many readout tables $\omega$;
• finitely many root sets;
• finitely many prefixes through $H$;
• finitely many subsets of each finite fiber $Q_s\times P(u)$.

Restriction of a depth-$H+1$ certificate to depth $H$ is again a certificate, because (5) is a forward local equality and the policy is checked prefixwise. The finite certificates therefore form a finitely branching tree with a node at every level. König’s lemma supplies an infinite compatible branch.

The stationary tables and every $K_u$ stabilize along that branch. Their union satisfies (5) and the policy at every finite prefix, hence defines one global relational controller. $\square$

The theorem also holds on a compact normalized core when the configuration fibers are compact hyperspaces and transition/readout legality is closed. That extension requires the relevant closed-image hypotheses and is not asserted here merely from compactness of the underlying physical space.

5. Deterministic and set-valued specializations

• If $K_\varepsilon$ is a singleton, every $\tau_{u,a}(p,\mu)$ on a reachable configuration is a singleton, and every reachable readout is singleton-valued, the theorem reduces to deterministic accessible sewing.
• If $\omega$ is singleton-valued but transitions branch, equation (6) retains all observable outcomes of nondeterministic physical evolution.
• If the presentation is a whole-sheet fold object with no prefix transition relation, it remains admissible to the M33 demand-field layer, but this causal theorem does not apply.

Thus “no deterministic physical outcome yet” is not a blocker. “No declared successor relation yet” is the precise boundary.

6. Unknown policy versus unresolved physics

The presentation doctrine may contain every finite causal policy $\delta$ built from the declared operation repertoire. FPRD may then minimize resources over that set without selecting a policy in advance. This is ordinary presentation search, not future advice.

Once one presentation is selected, however, its $\delta$ is fixed. It cannot choose one crease at a common prefix for suffix $v$ and a different crease at the same prefix for suffix $v'$ merely because each completed word has some successful branch. Only the physical successors in $\tau$ remain relational, and exact or safety semantics retain all of them.

If controlled choices and environmental outcomes are both put into one nondeterministic transition relation, a bare existential reading can select a lucky action separately for each completed word. That is an angelic lookahead model, not a causal folding controller. It is excluded here unless declared intentionally as a different doctrine.

7. Virtual creases as legitimate state

A virtual crease is not required to be an executed fold. It can be a finite mark carried in $P(u)$, for example:

$$(\text{substrate},\ \text{marked candidate creases},\ \text{current layer relation}).$$

The relation (1) can:

• add an allowed mark;
• retain several candidate marks;
• execute one permitted fold;
• merge physically indistinguishable outcomes;
• reject geometrically inconsistent outcomes.

Resources can separately charge:

$$\text{mark count},\quad \text{coordinate precision},\quad \text{layer depth},\quad \text{fold count},\quad \text{finite control roles}.$$

The proved finite theorem applies after crease positions and outputs are placed in a finite precision/candidate stratum. Continuous crease coordinates require a separate compact-hyperspace theorem with closed transition and readout relations; that theorem is not assumed here.

The prefix-indexed notation $P(u),\operatorname{obs}_u,\tau_{u,a}$ is typing convenience, not permission to hide the target in an oracle. A computational doctrine must have one uniform finite/effective local description, and its coordinate precision or other description resources must be charged.

This is enough to admit the proposed PEG/finger/origami comparison without claiming that PEGs already are folding machines. A correspondence becomes a theorem only after translations preserve the declared semantics and bound these resources.

8. What a meaningful comparison would prove

The first worthwhile bridge theorem would have the form:

Every presentation in a pinned finger/PEG fragment translates to a relational virtual-crease presentation with resource map $\Phi$, and conversely on a named subfragment with resource map $\Psi$.

The maps need not be equivalences. A one-way simulation, a separation, or a provable failure of sewing would already be informative.

The minimum controls are:

1. the right-sided-palindrome finger example;
2. the one-fold palindrome alignment construction;
3. the all-zero palindrome family, which rejects lineage-count inflation;
4. square length, which rejects constant exact-cut width as evidence of a finite global controller;
5. a nondeterministic fold with two observably different outcomes, to ensure that exact, may, and must semantics have not been conflated.

9. Current conclusion

Origami can be admitted before a successful crease-selection policy is known: the presentation field may quantify over every finite causal policy. Within each selected presentation, controlled action is causal and fixed, while the entire reachable set of physical outcomes is retained. This makes genuine physical uncertainty part of the mathematics without granting hindsight.

The gain is real but limited: relational admission does not itself establish a PEG–fold correspondence, a complexity result, or a causal sewing theorem for whole-sheet operations. Those require a pinned transition relation and resource-respecting translations.