FPRD cycle: derived controls, observable output cocycles, and exact archive height

Status. The proposed state-only right-congruence compression is refuted for the derived-control WFL wrapper. A two-state encoder realizes every late-indexed prefix-parity query with an optimal one-fold, linear-site construction. The correct compression object is a finite normalized residual congruence together with the accumulated output/archive datum. A pinned rooted-archive doctrine gives an exact gauge-invariant height-volume theorem.

1. Pinned derived-control doctrine

At grade $L$, the input has two typed phases

$$b_1\cdots b_L\;\#\;e_L\cdots e_1,$$

where $b_i\in\{0,1\}$, the separator is a zero-width control event, and $(e_1,\ldots,e_L)$ is promised one-hot. The encoder is a deterministic Mealy machine. On every material symbol it appends one unit cell with one Boolean site at its half-integer center. The separator marks the integer boundary between the equal old and late blocks. One all-layers fold at that marked boundary and one final Boolean WFL Look are allowed.

The relative marked crease and reversed unary query are load-bearing. Their strict alternatives are recorded below.

2. Exact two-state construction

Define the cumulative bits

$$z_i=b_1\oplus\cdots\oplus b_i,\qquad z_0=0.$$

The encoder has state $q\in\{0,1\}$, initially zero. On old symbol $b_i$, it performs

$$q\leftarrow q\oplus b_i$$

and emits the derived site value $z_i=q$. On late symbols it copies each arriving unary selector bit into its emitted site.

The old $z_i$-site lies at

$$x_i=i-\tfrac12,$$

and the late $e_i$-site, because the query arrives in reverse order, lies at

$$y_i=2L-i+\tfrac12.$$

Reflection about the marked integer crease $x=L$ sends

$$2L-x_i=y_i.$$

Thus the fold creates precisely the $L$ depth-two fibers $\{z_i,e_i\}$, and

$$\operatorname{Look}(b,e) =\bigvee_{i=1}^{L}(z_i\wedge e_i). \tag{1}$$

On the one-hot input selecting $i$, (1) returns $z_i$.

Theorem 1 (derived-prefix exact image). For every $L\ge2$, the late-indexed prefix-parity target has a uniform derived-control realization with $$\boxed{ q=2,\quad\lambda=1,\quad S=L,\quad M=2L,\quad D=2,\quad F=1. } \tag{2}$$ It uses one final Boolean Look and no query buffer.

Here $S$ is the number of effective fibers/terms, $M$ the number of material sites, $D$ maximum fiber depth, and $F$ the fold count.

3. Matching fiber and site lower bounds

After the triangular change of old coordinates $b\mapsto z$, the target’s answer matrix is unary INDEX:

$$I(z,i)=z_i.$$

The accepted pairs

$$\{(\mathbf e_i,i):1\le i\le L\}$$

form a fooling set. If $i\ne j$, both cross-pairs are rejected. A one-fiber DNF term is a $1$-rectangle in the old/query answer matrix, so it contains at most one fooling-set pair. Therefore $S\ge L$.

In the separated-site doctrine, every useful fiber needs an old-dependent site, or it would accept the all-zero old row. For $L\ge2$, it also needs a query-dependent site, or it could not distinguish an accepting query column from a rejecting one. Fibers partition physical sites, so $M\ge2L$. Split dependence forces $D\ge2$, hence at least one nontrivial overlay fold.

Theorem 2 (optimal one-Look resources). Within the separated old/query site doctrine, every realization satisfies $$\boxed{S\ge L,\qquad M\ge2L,\qquad D\ge2,\qquad F\ge1.} \tag{3}$$ The construction (2) is exact on all four coordinates.

This is not merely an upper-bound collapse. It pins the complete static one-Look frontier after derived prefix controls are admitted.

4. Exact state/archive entropy split

The map

$$(b_1,\ldots,b_L)\longmapsto(z_1,\ldots,z_L)$$

is triangular and bijective, because

$$b_i=z_{i-1}\oplus z_i.$$

The complete late-query row is $R=(z_1,\ldots,z_L)$. For uniform old bits,

$$\boxed{H(R)=H(Z_1,\ldots,Z_L)=L.} \tag{4}$$

If the final parity state $Q=Z_L$ remains at the cut, then

$$\boxed{ H(R)=H(Q)+H(Z_1,\ldots,Z_{L-1}\mid Q)=1+(L-1). } \tag{5}$$

Equivalently, each of the two final control states accompanies exactly $2^{L-1}$ archived response rows. Thus the weighted finite law is exact:

$$\sum_{q\in\{0,1\}} |\{Z:Z_L=q\}|=2\cdot2^{L-1}=2^L. \tag{6}$$

The construction therefore saturates the conditional residual-row capacity theorem. It does not contradict the linear fiber/site charge. It refutes only the claim that bounded finite control bounds retained history.

5. Two congruences and the observable-cocycle obstruction

Let

$$G(u)=\text{the word of prefix parities emitted while reading }u.$$

The subsequential transduction $G$ has an output-normalized residual that depends only on the current parity. Indeed,

$$G(uv)=G(u)\cdot T_{\operatorname{par}(u)}(v), \tag{7}$$

where $T_p(v)$ is the prefix-parity word of $v$, bitwise toggled by $p$. The residuals $T_0,T_1$ are distinct, so the normalized Myhill–Nerode congruence has exactly two classes.

Late WFL observation changes the semantics. It can query any coordinate of the already emitted factor $G(u)$. Since $G$ is injective on words of fixed length, equality under every legal late selector has $2^L$ classes at grade $L$.

For example, $00$ and $11$ have the same final parity state zero, but

$$G(00)=00,\qquad G(11)=10,$$

and selector $i=1$ separates them.

Theorem 3 (observable output-cocycle obstruction). A finite output-normalized right congruence can compress future transition dynamics while failing completely to compress legally addressable output history. Any sound atlas quotient for the derived-control WFL cascade must retain $$\boxed{ \text{normalized residual class} \quad+\quad \text{accumulated output/archive modulo the legal observation kernel}. } \tag{8}$$ In the prefix-parity construction the observation kernel on the archive is equality, so the second coordinate has $2^L$ grade-$L$ values.

The emitted archive is a transition-output cocycle: finite control determines the next emitted bit, but the product of those transition outputs is an unbounded word. Calling it an “output delay” is standard transducer language. Because late WFL queries address the delay itself, it cannot be cancelled from the physical configuration.

This gives a decisive negative answer to the state-only version of the derived-control/right-congruence attack. It also repairs the conjectural atlas statement: the candidate invariant is a divisor/residual pair, not a right congruence alone.

6. Exponential-to-linear resource substitution

If sites are controlled only by raw old literals, fixing the query to the last prefix exposes parity on $L$ bits. Exact DNF then needs $2^{L-1}$ minterms and $L2^{L-1}$ old-literal occurrences. The earlier variant with an additional selector bit needs $2^L$ minterms.

With two-state derived prefix sites, the complete indexed family has exactly $L$ terms and $2L$ sites. Thus one bit of causal control changes the fiber demand from exponential to optimal linear:

$$\boxed{ \text{raw/current-literal control: }2^{\Theta(L)} \quad\longrightarrow\quad \text{two-state derived control: }L. } \tag{9}$$

A one-state Mealy encoder cannot emit cumulative parity already at grade two; exhaustive enumeration confirms zero of four binary one-state machines. Exactly one of the 256 binary two-state Mealy tables realizes all prefix parities through grade five: XOR transition and transition-then-emit XOR output.

This is a sharp, nonconvex resource-substitution example. The saved resource has not disappeared: it moved into $L$ bits of spatial output history.

7. A pinned canonical-height entropy theorem

The obstruction identifies a normalized storage doctrine in which the height-volume bridge is exact.

Define a rooted binary archive configuration to be a finite ordered word $z\in\{0,1\}^*$ placed at consecutive rooted site positions, together with the residual control state determined by its last bit. Quotient global translation of the entire rooted strip, but preserve order and the legal index bonds. Define canonical archive height by

$$h(z)=|z|. \tag{10}$$

The reachable height-$m$ sphere contains exactly $2^m$ normalized configurations, and the ball has volume

$$\boxed{ V(m)=\sum_{j=0}^{m}2^j=2^{m+1}-1. } \tag{11}$$

Consequently,

$$\boxed{ m<\log_2 V(m)<m+1\quad(m\ge1), } \tag{12}$$

while the exact sphere law is

$$\boxed{\log_2 |S_m|=m=h.} \tag{13}$$

If $2^L$ pairwise distinguishable response rows are represented inside a height-$m$ ball, (11) forces $m\ge L$. The derived prefix construction has $m=L$, so the escape bound is exact.

This is a genuine gauge-invariant height theorem for the pinned rooted archive doctrine. It is not yet the repository’s gauge-reduced tropical canonical-height theorem. It establishes the necessary calibration: archive extent and indexed bond structure must contribute to height or to the associated volume.

Counting only open sites is insufficient. Words $100$ and $010$ have the same length, final state, and number of open sites, but a late selector distinguishes them. If substrate extent is uncharged, even configurations with one open site have infinite volume because that site may occupy arbitrarily many addressable positions.

8. Comparator status

The standard machinery is:

• Choffrut’s output-normalized Myhill–Nerode theory for subsequential functions: https://doi.org/10.1016/S0304-3975(01)00219-5.
• Bianchini and Puppis, Minimization of Streaming Transducers, LICS 2026, published 9 July 2026: https://doi.org/10.4230/LIPIcs.LICS.2026.18. Their configuration is explicitly a control-state/data pair; the data is write-only and separate from control. Their divisor vector and residual matrix are the exact standard blueprint for (8).
• Alur and Cerny’s streaming string transducers, which pair finite control with unbounded output registers: https://doi.org/10.4230/LIPIcs.FSTTCS.2010.1.
• Benalioua, Lhote, and Reynier’s 2026 state/register Pareto study for appending streaming string transducers: https://arxiv.org/abs/2604.11567.
• The static answer-matrix view of Gál and Miltersen: https://doi.org/10.7146/brics.v10i44.21816.

None of those sources proves the WFL construction or its geometric lower bound. The new FPRD content is the exact cascade from finite-state derived output to a fold-addressable archive, the $L$-fiber/$2L$-site optimum, the separation between normalized residual and historical Look congruences, and the rooted-archive height calibration.

The latest minimization theorem does, however, sharpen the next proof obligation. A computable WFL atlas compression requires:

1. associative composition of legal substrate updates;
2. a canonical common factor or epi-GCD for every vector of future Look updates;
3. right-extension closure of the observational quotient; and
4. finitary constrained domains for legal head-bond types.

Boolean Look and later fold incidence are noncancellative, so these conditions do not follow automatically from subsequential-transducer theory.

9. Hostile boundary and failed routes

• Reverse unary selector is load-bearing. A fixed finite-state encoder cannot expand a binary index into a reverse one-hot block. An ordinary-order unary selector needs reversal, buffering, or different geometry.
• Relative crease addressing is load-bearing. The construction assumes “fold at the marked phase boundary” or grade-indexed midpoint coordinates. A fixed finite absolute crease library cannot perform the construction uniformly. A grade-indexed coordinate costs $\Theta(\log L)$ description bits if midpoint is not primitive.
• A material one-cell delimiter breaks the integer-crease symmetry. A material repair uses two false padding cells around the crease, giving $2L+2$ physical cells while retaining $L$ effective terms.
• Strict pre-transition Moore output needs a delay/flush. The exact two-state, one-cell statement uses transition-then-emit Mealy semantics.
• Cross-phase derived sites are stronger still. If late emission may directly combine retained old state with the current query, the separated $2L$-site lower bound no longer applies and that computation must be charged.
• State plus aggregate site count is unsound. The indexed archive pattern, not only its size or Hamming weight, is observable.

10. Computational audit

check_derived_prefix_parity.py:

• verifies all 3,586 selected-prefix cases through $L=8$;
• enumerates all one- and two-state binary Mealy machines;
• confirms the unique XOR two-state realization;
• exhibits the strict-Moore timing obstruction; and
• counts all same-state pairs separated by a late selector.

check_derived_control_fold.py independently verifies the exact midpoint geometry, triangular row reconstruction, and the $2^{L-1}$-histories-per-state law through $L=8$.

11. Strongest result and exact next attack

Strongest current result. The derived-control WFL image for late unary prefix parity is exactly solved in the separated-site doctrine: $$(q,\lambda,S,M,D,F)=(2,1,L,2L,2,1),$$ with matching lower bounds, exact response entropy $L$, a two-class normalized residual congruence, and a $2^L$-class historical Look congruence. The rooted archive volume satisfies $\log_2 |S_h|=h$.

Still unproved. The analogous compression theorem for the restricted D30 atlas; existence and computability of epi-GCDs for actual WFL fold/site-update vectors; binary or ordinary-order late selectors without a charged reversal channel; and transfer to the repository’s tropical canonical height.

Exact next attack. Treat a rooted WFL atlas as a streaming transducer whose data are legal fiber-incidence carriers. Determine on the smallest nontrivial two-hinge subclass whether vectors of future Look updates admit a canonical greatest common divisor. A positive answer yields a divisor/residual right-congruence algorithm; a negative finite pair yields the first explicit dual obstruction to intrinsic atlas compression.