FPRD-D43 · three memory layers

Trace, meaning, and quotient

Let $X=\prod_{i\in I}X_i$ be a finite product of source coordinates and let $F:X\to Y$ be the complete answer that must survive a cut. For a continuation family, $F(x)$ is its response row.

Operational support records coordinates actually replayed by executions.
Semantic support records coordinates capable of changing some required answer.
Quotient memory identifies sources with the same complete answer.

The first depends on an implementation. The second and third depend only on the demanded map $F$ .

FPRD-T155 · support-quotient theorem

The three layers have exact comparison maps

A coordinate is essential when changing only that coordinate can change the complete answer. Write the essential set as

E(F)=\{i:\exists x,x'\text{ differing only at }i, \ F(x)\ne F(x')\}.

A coordinate set is sufficient exactly when it contains $E(F)$ . Thus $E(F)$ is the unique least semantic coordinate support. The exact summary quotient has $|F(X)|$ states, and

|F(X)|\le\prod_{i\in E(F)}|X_i|.

Equality holds exactly when the induced map on the essential product is injective. For a family $F=(F_c)_{c\in C}$ ,

E(F)=\bigcup_{c\in C}E(F_c).

Finally, any stable adaptive replay family bounds the semantic layer from above. If $Q(x)$ reproduces the answer at source $x$ , then

E(F)\subseteq\bigcup_{x\in X}Q(x).

Agreement on the union implies agreement on the particular replay support of the first execution, so the union is sufficient. The unique-minimum statement then gives the inclusion.

FPRD-D44 · residual tower

Every future horizon is a finite quotient

For a language $L\subseteq\Sigma^*$ , let $r_h(p)$ be the response row of prefix $p$ on every continuation of length at most $h$ , and put

Q_h=\{r_h(p):p\in\Sigma^*\}.

Each $Q_h$ is finite. Restricting rows gives surjections $Q_{h+1}\twoheadrightarrow Q_h$ . Reading a letter $a$ selects its response subtree:

D_{a,h}:Q_{h+1}\to Q_h, \qquad D_{a,h}(r)(z)=r(az).

Thus a symbol has a canonical semantic transport rule: it spends one level of future horizon. A word of length t maps $Q_{h+t}$ to $Q_h$ .

FPRD-T156 · canonical completion

The tower is a compact residual machine

The compatible finite quotients have an inverse limit

\widehat Q=\varprojlim_h Q_h.

It is compact and totally disconnected. The derivative maps induce continuous transitions $\widehat D_a:\widehat Q\to\widehat Q$ . Every prefix maps to its coherent response tower, and

\eta(pa)=\widehat D_a(\eta(p)).

Two prefixes have the same image exactly when they have the same full right residual. The Myhill--Nerode residual space therefore embeds equivariantly and densely in $\widehat Q$ .

A language is regular exactly when the tower stabilizes at a finite horizon. Equivalently, the sizes $|Q_h|$ are bounded, or some restriction $Q_{h+1}\to Q_h$ is bijective.

The completion need not contain only realized residuals. A unary sequence with increasingly long zero blocks has actual tails that converge to the all-zero response tower, although no actual tail is all zero. Density is exact; surjectivity is not assumed.

The audit checks all 32,768 binary language tables through total word depth three and all 5,898 binary DFAs with at most three states. It verifies 2,195,456 restriction and transport identities, 106,920 derivative-class checks, every DFA stabilization, and twelve ideal-residual witnesses.

Inverse limits and topological recognition are established theories. The FPRD use here is to make the response-horizon tower the substrate for stationary transport. No literature-priority or independent-review claim is made.

FPRD-D45 · physical transport invariant

Minimize over every stationary persistent presentation

A bounded stationary presentation has finite control and labels, fixed node degree, fixed descriptor radius, immutable old nodes, and one new node per input symbol. Its finite transition table reads a bounded unfolded neighbourhood and chooses the new label, control, and port targets. This is exactly a finite scaffolding automaton, viewed as a presentation of the residual shift.

Define the stationary transport spectrum by

\operatorname{Tr}(L)= \{(d,k,g,q):L\text{ has a presentation within these ceilings}\}.

The coordinates are node degree, descriptor distance, working alphabet size, and finite-control size. The spectrum minimizes over implementations inside the scaffold model rather than fixing one encoding.

FPRD-T157 · finite basis and Boolean calculus

Every nonempty transport spectrum has finitely many optimal profiles

Adding unused ports, permitted path length, labels, or states preserves a presentation. Therefore $\operatorname{Tr}(L)$ is upward closed in the componentwise order. Dickson's lemma gives a unique finite antichain of Pareto-minimal tuples:

\operatorname{Tr}(L)=\uparrow\operatorname{MinTr}(L).

Complement only flips accepting control states, so $\operatorname{Tr}(\overline L)=\operatorname{Tr}(L)$ . Two presentations can also be paired synchronously. One physical node stores a pair of component labels; its ports are divided into two blocks, and every component path is translated into its block. Hence, for any binary Boolean function b,

(d_1+d_2,\max(k_1,k_2),g_1g_2,q_1q_2) \in\operatorname{Tr}(b(L_1,L_2)).

A positive frontier theorem need only exhibit one feasible cone. A separation must prove that the entire spectrum is empty, not merely defeat one coordinate system or one resource tuple.

Nonempty spectrum is exactly recognition by a finite scaffolding automaton. FPRD-T110 then supplies an ordinary PEG for the reverse language. No converse PEG theorem is imported into this claim.

The audit reconstructs all 65,536 upward closures on a four-by-four grid and checks 4,096 four-dimensional families. It also runs 73,152 paired persistent computations, comparing 369,792 projected component rows and all 1,170,432 Boolean output cases.

Dickson's lemma is classical, and the exact presentation boundary unfolds the published scaffold model. The contribution is their organization as an FPRD transport invariant. No literature-priority or independent-review claim is made.

FPRD-T158 · finite future observers

Fixed future questions compile into finite control

Let a tuple $(d,k,g,q)$ present a language $L$ . Fix a nonempty finite suffix family $U\subseteq\Sigma^*$ and any Boolean function b. The observer language asks b to combine the membership answers after those suffixes:

\operatorname{Obs}_{U,b}(L)= \{p:b((1_L(pu))_{u\in U})=1\}.

Put $R_{k,h}=0$ when $h=0$ or $k=0$ , and otherwise put $R_{k,h}=k+(h-1)(k-1)$ . Then

(d,\max_{u\in U}R_{k,|u|+1},g,q\,2^{|U|}) \in\operatorname{Tr}(\operatorname{Obs}_{U,b}(L)).

The compiled machine appends exactly the node that the original machine would append. Its finite control also stores the current future-answer vector. On symbol a, FPRD-T153 says that the bit for suffix u depends only on the old control and the radius $R_{k,|u|+1}$ view, because it is the original answer after the fixed word au. One finite table therefore updates the original control, the unchanged graph, and every observer bit at once.

Every fixed right quotient $L/u$ , and every finite Boolean combination of fixed right quotients, preserves a nonempty stationary transport spectrum.

The audit checks 7,112 compiled runs, 21,336 observer coordinates, 1,820,672 Boolean outputs, and 235,008 finite-scaffold future evaluations. Twenty-four chain witnesses show that the stated locality radius is uniformly sharp: radius one smaller can lose the bit needed after the current symbol and suffix.

Closure under quotient-like operations is classical for many language families. The contribution here is an explicit shared-graph transport bound. No literature-priority or independent-review claim is made.

FPRD-T166 · causal recoding boundary

Finite-state causal preprocessing changes control, not memory

Let T be a deterministic synchronous Mealy transducer with s states, producing one output letter for each input letter. If T maps input words over $\Sigma$ to words over $\Gamma$ , then for every $L\subseteq\Gamma^*$ ,

(d,k,g,q)\in\operatorname{Tr}(L) \quad\Longrightarrow\quad (d,k,g,sq)\in\operatorname{Tr}(T^{-1}(L)).

The compiled control pairs the transducer state with the source scaffold control. On each input letter, it computes the one output letter and performs exactly that source transition. It copies the source node label and every source port descriptor. Consequently, the persistent graph is unchanged. Only finite control grows.

If T is a word bijection whose inverse is also computed by a finite synchronous transducer, applying the theorem in both directions gives

\operatorname{Tr}(L)\ne\varnothing \quad\Longleftrightarrow\quad \operatorname{Tr}(T^{-1}(L))\ne\varnothing.

Bounded stationary transport is invariant under finite-state synchronous causal changes of coordinates.

Forgotten-action projection is not such an inverse image. Given an observed word, it existentially chooses among all compatible hidden tag histories. FPRD-T164 can therefore fail without contradicting this theorem. The obstruction lies in merging hidden histories, not in deterministic finite-state recoding.

The audit compares 5,898,240 product runs and 16,256 inverse roundtrips. It checks control, local action traces, evolving abstract views, and acceptance.

This is a deterministic synchronous inverse-image theorem. It does not cover direct images, existential projection, erasing or expanding transductions, or nondeterministic transducers. Product constructions are classical; no literature-priority or independent-review claim is made.

FPRD-T167 · bounded hidden choice

A fixed finite menu of causal selectors is still harmless

Let $A\subseteq\widehat\Sigma^*$ have transport tuple $(d,k,g,q)$ . Suppose m synchronous finite-state selectors choose tagged lifts online and cover the projected language:

\pi(A)=\bigcup_{i=1}^{m}T_i^{-1}(A).

If selector i has $s_i$ states, then T166 compiles each branch and the Boolean product theorem combines them:

\left(md,k,g^m,q^m\prod_{i=1}^{m}s_i\right) \in\operatorname{Tr}(\pi(A)).

Call the least possible m the causal selection number $\operatorname{csel}(A,\pi)$ . A finite value is therefore a sufficient certificate for bounded stationary transport whenever A itself has bounded transport.

FPRD-T168 · hard visible lift

One persistent stack can become infinite transport when its actions are hidden

Apply FPRD-T91 to the Kim-Park language $C\in\mathsf{LIN}\setminus\mathsf{SCA}$ . It gives a deterministic visibly pushdown language $A_C$ with $\pi(A_C)=C$ . The tagged machine is a letter-synchronous one-stack machine, so the persistent-stack compiler gives it a degree-one, distance-two scaffold.

\operatorname{Tr}(A_C)\ne\varnothing, \qquad \operatorname{Tr}(\pi(A_C))=\varnothing.

If finitely many causal selectors covered all accepting lifts, T167 would put C back in SCA. Therefore

\operatorname{csel}(A_C,\pi)=\infty.

The hard resource is not the tagged stack. It is the unbounded family of incompatible online choices exposed when the action tags are hidden.

The audit checks 2,666,496 packed selector runs and 256 raw-cover growth cases. The SCA non-membership of C remains an imported dependency from Kim and Park.

Finite selector covers are sufficient, not necessary, for bounded projected transport. Infinite selection number alone is not a general SCA lower bound. T168 uses Kim and Park's independently established non-membership to obtain the conclusion for this lift.

FPRD-T169 · finite escape

Every fixed selector budget succeeds globally or fails at a finite horizon

Let $\operatorname{Sel}_n(A,\pi)$ contain the pairs $(m,s)$ for which at most m synchronous selectors, each with at most s states, cover all accepted observations through length n. These sets are upward closed and decrease as the horizon grows.

\operatorname{Sel}(A,\pi) =\bigcap_{n\ge 0}\operatorname{Sel}_n(A,\pi).

For fixed m and s there are only finitely many labeled finite-state selector families. The candidate families that work through horizon n form nested subsets of that finite universe. If one exists at every horizon, one family survives all horizons and works globally.

Every infeasible pair therefore has a least finite escape length $e_{A,\pi}(m,s) =\min\{n:(m,s)\notin\operatorname{Sel}_n(A,\pi)\}.$ For the hard visible lift from T168, this value is finite for every finite m and s. No fixed online strategy budget merely fails in the limit: some finite observed inputs already defeat every family within that budget.

The audit exhausts 32,768 bounded tagged-language relations and checks 196,608 nesting comparisons, 65,536 finite-intersection identities, 131,072 resource-upward cases, and 32,832 finite escapes.

This is a qualitative compactness theorem for finite-state synchronous selectors. It gives no computable bound on the escape length, and an empty selection spectrum is a property of a hidden presentation rather than an intrinsic language lower bound. The hard-lift application retains T168's imported Kim-Park premise.

FPRD-T170 · hidden-choice chain rule

Causal selection resources multiply through projection chains

Let $\Gamma\xrightarrow{\pi}\Sigma \xrightarrow{\rho}\Delta$ be letter projections, let $A\subseteq\Gamma^*$ , and put $B=\pi(A)$ . An outer selector chooses an intermediate history; an inner selector then chooses a top history. Their product-state composition remains a synchronous selector.

(m,s)\in\operatorname{Sel}(A,\pi),\quad (n,t)\in\operatorname{Sel}(B,\rho) \Longrightarrow (mn,st)\in\operatorname{Sel}(A,\rho\circ\pi).

The same implication holds for every finite-horizon spectrum $\operatorname{Sel}_h$ .

This localizes unbounded hidden choice. If the composite projection has empty selection spectrum, at least one stage must already have empty spectrum. With feasible budgets assigned escape value infinity, the horizon law gives

e_{A,\rho\circ\pi}(mn,st) \ge \min\{e_{A,\pi}(m,s),e_{B,\rho}(n,t)\}.

Sequentially applying T167 preserves the shared hierarchy. From $(d,k,g,q)\in\operatorname{Tr}(A)$ and uniform selector-state bounds, it obtains

(mnd,k,g^{mn},q^{mn}s^{mn}t^n) \in\operatorname{Tr}((\rho\circ\pi)(A)).

The audit exhausts 512 bounded top languages, 306,240 valid stage-cover pairs, 918,720 composed cover checks, and 983,040 product-state run comparisons.

This is an upper-bound calculus. Pairwise composition can be nonoptimal, and a later projection can erase a difficulty from an earlier stage. The theorem applies to deterministic synchronous finite-state selectors and makes no literature-priority claim.

FPRD-T171 · finite obstruction duality

Every failed finite selector budget has a finite test suite

Fix a selector-count ceiling m and state ceiling s. Because the alphabets are finite, there are only finitely many labeled selector families within that budget. Let $N_{m,s}(\pi)$ denote their number.

If $(m,s)\notin \operatorname{Sel}(A,\pi)$ , there is a finite set $W\subseteq\pi(A)$ with $|W|\le N_{m,s}(\pi)$ such that every admissible selector family misses at least one word in W.

For each accepted word w, collect the candidate families that miss it into a failure set $D_w$ . Budget failure is exactly

\bigcup_{w\in\pi(A)}D_w=\mathcal F_{m,s}.

The minimum number of test words is therefore an exact finite set-cover number. The T169 escape length has an equally exact certificate meaning:

e_{A,\pi}(m,s) =\min_W\max_{w\in W}|w|.

Thus obstruction cardinality and obstruction radius are separate resources. For the hard visible lift, every finite budget has both a finite witness suite and a finite optimal radius, although the theorem does not construct or length-bound those suites.

The audit checks 65,536 budgeted duality cases and 32,832 obstruction instances. In the bounded model, 25,392 instances need one witness word and 7,440 need two.

This is an existential finite certificate theorem. It does not make the certificate computable, and its explicit family count can overcount equivalent labeled machines. It bounds witness cardinality, not witness length.

Research note · September 24, 2026 · version 7

Pumping repeats a typed context, not necessarily a machine state

A pumping argument needs both a sound way to repeat a segment and a reason that a suitable segment must occur. The existing full-unfolding equivalence supplies the first, not the second. This is a standard congruence principle applied to scaffold computations, with no novelty claim.

Write $C(u)$ for the control and rooted graph after prefix u. If y is nonempty and $C(x)\sim C(xy)$ , meaning equal controls and bisimilar full rooted unfoldings, then

M(xy^jz)=M(xz)\qquad(j\ge0,\ z\in\Sigma^*).

A finite relation matching labels and all ordered successors checks the graph condition. Equal observations choose the same update; adjoining the fresh root pair preserves the relation. Induction proves safe repetition followed by any common suffix.

Two limits. Equal fixed-radius neighborhoods can hide different buried data that later cursor steps expose. Conversely, a one-state machine accepting every word can append a fresh zero-labelled chain node per symbol. Every cut has a different full unfolding, although its language is regular. Preserving an archive is stronger than preserving its client's answers.

A finite summary of complete configurations, closed under input updates and determining acceptance, is itself a DFA. To reach beyond regular languages, a cut-and-splice theory therefore needs structured or parameterized interfaces, not just a finite list of whole-machine states.

Let $F(C(w))$ be the minimal bisimulation quotient of the records reachable from the current root. If its size is uniformly at most B, then the scaffold language is regular, with a DFA of at most

N_B=q\sum_{b=1}^{B}b(g+1)^b(b+1)^{db}

states. Hence the ordinary pumping lemma holds with pumping length $N_B$ .

The reason is exact. Old records unreachable from the current root can never be reached again, while FPRD-T129 makes the reachable bisimulation quotient closed under every next input. There are only finitely many rooted labelled port graphs of bounded size, so control plus the minimal figure is a DFA state.

This gives a presentation-level characterization: a language is regular exactly when it has some scaffold presentation with uniformly bounded minimal figures. A DFA can keep all language state in finite control while allocating the same one-node self-loop figure. A different presentation of that same regular language may still retain an unbounded archive.

More generally, if configurations are well-quasi-ordered, every letter update is monotone, and acceptance is upward closed, then some uniform P makes every accepted word of length at least P factor as $xyz$ with $|xy|\le P$ , $|y|>0$ , and

xy^jz\in L\qquad(j\ge1).

This is the well-structured transition-system recurrence mechanism. It is one-sided: a later configuration covering an earlier one supports additional copies, but not deletion. Zero-copy pumping requires equivalence or a separate retraction. Applying this to scaffolds now requires an observationally stable extension order; ordinary graph inclusion is not compatible with missing-versus-present tests.

Exact unbounded stacks cannot carry such an order. If a preorder preserves the observable top symbol, including the empty marker, and makes pop monotone, then

s\preceq t\quad\Longrightarrow\quad s=t.

It is therefore not a WQO on the infinite stack space.

Repeated monotone pops would force every successive top observation to agree, which uniquely determines the entire stack. This is why visible-stack pumping needs two coordinated wings: the later wing removes exactly the frames introduced by the earlier one before the outer continuation resumes.

More generally, let finite observations and a finite family D of destructive probes define complete probe-tree equivalence $\equiv_D$ . Any preorder that preserves observations and makes every probe monotone satisfies

x\preceq y\quad\Longrightarrow\quad x\equiv_D y.

For scaffolds, the probes follow ports and the complete probe tree is the full labelled unfolding from FPRD-T129. Therefore a nonregular scaffold presentation cannot obtain a WQO by making exact native observations and every port traversal monotone. A viable pumping order must abstract navigation, act on whole updates, or protect a parameter behind paired interfaces.

Paired pumping does not require the first wing to be a loop. Let V, X, and Y be the quotient transformations induced by words v, x, and y. If

Y\circ X\circ V(p)=X(p)

for every p in the forward V-orbit of the cut c, then

Y^j\circ X\circ V^j(c)=X(c)\qquad(j\ge0).

Operationally, v may add a frame or move an unbounded parameter forward. The later wing y retracts that move around the middle computation x. The law must hold at every deeper parameter, not only at the first cut. This is the algebraic core of the two wings in context-free and visible-stack pumping; recurrence of such a certificate remains a separate obligation.

In any finite typed derivation system, a sufficiently deep derivation repeats a boundary type on one path. If the tree is $C[K[t]]$ , the repeated one-hole context gives valid derivations $C[K^j[t]]$ for every $j\ge0$ .

This classical tree-pumping principle contains ordinary finite automaton and context-free pumping as unary and branching cases. For visibly pushdown grammars, a repeated variable types an operation on an opaque stack tail. One pumped wing builds a frame and the paired wing later removes it. The persistent graph grows with every copy; only the parameterized stack operation recurs.

w=uvxyz\quad\Longrightarrow\quad uv^jxy^jz\in L \qquad(j\ge0).

LMR's Theorem 21rules out a universal total effective pumping procedure for PEGs. Checking a supplied certificate does not guarantee that one exists. Visible-stack scaffolds now give a concrete nonregular positive case by classical grammar recurrence. The remaining task is an intrinsic scaffold condition that extends beyond known grammar classes.

The audits check 106,929 graph pairs, 118,030 common-row extensions, 16,739 typed paths, 77,540 pumped Dyck runs, 23,270 no-resurrection instances, 22,500 live-quotient extensions, and 1,004,550 monotone one-sided pumps. Proof and scope · Checker · Typed-context checker · Live-quotient checker · WQO checker · Paired-retraction checker · Stack-WQO checker · Destructor-system checker · Destructor-separation output. Version 4 adds the WQO one-sided pumping theorem. Version 5 adds the paired-retraction theorem and its orbit-wide hypothesis. Version 6 adds the exact-stack WQO obstruction. Version 7 adds the destructor-separation theorem. No circulated paper or PDF changed.

Research note · September 24, 2026 · version 1

Distance-two scaffold locality does not bound graph width

There is a one-state, one-label scaffolding automaton with degree two and distance two that builds every finite rectangular grid exactly. For r rows and c columns, it reads

w_{r,c}=AH^{c-1}(VC^{c-1})^{r-1}.

The rows alternate direction. Port 0 follows the row being built. Port 1 reaches the cell above. At each C step, the two-hop address $(1,0)$ moves from the preceding new-row cell to its upper neighbor and then across the oppositely oriented old row. Induction gives

G(w_{r,c})\cong P_r\mathbin{\square}P_c.

Because square grids have unbounded treewidth, bounded degree and bounded update distance do not make the persistent archive tree-like. This is only a structural calibration: the sole state can be accepting, so the same machine may recognize the universal language.

This rules out one direct extension of typed-context pumping. Bounded-width graph contexts cannot be inferred from scaffold locality alone. A broader pumping interface must constrain how the controller uses the archive, rather than only the topology of everything retained in it.

The checker constructs every rectangle through 32 by 32 and verifies 278,784 cells, 523,776 undirected edges, and 246,016 length-two descriptor evaluations. Construction and proof · Checker · Audit output. No circulated paper or PDF changed.

FPRD-T175 · stack-clock dichotomy

Letter projection gives exact synchronization or unbounded skew

For an observed word w, let $\Delta_\pi(w)$ be the largest difference in final stack height among its compatible hidden lifts.

If calls exist, the clock diameter is uniformly bounded if and only if every visible-letter fiber has one action type. In that case it is identically zero. Otherwise, for every n, some word of length at most 2n has diameter at least n.

If an ambiguous fiber contains a call, repeating that visible letter separates a call lift from a noncall lift immediately. If the ambiguity is only between local and return, first use n calls to reach height n, then repeat the ambiguous letter n times. One lift stays at height n and the other returns to zero.

\sup_w\Delta_\pi(w)<\infty \quad\Longrightarrow\quad \Delta_\pi(w)=0\ \text{for every }w,

apart from the degenerate no-call case, where the stack is never populated and the model reduces to finite-state control. There is therefore no projection-level hierarchy of clock bounds between exact synchronization and unbounded divergence.

The audit exhausts all 399 nonempty action-fiber patterns on observed alphabets through size three. It verifies 52,084 zero-diameter words and 2,680 explicit linear-divergence witnesses.

This concerns all lifts compatible with the projection. A source language can forbid the divergent histories. Any genuine intermediate positive tier must therefore be source-conditioned, selector-conditioned, or use a context-dependent coordinate map.

FPRD-T174 · synchronized stack selection

A shared stack clock keeps hidden-choice coverage effective

Suppose A is deterministic visibly pushdown and the hidden-letter projection preserves calls, returns, and local actions. Every compatible hidden lift then pushes, pops, or stays local at the same observed positions.

For every fixed selector family, the accepted observations it misses form a visibly pushdown language. Coverage is decidable, and every fixed selector budget has a computable finite escape horizon.

The hidden source run and all selected runs may store different stack symbols, but their heights remain equal. Zip the symbols pushed at one position into a tuple. One product stack then simulates every history and recognizes exactly

\pi(A)\setminus\bigcup_{i=1}^{m}T_i^{-1}(A).

If A has q controls and selector $T_i$ has $s_i$ states, the direct construction uses at most $q^{m+1}\prod_i s_i$ controls and the product stack alphabet $\Gamma^{m+1}$ . Standard VPA emptiness and shortest-witness algorithms therefore compute the exact failure radius for each candidate family. Taking the maximum over the finite candidate universe gives

(m,s)\in\operatorname{Sel}(A,\pi) \quad\Longleftrightarrow\quad (m,s)\in\operatorname{Sel}_{B_{A,\pi}(m,s)}(A,\pi)

for a computable horizon B. This isolates the hard boundary: forgetting labels is compatible with one stack, but forgetting whether each position pushes, pops, or stays local allows histories with independent stack clocks.

The audit checks 4,096 deterministic visibly pushdown machines, eight selectors, 36 selector families, and 40 observed words. It verifies 5,898,240 direct-versus-zipped language cases across 15,040,512 synchronized input steps.

The shared-stack closure and emptiness results are established visibly pushdown theory. T174 applies them to acceptance-blind selector spectra. It does not cover action-erasing projection or claim the classical ingredients as new.

FPRD-T173 · effective regular-source horizon

Regular hidden sources make selector failure effectively finite

Let A be recognized by an r-state DFA. Fix m synchronous selectors with at most s states each. A single missed-word NFA can track one possible accepting hidden lift and, for every selector, the lift that selector actually emits. Its language is

\pi(A)\setminus\bigcup_{i=1}^{m}T_i^{-1}(A),

and it has at most $r^{m+1}s^m$ states.

The selector family covers every accepted observation if and only if it covers those of length at most $r^{m+1}s^m-1$ .

If coverage fails, a shortest accepting run in the product NFA cannot repeat a state. Its input word is therefore shorter than the product-state count. In the notation of T169,

(m,s)\in\operatorname{Sel}(A,\pi) \quad\Longleftrightarrow\quad (m,s)\in\operatorname{Sel}_{r^{m+1}s^m-1}(A,\pi).

This turns the qualitative compactness theorem into a direct bounded-budget decision procedure for regular hidden sources. If a budget is infeasible, its T171 obstruction suite can be chosen with both bounded cardinality and bounded radius:

\omega_{A,\pi}(m,s)\le N_{m,s}(\pi), \qquad e_{A,\pi}(m,s)\le r^{m+1}s^m-1.

The audit exhausts all 1,024 complete two-state source DFAs, 10,240 one-state selector-family cases, and 262,144 two-state singleton-selector cases. It checks 4,287,052 product transitions and directly validates 105,532 shortest missed words.

The product bound is uniform but need not be tight. The argument is elementary finite-automaton theory and makes no priority claim. It does not give an effective horizon for the nonregular visibly pushdown lift used in T168.

FPRD-T172 · uniformization boundary

Causal selectors choose histories without recognizing the language

For a tagged source language A, define its accepting-lift relation $R_{A,\pi}=\{(w,\widehat w): \widehat w\in A,\ \pi(\widehat w)=w\}.$ A standard sequential uniformizer must have domain exactly $\pi(A)$ . An FPRD selector is total on every observed word and lets A decide afterward whether the selected history accepts.

If A is regular, finite causal selector covers are equivalent, up to finite state overhead, to synchronous multi-sequential subrelations of $R_{A,\pi}$ whose domains cover $\pi(A)$ .

An s-state selector restricted by an r-state source DFA gives a partial sequential component with at most sr states. Conversely, a p-state partial component can be totalized with one sink to give a selector with at most p+1 states.

The equivalence separates strictly on nonregular languages. For any nonregular L under the identity projection, the one-state identity machine is an acceptance-blind selector:

(1,1)\in\operatorname{Sel}(L,\mathrm{id}).

But the relation $\{(w,w):w\in L\}$ has no standard sequential uniformizer because every finite sequential transducer has a regular domain. This is why T168 is not a disguised ordinary-uniformization obstruction: it rules out even finitely many selectors that are not charged for recognizing the projected language.

The audit checks 1,024 two-state source DFAs, 126,976 product-domain comparisons, 317,440 selector-cover comparisons, and totalization of every partial one-state selector table in the bounded model.

Sequential uniformization and multi-sequential relations are established theory. T172 records their exact boundary with the acceptance-blind selector invariant. It does not claim novelty for the classical transducer constructions.

FPRD-T159 · first transport threshold

Distance one is exactly finite-state

For a scaffold of degree d, working-alphabet size g, and q controls, the current control and radius-one root view form a complete finite-state summary whenever the machine distance is at most one. Every transition reads this summary, and every new port can name only missing, SELF, the old root, or one old neighbour. The next radius-one view is therefore a deterministic function of the current summary and input letter.

\#\text{DFA states}\le qV_1(d,g) =q\bigl[1+(g+1)^{d+1}\bigr].

Its Myhill--Nerode index obeys the same bound, and its residual tower stabilizes by one less than that bound.

Conversely, a DFA is a distance-one scaffold that ignores its graph and appends a dummy node at every step. Thus

\mathsf{SA}_{\le1}=\mathsf{REG}.

The persistent selector from FPRD-T151 has an explicit degree-two, distance-two scaffold, while its length-n data prefixes realize $2^n$ different residuals. It is not regular. Consequently,

\mathsf{REG}=\mathsf{SA}_{\le1} \subsetneq\mathsf{SA}_{\le2}.

Distance two is the first scaffold tier where persistent graph memory increases recognition power beyond finite control.

This is the constant-modulus case of the readable-cone law. For $k>1$ , graph agreement through radius $(k-1)h+1$ forces response agreement through horizon h. At distance one, that radius becomes constant, so one finite view determines the entire infinite residual.

The audit compares 12,264 scaffold runs with the compiled DFA, checks 78,336 finite-scaffold transitions and 78,192 pairs with the same summary, and rechecks the strict distance-two selector on 88,573 words and 2,046 residual profiles.

A targeted literature search did not locate this distance-one characterization. The proof is an elementary consequence of the native model and FPRD-T153, and no priority or independent-review claim is made.

FPRD-T162 · cell-probe transport obstruction

A data-structure lower bound can exclude every finite scaffold presentation

Kim and Park define an ((a,b))-SCA encoding of a three-stage Boolean problem: preprocessing builds a persistent state, update blocks have length at most (a(n)), and a final query suffix has length at most (b(n)). Their transfer theorem turns any SCA for the encoding language into an exact deterministic cell-probe data structure with proportional operation costs.

If no such data structure has word size (Theta(log n)), update time (O(a(n))), query time (O(b(n))), and polynomial space, then

\operatorname{Tr}(K)=\varnothing, \qquad \delta_{\mathrm{str}}(K)=\infty.

The proof is contraposition. Finite structural degree gives an SCA by FPRD-T161. The Kim-Park transfer would then produce the forbidden data structure. This excludes all finite resource tuples at once, rather than defeating one proposed encoding.

Their hard language encodes Multiphase Inner Product with one-symbol updates and a final query of length $O(\log n)$ . Ko's superlogarithmic cell-probe lower bound excludes the data structure that an SCA would provide.

This is a source-attributed corollary. The local checker covers only parameter arithmetic and implication direction. It does not reverify either external theorem or the Kim-Park Lean artifact.

FPRD-T163 · reversal asymmetry

Stationary transport can be finite in one direction and impossible in the other

For the record-first language C of Kim and Park, their main separation gives

C\in\mathsf{LIN}\cap\mathsf{PEG}\setminus\mathsf{SCA}, \qquad C^R\in\mathsf{LIN}\cap\mathsf{SCA}\setminus\mathsf{PEG}.

Translating SCA membership through FPRD-T161 yields

\delta_{\mathrm{str}}(C)=\infty, \qquad \delta_{\mathrm{str}}(C^R)<\infty.

The persistent representation is therefore genuinely directional, even inside the linear context-free languages. This confirms the Loff-Moreira-Reis conjecture that PEG languages are not closed under reversal. It does not settle the separate palindrome question.

FPRD-T161 · structural scaffold normal form

One label and two controls suffice

Let a source scaffold have tuple $(d,k,g,q)$ with $k\ge1$ . Combine the shortcut ports of FPRD-T160 with

m=\left\lceil\log_2(1+gq)\right\rceil

marker ports. Reserve the all-zero marker word for the distinguished base node. Encode every ordinary source label-control pair by a nonzero word: a one bit is SELF and a zero bit is missing. Every created physical node can now use the same neutral working label.

From the current top, its markers reveal the source control. One shortcut hop reaches the endpoint of every source path through length k, and one marker hop reveals that endpoint's source label. A radius-two transition therefore reconstructs the exact source state and radius-k view. Its two physical controls store only whether the simulated source state accepts.

(d,k,g,q)\in\operatorname{Tr}(L) \Longrightarrow \left(\sum_{\ell=1}^{k}d^\ell+ \left\lceil\log_2(1+gq)\right\rceil, 2,1,2\right)\in\operatorname{Tr}(L).

\operatorname{Tr}(L)\ne\varnothing \quad\Longleftrightarrow\quad \exists D\;(D,2,1,2)\in\operatorname{Tr}(L).

Thus scaffold recognizability has a one-dimensional canonical existence slice. Define $\delta_{\mathrm{str}}(L)$ as the least D on that slice, or infinity when none exists. Then L is scaffold-recognizable exactly when this structural degree is finite. The full four-coordinate spectrum remains meaningful for succinctness and Pareto tradeoffs.

The audit checks 3,060 compiled runs, 18,456 exact transitions and state/view reconstructions, 315,280 shortcut edges, 270,240 marker bits, and every one of 1,296 ordinary label-control codes for alphabet and control sizes through eight.

Structural encoding is a standard pointer-machine idea. A targeted search did not locate this exact scaffold normal form. No novelty, priority, or independent-review claim is made.

FPRD-T160 · distance-two normal form

Two hops give the full power of every finite distance

Let a scaffold have degree d and distance $k\ge2$ . Give every compiled node one shortcut port for each nonempty original port word of length at most k. The new degree is

D_{d,k}=\sum_{\ell=1}^{k}d^\ell.

The compiled radius-two view observes each shortcut endpoint with one further layer, distinguishing the base from a missing endpoint. It reconstructs the original native radius-k view, so the compiled transition chooses exactly the same control, label, and direct original ports.

To install a new shortcut, symbolically follow its original word from the new node. Repeated SELF choices stay at the new node. A missing choice stays missing. Otherwise the walk exits once by an old-root path t and then follows a remaining suffix s inside the old graph. The endpoint is named by at most two compiled hops, $[t][s]$ . Therefore

(d,k,g,q)\in\operatorname{Tr}(L) \Longrightarrow (D_{d,k},2,g,q)\in\operatorname{Tr}(L).

\mathsf{SA}_{\le1}=\mathsf{REG} \subsetneq \mathsf{SA}_{\le2}=\mathsf{SA}.

This completely classifies the distance coordinate for recognition power. Higher distance may give a much smaller degree, but it cannot recognize another language. Every scaffold-recognizable nonregular language has minimum distance exactly two.

The audit checks 6,132 compiled runs, 43,032 exact view reconstructions, 423,148 shortcut edges, and 18,697,728 symbolic endpoint equalities. The exhaustive symbolic cases include repeated SELF, empty old-root paths, old-node cycles, and missing paths.

Shortcut closure is a standard pointer-machine idea. A targeted search did not locate this exact scaffold normalization. No novelty, priority, or independent-review claim is made.

Strict calibrations

Every compression step can be arbitrarily large

A program may query all n bits and return only the first. Its trace envelope has n coordinates, while its semantic support and quotient need one bit.
Parity depends on all n coordinates but has two answers. Its semantic carrier has n bits, while its quotient has one.
The vector of all coordinate queries has n essential bits and exactly $2^n$ quotient states. The carrier bound is tight.

The audit exhausts 65,814 scalar Boolean maps and 65,812 two-component Boolean maps, checking 131,626 functions in total. It verifies every candidate support subset, the carrier bound, the component-union law, and strict examples through twelve coordinates.

Research strategy

The remaining resource is stationary transport

FPRD-T150 supplies operational replay certificates. FPRD-T155 extracts their semantic core. FPRD-T152 gives the exact quotient memory demanded by a future family. FPRD-T156 supplies its canonical inverse-limit dynamics. FPRD-T157 turns bounded physical realizations into a finite-basis spectrum. FPRD-T158 proves that fixed finite future observers stay inside that spectrum calculus. FPRD-T159 identifies its first exact threshold: distance one is precisely finite-state. FPRD-T160 proves that distance two already has the full power of every finite scaffold distance. FPRD-T161 then moves every finite working alphabet and source control into local marker geometry. FPRD-T162 now supplies an encoding-robust way to prove that a spectrum is empty: reduce bounded stationary transport to a dynamic data structure excluded by a cell-probe lower bound. FPRD-T163 provides the first externally established infinite-degree calibration and shows that structural degree is directional under reversal. FPRD-T166 separates deterministic causal recoding, which preserves bounded transport with the same graph, from existential hidden-history projection, which need not. FPRD-T167 extends the positive boundary to every fixed finite selector family. FPRD-T168 proves that the hard visible lift lies beyond that entire tier. FPRD-T169 turns this global failure into finite escape: every fixed selector-count and selector-state budget is defeated at some finite input horizon. FPRD-T170 gives the resulting spectra a chain rule, so a multi-stage hidden-history obstruction can be localized to a constituent projection. FPRD-T171 then gives every failed finite budget a bounded-cardinality obstruction suite and identifies escape length with the suite's optimal radius. FPRD-T172 locates this selector calculus relative to classical uniformization: the two agree on regular source presentations, while acceptance-blind selection remains meaningful on nonregular domains. FPRD-T173 makes the regular tier effective: for each fixed selector budget, global coverage collapses to one explicit finite horizon and both obstruction resources become computably bounded. FPRD-T174 extends that effective tier to deterministic visibly pushdown sources whenever all histories retain one shared stack clock. FPRD-T175 proves that letter projections have no nontrivial finite-skew tier between this exact synchronization and unbounded clock divergence.

The hard pattern is delayed, growing access demand: first build persistent state through cheap updates, then reveal which old information the final query needs. Any fixed finite menu of right-context questions can still be absorbed into bounded locality and finite control, and every deterministic finite-state recoding can be absorbed into finite control alone.

Persistent stacks solve transport when addresses move along a simple cursor. FPRD-T164 now closes the universal forgotten-action question negatively: projected deterministic visible languages are exactly CFL, and not every CFL is PEG. The remaining task is structural. A visible lift of the Kim-Park language C is the hard calibration for stationary frontier transport, while the existing allocator theorems describe positive subclasses.

The cell-probe separation and its language witness are due to Kim and Park, using Ko's lower bound. FPRD contributes the translation into transport-spectrum and structural-degree terms. No independent claim to their results is made.