Derived prefix control: exact WFL construction and audit

Model pinned for this statement

For 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 append-only. A cell contains one Boolean site at its half-integer center. The separator marks the integer crease between the two equal material blocks. One all-layers fold at that marked crease and one final WFL Look are allowed.

The phrase “marked crease” is load-bearing: see the hostile audit below.

Exact construction theorem

Define

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

There is a uniform two-state Mealy encoder realizing

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

It uses exactly

• $2L$ material cells and $2L$ sites, one site per cell;
• $L$ final fibers and $L$ DNF terms;
• stack depth and DNF width $2$;
• $2L$ literal occurrences;
• one all-layers fold.

Construction

The Mealy state is $q\in\{0,1\}$, initially $0$. On an old symbol $b_i$, update $q\leftarrow q\oplus b_i$ and emit one site with value $q=z_i$. On the typed phase separator, mark the current material boundary and optionally reset $q$ to zero. On the late phase, copy each arriving bit into one site. Because the late phase arrives as $e_L,e_{L-1},\ldots,e_1$, no counter, grade advice, or reversal buffer is used.

Number cells from left to right. The old $z_i$-site is at

$$x_i=i-\tfrac12,$$

and the late $e_i$-site is at

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

Reflection about the integer crease $x=L$ sends

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

Consequently the fold creates precisely the $L$ disjoint depth-two fibers $\{z_i,e_i\}$. WFL incidence normal form gives

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

Under the one-hot promise, if $e_i=1$, the answer is $z_i$.

The construction is genuinely uniform in the symbol processing: the old transition is XOR, the late transition is copying, and all placement roles are stationary. Its geometric uniformity requires a relative instruction “fold at the marked phase boundary” (or an equivalent grade-indexed midpoint fold).

Exact fiber/site lower bound

Consider any one-Look incidence construction in which every site predicate depends on only the old phase or only the late query phase. On the one-hot query domain, every fiber is a combinatorial rectangle

$$A_j(z)\wedge B_j(e).$$

The $1$-matrix of $Q_L$, after the bijective change of old coordinates $b\mapsto z$, is the unary INDEX matrix $I(z,i)=z_i$. It has the fooling set

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

Indeed $I(\mathbf e_i,i)=1$, whereas for $i\ne j$, $I(\mathbf e_i,j)=I(\mathbf e_j,i)=0$. A $1$-rectangle can contain at most one fooling-set element. Hence every incidence DNF needs at least $L$ effective fibers.

Every useful fiber must contain at least one old-dependent site: otherwise it would also accept the all-zero old row. For $L\ge2$, it must contain at least one query-dependent site: otherwise a row accepted at column $i$ would also be accepted at a zero column $j$. Physical fibers are disjoint, so at least $2L$ sites are necessary. Split dependence also requires depth at least two; in this geometry a nontrivial overlay requires at least one fold.

Thus the displayed construction is simultaneously optimal in effective fiber count, site count, depth, and fold count within the separated-site doctrine.

Entropy equality and the two different congruences

The triangular map

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

is bijective, with inverse $b_i=z_{i-1}\oplus z_i$. Therefore, for uniform independent old bits,

$$H(R)=H(Z_1,\ldots,Z_L)=L.$$

The construction has $S_x=M_x=L$, so with a constant post-separator control mode it attains equality in the residual-row capacity law

$$H(R)\le H(C)+\min\{S_x,M_x\}.$$

If the final parity state is retained rather than reset, then the exact state/archive decomposition is

$$H(R) =H(Z_L)+H(Z_1,\ldots,Z_{L-1}\mid Z_L) =1+(L-1).$$

There are two non-equivalent notions of quotient here.

1. Output-normalized sequential residual. After the already emitted output history is factored away, future prefix-parity output depends only on the current parity $q$. The normalized right congruence therefore has exactly two classes.

2. Raw late-query row. The complete row $i\mapsto Q_L(b,\mathbf e_i)=(z_1,\ldots,z_L)$ is injective in $b$. Raw late-query equivalence therefore has exactly $2^L$ classes.

The two-state quotient compresses continuation dynamics, not historical access. A rooted-atlas compression based only on that quotient is therefore unsound for late WFL queries. It must be enriched by the legal head-bond or fiber-incidence carrier that links each reversed selector cell to the corresponding archived prefix site.

Consequence for the previous parity obstruction

The literal-controlled theorem forced $2^L$ DNF terms after restricting the late index to $i=L$, because parity in the original $b$-literals has only minterm implicants. Derived prefix sites change the doctrine: $z_L$ is now one emitted predicate, and the full indexed family uses only $L$ terms. Thus the exponential literal-occurrence obstruction does not survive derived preprocessing. The linear residual-row charge does survive, and the construction attains it.

Hostile audit

• Reverse unary query is essential. A binary late index cannot be expanded into a length-$L$ reverse one-hot block by fixed finite control. An ordinary-order unary block requires reversal, buffering, or a different fold construction.
• The separator cannot silently consume one material cell. Sites are at half-integers and legal creases at integers. One separator cell would put the desired reflection axis at a half-integer. A material version may instead use two false padding cells straddling the crease. Its physical counts are $2L+2$ sites and $L+1$ stack locations, while the simplified incidence DNF still has $L$ effective terms.
• Relative crease addressing must be legal or charged. If WFL permits only a fixed finite set of absolute crease coordinates, independent of grade, then the fold $x=L$ is not a uniform program. A marked-boundary fold repairs this. A nonuniform grade-indexed coordinate carries $\Theta(\log L)$ numerical description unless midpoint operations are primitive.
• One site per cell is respected. No cell contains both an old and a query site before folding; the conjunction arises only from the final physical overlay.
• Cross-derived late sites are a stronger doctrine. If a late site may directly compute a predicate of retained old state and the current query, the two-sites-per-fiber lower bound no longer applies. Such cross-phase preprocessing must be charged as state/query computation.