Version 7 generalizes the stack obstruction. For any finite observation and finite family of destructive probes, an observation-preserving preorder that makes every probe monotone can compare only configurations with identical complete probe trees. Scaffold port probes generate full labelled unfoldings, so every nonregular scaffold presentation has too many residual figures for such an order to be a WQO. A viable pumping theory must abstract navigation, order whole updates, or use paired opaque interfaces.
Version 6 proves that an unbounded exact stack cannot use the WQO mechanism from version 4. Any preorder that preserves the observable top, including emptiness, and makes pop monotone collapses to equality, so it is not a WQO. This explains why stack pumping needs the paired-retraction mechanism: a later wing must remove exactly the frames introduced by the earlier wing before the outer continuation resumes.
Version 5 isolates the operational algebra behind the two wings of context-free pumping. The first wing need not loop: it may advance an unbounded parameter, provided the second wing retracts that move around the middle computation at every point of the forward orbit. The resulting sandwich identity proves uv^jxy^jz equivalent to uxz for all j>=0. This supplies soundness, not recurrence, and checking only the first cancellation is insufficient.
Version 4 connects FPRD pumping to well-structured transition systems. A finite-alphabet deterministic system with a well-quasi-ordered configuration space, monotone letter actions, and upward-closed acceptance has a uniform pumping length, but only for one or more copies of the repeated block. Deleting the block requires equivalence or a separate retraction. The remaining scaffold problem is to find an observationally stable extension order; ordinary graph inclusion fails.
Version 3 proves that unreachable scaffold records can never return and that a uniformly bounded minimal bisimulation quotient of the live records compiles to a finite automaton, with an explicit state and pumping bound. A language is regular exactly when it has some scaffold presentation with this property. Nonregular pumping must therefore repeat parameterized operations on an unbounded live quotient, as visible-stack contexts do.
A one-state, one-label, degree-two, distance-two scaffold builds every finite rectangular grid by alternating row direction. Square grids have unbounded treewidth, so local update bounds do not imply tree-like persistent memory. This is a structural calibration, not a language separation: the machine may accept every word. Any graph-context pumping theorem must constrain controller use or reachable interfaces, not locality alone.
Version 2 identifies the classical unifying mechanism behind regular, context-free, and visible-stack pumping: a repeated finite boundary type in a derivation tree yields an iterable one-hole context. For persistent visible stacks, paired wings build and discharge a frame above an opaque tail even though the allocated scaffold graph never repeats. This supplies a concrete nonregular positive case while leaving unrestricted PEG recurrence impossible.
FPRD-T175 sharpens the synchronized-stack boundary. If calls exist, a letter projection either preserves one action type in every visible-letter fiber, giving exact stack-height agreement, or compatible hidden lifts diverge by n after at most 2n observations for every n. Thus there is no projection-level hierarchy of finite skew bounds between T174 and forgotten-action projection. Any intermediate positive tier must use source-language or selector structure.
FPRD-T174 extends effective selector compactness from finite automata to deterministic visibly pushdown sources when projection preserves call, return, and local actions. All source and selected runs then have equal stack height and zip into one visibly pushdown product. Fixed-budget coverage, shortest missed words, and obstruction horizons are computable. The hard forgotten-action tier begins precisely when projection destroys this common stack clock.
FPRD-T173 makes the selector obstruction theory effective for regular source presentations. For an r-state hidden-source DFA and a family of m selectors with at most s states each, global coverage is equivalent to coverage through length r^(m+1)s^m-1. The same bound controls obstruction radius and gives a direct fixed-budget decision procedure.
FPRD-T172 calibrates the selector spectrum against established transducer theory. For regular source languages, finite selector covers are exactly domain-covering multi-sequential subrelations of the accepting-lift relation, up to finite state overhead. On nonregular domains the notions separate: a one-state identity selector exists for every language, while a standard sequential uniformizer must have a regular domain.
FPRD-T171 turns hidden-choice failure into an exact finite duality. For a fixed selector-count and state budget, accepted words define failure sets over the finite candidate-family universe. Infeasibility is exactly a set-cover condition, giving an explicit bound on certificate cardinality. The escape length from T169 is precisely the smallest possible maximum word length of such a certificate.
FPRD-T170 proves a chain rule for causal selection spectra. Selector profiles (m,s) and (n,t) for two successive hidden-coordinate maps compose to (mn,st), both globally and at every finite horizon. Therefore an empty composite spectrum forces an empty spectrum at some individual stage. Sequential transport compilation also retains the hierarchy's shared outer states and improves the resulting control bound.
FPRD-T169 defines the causal selection spectrum by selector count and selector-state size. Its finite-horizon spectra decrease exactly to the global spectrum: a fixed budget works at every finite horizon only if one finite selector family works globally. Thus the hard visible lift from T168 defeats every fixed budget on some finite input horizon, although the theorem gives no computable bound on that escape length.
FPRD-T167 shows that any fixed finite family of synchronous finite-state selectors preserves bounded stationary transport, with an explicit product bound. FPRD-T168 applies the contrapositive to the Kim-Park witness: its deterministic visible lift has a degree-one, distance-two scaffold, but after action tags are hidden its projection has empty transport spectrum. No finite family of online finite-state selectors can cover all required accepting lifts.
FPRD-T166 shows that the inverse image under an s-state synchronous Mealy transducer changes a transport tuple (d,k,g,q) only to (d,k,g,sq), while building exactly the same persistent graph. Finite-state synchronous coordinate changes with finite-state inverses therefore preserve bounded transport. This isolates why forgotten-action projection can fail: it existentially merges hidden histories rather than deterministically recoding the observed input.
FPRD-T91 identifies forgotten-action projections of deterministic visibly pushdown languages with CFL. Kim and Park's linear context-free language outside PEG therefore closes FPRD-C02 negatively. Applying the visible-lift theorem to both sides of their reversal pair separates two obstructions: C is PEG but not SCA and is the stationary-transport-hard frontier, while C^R is SCA but not PEG and is the direct counterexample to the universal PEG claim.
Kim and Park's SCA-to-cell-probe transfer gives an encoding-robust obstruction to every finite scaffold presentation. In FPRD terms, a matching data-structure lower bound forces an empty transport spectrum and infinite canonical structural degree. Their linear witness C has infinite degree while C^R has finite degree, showing exact directional asymmetry. This is a source-attributed translation, not an independent reproof of their separation or Ko's lower bound.
The Scaffold proof review pins current editions and follows shared assumptions across the transfer, direct palindrome, causal algebra, and unary coherence papers. Run 01 records the native base-node example and edition baseline. Next is the exact relation between finite views and full behavior. Findings accumulate in the review activity before justified changes become matching numbered PDF and HTML editions.
Shortcut ports and local marker geometry jointly compile every tuple (d,k,g,q) to (sum d^i + ceil(log_2(1+gq)),2,1,2). The distinguished base uses the zero marker code; every ordinary label-control pair uses a nonzero SELF/missing code. Thus a language is scaffold-recognizable exactly when its transport spectrum meets the one-dimensional slice (D,2,1,2). Degree is the only unbounded resource needed for existence, although all four coordinates still matter for succinctness.