From Observable States to Acquirable Predictive Models
Brian Theory
Revised technical report, 7 October 2026
Abstract
A predictive representation can be defined from known dynamics yet remain impossible to acquire
through the available observations. We examine this distinction for a finite hidden state on
P ={a}+M +{z}, withM an antichain, and a supplied library of oriented binary queries. A
query is admissible when its pre-answer order-complex inclusion is a homotopy equivalence. This
is a formal constraint on sensing; no physical safety guarantee is assumed.
The structural result, consolidated from earlier research notes, characterizes every reportable
target and gives a finest attainable transcript partitionR. For known deterministic actions
and output, letE∗identify states with the same output after every action word. Fixed-state
acquisition of an exact predictive label is possible precisely whenR⊆E∗. Standard congruence
constructions then describe feasible predictive representations and the finest feasible coarsening
of the original output labels. A reported class need not be one exact transcript leaf.
When queries must alternate with actions, finite-horizon support dynamic programming gives
a different feasibility test. We state its common-protocol quantifier, query-first phase, branch-
dependent actions and stopping rules explicitly. A reset can give current predictive certainty
without revealing the initial state; a complementary invariant trap shows that fixed-state feasibility
need not survive mandatory alternation. The report supplies self-contained proofs, worked
examples and bounded computational corroboration. It establishes neither external priority
for the specialized classification nor a new general minimization, planning or synchronization
method.
1. The acquisition problem
Suppose the output and transition tables of a finite system are known, but its actual state is
hidden. Computing which states have the same future behavior is an offline problem. Obtaining
that behavior class from the observations available at runtime is a separate task. This report
is for readers of finite-state systems and adaptive testing who want an exact example of that
separation.
Our sensing constraint makes the orientation of a supplied query matter. On the three-state
chaina<x<z , the tests110 and 001 carry the same unrestricted binary information, but only
the latter preserves the stipulated order-complex inclusion on{a,z}. Computing the complement
of an answer already obtained cannot make an inadmissible physical query executable. Example
8 develops this running example with an output and a known transition.
Throughout, P ={a}+M +{z}is the ordinal sum of two distinct endpoints and a finite
antichain, possibly empty:a<x<z for eachx∈M. Every nonempty support has the order
1induced from this specifiedP. Height counts strict inequalities, so the height is at most two. We
do not claim a theorem for every poset of that height. The proofs start with this order and the
supplied truth tables; a background observation model is motivation, not an additional hidden
premise. Appendix A explains positive observation topology and complete signatures.
The central structural result is the endpoint-sensitive classification in Theorem 2 and its finest
attainable partition in Corollary 3. These results are reproduced from the dated notes [1,2],
with their local admissibility proof included. Ordinary predictive refinement and generated
congruence then give Propositions 4 and 6 and the compatibility criterion. Corollary 7 is a short
output-quotient consequence. Proposition 10 specializes standard support planning to an explicit
alternating protocol. The purpose is a consolidated technical report with a complete operational
contract. External priority of the specialized classification remains unresolved; writing out the
later consequences does not establish a separate research increment.
We distinguish four objects: a reported label may combine several terminal transcripts; an exact
leaf records the whole transcript; a current predictive class describes the state after any executed
actions; an origin is the state before the experiment. Neither changing the requested label nor
changing the state should be mistaken for learning a previously inaccessible distinction. Section
8 places these tasks beside equivalence-class determination, automata equivalence, contingent
planning and adaptive homing. Appendix B records which results already occur in the source
notes and which source identities remain unconfirmed.
2. Fixed-state experiments and local admissibility
Let Q be an explicitly supplied finite set of deterministic mapsq : P →{0,1}. Fixed-state
histories contain query-answer pairs, and their support is
Sh ={x∈P :q(x) =d for every recorded(q,d)}.
There is no noise, query-dependent state change, shared consumption across counterfactual
branches, or additional history-dependent restriction. All legality tests precede answer con-
ditioning. Following the formal model [1], a query on nonempty supportS refines the order
by
x≤q y ⇐⇒x≤P y and q(x)≤q(y).
It is legal if the inclusion of order complexes∆((P|S)q)↪→∆(P|S) is a homotopy equivalence.
The order complex has chains as simplices. Legality concerns this inclusion, not an isomorphism
of finite posets. We retain the source abbreviation RF for this formal admissibility rule. The
term “repair-safe” in the notes does not establish physical safety. The test is applied before the
answer: conditioning itself may change topology. For example, on{a}+{x,y}+{z}, query 0110
is legal, but its answer-one support is the disconnected antichain{x,y}, whereas the original
order complex is contractible.
After an answer, all remaining states have the same value of the query just asked. Therefore that
query imposes no extra order restriction within its answer support. The same holds for every
previously recorded query. This is branch cancellation: along a fixed-state branch, the current
order is precisely the original induced order on its support. Repeated queries are constant there
and add no information. Constants are legal, but can be removed from useful fixed-state decision
trees.
Lemma 1 (local RF admissibility; [2, P1-L1]).If both endpoints are present, legality
is equivalent toq(a)≤q(z). If onlya is present, legality meansq(a) = 0 or the restriction is
constant one. If onlyz is present, legality meansq(z) = 1 or the restriction is constant zero. If
2neither endpoint is present, every query is legal. These rules include singleton supports and the
two-endpoint support without middle states.
Here is the complete local argument. With both endpoints, the only triangles areaxz. Endpoint
values 01 delete no simplex. For endpoint values00, each middle state valued one loses its
edge xz and triangleaxz. That edge is a free face of its unique triangle, so deleting these pairs
gives elementary collapses onto the refined complex. Endpoint values11 give the dual collapses,
deletingax for middle states valued zero. These explicit collapses prove homotopy equivalence of
the inclusion in the safe cases.
For endpoint values10, letk =|S∩M|. There arek + 1 deleted edges, includingaz, andk
deleted triangles. In relative chains overF2, each triangle boundary isaz +ex, whereex is its
unique deleted spoke. The columns are independent because the spokes are distinct. There
are no relative vertices, so relative first homology has dimension one. A homotopy equivalence
would induce zero relative homology, which is impossible. This uses homology only to obstruct
equivalence, not to infer equivalence from acyclicity.
On a one-endpoint support, the order complex is a star. Any descending spoke is deleted
without a triangle and disconnects the complex; otherwise nothing is deleted. On a middle-only
antichain there are no edges to delete. This proves all four cases. In this particular family,
legality is also equivalent to preserving the number of connected components. With endpoint
word 10, the refined complex consists of two disjoint nonempty stars; with one endpoint, a
deleted descending spoke isolates its middle vertex. The legal cases have the explicit collapses or
unchanged complexes above. This component criterion is a consequence of these cases, not a
criterion for homotopy equivalence of general complexes.
DefineQs ={q∈Q :q(a)≤q(z)}. An observation outsideQs has endpoint word10. It cannot
be both useful and legal on any support containing an endpoint: with both it is illegal, and with
one it is legal only when constant. Such a query can nevertheless be useful on a middle-only
branch. This asymmetry is the mechanism behind the classification criterion. Membership inQs
means legality on the full carrier and on supports containing both endpoints. It does not ensure
legality on every smaller support:010 is legal ona<x<z but illegal on{x,z}. Each support
must pass its own test.
3. Reportable targets and the finest transcript partition
Theorem 2 (target classification; [2, P1-T1]).Starting with supportP, for every finite-
valued targetc :P→C, a finite legal supplied-query tree reportsc(x) for every hidden state
if and only if two conditions hold: fullQ separates every differently labelled pair of middle
states, andQs separates every differently labelled endpoint-state pair. Equal endpoint labels are
permitted. In particular, the theorem does not require an ascending endpoint query when the
target does not need one.
Necessity follows by following transcripts. Two states with identical full signatures receive
identical answers to every supplied query. For an endpoint and another state with different
required labels, consider their first separating query in a successful tree. Before that answer,
both states remain in the support. The query is useful and legal there, so the preceding lemma
puts it inQs. Separation by an unavailable or inadmissible query cannot substitute for this first
legal separator.
For sufficiency, divideQs intoQ=, the queries with equal endpoint values, andQ↑, those with
endpoint values01. While the branch containing both endpoints has a nonconstantQ=-query,
ask it. The answer opposite to the common endpoint value detaches a middle-only branch; the
other answer retains the endpoints. On a middle antichain every query is legal. Full-signature
3classes there are target-homogeneous by the first condition, so querying separating members of
Q classifies every detached branch.
After these peels, all remaining middle states agree with both endpoints on everyQ=-query. If
Q↑is empty, the second condition forces this entire core to share one target label. If the core is
already homogeneous, stop regardless ofQ↑. Otherwise ask any member ofQ↑. Its zero child is
a lower star witha, and its one child is an upper star withz.
On the lower star, a state labelled differently froma has aQs-separator froma. It cannot be
in Q=, after the completed peels, so it belongs toQ↑, has value zero ata, and value one at
that state. It is legal throughout the lower star and detaches a middle-only branch. Repeating
removes every incorrectly labelled state. The upper star is dual, using queries with value one
at z and zero at the state being detached. All detached branches are classified as above. Each
informative query strictly shrinks both children, so this finite construction terminates.
Corollary 3 (acquisition equivalence and exact leaves; [2, P1-T2]).The pair conditions
compress into a single equivalence. LetK partition states by theirQs-signatures. Keep each
K-class containing an endpoint intact, and refine each endpoint-freeK-class by fullQ-signatures.
Call the resulting equivalenceR. Equivalently, take the equivalence closure of endpoint-state pairs
sharing their root-admissible signatures and middle-middle pairs sharing their full signatures.
Every generator lies in oneK-class. If that class contains an endpointe, its endpoint-state
generators join every member toe; hence the entire class stays together, including middle states
with distinct full signatures. If it contains no endpoint, only full-signature equality generators
occur, and no endpoint path can enter it. Its equivalence classes are therefore exactly the
full-signature fibers. This proves that the two descriptions ofR agree.
The theorem becomes
c is RF-reportable ⇐⇒R⊆ker(c).
Here and throughout, inclusion means inclusion of equivalent pairs: everyR-class lies inside
one target fiber. This convention avoids reversing the comparison between finer partitions and
smaller equivalence relations.
There is also a legal tree whose exact transcript leaves are precisely theR-classes. Apply the
classification theorem to theR-class label. Every legal tree must keep each generating pair,
and hence eachR-class, together; otherwise its own leaf labels would violate necessity. The
constructed classifier must also separate differentR-classes. Thus R is the finest attainable
transcript partition. Its coarsenings can all be reported, but need not be exact transcript
partitions. A reduced fixed-state tree has no constant internal query, so every internal node has
two nonempty children. An exactR tree consequently has|P/R|leaves and|P/R|−1 internal
nodes. Its depth is at most|P/R|−1; this is an existence bound, not an optimal-depth algorithm.
The formula forR concerns acquisition starting from all ofP. It cannot in general be restricted
to a fresh smaller support: forQ ={101}on a<x<z , globalR ={a,z}|{x}, but on fresh
{a,x}the only query is descending and illegal, although the restricted relation is discrete. A
new starting support is a new admissibility problem.
4. Predictive equivalence and feasible quotients
Fix a finite action setU, known deterministic mapsTu :P→P, and a finite-valued outputb.
In fixed-state acquisition these tables define the prediction problem; actions are not executed
during sensing. For a wordw =u1···uk, setTw =Tuk◦···◦Tu1; the empty word acts as the
identity. The predictive equivalence is
xE∗y ⇐⇒b(Twx) =b(Twy) for every finitew.
4No value ofb is observed for free. If its measurement is needed, an appropriate query must
actually belong to the library and be legal.
Proposition 4 (standard predictive refinement; [2, P1-T3; 4]).Compute this equivalence
by ordinary finite refinement:
E0 = ker(b),
Ek+1 =Ek∩
⋂
u∈U
(Tu×Tu)−1(Ek).
Induction shows thatEk means agreement after every word of length at mostk. The induction
step includes the empty word throughEk, and longer words through their first action and
remaining suffix. Each strict refinement increases the number of classes, so there are at most
|P|−|b(P )|strict refinements. Once consecutive relations agree, the recurrence remains fixed,
giving E∗.
A transition congruence is an equivalenceF satisfying xFy⇒TuxFT uy for every action. The
fixed relationE∗is such a congruence and lies insideker(b). Every congruence insideker(b) lies
inside everyEk, by induction, hence insideE∗. ThereforeE∗is the largest such relation, or the
coarsest exact predictive partition. Output and transition maps on its classes are well-defined.
Knowing the current class suffices to update it and predict outputs indefinitely without additional
observations.
Corollary 5 (fixed-state compatibility; [2, P1-T3]).RF-admissible acquisition reportsE∗
exactly whenR⊆E∗, by classification. More generally, suppose some RF-reportable transition
congruence F⊆ker(b) is acquired. Reportability impliesR⊆F, while predictive maximality
implies F⊆E∗. Hence compatibility is necessary. Conversely, when compatibility holds,E∗
itself is a reportable predictive representation.
This also explains a failed repair strategy. IfxRy but notxE∗y, every finer predictive equivalence
still separates these states. No additional subdivision of the desired labels can make that blocked
pair observable. Failure can be certified by the pair together with a finite wordw for which the
two outputs differ. Such a word exists by the characterization of the refinement sequence; tracing
a separating refinement step supplies one.
Proposition 6 (generated transition congruence; [2; 10]).Start from acquisition rather
than output:
G0 =R,
Gk+1 = Eq
(
Gk∪
⋃
u∈U
(Tu×Tu)(Gk)
)
.
Here Eq denotes equivalence closure. The increasing sequence stabilizes at the least transition
congruence G containing R. Indeed a fixed point is forward invariant, and every congruence
containingR contains every iterate. There are at most|P/R|−1 strict class mergers.
Compatibility is equivalentlyG⊆ker(b). If R⊆E∗, forward invariance ofE∗contains every
closure iterate. Conversely, ifG⊆ker(b), maximality putsG⊆E∗. When compatible, the
reportable predictive equivalences are precisely the transition congruences satisfyingG⊆F⊆E∗.
Thus G is the finest and E∗the coarsest reported representation in this class. Arbitrary
intermediate equivalences need not be congruences. These are standard refinement and closure
operations used with P1’s acquisition obstruction, not new automata algorithms.
Corollary 7 (finest feasible output coarsening).If exact prediction of the original output
is infeasible, relabeling inaccessible states cannot restore it. A different repair is available only if
the prediction requirement itself may be weakened. LetB =b(P ), letG be the least transition
congruence containingR, and letD be the least equivalence onB containing every pair(b(x),b(y))
5with xGy. Writeπ:B→B/D for the quotient map and defineb′=π◦b. ThenG⊆ker(b′) by
construction. SinceG is a transition congruence, it is contained in the predictive equivalenceE′
∗
forb′. ThereforeR⊆G⊆E′
∗: the coarsened future outputs are safely acquirable and predictable.
This is the finest feasible coarsening of the specified output labels, by the following elementary
quotient argument. Letf :B→C be any relabeling whose future outputs are safely acquirable.
Its predictive equivalenceEf is a transition congruence containingR, so minimality givesG⊆Ef.
The empty action word givesf(b(x)) = f(b(y)) whenever xGy. Every generating pair ofD
thus lies inker(f), and equivalence closure givesD⊆ker(f). Consequentlyf factors through
π. Conversely, a relabeling factoring through πis constant on everyG-class, so the same
congruence argument proves feasibility. This statement concerns output-label coarsening, not
optimal protocol depth. It discards the original output distinctions merged byD, and does not
establish exact prediction of that unchanged output. Changing the query library is a separate
intervention: it can changeR, whereas this construction fixes the library and transitions.
5. A running example: orientation and timing
Example 8 (three-state obstruction and controls; [2, P1-C1]). This is a smallest
obstruction under the conditional assumptions stated below. Ona<x<z , write binary tables
in that order and take
Q ={010,110}, b = 010, T = (x,x,z).
The map is order preserving. The second supplied query is already the pullbackb◦T = 110;
this is not an example where the needed truth table was omitted. The base-output partition is
{a,z}|{x}, buta and z have different outputs after one action. ConsequentlyE∗is discrete.
Only 010 belongs toQs, soR ={a,z}|{x}. The base query is legal initially, with endpoint
values 00. Its zero answer leaves{a,z}, where the other query is the descending test10, hence
illegal. Asking that other query first is also illegal. Repetition yields no new information. Thus
RF-admissible acquisition of the predictive class fails, although the base output is safely obtained
in one query. Ignoring legality, ask010, then 110 on the zero branch: ordinary identification
succeeds in depth two.
Actually supplying the complement001 in place of 110 repairs this example. Ask 010, then
001 on{a,z}; the latter is ascending and legal. The repaired tree identifies the state in depth
two. The two libraries induce the same unrestricted Boolean information, but not the same
permissible physical experiments. Computationally flipping an answer from110 would first
require performing that illegal query; it is not a substitute for supplying001.
The conditional minimality argument is elementary. With two states, a nonconstant base output
is already injective, so it cannot have a strict predictive refinement. A constant output remains
constant after every word. At least three states are therefore required. If the base is nonconstant
and safely obtainable, while ordinary acquisition can obtain its strict predictive refinement, one
binary query is insufficient: every obtainable nonconstant base must distinguish its two fibers,
and repetitions cannot split either fiber. At least two distinct supplied binary query maps are
necessary under those conditions. This example attains that library-size lower bound and the
three-state lower bound; its exhibited worst-case query depth is two. This says nothing about
other sensing or control contracts.
Two timing controls are also inherited from [2]. In the monotone example above, passive
observations ofb at times zero and one produce histories01,11,00 froma,x,z, respectively. They
retrospectively distinguish initial states only after the second observation. Under the moving-state
rule used below, however, the initial zero branch{a,z}maps to{x,z}. On that image support,
6b is descending and illegal. The passive-history success therefore does not implement a safe
two-sample experiment.
For the swap control, useQ ={010,100}, keepb = 010, and letT fix a while swappingx,z.
Again Qs contains only the base query,R joins a,z, and the predictive equivalence is discrete, so
fixed-state acquisition fails. But queryb, executeT, and queryb again. The zero branch becomes
{a,x}, where the base query is ascending; the one branch becomes a singleton. Histories00,10,01
distinguish initial states, and therefore also current states because this action is bijective. The
report is after two queries and one action, not at time zero. These controls motivate the changed
interaction model of Section 7.
The unsuccessful approaches are now explicit: ignore orientation, subdivide blocked labels, treat
an unsupplied complement as available, or postpone an observation without checking its new
support. Each changes or violates an assumption rather than overcoming the fixed-state theorem.
6. A reported class need not be one transcript leaf
Example 9 (no coarsest exact-leaf refinement; [2, P1-C2]).Let P ={a}+{x,y}+{z},
with tables ordered as(a,x,y,z), identity dynamics, and desired partition
B ={a,x,z}|{y}.
Supply 0001,0011,0111,0101. Every root query has endpoint word01, so every root query is
legal. Their full signatures distinguish all four states, giving discreteR. Identity dynamics makes
E∗=B when b is the class label ofB. Compatibility therefore guarantees reportability.
One legal tree asks0001, then 0011 on{a,x,y}. Its exact leaves are
Π 1 ={a,x}|{y}|{z}.
The second test is legal on the lower star because its value ata is zero. Another legal tree asks
0111, then 0101 on{x,y,z}, producing
Π 2 ={a}|{y}|{x,z}.
The second test is legal on the upper star because its value atz is one. Both trees reportB by
assigning the same reported label to multiple leaves.
No tree has exactlyB as its transcript partition. The target is nonconstant, so a successful tree
must ask a root query. Every supplied query splits{a,x,z}; once states have different recorded
answers, later queries cannot merge their transcripts. Moreover, any common coarsening of
Π 1,Π 2 that still refinesB must joina with x and x with z, hence must equalB. SinceB is
unattainable as exact leaves, there is no coarsest attainable exact-leaf refinement of this target.
The distinction is substantive rather than terminological. A report may discard acquired
information, whereas an exact transcript records it. The finest attainable transcript partitionR
still exists, and the coarsest predictive reported partition still exists here. Neither fact provides a
coarsest feasible transcript refinement of every target. Any implementability statement must
specify which object it minimizes.
7. Alternating observations and actions
We now change the acquisition contract explicitly. The same finite carrier, supplied queries,
output, and known deterministic actions are used, but actions may change the hidden state
7during acquisition. A persistent history records every chosen query, observed bit, and chosen
action. Queries read the current pre-action state without changing it. The next action may
depend on that answer. After the action, the order is reset to the original order induced on the
image support. Actions need not be injective or order preserving; no additional action-safety
condition is imposed. These are modeling assumptions, not consequences of [1]. The syntax
is strict: no action-only start and no consecutive queries without an intervening action. An
identity action, if supplied, can embed a fixed-state query tree in this syntax; no identity action
is assumed.
At a pre-query supportS, a legal queryq gives the nonempty answer supportSd ={x∈S :
q(x) =d}. If actionu is then chosen, the next current support isTu(Sd), with duplicate images
identified. The RF test is applied onS before learningd, not separately on hypothetical favorable
outcomes. There is no free output observation, complemented query, pullback oracle, or transition
learning. A constant query is legal and is allowed to precede an action even though it yields no
information.
Reporting targets the currentE∗-class, computed from the same known future dynamics and
output. A support is terminal, denotedH(S), when it is contained in a singleE∗-class. After a
correct report, all subsequent output predictions must be produced by quotient updates without
further sensing. This target does not require identifying the initial state or recovering distinctions
that actions have erased.
Proposition 10 (finite-horizon support planning).For each nonempty supportS and
integerh≥0, letWh(S) mean that there exists one deterministic legal history-dependent protocol,
starting at a pre-query checkpoint with current supportS, that reports the correct current
E∗-class for every possible state inS. On each executed branch it uses at mosth actions and at
most h + 1 queries. It alternates query then action and may stop before a query, immediately
after its answer, or immediately after an action. Then
W0(S) =H(S)∨
∃q∈Q legal onS ∀d∈{0,1}with Sd̸= ∅ :H(Sd),
and, forh≥1,
Wh(S) =H(S)∨
∃q∈Q legal onS ∀d∈{0,1}with Sd̸= ∅ :
[H(Sd) ∨ ∃u∈U Wh−1(Tu(Sd))].
This is finite-horizon AND-OR planning on supports [9], with RF deciding which query choices
are enabled. The formula permits different actions, or a decision to stop, on different answer
branches. All existentially chosen queries belong toQ. Empty answer branches impose no
obligation.
Proof proceeds by induction on the action allowance. With none available, a protocol either
reports immediately or asks one legal query and reports after every possible answer. This
is exactly the first equation. With positive allowance, an immediate report again requires
homogeneity. Otherwise the first query must be legal on the full current support. After its
answer, the protocol either reports a homogeneous class or chooses an action and continues
with one fewer action and at most one fewer query available. Known determinism gives exactly
the image support appearing in the equation. Inductive necessity follows by considering every
possible answer.
Conversely, choose witnesses to the displayed recurrence. A homogeneous branch receives its
unique predictive label. On every other branch execute its witness action and attach the finite
protocol supplied by induction. Legality holds before the first answer by construction and
thereafter by the inductive protocols on the reset image orders. The horizon decreases after
8each action; stopping after the final query is already covered byW0. This produces a finite legal
experiment with the stated bounds and proves sufficiency.
Non-injective actions cause no exception. Every image state has at least one predecessor in the
conditioned support, and every such predecessor can generate the recorded branch. Coalescing
predecessors therefore preserves the exact set of possible current states. At pre-query checkpoints,
the current support and remaining action allowance determine whether a successful continuation
exists: the available queries, actions, induced order and predictive target are then identical.
A controller can maintain this support from its recorded history and choose branch-specific
witnesses; this is not a memoryless strategy on the hidden physical state. At a post-answer
checkpoint, the next permissible instruction is instead an action or a report. This sufficiency
depends on the absence of consumption constraints or extra history-dependent syntax. It would
not justify discarding memory in a different model.
A separate relationI tracks initial reconstruction. Start withI ={(x,x) : x∈P}. A query
keeps pairs whose second coordinate has the received value; an action replaces(x0,x) by (x0,Tux).
Induction identifiesI with precisely the initial-current pairs consistent with the history. Its
second projection gives current support, while its first projection gives possible initial states.
This explains exactly what information a support-only description omits.
Example 11 (reset, current certainty and origins).Consider the following explicitly
refuted transfer claim: moving-state prediction is possible only if the original fixed-state inclusion
R⊆E∗holds. This is a claim tested here, not an assertion attributed to a publication. On
a<x<z , supply justq = 000, letb = 010, and supply one actionT sending every state toa.
Every nonempty action word ends ata, soE∗= ker(b) ={a,z}|{x}. But the constant library
gives universalR, violating the fixed-state criterion.
Ask the constant query once, receive zero, execute the reset once, and report immediately after
that action. The query is legal, the image support is{a}, and the current predictive class is known.
This uses exactly one query and one action, with report time one measured in executed actions
and no post-action query. Every initial state produces the same entire record. The pair relation is
now{(a,a),(x,a),(z,a)}, so initial identification fails. Indeed no later experiment can distinguish
those origins: all subsequent states and query answers coincide under the same history-dependent
choices. The output is nonconstant on the original carrier, making the separation nontrivial.
This proves the negative transfer result while explaining its mechanism: control can acquire
current predictive certainty by changing the state, rather than by learning its past.
Example 12 (fixed-feasible but alternating-impossible).Conversely, ona<x<z , take
Q ={010,011}, b = 010, and the sole actionT = (x,x,z). Both queries lie inQs and their
signatures separate all states;a,zdiffer after one action and the other pairs differ already under
b. ThusR =E∗is discrete, and fixed-state acquisition succeeds.
No alternating protocol succeeds at any finite horizon. If the first query is010, its zero branch
{a,z}is not terminal and the required action takes it to{x,z}. If the first query is011, its one
branch is already{x,z}, is not terminal, and the required action fixes it. On this support,010 is
illegal, 011 is constant, andT is the identity. The support contains two predictive classes forever.
If no action allowance remains, the nonterminal branch fails immediately; with any positive
allowance it remains trapped. HenceWh(P ) is false for every finiteh. This invariant argument,
not a finite extrapolation, proves the claim. Together with Example 11 it shows that neither
direction of fixed-state compatibility transfers under the specified mandatory alternation.
Even winning supports need not be downward closed. WithQ ={010}, b = 010, and no actions
on the three-state chain,W0(P ) holds butW0({x,z}) fails because its sole informative query is
illegal. Support planning still applies; an optimization that assumes every subset of a winning
support is winning would not be justified.
98. Relation to earlier work
The terminal target in the fixed-state problem is equivalence-class determination: an adaptive
test policy may stop when all consistent hypotheses belong to one required class [7, §3, expanded
version p4]. Decision-region determination generalizes this condition to overlapping regions [8,
§2, Eq.(2), p432]. Thus reporting a label need not identify the hidden state or make the terminal
support equal the whole label fiber. The additional problem here is that the enabled physical
query depends on the entire current support through an oriented topological inclusion. The RF
classification and the exact realization of its signature partition address that constraint; the
expected-cost algorithms of [7,8] are not imported.
The predictive equivalence is ordinary finite deterministic behavioral equivalence. Hopcroft’s
introductory account describes repeated output partition refinement by successor blocks [4, p1],
and the formal word-equivalence definition appears on p2. Here we use the routine finite-output
(Moore) version; the formal automaton in [4] has a binary final-state output. Treating each action
as a unary operation also makes G a standard generated congruence [10, II§5]. The output-
label result is the elementary quotient property applied after this closure. These constructions
organize the consequences of the acquisition obstruction; they are not new minimization or
closure algorithms.
The moving-state recursion is finite-horizon AND–OR dynamic programming on exact belief
supports, with RF legality as its enabled-query predicate. Bonet and Geffner give belief images,
observation conditioning and a worst-case Bellman formulation [9, §2, Eqs.(1),(5),(6)]. The
present contract fixes a separate query-first phase, answer-dependent actions, a remaining horizon
and the current predictive-class goal. Support-dependent admissibility need not survive taking
subsets, so hereditary-support optimizations require additional proof. The source also explicitly
distinguishes setting a known current value from recovering its initial value [9, p54, footnote3].
Adaptive homing determines the state after an experiment; the finite adaptive-test definition
in [6, §2, p75] makes this timing explicit. The target here is a current predictive class, with
separate pre-action queries and RF constraints. The constant reset example is synchronization:
one action maps all states to a common state [11, §1]. Its value here is to expose an invalid
transfer of the fixed-state criterion, not to introduce synchronization.
The order-complex and elementary-collapse methods are standard [12, §2]; the particular binary
RFclassificationisspecializedtothetwo-endedantichainfamily. Thepositive-topologyconvention
uses upward sets, whereas [12] uses the dual finite-space order. Finally, computational mechanics
groups stochastic histories by conditional future laws [5, Def5]. Its predictive-sufficiency and
minimal-complexity statements [5, Thms1–2] concern a different stochastic and almost-sure
setting; they neither identify the present controlled quotient with causal states nor establish
lawful acquisition of it.
The fixed-state theorems and examples are reproduced from the matching dated P1 note [2], with
the local RF argument inherited there from the RF foundation. The manuscript consolidates those
arguments and adds an elementary output quotient and an explicit moving-state specialization.
These are provenance statements about the inspected snapshots, not established publication
histories. The broader priority and significance of the RF classification remain open.
9. Reproducibility, limits and conclusions
The structural content is the classification of admissible fixed-state information on the specified
two-ended antichain poset. Corollary 3 proves that all reportable targets are coarsenings of an
attainable partitionR. Standard predictive equivalence describes which distinctions matter for
10every future action word; comparing the two relations gives the exact acquisition criterion. A
report may discard transcript distinctions, so it is essential to specify whether a desired partition
is a label or an exact leaf partition.
The accompanying verification material contains the independent audit’s retained checker, actual
environment, stdout and results. That prior run reported 5,332 direct-complex legality cases
for all supports and binary queries at 2–6 states; 984,352 distinct full-support library/target
classification cases through four states; and 754,080 bounded moving-policy comparisons at 2–3
states, all query libraries, zero or one arbitrary action, every output partition and support, and
horizons 0–2. These are separate metrics with different ranges, not an additive total. Larger
moving checks were seeded samples. The legality path uses actual collapses or nonzero relative
homology; the moving comparator explicitly tracks histories, phases, budgets and initial-current
pairs independently of the support recurrence. Full ranges and implementation limits are recorded
in the accompanying verification report. These retained audit results are not relabeled as a new
full revision run. Targeted revision checks and their actual logs are recorded separately. Finite
checks corroborate the arguments and do not prove the unbounded theorems.
Known actions change the state whose class is sought. Synchronization can remove current
uncertainty while preserving uncertainty about every origin. Mandatory query/action alternation
can also obstruct an experiment that succeeds with a fixed state. Proposition 10 handles both
effects by checking legal queries before answers, selecting actions on each answer branch, and using
the stipulated induced order on each image support. Its correctness is ordinary finite-horizon
support reasoning; its operational content depends on this precise contract.
The report does not justify RF as a physical safety condition. Within this poset family its local
homotopy test reduces to preservation of component number, and neither answer conditioning
nor actions are required to preserve topology. Noise, unknown transitions, consumed queries,
restricted memory, action admissibility and sensing costs would require changed models and
proofs. No general-poset classification, optimal-depth formula, new automata minimization
bound or general infinite-horizon characterization is established.
The consolidated account makes the fixed-state theorem, standard consequences and timing
counterexamples independently checkable. The source notes already contain most of the structural
mathematics. Broader priority and research significance of that classification remain open, as
do the source-history details specified in Appendix B. The appropriate present claim is a self-
contained technical report, not evidence of a newly established research contribution merely from
clearer exposition.
Appendix A. Positive observations and full signatures
The distinction between positive observation topology and complete signatures is important.
The observation-topology manuscript [3, v0.9, §3] defines the topology generated by positive
truth regions of background observationsΦ. On our finite carrier, writeoΦ (x) = (φ(x))φ∈Φ,
with binary coordinates ordered by0< 1. The signature representation identifies opens with
inverse images of upward-closed subsets of realized signatures. This construction records oriented
positive information, not unrestricted Boolean access to all coordinate values at the hidden state.
For completeness, finiteness makes the signature description direct. For a realized signature
s, select, for each realized signature not aboves, a coordinate witnessing that failure. There
are finitely many such signatures. Intersecting the corresponding positive coordinate regions
selects exactly those realized signatures aboves. Unions of these principal regions give all
upward-closed sets. Conversely, each positive region is upward closed, and this property survives
finite intersections and arbitrary unions. This argument is confined to the finite carrier used here.
11Complete-signature equivalence instead identifiesx,ywhen all coordinates agree. Its cells
belong to the Boolean algebra generated by the observations, using negative as well as positive
information. For a two-state carrier{p,r}with one coordinate 01, the positive topology is
{∅,{r},{p,r}}, whereas the complete-signature partition is discrete. The Boolean algebra
contains{p}, although that singleton is not positive open. Accordingly, distinguishability by
complete signatures is not identical to availability of an oriented query.
Background signatures may be assumed injective as in [1], but their actual values at the hidden
state are not supplied to the agent. The runtime libraryQ can be smaller and need not separate
states. Known truth tables permit offline computation; they are not answers. Negating an
acquired bit is postprocessing, not permission to perform an unsupplied complemented query.
These observations explain the motivation for the stipulated order; the classification proofs do
not require reconstructing it from background observations.
Appendix B. Provenance and result overlap
The matching local snapshots of [1]–[3] and the RF foundation note were recovered and hash-
checked in the independent audit. They are preserved unchanged in the accompanying materials.
Reference [1] supplies the fixed-state contract. Reference [2], P1-L1, contains Lemma 1; P1-T1
contains Theorem 2; P1-T2 contains Corollary 3; P1-T3 and its dual-closure section contain the
predictive refinement, compatibility and generated-congruence results. P1-C1 contains Example
8 and its timing controls, and P1-C2 contains Example 9. Appendix A draws on [3, v0.9, §3].
Proofs needed here are reproduced rather than delegated to inaccessible premises.
The output-label quotient and formal moving-support proposition are additions relative to
that inspected P1 version, but use standard quotient and planning principles. The reset is a
synchronization illustration. Example 12 and the component-count observation were added
during this revision after proof rechecking. The previous audit and the revision’s specialist checks
are AI assessments, not external human peer review.
Confirmed public identifiers and authorship for [1] and [2], permanent locators for the unpublished
materials, and their dissemination/submission histories remain unresolved unless supplied in the
accompanying provenance record. A local hash identifies bytes; it does not authenticate priority
or establish public availability. No duplicate publication is inferred from overlap with private
notes. No public deposit is made by this local revision.
References
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