Derived-control prefix parity: hostile audit

Result

The proposed q=2, lambda=1, reversed-one-hot, one-midpoint-fold construction is correct in the natural transition-then-emit model. It is also a decisive obstruction to any atlas compression that quotients only the finite controller state: the controller has two states, but the emitted substrate has 2^L distinguishable late-query rows at grade L.

This does not contradict the literal-controlled WFL parity lower bound. Each emitted site is no longer a raw input literal. It is the derived predicate

p_t = b_1 xor ... xor b_t,

computed causally by the controller. The construction transfers the exponential raw-literal complexity into an append-only output word.

Exact construction

For data b_1...b_L, maintain the controller state

p_t = p_{t-1} xor b_t, with p_0=0,

and emit one site value a_t=p_t after reading b_t. Append a selector block y_1...y_L with

y_{L+1-i}=1

and every other selector site zero. A fold at the boundary between the two length-L blocks pairs data cell t with selector cell L+1-t. Therefore final Look is

OR_t (a_t AND y_{L+1-t}) = a_i = p_i.

The construction uses two states, one emitted site per source symbol, one fold, and one final Boolean Look.

The checker exhaustively verified every word and every selected prefix through L=8 (3,586 query cases).

Small-resource audit

Within this exact streaming/pairing model:

• q=1 is impossible already at L=2: the emitted value at position two can depend only on b_2, but the requested value is b_1 xor b_2. Exhaustive enumeration finds zero of four one-state binary Mealy machines.
• q=2 is sufficient. Of all 256 binary two-state Mealy machines with initial state zero, exactly one realizes every prefix parity through L=5: both its transition and emission tables are XOR.
• lambda=0 cannot realize a nonconstant selected response, so one emitted site is minimal.
• The midpoint fold is the only resource that couples an old answer site to a late selector site in this architecture. With no such coupling, final Look has no selected conjunction. Thus one fold is minimal for this compilation (this is a claim about the stated paired-site architecture, not every possible enriched WFL doctrine).
• Reversal is forced by the midpoint pairing: data cell t meets selector cell L+1-t. A forward one-hot selector asks for the reversed data position and cannot uniformly repair the last-position dependency causally.

There is one timing caveat. A strict pre-transition Moore output cannot emit p_1 for both first symbols because its initial output is independent of b_1. Transition-then-emit Mealy semantics works exactly. A strict Moore convention needs a one-cell delay plus a terminal flush, or an equivalent indexing shift. This is an implementation convention, not the semantic memory resource.

Right-congruence obstruction

Let

E(u) = (p_1(u),...,p_|u|(u))

be the already emitted substrate word and let s(u) be the two-state controller parity.

The relation u ~state v iff s(u)=s(v) is a right congruence for future incremental emissions: after the old output is factored away, equal states have equal residual transductions. It is not a congruence for the permitted late-query semantics. For example,

u=00, v=11

have the same controller state zero, while

E(u)=00, E(v)=10.

A later selector for i=1 distinguishes them.

More generally, E is injective:

b_1=p_1 and b_t=p_{t-1} xor p_t.

Hence, at grade L, the semantic relation that equates prefixes only when all legal late selectors agree has exactly 2^L classes. Every pair of distinct same-state words is separated by some late selector. The checker confirms:

Thus “future-row equivalence” has only two possible readings here:

1. If it ignores selectors that can address the old substrate, it has two classes and is right-invariant, but is unsound.
2. If it includes every legal late selector, it is sound and right-invariant, but is equality of the full emitted trace and has exponential grade index.

There is no finite two-class semantic quotient between these readings.

Nor can an unindexed aggregate of the emitted substrate repair the quotient. For example, data words 110 and 011 both finish in state zero and emit, respectively, 100 and 010. These substrates have the same length and the same number of true sites, but selector i=1 distinguishes them. More generally, any complete scalar or finite-profile substitute must itself separate all 2^L indexed traces at grade L; site count or another permutation-invariant “height” cannot do so.

Exact missing coordinate

The missing coordinate is the emitted substrate word E(u).

In subsequential-transducer language, the mismatch between E(u) and E(v) can be called an output delay. Here arbitrary late indexing prevents useful cancellation, so retaining only a bounded delay statistic is insufficient: the complete word-indexed profile is observable. Grade advice is not the semantic memory source. Given an allowed affine midpoint fold and a reversed selector block, the grade supplies alignment only. If a doctrine permits only absolute crease positions, grade advice would be separately necessary, but it would not repair state-only equivalence.

Accordingly, a rooted-atlas/right-congruence compression theorem must either:

• include the emitted substrate (or an exactly equivalent legal-head-bond profile) in its configuration;
• charge the number of distinguishable emitted words/canonical height; or
• restrict legal future bonds so that old substrate positions are no longer independently addressable.

For this construction the legal future bond profile is precisely the indexed word E(u), so bounded head-bond summaries cannot be complete.

Checker

check_derived_prefix_parity.py verifies the construction, performs the one- and two-state machine census, exhibits the strict-Moore timing issue, and counts the state-versus-semantic congruence classes through grade eight.