INDEPENDENT AI PROOF REVIEW — ITERATION 02
Reviewer: child agent /root/review_iteration02
Date: 4 November 2026 (Asia/Bangkok)
Scope: parent-supplied theorem proposals, independent exact checks, and targeted
primary-source inspection. This is AI review, not external peer review. No Git
commands or inherited-file writes were performed.

VERDICT

The main characterization is correct. For a finite transformation monoid S on
nonempty X, robust decodable memory partitions are exactly S-congruences whose
quotient action is a permutation group. A single minimum-rank idempotent suffices
to construct the finest such quotient, provided its kernel is closed under the
whole S-action. The construction must not be described as taking the raw kernel
alone, nor as simply choosing a quotient of size equal to the minimum rank.

The three-state irreversible example is correct, with an important refinement:
its only state automorphism is identity, while a nonbijective endomorphism induces
the nontrivial swap on the two memory classes. Irreversible physical-state maps
can therefore implement reversible recodings of retained memory.

1. FINITENESS AND THE FIRST EQUIVALENCE

Let r:X->R have attained alphabet K=r(X), and let R_r=ker(r). Recovery after F
means r=d_F composed with r composed with F. This implies ker(rF) subset R_r.
Since |(rF)(X)|<=|r(X)| and the kernel inclusion gives the reverse inequality on
numbers of blocks, the two finite partitions are equal. Thus recovery is
equivalent to BOTH forward invariance of R_r under F and injectivity of its
induced map on X/R_r. The latter map is then bijective because the quotient is
finite. Conversely its inverse supplies d_F on K, extended arbitrarily outside K.

This argument depends on using the same register r before and after and on
finite X. It cannot be imported unchanged for enrichment into a larger register
alphabet or for infinite state sets. The first iteration's general relation
between different registers is still a preorder; this special finite endomorphism
setting forces preservation up to a permutation of the attained symbols.

For all F in S, the induced permutations form a finite submonoid of Sym(K), hence
a group. The correspondence is a representation of the replacement monoid. Its
congruence is for S=End(A), not automatically for the task transition monoid.
An S-invariant partition need not be a predictive/task congruence.

2. FINEST QUOTIENT AND A SINGLE MINIMUM-RANK IDEMPOTENT

Choose a minimum-rank element of S. A positive power gives an idempotent e still
of minimum rank: the rank cannot drop below the global minimum. Let rho be the
least S-invariant equivalence containing x~e(x) for every x. Equivalently take
the S-congruence generated by ker(e), since e is idempotent.

In the quotient Q=X/rho, e acts as identity. For any s in S, ese has minimum rank
(it belongs to S and its image is contained in eX), and its restriction to eX
is a permutation: eX has the minimum number of states and ese(eX)=ese(X).
Every Q-class has a representative in eX because x~e(x). Therefore ese induces
a surjection on Q, and so does s, whose quotient action equals that of ese.
Since Q is finite, every s acts bijectively. This proves that rho gives a
permutation quotient without invoking an unproved minimal-ideal theorem.

Conversely any robust partition P gives a permutation quotient. An idempotent
permutation is identity, so x P e(x) for every x. As P is S-invariant it contains
rho. Hence rho is the unique finest robust partition and does not depend on the
choice of minimum-rank idempotent. The equivalent construction using all
idempotents is valid, but computationally unnecessary once a minimum-rank one is
available.

All robust partitions are the pullbacks of invariant equivalences on Q under
the resulting image group. Merely being coarser than rho is NOT sufficient;
invariance under the image group remains necessary. For the regular C3 action,
rho is discrete but the two-block partition {0,1}|{2} is not invariant.

For End(A), every F preserves all task-output words because it commutes with
every transition and preserves b. Thus x E F(x), where E is predictive
equivalence. E is itself an End(A)-congruence on which End(A) acts trivially.
It follows that rho subset E. The finest robust quotient can distinguish inside
predictive classes, but does not merge distinct predictive classes.

Additional verified distinction: rho itself IS also a task congruence. It is
the equivalence closure of the pairs (s(x),s(e(x))) for s in S. Applying a task
transition T sends such a pair to (s(T(x)),s(e(T(x)))) by commutation, another
generator. Therefore every T descends to Q, and b does too since rho subset E.
The inherited image group acts by automorphisms of this quotient Moore machine.
It need not equal the entire automorphism group of the quotient.

An arbitrary robust memory partition need not be a task congruence. For example,
T=(1,2,2), b=(0,0,1) has End(A)={id}. Thus r=(0,0,1) is robust, but T sends the
equivalent states 0,1 to nonequivalent states 1,2. This was independently checked.

3. MINIMAL EXAMPLES AND A FAILURE OF THE MINIMUM-RANK SHORTCUT

Two states suffice for nonconstant robust memory inside a predictive class:
constant output and the transposition task transition give End(A)=C2; the
injective register is recovered by identity or swap. One state cannot contain
two distinct states in the same predictive class. This proves global state-count
minimality without reliance on enumeration or a fixed alphabet size.

For the irreversible case let T=(1,0,0) on states 0,1,2, with constant output.
Direct enumeration gives exactly
  id=(0,1,2), e=(0,1,1), t=(1,0,0).
Here e^2=e, t^2=e, and et=te=t; ranks are respectively 3,2,2.
Aut(A)={id}. The finest robust partition is {0}|{1,2}; r=(0,1,1).
The id and e induce identity on this register, while t induces complement.
The full predictive equivalence is universal. There are exactly two robust
partitions: this two-class partition and the universal partition.

This is state-count minimal among machines having a noninjective endomorphism
and a nonconstant robust register. On at most two states a noninjective map is
constant, and recovering a nonconstant register after a constant map is
impossible. This lower bound is independent of task alphabet and output choice.

Minimum rank is only an upper bound on the number of robust memory classes.
Take T=(1,0,3,2), two disjoint 2-cycles, with constant output. End(A) has 16
elements and minimum rank 2. Choose e=(0,1,0,1), so ker(e) pairs 0~2 and 1~3.
The automorphism g=(0,1,3,2) rotates only the second cycle. Congruence closure
adds 0~3, and the two original pairs then merge every state. Hence rho is
universal and no nonconstant robust register exists despite minimum rank 2.
The reviewer checks verified the full endomorphism monoid and this conclusion.

Smaller rank-gap example supplied by the parent and independently verified:
on three states take two task actions, constant 0 and constant 1, with constant
output. End(A) fixes 0 and 1 while sending 2 to any state. Its three elements are
id=(0,1,2), e0=(0,1,0), e1=(0,1,1). The minimum rank is 2, but the two kernel
identifications 0~2 and 1~2 force rho universal. This is state-count minimal for
a strict gap between minimum rank and |Q|: on two states, minimum rank 1 forces
|Q|=1, while minimum rank 2 means all endomorphisms are permutations, giving
rho equality and |Q|=2. The four-state example above remains a useful unary case.

4. SPECIAL CASES

If only automorphisms G are allowed, rho is discrete and robust partitions are
exactly G-invariant equivalences (systems of imprimitivity). The regular cyclic
action C_n has subgroup-coset partitions: the possible numbers of classes divide n.
This is an established group-action fact, not a novel automata classification.

If S is aperiodic, its finite group image on any robust quotient is trivial.
Every s must therefore fix every memory class. Robust partitions are precisely
those coarser than the equivalence generated by all pairs (x,s(x)), the weak
orbit/component relation. A constant map forces this relation to be universal.
The aperiodicity statement concerns S, not necessarily the task transition
monoid. These are different transformation monoids.

5. KNOWLEDGE OF THE REPLACEMENT IS A REAL ASSUMPTION

The quantifier 'for every F there exists d_F' permits an F-dependent decoder.
It does not give the observer F, nor identify which decoder should be used.
The report should say explicitly that the replacement or its induced memory
permutation is known when reconstruction is claimed operationally.

With one common decoder d for every F including identity, identity forces
d(k)=k on r(X). Consequently the stronger condition is simply rF=r for every F.
Its finest partition is the weak-orbit relation, which can be strictly coarser
than rho. The two-state swap and the three-state irreversible phase example
both demonstrate the difference. No active diagnosis is needed to state this
quantifier boundary.

6. SELF-REPORTS

Give the narrative channel a declared map n:X->N. Equality of current n-values
is a readout equivalence, not a causal provenance relation. Even equality of
all future narrative traces cannot distinguish two interface-equivalent histories
whose survivor labels differ; the first iteration's coupled-history argument
applies directly to this enlarged interface.

Matching narrative output also need not preserve another register. For example,
on X={(v,m):bits}, narrative n(v,m)=v and r(v,m)=m, erasing m retains the entire
narrative when task dynamics leave v independent of m, but makes old m
undecodable. A constant sentence such as 'I remember' is an even simpler case,
but a nonconstant narrative makes the counterexample less vacuous.

If n is included in b, End(A) is defined to preserve n. Then r=n is literally
retained by every allowed F. This follows from the stipulated model constraint;
it does not authenticate the sentence's semantic claim or establish that an
unobserved history occurred. Do not conflate observed report agreement with a
verified provenance port.

7. PRIMARY LITERATURE CHECK AND NOVELTY STATUS

Benjamin Steinberg, A Theory of Transformation Monoids: Combinatorics and
Representation Theory, Electronic Journal of Combinatorics 17(1), R164 (2010).
Primary publisher PDF retrieved on 2026-11-04:
https://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v17i1r164/pdf/
https://doi.org/10.37236/436

Inspected locations: Section 2.1, printed page 7 (congruences and quotient
M-sets); Theorem 3.7, printed page 16 (minimum-rank elements form the minimal
ideal), and Proposition 3.11 nearby (permutation action at a minimal idempotent);
Section 7.1, printed page 34 (maximal group image and idempotents). This is a
direct primary antecedent for the algebra used. The paper itself labels the
minimum-rank description as folklore. The present finite-memory interpretation
and examples do not establish a new general semigroup theorem. The one-idempotent
quotient proof above is self-contained; no publication priority is certified.

8. INDEPENDENT EXACT CHECKS

reviews/reviewer_checks.py uses explicit decoder enumeration as its oracle and
separately constructs the minimum-idempotent congruence and checks invariance.
Actual run: PASS, Python standard library, CPU only, less than one second here.

Coverage:
  - All 729 ordered generating pairs of maps on three states, deduplicated into
    206 transformation monoids (monoids generated by at most two maps).
  - All five partitions of three states: 1,030 monoid/partition comparisons.
  - All unary machines with binary outputs on 1,2,3 states: 234 machines and
    1,114 machine/partition comparisons.
  - Every minimum-rank idempotent in each tested monoid gives the same rho.
  - Full End(A), units, rho and robust partitions for the three-state tail and
    four-state disjoint-cycle examples.

These are bounded independent corroborations, not proofs of the general theorem.
reviewer_checks.json records exact lists and counts.

RECOMMENDATION

The main theorem is ready to include with its finite-domain, same-register,
full-congruence-closure and known-replacement qualifications. Keep the report's
contribution at the level of a carefully specified operational interpretation,
complete proof, and checked examples. A useful final-iteration question concerns
how much trusted replacement information is needed to select d_F; that would
close the gap between algebraic recoverability and an observer able to recover.
