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Causal Self-Diagnosis with Noisy Tests
Observation Laws, Irreversible Access, and Task Recovery
Research development draft prepared for Brian Theory
AI-assisted derivation, computation, and independent internal review
5 November 2026 Version 0.3
Abstract
A controller can compensate for a malfunction while remaining unable to attribute it to its
actuator or environment. We study this distinction in finite diagnostic models with specified
experiment permissions. Positive readout likelihoods preserve exact posterior support, so a fixed
support-only permission rule restricts all acquisition to the initially available library.
For history-dependent access, initial statistical separation is insufficient. With static known
finite-alphabet channels and always-legal idempotent repetition, an oracle revealing each test’s
complete observation-law class characterizes arbitrarily reliable finite diagnosis. The closure of
attainable randomized terminal decision channels is the convex hull of finite legal oracle-tree
channels, giving Bayes and minimax limits. Four hidden states are minimal for the exhibited
gap between reliable finite-policy families and one ongoing consistent policy. Known persistent
nuisance and sampling prices have distinct, explicitly delimited extensions.
We compare order-complex homotopy preservation with a specified recovery task. Neither
implies the other; minimal three-state witnesses and a task-dependent availability predicate
make this distinction concrete. A conditional implementation bridge states the additional phys-
ical obligations. An anchored motor model and independent stateful calculations expose effects
of initial motion, persistent preparation error, heat, and depleted resources. The original 48,000
episodes are supplemented by 240,000 independent simulated episodes and exact confirmatory-
power calculations.
Five simulated specialist reviews and two cross-reviews sharpen the proofs, source attribution,
and experimental scope. A reconstructed historical witness challenges one printed necessity
claim under its exact definition. The paper concerns functional diagnosis. It claims neither
subjective awareness, hardware validation, external peer review, nor publication priority.
Status. A working paper, not a submitted or externally reviewed manuscript. Author attribution
for submission remains undecided. Proved results refer to the arguments and internal checks here,
not formal proof-assistant certification.
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Contents
1 Useful control without causal self-diagnosis 3
2 Related work and limits of the novelty claim 3
3 A fixed-state diagnostic contract 4
4 What noisy support permits 5
5 Cumulative orders: a finite oracle characterization 6
6 Complete observation laws and attainable decisions 9
7 Self, world, and the decision that needs the distinction 12
8 Cost and risk as parts of the contract 14
9 Exact experiments and independent checks 15
10 From formal permission to physical experiments 17
11 Preserving a specified recovery task 19
12 Observation fidelity and finite-horizon robustness 22
13 An anchored motor experiment: model and test protocol 23
14 Stateful experiments, resource limits, and confirmatory power 26
15 Conclusions and remaining boundaries 30
A Assumption failures and boundary cases 30
B V erification and novelty ledger 32
C Reproduction 33
D A source-audit witness for a printed necessity claim 33
E Historical coverage and remaining originals 35
F Simulated review and revision ledger 35
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1 Useful control without causal self-diagnosis
Suppose an action produces the opposite of its intended outcome. The actuator may have reversed,
or the environment may have reversed its effect. A controller can learn the resulting input–output
relation while remaining unable to attribute the change. That ambiguity matters only for particular
downstream tasks: repair, forecasting after an intervention, or transfer may require distinctions that
immediate control does not.
We use causal self-diagnosis operationally: estimating a variable assigned to the system’s own
mechanism within a specified causal model and intervention interface. The assignment to self is
supplied by the physical model, not recovered from a statistical partition alone. No criterion for
consciousness is proposed.
Two B-Theory manuscripts motivate the study. R4 compares actuator-tracking controllers
under a known finite model. Its explicit, compiled, and external trackers implement the same
mapping; gains over restricted baselines concern evidence processing, not the self-model label [ 25].
The acquisition report distinguishes a sufficient predictive representation from one that permitted
observations can actually acquire [ 31]. We connect these questions by adding noisy, potentially
confounded diagnostic channels and explicit losses.
The central modeling issue is what a test changes. A noisy answer changes belief weights but
need not eliminate a state. A test may also change a formal observation order, or an intervention
may change the physical system. These are different updates. Treating them as interchangeable
can produce an incorrect theorem.
The paper studies diagnostic access under explicit constraints. It gives a support-only reduction,
develops an ideal-oracle characterization for cumulative availability, connects identifiable distinc-
tions to regret and interventions, and tests the physical interpretation with counterexamples, an
implementation contract, and a proposed motor experiment. Exact computations and internal in-
dependent checks accompany the arguments. The support-only theorem and statistical tools are
elementary specializations of established ideas. We give a finite decision-tree characterization for
static diagnostic models with irreversible query availability and idempotent repeats, combining
adapted simulation, finite-state common randomness, and amplification. Section 6 extends this
reduction to complete observation laws and terminal channels. The cumulative-order results are
derived here as a concrete extension; this does not assert publication priority.
2 Related work and limits of the novelty claim
Model-based diagnosis already treats competing fault explanations and discriminating measure-
ments. Reiter’s consistency-based account [ 27] and de Kleer and Williams’ probabilistic measure-
ment selection [ 8] are direct antecedents. Naming an actuator variable does not create a new
diagnosis paradigm.
Controlled sensing and active sequential hypothesis testing select informative actions under error
and cost criteria [ 7, 21, 20]. Pairwise separation by action-conditioned laws is an established identifi-
ability condition. We restrict the executable library and ask for labels grouping hypotheses. This is
closely related to equivalence-class determination [ 11] and decision-region determination [ 15]. Exact
version-space termination eliminates incompatible hypotheses. Our repeated full-support setting
permits arbitrarily reliable finite reporting while generally forbidding zero-error finite certification.
Stochastic diagnosis already distinguishes exact fault certainty from asymptotically high con-
fidence [ 33, 2]. The recovered 2005 original gives a sufficient AA-diagnosability condition and
examples showing it is not necessary. Active information acquisition and information-state dy-
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namic programming are also established [ 34]; its erratum makes policies measurable relative to the
observations they generate. The recovered unreliable-observation paper [ 35] motivates the focused
source-level check in Appendix D. Later criteria require care about the fault-prefix quantifier [ 6];
the relevant 2017 full theorem remains unavailable [ 32]. We claim neither the support/confidence
distinction nor active diagnostic sensing as new.
Qualitative POMDP safety uses belief supports rather than positive probability magnitudes [ 16,
5]. High-probability safe exploration instead uses confidence bounds under additional assumptions
[29]. A zero-risk support rule and a small failure-probability rule are different problems. Our
topological predicate is not a physical safety theorem.
Identifiability relative to observations and interventions is standard causal reasoning [ 23, 13].
Our XOR model concerns parameter attribution, not graph discovery or an average causal effect.
Predictive state representations explain why useful prediction need not recover a privileged latent
description [ 19]. Robotic self-modeling and damage recovery have substantial experimental prece-
dent [ 4]; we supply no robot data or architectural superiority claim.
Repetition and majority voting to simulate a noiseless decision tree are established [ 10]. Com-
paring a noiseless experiment with its noisy garbling is classical [ 3]; sequential versions also exist
[9]. Renou and Venel explicitly treat restrictions on feasible histories, controlled information, and
early actions constraining later access [ 28]. Their Section 4.3 is a particularly close antecedent.
This is a close antecedent to constrained experiment comparison; our claim is a finite specialization
with legality-preserving simulation. Recent precedence-constrained decision trees are particularly
relevant [ 30]. Their monotone prerequisites differ from the irreversible incompatibilities in our
cumulative inclusion rule. This distinction is not a completed priority assessment.
3 A fixed-state diagnostic contract
Let H be finite and nonempty, with known prior (h)> 0 for every state. The hidden diagnostic
target is fixed during the experiments in Sections 4–5. Concrete apparatus coordinates may change,
as modeled separately in Section 13. A finite library E contains repeatable tests. Test e has finite
alphabet Ye and known kernel Ke(y jh), with
Ke(y jh)> 0 ( e 2 E; y 2 Y e; h 2H): (1)
The conditional law of each observation given the entire past, selected test, and state is this kernel.
Repetition therefore supplies fresh information; one-time marginal distributions alone are not the
assumption.
A policy is one history-dependent selection, stopping, and reporting rule used under every state.
Private randomness is independent of the state and noise. All tests and answers are recorded. The
target is ` :H ! L. States with the same label need not share observation laws.
Letbt be the posterior and St = fh :bt(h)> 0g its exact support. A support-only admissibility
rule is any fixed predicate
L : E  (2H n f∅g) ! f0; 1g:
Teste is allowed only if L(e;S t) = 1 , evaluated before its answer. No downward monotonicity is
assumed. Robust physical safety L(e;S ) = 1fS  Aeg, for a set of safe states Ae, is one special
case; formal structural rules may mean something else.
Definition 3.1. A label is reliably acquirable if for every  >0 there is a legal policy with a finite
deterministic observation bound and maxh Ph(b` 6=`(h)) . This is a family of finite procedures,
not a zero-error stopping certificate.
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For finite H and positive prior, vanishing maximum, statewise, and Bayes errors are equivalent
for a sequence of policies:
min max
h
errh 
X
h
(h)errh  max
h
errh;  min = min
h
(h)> 0:
Guarantees concern the supplied model, not unspecified noise, omitted faults, or a continuum of
arbitrarily close alternatives.
4 What noisy support permits
Theorem 4.1 (Support persistence) . After every finite reachable history, St =H. The executable
library is always E0 = fe :L(e;H ) = 1 g. Conversely, every adaptive policy using only E0 is legal.
Proof. Posterior mass is proportional to (h)Qt
s=1Kes(ys jh), a product of positive terms. Com-
mon policy factors cancel because identical histories receive identical selection probabilities un-
der every state, including randomized policies. Substitute St = H into L. No monotonicity is
needed.
Define h 0 g when Ke(
 j h) = Ke(
 j g) for every e 2 E 0. This equivalence of experimental
laws is different from posterior support.
Theorem 4.2 (Labels and residual error) . A label is reliably acquirable exactly when it is constant
on every 0 class. If h 0g but`(h) 6=`(g), every legal policy satisfies errh +err g  1. The limiting
optimal Bayes error is
R∗ = 1 
X
C∈H/≡0
max
l∈L
X
h∈C
ℓ(h)=l
(h): (2)
Proof. For an equivalent pair every legal test has the same response law. Induction gives identical
adaptive transcript laws, including stopping and reports. The two correct labels are disjoint events
under this common law, proving the error bound and necessity.
For sufficiency, cycle through all tests in E0, use maximum likelihood over H, and report the
selected state’s label. The bound below tends to zero when all differently labelled pairs are separated.
A constant target needs no tests; an empty library can acquire only a constant target.
An oracle revealing the equivalence class cannot worsen performance. Within a class, all tran-
script likelihoods agree and posterior ratios remain the prior ratios. Its best report is the largest
prior label mass, giving ( 2). Cycling consistently learns the class and approaches this oracle er-
ror.
4.1 Sample bounds and stopping distinctions
For each pair define affinity e(h;g ) =P
y
p
Ke(y jh)Ke(y jg). With deterministic sample counts
ne, maximum likelihood satisfies
Phf`(bh) 6=`(h)g 
X
g:ℓ(g)̸=ℓ(h)
Y
e∈E0
e(h;g )ne: (3)
A wrong-label maximizer has likelihood at least that of h. Markov’s inequality for the square root
of its likelihood ratio gives the product of affinities; apply the union bound. Affinity equals one
exactly when the two laws agree.
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If each test is used m times, define Cmin = min ℓ(h)̸=ℓ(g) P
e loge(h;g )> 0 for a nonconstant
separated label. Maximum error is at most (jHj  1)e−mCmin. Summability shows that reports
under one infinite cycling policy eventually remain correct almost surely. The agent need not know
when its last error occurred.
In contrast, no nonconstant label has a zero-error procedure terminating almost surely under ev-
ery state. Finite transcript laws are mutually absolutely continuous, as are finite stopped transcript
laws by decomposing an event over the possible stopping times. A label reported with positive prob-
ability under one state therefore has positive reporting probability under a differently labelled state.
Unbounded expected time does not remove this obstruction when termination remains almost sure.
The results extend to standard Borel observation spaces when each test’s laws are mutually
absolutely continuous across states. Use a common dominating measure and positive densities
almost everywhere. Support persistence then holds almost surely simultaneously over finite times.
Equal topological support alone is insufficient (Appendix A).
4.2 A noiseless RF example and an explicit noisy lift
The acquisition manuscript thins a finite order with an oriented Boolean test: retain u v only if
q(u)  q(v). The test is admissible when the order-complex inclusion is a homotopy equivalence
[31]. For P = fag + fx;y g + fzg, root admissibility is q(a) q(z); every test is legal on the middle
antichain. This is a structural rule, not a physical safety guarantee.
In coordinate order (a;x;y;z ) take
q = 0110; r = 1100; ` = (0; 1; 2; 0): (4)
Noiselessly, legal query q returns zero on a homogeneous label-zero branch, or one on fx;y g, where
r is legal and separates labels one and two.
Now define a support-reset noisy lift : after each noisy answer, evaluate the predicate on the
original order induced on exact posterior support. Every query returns its nominal bit through a
fresh known BSC with 0 < " <1=2. Support remains P , only q is available, and its laws at x;y
agree. Limiting uniform-prior Bayes error is 1=4; limiting minimax error is 1=2. Arbitrarily small
positive noise changes feasibility under this exact support convention. No general impossibility of
noisy active learning follows.
An additional supplied test s = 0100 is root legal. The q;s signatures separate the labels.
Repeating each n times and using majority decoding gives error at most 2 exp[2n(1=2 ")2]. At
" = 1=5;n = 21 the bound is 0:0456454; using the exact binomial tail gives the sharper union
bound 0:00193939.
This lift is a choice: a noisy answer does not make prior tests constant on its support, so the
noiseless branch-cancellation reasoning does not apply. Retaining previous structural changes gives
a different model.
5 Cumulative orders: a finite oracle characterization
Let P be a finite poset with carrier H, and E a finite set of nominal Boolean tests qe :H ! f0; 1g.
The physical hidden state is fixed. Let J  E record the distinct executed tests. Define an order
on the full carrier by
u J v ( ) u P v and qe(u) qe(v) for every e 2J:
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Teste is legal at J when
A(J;e ) = 1f∆(PJ∪{e}),! ∆(PJ ) is a homotopy equivalence g = 1: (5)
Repeating an executed test is structurally idempotent and legal. Each repetition returns a fresh
independent BSC observation with common known 0<"< 1=2.
The structural carrier is never reduced by a noisy answer. Initial-state support remains full,
but the order can change. Theorem 4.1 therefore does not freeze A(J;e ).
Example 5.1 (Root separation is insufficient) . On the diamond a<x;y <z , supply q = 0110;s =
1101 and target ` = (0; 1; 2; 0). Both tests are legal initially, with endpoint words 00 and 11, and
their joint signatures separate all different labels.
After q, the strict edges are ax;ay;az , a tree. After s, the edges are ax;az;xz;yz ; the filled
triangle axz and attached edge make a contractible complex. After both, only ax;az remain and
y is isolated. The second distinct test increases the component count from one to two, so is illegal
in either order. Repetitions cannot create the missing distinction; neither first test alone resolves
every label.
a
x y
z
Initial order
a
x y
z
After q = 0110
a
x y
z
After s = 1101
a
x y
z
Both: inadmissible
Figure 1: Cumulative-order counterexample. Lines show all strict comparabilities, not only Hasse
edges. Chain triangles are filled in the order complex. The final inclusion is inadmissible because
y becomes a separate component.
5.1 Keep structural history separate from candidate states
An ideal-oracle problem has state (J;C ): J records structural history andC records states consistent
with exact nominal answers. Legality remains A(J;e ) on the full carrier; it is not recomputed on
C. A new test returns qe(h) and updates J and Cb = fh 2C :qe(h) = bg.
Let T (C) mean that ` is constant on C. Define
W (J;C ) = T (C) _
9e 2 E n J :A(J;e ) = 1 ^
^
b∈{0,1}
Cb̸=∅
W (J [ feg;C b): (6)
Each recursive step enlarges J, so the recursion terminates. A test constant on C is allowed: it can
change structure and unlock a later test without adding nominal information on that branch.
Theorem 5.2 (Arbitrarily reliable finite diagnosis) . The noisy cumulative-order model reliably
acquires ` if and only if W (∅;H ) holds. The equivalence also holds if noisy policies may have
almost surely finite stopping times under every state.
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Proof. If W holds, select its finite ideal tree. Every path uses at most m = jEj distinct tests. At
a node, execute its prescribed test an odd number k of times and branch on the majority. First
execution is legal by the tree; repetitions are idempotent. Even after an erroneous earlier majority,
future structural choices remain legal because legality depends on J, not the true state or inferred
C. If a majority selects a nominally empty child, stop and return an arbitrary label. This makes a
total policy on every possible noisy history.
Conditioned on the past and true state, a majority error has probability at most exp[2k(1=2 
")2]. If no majority is wrong, the ideal tree is followed and returns the correct label. Thus maximum
error is at most m exp[2k(1=2 ")2], using at most mk samples.
Conversely, take a noisy policy with maximum error  < 1=jHj. Simulate it using the ideal
oracle: on the first request for e, obtain and cache qe(h); every requested noisy response is then
generated from the cached bit and a fresh BSC coin. Repetitions require no additional oracle query.
Structural histories and legality match the noisy execution.
Run this simulation separately for all finitely many states using one common random tape.
Each run has its correct marginal law; runs need not be independent. The union bound gives
positive probability that all state runs are correct. Almost sure termination under every state gives
simultaneous termination with probability one. Fix a tape on which every run terminates correctly.
Prune cached repetitions and deterministic computation. This leaves a deterministic ideal tree
with at most jEj new tests per path. Each nonempty branch corresponds to an actual state and
terminates. The remaining legal choices witness ( 6).
This is a characterization given a legality predicate, not an efficient general homotopy-
equivalence algorithm. Its (J;C ) state space is exponential. The theorem concerns a family of
finite policies with arbitrarily small error. It does not automatically imply a single indefinitely
running consistent policy: a mistaken irreversible branch can permanently remove useful access.
Corollary 5.3. IfW (∅;H ) fails, every almost surely terminating noisy policy satisfies P
h errh  1,
so maximum error is at least 1=jHj and Bayes error at least min.
Proof. A sum below one would give a positive-probability common tape correct for all states, and
hence a successful ideal tree.
5.2 Limiting Bayes decision value
Let D be finite and g(d;h ) bounded. In this subsection samples have no cost, and the allowed
sample budget grows. Define the unnormalized value
V (J;C ) = min

min
d∈D
X
h∈C
(h)g(d;h );
min
e /∈J
A(J,e)=1
X
b∈{0,1}
Cb̸=∅
V (J [ feg;C b)

:
(7)
An empty inner minimum is omitted; priors are not renormalized at branches. Let BN be the
optimal noisy Bayes terminal loss with at most N samples.
Theorem 5.4 (Exact limiting value) . limN →∞BN = V (∅;H ). With G = max d,hg(d;h ) 
mind,hg(d;h ), an amplified ideal policy using at most mk samples has excess Bayes loss at most
Gm exp[2k(1=2 ")2].
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Proof. An ideal oracle can simulate any noisy policy using cached bits and private noise. Random-
ization cannot improve a finite Bayes decision-tree optimum; conditioning on the random tape and
pruning cached queries gives the recursion ( 7). Thus V lower-bounds every noisy risk.
Take an optimal deterministic ideal tree and amplify as above. Couple it to the exact ideal run.
Until a majority error their decisions coincide. The loss difference is at most G on the failure event.
The same union bound gives the upper bound. The sequence BN is nonincreasing, completing the
limit argument.
In Example 5.1, uniform-prior ideal Bayes error is 1=4. The limiting randomized minimax error
is instead 1=3. Every deterministic ideal policy errs on at least one representative of fa;x;y g, so
the sum of those three error probabilities is at least one under any mixture. For attainment, mix
equally three ideal policies: query q and choose label one on its middle branch; query q and choose
label two there; or query s and choose label one on its fa;x;z g branch. Unique-label branches are
correct. These policies err respectively only at y, only at x, or at a;z . Repetition approximates the
mixture. A fully deterministic ideal policy has worst-state zero–one error one; randomized terminal
reporting and randomized test selection are substantive in a minimax comparison.
5.3 A general availability rule and a strict quantifier gap
The proofs of Theorems 5.2 and 5.4 use only a finite predicate A(J;e ), the update J 7! J [ feg,
static nominal bits, and always-legal idempotent repetition. They therefore apply to any such
availability rule. Homotopy inclusion is one specialization. A resource-consuming or arbitrary
history-dependent rule is not automatically covered.
For this general class, the finite-policy-family distinction can be strict. Let H = f(b;c ) : b;c 2
f0; 1gg and let the target be c. A test g returns nominal bit b. Test e0 returns c when b = 0
and zero otherwise; e1 returns c when b = 1 and zero otherwise. Selecting either specialized test
permanently blocks the other; g and repetitions of the selected test remain available. All responses
have fresh common-support BSC noise with known error below one half. An ideal policy reads g,
then eb, so finite majority-amplified policies achieve arbitrarily small error.
No single ongoing policy can have statewise error tending to zero. If it never chooses a specialized
test, its observations are independent of c. Otherwise some first choice ei occurs by a finite time n
with positive probability. Before that choice only g is observed. Finite-history absolute continuity
gives positive probability to the same event under the wrong value b = 1 i. On that event the
two states with c = 0; 1 have identical available future observation laws: the correct specialized
test is blocked forever. The sum of their error probabilities is bounded below by this positive event
probability for every t  n. Thus they cannot both have vanishing error. The lower constant
depends on the policy; there is no contradiction with arbitrarily accurate finite families.
This last availability example has not been shown here to arise from order-complex inclusions.
It substantiates the general quantifier warning, not a separate topological realization claim.
6 Complete observation laws and attainable decisions
The nominal bit may omit distinctions carried by the readout. For known channels, a more general
oracle returns each test’s complete observation law.
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6.1 The actual-law oracle
Let H be finite and nonempty, E finite, and D finite and nonempty. The hidden state is fixed.
Teste has finite alphabet Ye and a known probability vector Ke(
 j h); zero entries are allowed.
Conditional on h, the selected test, and the controller’s entire available past, including its private
randomization, the next response has this same vector. Private coins are independent of the
hidden state and response innovations. A fixed A(J;e ) governs access, and execution updates J to
J [ feg. Every executed test remains legal to repeat without further structural change. Stopping
and reporting are always allowed.
Defineh eg when their complete test-e vectors agree, and write re(h) = [h]∼e. The law-oracle
answer isre(h). For candidate set C, its children are Ce,r = fh 2C :re(h) = rg. Replace the binary
children in ( 6) and ( 7) by these nonempty children, retaining the same full-carrier legality predicate.
Empty disjunctions mean failure; a minimum over no legal new tests is +1. Empty children are
homogeneous by convention and have unnormalized loss zero. Their noisy implementations stop
with an arbitrary decision.
Theorem 6.1 (Actual-law characterization) . A label is arbitrarily reliably acquirable by bounded
finite policies exactly when the law-oracle recursion succeeds. Allowing statewise almost surely finite
stopping does not change feasibility. For a full-support prior and bounded terminal loss, the limiting
finite-horizon Bayes value equals the law-oracle value.
Proof. A law answer determines its response vector, so it permits state-independent simulation
of each response from a fresh uniform number. Cache answers on first use. The common-tape
argument of Theorem 5.2 preservesJ and legality; pruning cached repetitions gives at most m = jEj
new queries on any path. This proves necessity and the terminal-loss lower bound, including
statewise almost sure stopping.
For sufficiency, decode law classes by maximum likelihood from k fresh readings. For distinct
class vectors pe,r and pe,s,
Pr(br 6=r) 
X
s̸=r
 X
y
q
pe,r(y)pe,s(y)
!k
:
On a sequence where a competing likelihood exceeds the true one, the true likelihood is bounded by
their geometric mean. Summing proves the bound without division by zero. Distinct vectors have
affinity below one by the equality case of Cauchy–Schwarz. Thus maximum local error k tends
to zero. Amplify each node of a legal ideal tree; total report error is at most mk in at most mk
samples. Wrong class decisions preserve structural legality, since it depends on J, not the inferred
candidate set. Empty children are totalized by immediate reports. Loss range width G gives excess
terminal Bayes loss at most Gmk.
If any test has distinct vectors, let ∗ < 1 be the largest affinity over those pairs. Then
k  (jHj  1)k
∗ for k  1. If no such pair exists, take k = 0; if ∗ = 0, one reading distinguishes
each class exactly. The earlier support-persistence and finite-certification impossibility require
positive likelihoods or mutual absolute continuity. This theorem does not. Its permission rule is
stipulated independently of posterior support: changing that rule, or the carrier of an RF poset, is
a new model.
6.2 A finite polytope of terminal channels
A deterministic legal law-oracle tree realizes a function f :H ! D . There are finitely many such
functions, since no branch repeats a test. Let their set be F, including immediate reports, and
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define
Qf (d jh) = 1ff (h) = dg; P = convfQf :f 2 F g:
For a noisy policy Π, put QΠ(d j h) = Ph(Π reports d). Use the metric kQ Q′k = max h kQ(
 j
h) Q′(
 j h)kTV.
Theorem 6.2 (Terminal-channel closure). Every legal statewise almost surely terminating random-
ized noisy policy has QΠ 2 P . Moreover,
[
N <∞
fQΠ : Π legal, randomized, and stops by N g = P:
Proof. Simulate the policy with cached law answers and a state-independent tape containing private
coins and response uniforms. Almost surely, all finitely many hidden-state runs terminate. Partition
this set by their terminal report vector f 2 D H :
Ef =
\
h∈H
fthe h-run terminates and reports f (h)g:
These events are measurable, since each finite report event is a countable union over stopping times.
Every positive-mass Ef has a representative tape whose pruned legal tree realizes f . Consequently
QΠ =P
f P(Ef )Qf 2 P . This avoids any measurable selection of entire trees.
Conversely, amplification approximates each tree channel within mk. A state-independent
private choice of tree implements any finite convex mixture with the same bound. The finite
polytope is closed, proving equality.
Bayes loss is linear in Q; randomized minimax loss maxh
P
dQ(d j h)g(d;h ) is continuous.
Their finite-budget limits are minima over P; minimax is a finite linear program over tree mixtures.
Private randomization is essential for the full convex hull. Closure does not imply finite attainment
of a zero-error vertex, consistency of one ongoing policy, or convergence of arbitrary constrained
optima on unattainable boundaries.
Adapted simulation and controlled experiment comparison are established [ 9, 28]. The claim
is a finite reduction for the stated idempotent-access class. Recent sequential stopping/budget
comparisons [ 36, 18] use different decision classes; they further caution against a broad priority
claim.
6.3 Four states are necessary for the consistency gap
Proposition 6.3. For jHj  3, a successful law-oracle tree implies a legal deterministic sequence
of distinct tests whose joint law signatures separate all differently labelled states. One ongoing
legal policy then has reports eventually permanently correct almost surely. Four states are therefore
minimal for the latch’s finite-family versus ongoing-consistency gap.
Proof. Stop a successful tree at homogeneous nodes. With at most three candidates, a nonconstant
answer partition has at most one child containing two or more states; all other children are singleton
leaves. Follow the sole potentially nonterminal child, with constant-answer queries having one
nonempty child. This gives a legal root-to-leaf test sequence. Every differently labelled pair either
separates on the sequence or remains together down its sole nonterminal branch; it cannot persist
to a homogeneous leaf. Legality of the entire sequence is independent of the true state. Execute
it and cycle its now-legal tests. Affinity bounds give summable errors at completed cycles; Borel–
Cantelli gives eventual permanent correctness. Hold the latest completed-cycle estimate between
cycle ends. Constant targets need no tests.
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The four-state latch supplies sufficiency for the gap under common BSC noise. This is a cardi-
nality result for general availability, not a topological realization of that latch.
6.4 Persistent nuisance, small separation, and prices
Known finite persistent noise can be included in the static state. If a vector Z = (Ze) is drawn
once jointly with h and repeated queries return Ze, use state (h;Z ), target `(h), and deterministic
kernels. Retain the declared A(J;e ); do not silently rebuild the RF order. The law oracle is exactly
the cached-response experiment. For fair  and independent N  Bernoulli(0:2), repeated  N
has Bayes error 0:2, attained after one read. The augmented prior on (;N ) is (0:4; 0:1; 0:4; 0:1);
each observed class has mass 0:5 and maximum target mass 0:4. This represents finite known static
nuisance, not arbitrary temporal dependence. Persistent-response active diagnosis is established
[1].
Known state-dependent noise is different: rates 0:1 and 0:2 split actual-law classes even when
the nominal bits agree. The law-oracle value is zero, matching the observed limiting risk. Tiny
known gain or noise differences can remove a qualitative obstruction while demanding impractical
budgets. For fixed 0<p< 1,
(p;p +d) = 1  d2
8p(1 p) +O(d3):
Sample requirements diverge as laws approach equality. Estimated kernels need quantitative un-
certainty bounds.
For nonnegative reading prices ce, ce cmax, the optimal total objective satisfies
0 C() V Gmk +mcmaxk (k  1):
The lower bound is oracle domination; amplifying an optimal ideal tree proves the upper bound.
Hence C() ! V as  # 0. Exponential decoding gives an O( log(1=)) upper rate, not an
optimality claim. Fixed positive prices do not vanish as a cap grows.
First-use resource registers can be included if repeated tests remain legal stutters and their
kernels stay unchanged. Per-read depletion does not qualify. A fair binary target, one BSC test
of error 0:2, and a battery allowing one read has error 0:2 at every positive horizon, although the
noiseless query identifies the target.
Independent exact checks cover 8,192 oracle/report-map comparisons, 5,120 noisy Bellman val-
ues, 640 general-channel values, and 8,704 small-state path checks. These are bounded corrobora-
tions, not proofs of priority or unrestricted correctness.
7 Self, world, and the decision that needs the distinction
Let h = (;w ) 2 f 0; 1g2, where  is actuator reversal and w environmental reversal. Command a
produces physical output y =a  w. A routine measurement is
o =y E; r =o a = w E; E  Bernoulli():
Here E is measurement error, not physical execution error. The numerical experiments score
classification of the noiseless effective mechanism.
Proposition 7.1 (Routine symmetry) . Every adaptive command policy has identical routine tran-
script laws at (;w ) and (1 ; 1 w). For known  6= 1=2, the two parity classes  = w are
distinct in distribution.
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PAGE 13
Proof. The response kernel depends only on . Equal histories give identical action distributions
and then identical response distributions. Induction includes randomized commands. Different
parity values give different Bernoulli laws when  6= 1=2.
Once  is learned, a = g  attains target g in the original environment. After transfer to
a known environment wnew = 0 , the correct command is a = g . With independent fair ;w ,
routine histories leave  fair, so routine-only transfer error is 1=2 even with unlimited data.
Assign the RF witness ( 4) to a = 00;x = 01;y = 10;z = 11 . Its label distinguishes single-
reversal cases while merging both parity-zero states. It is conditional discrepancy attribution, not
full pair identification or actuator identification everywhere. This interpretation does not validate
RF as physical safety.
7.1 Task sufficiency and regret
For a finite decision set and loss g, put g∗(h) = min dg(d;h ). A class C admits a decision with zero
regret at every state exactly when
\
h∈C
arg min
d
g(d;h ) 6= ∅: (8)
A common minimizer proves sufficiency. Conversely, regrets are nonnegative; zero expected regret
requires every positive-probability decision to minimize loss at every state. Randomization does not
relax the criterion. Pairwise intersections are insufficient with ties: fu;v g; fv;z g; fu;z g intersect
pairwise but not jointly.
For a positive distribution p on C, residual Bayes regret is
R(p) = min
d
X
h
p(h)g(d;h ) 
X
h
p(h)g∗(h):
It is zero exactly under ( 8). A nondestructive observation Z has information value R(p) 
EZR(pZ)  0 before its cost. This is finite decision theory, not a consciousness score. If the
experiment changes the physical state, this formula alone does not isolate its value.
Repair need not require attribution. Suppose leaving an actuator incurs 10 and replacement
costs 3. Repair is optimal when p( = 1) > 0:3. At prior 1=2, always repairing costs 3. A globally
admissible nondestructive diagnostic Z = N , with fresh N  Bernoulli(0:1) and cost 0:2, yields
posterior 0:9 or 0:1 with equal probability. Repair on the first outcome and leave on the second:
terminal loss (3+1)=2 = 2 , or 2:2 including testing. Clairvoyant expected loss is 1:5. These numbers
describe one optional test, not unrestricted sequential optimality. Free repair can remove all regret
without diagnosis; cost-free compensating commands can also make physical repair irrelevant.
7.2 Interventions that preserve or break ambiguity
Known toggles of orw change parity by a known offset but preserve the symmetry. More generally,
if a symmetry preserves observation kernels and commutes with every allowed transition, adaptive
experiments preserve its transcript equivalence. The name intervention does not imply informative
attribution.
A trusted reset w := 0 preserves  and provides noisy direct measurements of it through the
routine channel. Resetting  := 0 instead preserves w. Combined with pre-reset parity information,
that can reveal the original actuator state; without such a record it may erase the initial coordinate.
Knowing the current reset coordinate is different from learning its origin.
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PAGE 14
For a one-use known environmental shift, preserve  and let qw = P(wnew = 1 j wold = w).
Within original parity class , the two candidates are (0; ) and (1; 1 ). Under a = 0 their
observed probabilities of one differ by
(1  2)(q0 +q1  1): (9)
Thus the pair is separated in distribution exactly when q0 +q1 6= 1 and 6= 1=2. Identity, toggling,
and independent flipping with known probability have sum one and fail. Reset to zero has sum
zero. A fresh independent environment with known bias  has sum 2, so helps unless  = 1=2.
Repetition requires a specified replenishment protocol; one separated distribution is not unlimited
independent data.
For a fixed known finite schedule of affine deterministic interventions over F2, write ht =Ath0 
bt. A residual has coefficient row [1 1]At on the initial state. Stack these rows into M . Noiseless
observations identify a label precisely when it is constant on fibers of M ; both initial coordinates
are identified precisely when rankM = 2 . Toggles change offsets, not rank. A baseline row [1 1]
plus reference row [1 0] has rank two. With fresh informative BSC noise the same criterion concerns
separation of joint laws, not exact finite recovery. Adaptive schedules require the corresponding
branchwise analysis.
8 Cost and risk as parts of the contract
8.1 Diagnostic value and information lower bounds
For nondestructive tests with costs ce> 0 and finite terminal decisions, the Bayes recursion is
U0(b) = min
d
X
h
b(h)g(d;h ); (10)
Un(b) = min
(
U0(b); min
e legal
"
ce +
X
y
p(y jb;e )Un−1(be,y)
#)
: (11)
Cumulative structure requires J as an extra argument; physical actions require transitions and
possibly a joint origin–current belief. This is standard dynamic programming.
In the support-only model, uniform label error at most  < 1=2 implies for each differently
labelled pair, under finite expected stopping time,
X
e∈E0
EhNeDKL(Kh
e kKg
e )  kl(1 ; ): (12)
Here kl is binary relative entropy. The stopped likelihood ratio accumulates observation information;
policy factors cancel. Data processing to the event of reporting `(h) gives the bound. For unbounded
stopping, use the stopped change-of-measure theorem [ 17, Lemma 1], not optional stopping without
hypotheses. Finite alphabets and positive kernels make increments bounded; positive costs and
finite expected cost give the needed integrability. Minimizing P
ecexe over xe  0 subject to
these pairwise information constraints gives an expected-cost lower bound under h. No attaining
algorithm is claimed.
8.2 A probabilistic gate changes the rule
If ze(h) is hazard probability, a one-use Bayesian chance rule permits the experiment whenP
hb(h)ze(h)  . It cannot override a failed exact predicate without changing the contract. Its
guarantee depends on the specified prior, likelihood, and hazard model.
14

PAGE 15
Take states 0; 1; 2, prior (1=2; 1=4; 1=4), and a harmless screen whose zero probabilities are
(0:9; 0:1; 0:1). A diagnostic has hazard zero at state zero and one at states one and two. After n
zero screens,
pbad = 1
1 + 9n:
Two zeros give 1=82  0:012195, below an explicitly allowed 0:02 threshold, while all three states
remain supported. Under an actually bad state the diagnostic remains hazardous with probability
one.
A policy screening exactly twice and unlocking only after two zeros has gate probability 0:41
and joint bad-and-gate probability 0:005. Gate probabilities conditional on safe and bad states
are respectively 0:81 and 0:01. The conditional risk is 0:005=0:41 = 1 =82. These are different
quantities.
If, for every allowed screening test and outcome, every good-state to bad-state likelihood ratio
is at most Λ> 1, prior bad mass p0 satisfies after any n screen observations
pbad(Hn)  p0
p0 + (1 p0)Λn:
Bound each state-pair transcript ratio and then mix over states. Reaching <p 0 therefore requires
at least dlog[p0(1 )=((1 p0))]= log Λe screens. This is necessary, not a guarantee that favorable
data occur.
For a statewise gate guarantee, suppose a finite screening policy unlocks with probability at
least 1  under safe h and at most 
 under unsafe g, with 1  >
. The unlock event gives
X
e
EhNeDKL(Kh
e kKg
e )  kl(1 ;
 ):
Identical screen laws force  +
  1. Zero false unlocking cannot coexist with positive finite-time
unlocking power under mutually absolutely continuous finite transcript laws.
A per-use threshold does not bound episode risk by the same threshold. If conditional budgets
t sum to at most ∆ on every path, conditional expectation and a union bound give any-hazard
probability at most ∆ under the Bayesian prior-predictive law. A time-uniform confidence set
with statewise simultaneous coverage 1  ∆ also bounds unsafe selection by ∆ if actions are safe
throughout that set [ 14]. It is not exact posterior support; applying it to an arbitrary topological
rule creates no physical guarantee.
9 Exact experiments and independent checks
All experiments are synthetic consequences of declared finite models. Exact fractions are used
for posterior updates, decision values, and enumeration. Simulations are separate implementation
checks, not empirical evidence about robots or consciousness.
9.1 Task-specific optimal policies
Use independent fair ;w and two globally admissible nondestructive ports. Routine sensing re-
turns  =  w through error 0:1; body sensing returns  through error 0:2. The body port is
a trusted measurement assumption. Each sample costs c, at most six are allowed, and early stop-
ping is permitted. Terminal zero–one losses concern  (present control),  (reference-environment
transfer), or (;w ) (full causal diagnosis). These objectives exclude execution noise and additional
intervention costs.
15

PAGE 16
0 10 20 30 40
Repetitions per available test
0.0
0.1
0.2
0.3
0.4
0.5Label error / bound
A. Missing admissible distinction
Original library: Bayes error
Proved asymptotic floor 1/4
Added legal test: error upper bound
1 2 3 4 5
Consecutive favorable screens
10 4
10 3
10 2
10 1
Conditional probability
B. Confidence without elimination
Posterior unsafe-state mass
Explicit relaxed threshold 0.02
Figure 2: Left: exact uniform-prior Bayes error for the original RF library and an exact binomial
union bound for the augmented library, at noise 0:2. This compares an error with an upper bound,
not two measured policy errors. Right: risk along favorable screen histories; no original state
reaches zero likelihood.
The Bellman solver evaluates both responses to every test. Ties favor stopping, then port order;
terminal label ties favor the lowest label. At c = 0:03:
Task First test Label error Mean tests Total cost
Present control routine 0.015040 2.3924 0.086812
Transfer body 0.073280 2.7424 0.155552
Full causal state routine 0.119872 4.5448 0.256216
Total cost is error plus 0:03 times expected tests. These are optima only within the stated horizon,
ports, known model, and losses. With routine sensing alone, transfer optimally stops immediately:
error 1=2 and zero sampling cost. Routine symmetry proves the stronger unlimited-data attribution
obstruction.
Under the fair prior,  and are independent. Their posterior odds are 4dθ and 9dϕ, where each
d is observed ones minus zeros for its port. This supplies a distinct sufficient-statistic simulation
and shows that a matched compiled or external controller can reproduce the same policy.
9.2 Bounded exhaustive verification
For the diamond, all 16 Boolean tables, all sublibraries of size zero through three, and all 15 target
partitions give 697 libraries and 10,455 library–target cases. A noiseless tree recursion agrees with
the acquisition report’s specialized classification in every case: 3,916 are noiselessly reportable.
Under the support-reset noisy lift, 3,744 satisfy initial-test separation; 172 noiselessly feasible cases
are lost. Under cumulative admissibility, the oracle recursion reports 3,632; 112 initially separated
cases are blocked. These counts describe one bounded family, not statistical estimates or proofs for
larger posets.
Additional checks cover 240 support/query legality cases by directly constructed comparability
graphs; 1,023 binary histories through length nine for positive support in the RF witness; and
4,096 paired paths across all 128 deterministic adaptive command policies of horizon three for
16

PAGE 17
.005 .03 .08 .2
Cost per observation
0.0
0.2
0.4
0.6Optimal policy classification error
Present control
.005 .03 .08 .2
Cost per observation
Transfer to known world
.005 .03 .08 .2
Cost per observation
Full causal state
routine
body
routine+body
Figure 3: Label error of the optimal cost-minimizing policy with six tests available. Only the
four displayed costs were computed; lines guide the eye. High costs can favor immediate stopping.
Useful diagnostic access depends on the terminal task.
XOR symmetry. The graph test is justified here because components of these diamond suborder
complexes are contractible. It is not a general homotopy-equivalence test.
An independent evaluator enumerates all 723 deterministic sensing/stopping trees of depth at
most three for two ports. Across three tasks and four costs, its unnormalized path calculations
match all 12 Bellman values. A Monte Carlo implementation instead reconstructs beliefs from
integer sufficient statistics and runs 100,000 episodes per task with seed 20261103. Estimated total
costs are 0:086650, 0:1555168, and 0:2557106, with approximate 95% normal half-widths 0:0007791,
0:0016509, and 0:0020388. Exact values lie within the intervals. These intervals describe simulation
variability, not uncertainty in exact model values.
The previous report and supporting calculations were recovered and byte-hashed. Its R4 checker
is preserved separately and rerun for continuity. That is not an audit of the historical R4 imple-
mentation or validation of physical assumptions.
10 From formal permission to physical experiments
The general availability model describes diagnostic access under irreversible constraints. A finite
register can implement it, but that observation alone does not make the constraint useful for a
physical device. We separate three questions: whether a test is formally permitted, whether it is
executable, and whether its execution preserves the physical conditions needed by the task.
10.1 Homotopy does not preserve directed recovery
Section 11 proves a minimal three-state example: a homotopy-preserving query deletes the only
directed recovery from one start to its target. The following limits explain why an implementation
contract is necessary.
If every directed start–target reachability pair must be preserved, a suborder R  P of a
transitively closed order must equal P : every pair in P must remain in the reachability relation,
which is R. For query thinning this forces q to be monotone on P . Universal directed recovery
therefore forbids any deletion. A specified target or task may admit a weaker, useful condition.
17

PAGE 18
A viability example sharpens the difference. Add an absorbing damage state d. Before refine-
ment, x can repair to z or wait and enter d; afterward only the latter action remains. From a,
moves to x andz are available; z can hold. The controlled invariant subset of the safe set fa;x;z g
changes from fa;x;z g to fa;z g. The finite-state calculation accompanies the paper. The added
dynamics are declared separately from the order.
There is a limited positive interpretation. Suppose each retained undirected edge is an ex-
ecutable, reliable, safe motion in both directions, all vertices are safe, and time and resources
impose no constraints. A homotopy-equivalent inclusion preserves connected components, hence
existence of a route between a start and goal. Connected-component preservation already suffices
for that task. It does not preserve cost: if the removed edge xz costs one and xa;az each cost 100,
the shortest route costs 200 after refinement.
10.2 A sufficient implementation contract
Identical abstract labels, kernels, and orders can describe two devices when one test also causes
unmodeled damage in one device. No statement using only that tuple can distinguish them. A
sound physical interpretation must include the relevant transitions.
Let X be concrete apparatus states,  : X ! H the diagnostic-state map, G  X a safe set,
and R  X  2E a relation to abstract test histories. Let Fe(x) contain every possible concrete
successor when test e executes. Require related safe initial states and, whenever (x;J ) 2 R and
A(J;e ) = 1 , require executability, nonempty successors, and
∅ 6=Fe(x) G \ fx′ : (x′;J [ feg) 2 R; (x′) = (x)g: (13)
The hidden diagnostic target must be preserved, and the full conditional observation law must
match the model.
Proposition 10.1 (Conditional implementation bridge) . These assumptions transfer finite-step
safety to every legal abstract policy. With the declared observation law and target preservation, they
also transfer the corresponding finite diagnostic guarantee.
Proof. Initially the concrete state is safe and related. Given a related safe state, the selected action
executes, and every possible successor is safe and related by ( 13). Induction proves the safety
statement. The observation assumption gives the same adaptive transcript law, by induction on
histories, and hence the same terminal diagnostic error.
Recovery requires an additional progress argument. An unchanged J during repetition does not
prevent heating, wear, or depletion. Equation ( 13) must be justified from independently specified
dynamics, not by defining safety to mean formal permission. Feedback refinement relations provide
an established framework for action and successor conditions, behavioral inclusion, and specification
transfer [ 26]. That framework is distinct from B-Theory’s RF condition.
10.3 A latching diagnostic fixture
The general availability example in Section 5 has a direct finite implementation. Keep the hidden
target h = (b;c ) fixed and use a known apparatus mode m 2 f free; 0; 1g. Probe g reads b and
preserves the mode. Probe e0 reads (1 b)c, is allowed in modes free or 0, and sets mode 0. Probe
e1 reads bc, is allowed in free or 1, and sets mode 1. The selected specialized probe and g remain
repeatable; switching specialized modes within an episode is blocked. Independent BSC readouts
leave the hidden target unchanged.
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PAGE 19
A digital guard implementing this table realizes the availability predicate. More generally, a
register storing J can encode any finite A(J;e ) with always-legal idempotent repeats. This is a
synthetic realization, not a proof that such constraints naturally describe a motor or arise from
RF homotopy. A disposable reagent or a probe that consumes its resource on every read does not
satisfy this repetition premise.
For flip rate 0:1, repeat g an odd number k of times, use its majority to choose the specialized
mode, and repeat that probe k times. Write
ak =
kX
j=(k+1)/2
k
j

(0:1)j(0:9)k−j:
For target c = 0 , the error is ak. For c = 1 , the correct and incorrect mode branches give error
(1 ak)ak +ak(1 ak) = 2ak(1 ak). Thus maximum error is 2ak(1 ak) and uniform-prior Bayes
error is 3ak=2 a2
k. At k = 5 these are 0:0169734528 and 0:0127667264. They are policy values,
not finite-budget optima.
After a fixed k-probe mode decision, infinitely many subsequent readings can identify whether
the wrong mode was selected, but cannot recover c on that branch. Its limiting Bayes floor is ak=2,
or 0:00428 at k = 5. This makes the finite-family versus single-policy distinction operational. The
implementation was checked on 72 reachable history/request/hidden-state combinations, and the
error formula was independently enumerated for k = 1; 3; 5; 9; 15. Neither is a hardware experiment.
11 Preserving a specified recovery task
Ordinary homotopy is not a directed control specification. A more direct alternative is to declare
which recovery outcomes must remain possible. This subsection first states a finite graph property.
Its interpretation as apparatus behavior requires the implementation assumptions below.
LetP be a finite poset on H,T H a nonempty set of acceptable recovery configurations, and
D H a fixed set of protected starts. For a suborder R P , put
VT (R) = fx 2H : 9t 2T; x R tg:
This is a reachability set, not an infinite-horizon controlled invariant set. The order is reflexive, so
starts already in T satisfy the objective. Assume D VT (P ), and define
FT = fJ  E :D VT (PJ )g; A T (J;e ) = 1fJ [ feg 2 F T g (J 2 F T ):
Thus a diagnostic test may remove recovery options while retaining at least one acceptable outcome
from every protected start. Unlike preservation of every ordered pair, this requirement permits
nonmonotone tests. The protected set D is fixed independently of noisy answers; substituting an
inferred candidate set would need a different safety argument.
Proposition 11.1 (Task feasibility and the oracle) . Forx P t, define B(x;t ) = fe 2 E :qe(x) =
1; qe(t) = 0 g. Then
J 2 F T ( ) 8 x 2D 9t 2T : x P t and J \B(x;t ) = ∅:
In particular, FT is closed under taking subsets, and an executed test is always legal to repeat.
With static nominal bits and fresh BSC readings, Theorems 5.2 and 5.4 hold with AT in place of
homotopy admissibility.
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PAGE 20
Proof. The displayed condition is exactly the definition of x J t, with the quantifiers written out.
Removing tests cannot destroy it. Repeating a test leaves J unchanged. The availability predicate
depends on J, not the answer, so both the cached-oracle simulation and majority amplification
preserve legality, including after a mistaken majority. These are precisely the premises used in the
two earlier proofs.
Here later execution cannot unlock a previously unavailable test: if J K 2 F T andK [ feg 2
FT , then J [ feg 2 F T . A query constant on a candidate set can therefore be pruned from a
successful ideal tree for this particular rule; deleting it enlarges future availability and loses no
branch information. This pruning need not be valid for arbitrary availability predicates.
Proposition 11.2 (Smallest separation from RF) . Even with D = H, ∅ 6= T ⊊ H, and a
nonconstant Boolean test, RF and preservation of VT imply neither one another. Three states are
necessary and sufficient for each failed implication.
Proof. For RF without task preservation, use a<x<z , T = fzg, and q = 010. The triangle axz
collapses onto the retained tree ax [az by removing the triangle and its free edge xz, but x no
longer reaches z.
For task preservation without RF, use a<x;a<y , T = fx;y g, and q = 101 in order (a;x;y ).
The edge ax is removed and ay remains. Every state still reaches T , but the number of connected
components increases from one to two. Both examples use exactly the original maximal elements
as acceptable terminal configurations.
There is no nonempty proper T on one state. On two states, universal initial reachability of
a proper T forces a two-element chain with its maximum as the sole target. Retaining its edge
preserves both properties; deleting it destroys both. This proves minimality without relying on the
finite computation.
There is a useful boundary. If T = ftg and VT (P ) = H, then t is a greatest element. Task
preservation keeps it greatest in PJ , so both order complexes are cones and their inclusion is a
homotopy equivalence. For one Boolean test, preservation is equivalent to q(t) = 1 or q  0.
Consequently all individually legal tests are jointly legal in this special case. Pure reachability to
a single universally reachable target does not produce cumulative exclusions.
It still permits informative strict thinning. On a < x < zwith target z, the test q = 101
removes a < x but preserves recovery. Together with r = 011 it produces the three distinct
signatures 10; 01; 11, while the joint order retains a < zand x < z. Thus a task constraint can
permit full diagnosis without preserving every directed pair.
A three-state cumulative obstruction. On a < x;a < y, take T = fx;y g and tests p = 110 ,
s = 101. Each test alone keeps recovery from all three states, through x ory respectively. Together
they remove both outgoing edges from a. Their joint signatures 11; 10; 01 separate the three identity
labels, but either first test blocks the other, and its nonsingleton answer cell is unresolved. Hence
the ideal oracle predicate is false. Under the fresh-noise assumptions, the limiting uniform-prior
Bayes error and randomized minimax error are both 1=3. Every deterministic ideal policy makes
at least one error. Policies erring only at a, only at x, or only at y are attainable, and their equal
mixture attains the minimax bound. Repeated readings approximate these ideal policies. This is a
failure of initial separation to imply cumulative feasibility; it is not a separation between arbitrarily
accurate finite families and a single ongoing consistent policy.
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PAGE 21
Cost and robust control refinements. Reachability does not preserve the cost of recovery.
Give the chain’s separately executable edges a ! x, x ! z, a ! z costs 1; 1; 10. The test 101
preserves RF and reachability to z, but increases the best cost from a from 2 to 10. The direct
edge here is an independent primitive action; it is not a shorthand for the now unavailable cheap
path. Given independently specified edge costs, one can instead require
dJ∪{e}(x;T ) Bx (x 2D);
where dJ is the optimal recovery cost and the budgets Bx are fixed before diagnosis. Exact cost
preservation uses Bx = d∅(x;T ). Per-step allowances should not be confused with a fixed total
allowance. These budgets concern subsequent recovery; they do not make repeated diagnostic
samples free of physical cost.
For a finite fully observed recovery system with safe set G, target T  G, enabled recovery
actions UJ (x), and successor sets FJ (x;u ), robust finite reachability is computed by
Z0 =T; Z k+1 =Zk [ fx 2G : 9u 2UJ (x); ∅ 6=FJ (x;u ) Zkg:
The stabilized set WJ is exactly the set from which a controller can force safe arrival in T in finitely
many steps: the first entry stage supplies a strictly decreasing rank. Preserving D  WJ gives
another finite availability predicate when the supplied dynamics depend only on J and repeated
diagnosis leaves that dynamics unchanged. Infinite-horizon safety uses instead the greatest fixed
point of Z 7!G \ fx : 9u; ∅ 6=FJ (x;u ) Zg. These are standard finite reachability and invariant-
set constructions, not a new control-synthesis method. If the physical state is not observed, the
controller must use a common action on each information set; pointwise existence of a recovery
action is insufficient.
What a physical use would require. The symbols in the preceding poset are counterfactual
recovery configurations. During diagnosis the unknown origin or mechanism label h can remain
fixed while the apparatus mode J removes options for a later recovery phase. The actual recovery
configuration must be represented separately when it changes. A transitive comparability is not
by itself an executable action, and the thinning rule does not establish that probing a real device
disables exactly those actions.
A physical implementation must independently supply the diagnostic transition law, recovery
actions and their successors, safe and target sets, costs, and the observation interface. Each legal test
must execute without changing the diagnostic label, leave the apparatus in the protected winning
region for its new mode, and produce the declared conditional observation law. Repetitions must
satisfy the same obligations despite wear, temperature, and resource use. Recovery must have an
implementable policy at the controller’s actual information level. A relation from the concrete state
to the abstract state must preserve enabled inputs and contain all concrete successors in the abstract
successors, with compatible safety and target labels. These are the familiar local obligations behind
controller refinement; see Definition V.2 and Theorem VI.3 of [ 26]. Establishing them for a device
would supply the missing physical content. Defining safety as “the chosen abstract test is legal”
would not do so.
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12 Observation fidelity and finite-horizon robustness
12.1 Extra information hidden in the noise
The oracle lower bound requires simulation of the complete readout law from the nominal answer
and public history Ht:
Ke(y jh; Ht) = eKe(y jqe(h); Ht): (14)
There may be no additional hidden-state information within a nominal-answer class. The original
common BSC satisfies this condition. Having every error rate below one half is insufficient.
For two equiprobable states let q = 0 in both, but let the probability of readout 1 be 0:1 and 0:2,
respectively. Every finite history has positive probability under both states. The nominal oracle
always returns zero and has label error 1=2. Independent actual readings identify the state by their
frequencies. The equal-prior MAP error after n readings is
Rn = 1
2
nX
k=0
n
k

minf0:1k0:9n−k; 0:2k0:8n−kg  !0: (15)
The convergence follows, for example, by thresholding the empirical mean at 0:15 and applying
Hoeffding’s inequality under both states. Any distinct fixed rates, however close, produce the same
asymptotic obstruction to nominal-oracle domination.
Samples n 1 21 101 501 1001
Rn 0.450000 0.261171 0.0757204 0.000721426 3:33741 10−6
At n = 2001 , Rn  9:66883 10−11. These are exact integer binomial sums converted to decimals.
Independent binary-path enumeration matches the count formula in 39 cases: lengths 0–12 for
rate pairs (0:1; 0:2), (0:10; 0:11), and (0:2; 0:2). They do not contradict the fixed-library separation
theorem, which uses actual laws rather than nominal bits.
12.2 What a small model discrepancy does guarantee
Proposition 12.1 (Finite-horizon perturbation bound) . Use the same fixed hidden-state space,
prior, policy, independent private randomization, terminal loss, and admissible policy class in two
models. The policy stops by N . Conditional on every fixed h, matching relevant controller history,
and selected action, suppose the complete next-step laws differ by at most  in total variation. For
padded transcript TN ,
max
h
kLP (TN jh)  LQ(TN jh)kTV dN = 1  (1 )N  minf1;N g:
The joint laws of (H;T N ) also differ by at most dN under the common prior. Statewise and Bayes
classification errors differ by at most dN . Expected terminal losses in a common interval of width
G differ by at most GdN . The value bound persists when taking infima over the same admissible
policy class.
Proof. Fix h and couple private coins identically. While histories agree, actions agree. A maximal
coupling keeps each next step equal with conditional probability at least 1 ; multiply through
N . The coupling inequality gives the statewise bound. Couple H identically and integrate for the
joint bound. Couple randomized reporting too, so classification error is a common event of target
and report. Rescale losses for the bounded-loss result, then take infima.
22

PAGE 23
Stopping can be represented by a common absorbing symbol through time N . An observation-
only discrepancy bound does not control hidden damage or changed interlock transitions: include
those outcomes in the one-step law or justify them separately. Likewise, the proposition does
not compare optima under different admissibility predicates. For  = 0:01, d10  0:095618, d100 
0:633968, andd1000  0:999957. A fixed small discrepancy supplies no useful infinite-budget transfer
by itself.
12.3 A weaker premise for amplification, but not domination
LetEj indicate an erroneous binary readout. Independence can be weakened for the majority upper
bound to
E[Ej j F j−1;h ] "< 1=2
at every step, where Fj−1 includes the entire past. For > 0, the conditional exponential moment
is at most 1 " +"eλ. Iterating this bound and applying the usual Bernoulli Chernoff bound gives
majority error at most exp[2k(1=2 ")2]. This permits some dependent readings. It does not
imply ( 14): the state-dependent-rate example satisfies the conditional error bound while defeating
oracle necessity. A single reused noise bit also fails the conditional error premise after informative
observations.
Observation marginals alone are insufficient. For fair H, compareY =H E withY = 1H E,
where E  Bernoulli(0:1). Both Y marginals are fair, yet reporting Y has errors 0:1 and 0:9. The
relevant target-conditional or joint laws differ. The product bound is sharp: an all-zero record
versus independent Bernoulli contamination of rate  has transcript TV exactly 1  (1 )N .
13 An anchored motor experiment: model and test protocol
13.1 Measurement model and scope
A proposed bench study uses an enclosed rotary motor with configurable command polarity, an
external direct or reversing transmission, upstream shaft and downstream endpoint angle sensors,
and an independently established reference transmission. The motor/transmission interface defines
the self/world boundary. Mechanical reversal may change inertia, gain, friction, or backlash; those
changes should be measured rather than assumed away. A camera-coordinate flip is a measurement-
interface control, not a physical transmission reversal.
Let;w;
; denote actuator reversal, transmission reversal, body-sensor orientation error, and
endpoint-sensor orientation error. For a signed command u 2 f 1; +1g, an initially resting first-
order model is
˙! =

(1)θKu !


=; ˙ =!; (16)
∆ = (1)θKu[T  (1 e−T /τ)]; ∆ = (1)wr ∆; (17)
B(t) = cB + (1)γ(t) +B(t); Y (t) = cY + (1)β (t) +Y (t): (18)
HereK is gain,  the time constant, r the transmission ratio, and cB;c Y fixed measurement offsets.
Define signed differencesDB =u[B(T )B(0)] andDY =u[Y (T )Y (0)]. Offsets cancel; orientation
errors do not.
23

PAGE 24
With anchored orientations 
 = = 0, r = 1, matched gain across hidden states, and indepen-
dent Gaussian frame errors of standard deviations B; Y ,
DB j N ((1)θ; 22
B); (19)
DY j;w N ((1)θ⊕w; 22
Y );  =K[T  (1 e−T /τ)]: (20)
Thresholding at zero gives a BSC with error Φ(=(
p
2)), where Φ is the standard-normal CDF.
Thus the binary model follows from an explicit measurement interface. For k independent differ-
ences of a port measuring  with equal prior,
P( = 1 jD1;:::;D k) = logistic
0
@ 
2
X
j
Dj
1
A:
This decoder uses signed differences; it is not claimed optimal among all processing of every raw
frame.
The fault parameters are fixed during an episode, while angle and velocity move. Repeated
probes need verified settling/reset or an initial-motion model. Wear, heat, and changing polarity
must be modeled when relevant. The originating R4 polarity-shock task [ 25] is a different contract
from this fixed-target episode. The present equations and parameters are not hardware operating
limits.
13.2 Orientation anchors and a remaining gauge ambiguity
Without orientation calibration, the nominal measurements identify
b = 
; y = w ; y ref =  after setting w := 0:
Flipping ;
; together preserves all three observations. A reference transmission alone therefore
cannot identify . However y yref =w identifies the original transmission bit.
Measurements and independent anchors  Original w
Body and endpoint; neither sensor anchored No No
Add known reference transmission; neither anchored No Yes
Anchor body sensor; no reference Yes No
Reference transmission and anchored endpoint sensor Yes Yes
Anchor both sensors; no reference Yes Yes
Enumeration of all 16 orientation configurations verifies the table. Identification means dis-
tinction in observation law, not finite noisy certainty. Calibrating one unanchored sensor against
another does not remove the ambiguity. An independent displacement or equivalent physical refer-
ence is required. If the original environment must be restored after reference testing, restoration
reliability and cost belong in the task model.
13.3 Executed simulation and negative controls
Six conditions each contain 8,000 episodes, balanced at 2,000 per (;w ) cell, with five probes per
port. Parameters are  = T = 0:08 seconds, normalized K = 1 , B = 0:025 and Y = 0:016
radians. Thus   0:0294303553 radians. Condition i = 0;:::; 5 uses seed 20261105 + 1009i. A
24

PAGE 25
self-contained JavaScript implementation supplies raw-frame examples, statewise results, and all
seeds.
Under nominal independent noise, analytical transfer errors are approximately 0:059926 for five
body signs, 0:031348 for five body differences, and 0:001817 for five reference differences. Simulated
errors are respectively 0:061250, 0:032625, and 0:001500. Routine sensing predicts present effective
polarity with error 0:001625, but its parity-only actuator transfer decoder has error 0:5.
Condition Body Reference Gain-aware routine
Nominal 0.032625 0.001500 0.492500
Body sensor reversed 0.967125 0.002125 0.503875
Endpoint sensor reversed 0.028875 0.997750 0.498750
Common body noise 0.198875 0.001875 0.499750
Reference wrong in 15% of episodes 0.029500 0.155000 0.505875
Unequal actuator gain 0.017625 0.000625 0.185875
Entries are transfer errors; body and reference columns use continuous differences. The gain-aware
routine decoder assumes K = 1 for  = 0 and K = 1:6 for  = 1 ; it is correctly specified only in
the final row. That row demonstrates information about  through magnitude. The parity-only
routine decoder remains at 0:5 in every row. Even quantizing signs may retain leakage, because
gain changes sign-error probabilities.
The common-noise condition reuses a command-aligned Gaussian difference, so repetition adds
no body information; error remains near the one-probe value 0:202587. Sensor-reversal controls
deliberately leave the decoder unaware of the new orientation. They test reliance on calibration.
They do not show unavoidable failure for a decoder told the reversal. The imperfect-reference
condition selects a reversed reference in 15% of episodes and retains that selection throughout the
episode.
For an illustrative repair decision, leaving a reversed actuator costs 10, successful replacement
costs 3, each measurement costs 0:05, and reference setup costs 0:4. Replace when posterior re-
versal probability exceeds 0:3. The nominal body and reference scores are 1:893875 and 2:156750,
compared with 3 for always replacing and 5 for never replacing. Body reversal raises its score to
6:516250. These are counterfactual terminal scores: physical repair and replacement success were
not simulated.
All 48,000 episodes were reproduced from the saved code after reconnecting the local executor.
Eighty-eight paired noiseless trajectory points also match under (;w ) 7! (  1;w  1); adding the
same observation noise couples the paired routine laws. The finite trajectory check corroborates
the algebra, not an unrestricted simulation theorem. No hardware measurements are included.
13.4 A proposed confirmatory protocol
The following protocol is a research design, pending apparatus selection and engineering review.
1. Commission independently: establish sensor orientations, gain, time constant, backlash, set-
tling, and noise using separate calibration data. Actual apparatus design supplies mechanical
and electrical limits.
2. Fix the observation interface. Compare sign-only readings with continuous differences; audit
current, timestamps, fixture identity, settling duration, and auxiliary channels. Expose those
only in explicitly defined interface conditions.
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PAGE 26
3. Freeze target labels, likelihoods, policies, exclusions, loss functions, setup/restoration costs,
and stopping rules before confirmatory outcomes.
4. Blind the configuration schedule, balance all four hidden cells, randomize trial order and
command direction, and lock predictions before revealing configurations. Balance handling
and timing to limit setup cues.
5. Score present control, reference-environment transfer, and repair separately. Include the
reversal, common-noise, imperfect-reference, and unequal-gain controls above.
6. Implement actual enclosure, travel, current, settling, fixture, and resource checks. Posterior
confidence cannot override these interlocks. An adapter latched for an episode can instantiate
the separate availability model; reset occurs outside the episode.
Section 14 quantifies the proposed confirmation, its decoder-dependent power, cost criteria,
resource limits, and calibration precision. The 4,800 nominal episodes do not include perturbation
controls. The acquisition arms and independent sampling unit must be frozen after a separate
pilot.
14 Stateful experiments, resource limits, and confirmatory power
The motor calculation and the irreversible latch address different claims. The motor supplies an
explicit observation model for attribution and decision value; the latch supplies a stipulated restric-
tion on future measurements. The original motor simulator does not implement that restriction.
It generates each port from rest and does not execute reference installation, restoration, or repair.
Imposing a latch on a motor adapter would define a deliberately constrained benchmark, whose
physical implementation would need its own transition contract.
An independent simulation study examines several assumptions omitted by that static calcu-
lation. It uses standard-library Python, distinct random-number generation, and explicit motion
and resource state. Its 240,000 episodes supplement the original JavaScript study’s 48,000 episodes;
they are not a recount of those results. Neither study contains hardware observations.
The diagnostic target must also be distinguished from current apparatus state. Write h =
(origin;w original) for the original configuration, while tracking current actuator condition and trans-
mission setting separately. Reference installation changes wcurrent; repair may change current. These
interventions do not rewrite the original label. Attribution of the original fault, present control,
and successful repair consequently require separate scores.
14.1 Initial motion changes the experiment
For constant target speed v = (1)θKu, the first-order model has the exact transition
!′ =v + (! v)e−T /τ;  ′ = +vT + (! v) (1 e−T /τ):
A signed displacement therefore has mean
u(′ ) = ( 1)θK[T  (1 e−T /τ)] +u! (1 e−T /τ):
The second term disappears only when the initial velocity is zero or its effect is explicitly incorpo-
rated into the likelihood. Zero-drive settling for duration S multiplies velocity by e−S/τ , but also
changes position.
26

PAGE 27
Using the paper’s parameters, independent enumeration of all 32 five-command sequences gives
body-decoder error 0:070783 when independent random command signs are applied without settling.
Alternating signs give 0:273952, compared with 0:031348 for pulses independently starting at rest.
The second alternating pulse even has negative signed displacement. Repeating the same command
instead gives error 0:000025117, because accumulated velocity increases the signal. These are
different physical interventions, despite having the same nominal number of pulses.
A separate Runge–Kutta integration checks the analytic transition in 108 cases. Raw-frame
Monte Carlo results provide a further implementation check: each condition contains 40,000
episodes, balanced between the two actuator labels.
Condition Analytical error Simulated error
Independent starts at rest 0.031348 0.032375
No settling, random commands 0.070783 0.070975
No settling, alternating commands 0.273952 0.274125
Settling for 0:4 seconds 0.031350 0.029800
Persistent difference offset 0.124178 0.124625
Temperature-dependent gain — 0.065250
Condition i = 0;:::; 5 uses seed 20261105031 + 104729i. The final condition stipulates gain
K = 1  0:5H before each pulse, held constant during that pulse, with the thermal state defined
below. This extension isolates body sensing; it does not simulate a reference installation or an
additional transmission factor.
Given the conservative pulse-end bound j!j  K, a sufficient settling duration for subsequent
mean contamination below 0:01 is
S  log
 
K (1 e−T /τ)
0:01
!
 0:41172 s:
This is a model calculation, not a hardware operating prescription. Settling does not reset cumu-
lative travel, backlash, temperature, or wear. Repositioning must therefore be represented and
charged separately.
14.2 Preparation error and correlated readings
Fresh observation noise and persistent preparation error have different consequences. Suppose the
selected latch route stores qe(h) Z, where Z  Bernoulli(0:02) is sampled once. Subsequent
readings independently flip that stored bit with probability 0:1. The complete observation process
still factors through qe(h), so the nominal oracle can simulate it. Repetition, however, learns the
corrupted stored value.
For the two-block majority policy, let ak denote the majority error of k fresh 0:1 flips and set
bk = 0:02 + 0:96ak:
Its errors for c = 0 and c = 1 are respectively bk and (1 ak)bk +ak(1 bk). Exact rational
calculation gives Bayes error 0:032256057344 at k = 5 , tending to 0:02. Thus this modification
defeats amplification while preserving the oracle’s simulation premise.
A continuous analogue is
Dj = (1)θ +S +j; S N (0;s 2);  j N (0; 22
B);
27

PAGE 28
with S fixed throughout an episode. The sign-of-sum decoder has error
Φ
0
@ q
s2 + 22
B=k
1
A  !Φ(=s):
Fors = 0:02 radians, five-read error is 0:124178 and the limiting error is 0:070576. Calibration or
preparation uncertainty cannot be made fresh merely by acquiring another frame.
For exchangeable Gaussian differences with unchanged marginal variance and correlation ,
five-read error instead becomes
Φ
 

p
5p
2B
p1 + 4
!
:
It rises to 0:057846 at  = 0:10 and 0:094059 at  = 0:25. Even reusing a single noisy baseline
frame changes the dependence structure. Acquisition specifications must identify shared frames,
calibration variables, and reference-setting errors, rather than reporting only marginal noise.
14.3 Finite resources and the implementation bridge
Consider the illustrative, dimensionless thermal recurrence
Hpeak =e−T /2Hpre + 0:12; H next =e−S/2Hpeak; T = 0:08:
Its stationary peak is
0:12
1 e−(T +S)/2:
Starting cold, repetition without waiting crosses the stipulated limit one on pulse ten. Waiting
0:16 seconds crosses it on pulse twenty-four. The cold-start sequence remains below the limit when
S  0:175667 seconds. A separate thirty-use resource still blocks pulse thirty-one, irrespective of
cooling.
These checks expose an abstraction issue. A guard correctly refusing an exhausted or overheated
test may preserve physical safety while violating the abstract requirement that every permitted
repeat executes. One remedy is to fix a finite horizon N and establish adequate initial reserves and
transition bounds through N . Another is to include resources and motion in the abstract state and
establish an appropriate new result. The always-repeatable theorem does not automatically cover
that enlargement.
Actual engineering must supply the safe set, disturbance bounds, travel and current limits,
shutdown behavior, and reliable setup transitions. Rejection, timeout, and abort are outcomes with
information and cost; silently discarding them changes the experiment. None of the dimensionless
thresholds above validates an apparatus.
14.4 Operating characteristics of the proposed confirmation
Specify three acquisition arms—routine, body, and reference—and freeze one primary decoder per
arm. With twelve cell-by-arm error rates, one-sided exact binomial upper bounds at  = 0:05=12
have simultaneous coverage at least 0:95. At 400 episodes per cell, the requirement U < 0:10
is equivalent to observing at most 24 errors. The bounds for 24 and 25 errors are 0:098392 and
0:101447. Integer arithmetic verifies the corresponding binomial tail inequalities.
For four independent cells with common error p, the probability that all pass is
[PrfBinomial(400;p )  24g]4:
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PAGE 29
It equals 0:995928 for the continuous body model, but only 0:095909 for the five-sign body decoder,
whose stipulated error is 0:059926. Exhaustive sample-size searches using finite binomial sums give
minimum sizes of 266 and 963 episodes per cell, respectively, for at least 90% four-cell power at the
same confidence level. The reference model requires 73 per cell under its stipulated error 0:001817.
These planning results depend on the model and should not replace an independent pilot.
Bonferroni coverage does not require independence between arms. Each binomial bound nev-
ertheless requires its marginal sampling model, and the power product additionally assumes inde-
pendent cells. Persistent apparatus or session effects may invalidate episode-level independence.
Fixed-sample intervals also do not automatically permit optional stopping.
The stated 4,800 episodes cover the three nominal arms only; negative controls require additional
episodes. Two decoders evaluated on shared recordings form a paired comparison and expand the
confirmatory family if both are primary. Routine attribution is a negative control, rather than an
arm expected to satisfy the attribution threshold.
14.5 Joint error and decision-cost criteria
A possible joint 95% confidence family allocates 0:04 across the twelve error bounds and 0:01
across two anchored-arm cost comparisons. At 400 episodes the error cutoff remains 24. For 1,600
independent episodes per arm and the stipulated loss range of width ten, Hoeffding’s inequality
supplies the upper cost bound
L + 10
s
log(1=0:005)
2(1600) =L + 0:406906:
Predeclare success when this bound is below the uninformed baseline three.
Independent Gaussian integration gives nominal expected costs 1:890983 for body and 2:158251
for reference. Under these stipulated losses, sufficient lower bounds on passing the cost criterion
are 0:999999859 and 0:997644, respectively. Combining cost and error success probabilities requires
a union bound or a joint calculation, not an assumption that the statistics are independent.
Reference sensing improves terminal decision loss over body sensing by only 0:132732; its addi-
tional setup charge 0:4 outweighs that improvement. Actual settling, restoration, failed trials, and
repair expenses must be included. If replacement succeeds with probability s, its expected loss is
3 + 10(1 s)p, giving threshold p > 3=(10s). The baseline and confidence calculation must also
change; unsuccessful replacement may enlarge the loss range. These results concern counterfactual
decisions under declared prices, not demonstrated repair performance.
14.6 Calibration precision and finite claims
The earlier 1,293-per-stratum Hoeffding count estimates sixteen Bernoulli rates to within 0:05
simultaneously. It bounds a pairwise difference only by its observed difference plus 0:10; it does
not certify a continuous conditional kernel. To transfer a ten-reading error guarantee within 0:01
using Proposition 12.1, require
  1  0:991/10 = 0:0010045287
at every relevant conditional step. Even with observed Bernoulli difference zero, the same con-
servative method with individual radii =2 needs 12,806,679 samples per stratum. More efficient
parametric calibration requires justified distribution, variance, drift, and dependence assumptions.
Finite data cannot establish exact infinite-horizon kernel equality against arbitrary alternatives.
Tie practical equivalence margins to the intended finite-horizon risk claim rather than treating
failed significance tests as evidence of equality.
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PAGE 30
15 Conclusions and remaining boundaries
Functional self-diagnosis is task-relative diagnostic access: which distinctions about a system’s
mechanisms can permitted experiments acquire, and which decisions benefit? Present control,
original-cause attribution, transfer, and repair need different labels and losses.
The complete-law oracle characterizes limiting information under static known finite channels
and idempotent access. Its terminal-channel polytope captures Bayes and randomized minimax
decisions. Four states are minimal for the displayed gap between reliable finite families and one
ongoing consistent policy. Tiny known law differences can remove qualitative ambiguity while
making finite budgets decisive; estimated kernels and changing faults require further models.
A specified recovery task gives a direct availability criterion. It and ordinary homotopy im-
ply neither one another. Pointwise recovery opportunity still needs an implementable policy at
the controller’s actual information level. No naturally occurring apparatus satisfying the exact
repeatable-read latch or RF implementation contract has been established.
Stateful motor calculations quantify initial motion, preparation error, heat, depleted resources,
and confirmation power. They improve a falsifiable protocol but provide no hardware observations
or engineering approval. Actual calibration, transition envelopes, resets, resource reserves, reference
restoration, and repair success remain empirical obligations.
Five simulated reviews and two cross-reviews completed the requested internal assessment. Four
original historical full texts remain unavailable (Appendix E). The paper is a finite-model synthesis
with proofs and counterexamples; publication priority remains unassessed.
Video exposition can show parity compensation, failed attribution, anchored sensing, and an
irreversible mistaken choice, followed by the three-state recovery examples and stateful pulse results.
Use checked values and identify simulation. No video publication or website changes are included.
A Assumption failures and boundary cases
T opological support. Enumerate the rationals (rk)k≥1 and assign mass 2−k to rk under state
zero; use a Gaussian under state one. Both have topological support R, but rational versus irrational
outcomes identify the state almost surely. Equality of null sets, not support closure, matters.
Reused noise. Draw one N  Bernoulli(0:2) and return Yt = N forever. One-time marginals
are positive and separate , yet the entire record contains one noisy bit; equal-prior error stays 0:2.
Conversely, correlation can identify a state even when one-time marginals agree.
Noise endpoints and unknown noise. For a known BSC, " = 1=2 is uninformative. " = 0 or 1
is deterministic and can shrink support. Known "> 1=2 is informative after reversing decoding. If
unknown error rates span both sides of 1=2, ( = 0;" ) and ( = 1; 1 ") have the same law. Finite
unknown parameters may be included in H; a continuum without a separation gap need not admit
a uniform sample bound.
Prior exclusions and numerical pruning. Guarantees concern initial prior support. Assigning
a true state probability zero does not make it physically impossible. Replacing small positive weights
with zero changes admissible policies. Log-domain arithmetic can prevent underflow; pruning
requires a declared approximation or risk contract.
30

PAGE 31
State changes. A trusted reset can put every current state into a safe state while leaving every
origin possible. The static theorem does not forbid this. Track origin and current state jointly
when both matter. Resource depletion, time-dependent permission, and structural changes require
additional state.
Finite oracle arguments. The common-tape proof uses finite H and  < 1=jHj. It estab-
lishes no infinite-hypothesis analogue and no automatic single-policy consistency under irreversible
choices. With nonzero sample prices, the terminal-risk limit does not describe the cost-optimal
value: amplification may become arbitrarily expensive. If repetitions consume permission, it can
fail.
31

PAGE 32
B Verification and novelty ledger
Claim Evidence Status and limitation
Support persistence and
separation
Proofs, independent audit, finite
examples
Elementary synthesis; no priority
claim.
Cumulative
counterexample
Explicit complexes and edges; two
mathematical reviews
Correct for the declared contract;
physical use needs the bridge in
Section 10.
Oracle feasibility and
limiting value
Simulation/amplification proofs;
reviewed quantifiers and empty
branches
Derived specialization; priority
and external review open.
XOR and intervention
criteria
Symmetry and rank proofs; 121
rational shift checks
Known identifiability principles,
not causal graph discovery.
Optimal-policy table Exact fractions, 12 exhaustive tree
comparisons, 300,000 episodes
Finite supplied model only; no
physical data.
Risk gate Exact posterior/event probabilities
and information bounds
Relaxed guarantees; no
zero-hazard claim.
Self-model architecture Matched-information equivalence
retained
No superiority or consciousness
result.
Physical interpretation Directed counterexample;
conditional bridge; 72 interlock
checks
Homotopy alone insufficient; no
hardware validation.
Kernel robustness 39 exact leakage comparisons;
coupling proof
Finite-horizon bound;
factorization remains an
assumption.
Motor proposal 48,000 reproduced episodes; 16-state
orientation audit
Synthetic study and proposed
protocol only.
Historical necessity check Three-state model and explicit
probability bound
Source-defined witness; two
simulated source reviews; no
official correction.
The reviews are AI-assisted internal assessments, not external peer review. Their objections
changed the draft: general support predicates replaced an unnecessarily narrow safety-set model;
mutual absolute continuity replaced ambiguous continuous full support; cumulative structure was
separated from belief support; empty majority branches were totalized; and Bayes, minimax, zero-
error, and finite-family guarantees were distinguished. Code and checks corroborate proofs; they
do not prove unrestricted claims.
32

PAGE 33
C Reproduction
Current v0.3 build and verification commands are in README-v0.3.txt. Its manuscript is under
review_v03/paper; reviewer scripts and results are in the corresponding review folders. The
original v0.1 files are preserved. The v0.2 source is under continuation/paper; recovered Page
snapshots retain the earlier disconnection notices as historical records. Current delivery status is
in README-v0.2.txt. From the package root:
python code/experiments.py
python code/independent_checks.py
python code/cumulative_oracle.py
python code/figures.py
python reviews/causal/verify_model.py
python reviews/causal/verify_oracle.py
node continuation/code/robustness_checks.js
node continuation/code/interlock_checks.js
node continuation/code/motor_simulation.js
node continuation/code/historical_counterexample.js
python review_v03/oracle/verify_oracle.py
python review_v03/diagnosis/verify_source_witness.py
python review_v03/topology/enumerate_task_orders.py
python review_v03/experiment/independent_checks.py
python review_v03/code/build.py
The first three commands use the Python standard library. Figures also require NumPy and
Matplotlib. The simulation is separate from exact calculation. The predecessor checker writes
into its current directory; the README specifies a dedicated reproduction directory. The LaTeX
source, figure sources, exact JSON outputs, bibliography, source inventory, and independent reviews
accompany the PDF.
D A source-audit witness for a printed necessity claim
Under Definition 3.5 and the operational diagnoser construction (17)/(22) of Thorsley, Yoo, and
Garcia [ 35], the necessity direction printed in Theorem 6 fails for this three-state example. Two
simulated source reviews independently checked its premises and reconstruction. This is an internal
mathematical audit, not an official correction or external peer review. The sufficient assertion
is separate; none of our theorems uses its converse. Frequency separation is already explicit in
Example 5 of [ 33]; this is a noisy-mask adaptation.
Start in state 0. Invisible normal event u enters persistent mode 1, or invisible fault f enters
mode 2, each with probability 1=2. Mode 1 self-loops on a;b with probabilities 1=5; 4=5; mode
2 reverses them. The independent sensor reports  from a with probability 9=10, from b with
probability 1=10, and  otherwise. There are no later u;f events. All states are live, enabled
state/event successors are unique, and there is no invisible cycle. Then
p = P( jN ) = 13=50; q = P( jF ) = 37=50:
D.1 Source-defined matrices and recurrence
The sensor-output automaton has a normal invisible transition from 0 to 1 and a fault-suffixed
invisible transition from 0 to 2, each of probability 1=2. Subsequent visible events are ordinary
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events in both modes; the fault label persists. In augmented order (0N; 1N; 2N; 0F; 1F; 2F ), the
null-output matrix U has U 2 = 0. The absorption equation gives
Ry = bQy +UR y; R y = (I +U )bQy:
The first-observation rows restricted to reachable (1N; 2F ) are [13=100; 37=100] for  and
[37=100; 13=100] for . The later cell is f(1;N ); (2;F )g, with
Tα = diag(p;q ); T β = diag(q;p ):
In embedded order (q0; 0N ); (q1; 1N ); (q1; 2F ), the component chain is
Π =
0
@
0 1 =2 1 =2
0 1 0
0 0 1
1
A:
Both conflicting components are recurrent in the source-defined chain, so the recurrent-certainty
criterion fails. Posterior vanishing on records from the other mode does not make a component
transient in this chain.
The exact checker compiles all six output matrices, augmented matrices, and the 18-state one-
observation chain; solves the absorption equation rationally; extracts reachable cells; and classifies
recurrence. It independently sums all 254 visible histories of lengths one through seven from the
original event/mask model, matching their likelihoods. All checks pass.
D.2 Posterior and uniform quantifiers
Forn visible symbols containing k occurrences of , every compatible underlying word has exactly
n + 1 events, with no invisible suffix or ambiguity about event length. Posterior odds are
P(F jY1:n)
P(N jY1:n) =
 37
13
2k−n
:
Definition 3.5 quantifies over prefixes ending in a fault. That set is ff g here, and each subsequent
event emits one observation. For 0<< 1, the confidence-failure probability is
PF
Kn
n  1
2 + log(=(1 ))
2n log(37=13)

; K n  Binomial(n; 37=50):
Choose N so the threshold is at most 31=50 for every n  N . Hoeffding bounds the event by
exp(18n=625). Increasing N makes it smaller than any positive tolerance for every later horizon
and every required prefix. For   0 the failure event is empty. This checks the uniform quantifiers,
not only almost sure convergence. At  = 0:99, exact failure at n = 160 is PfBinomial(160; 37=50) 
82g  6:0400819 10−10. The earlier uniform envelope below 0:01 for n  160 remains valid.
Both inspected copies retain the printed if-and-only-if. Equation (21) appears to reverse absorp-
tion indices; this reconstruction uses the consistent operational definitions (17)/(22). Referenced
appendices were absent from the eight-page copies. Neither review found an excluding premise. A
bounded search located no official correction; that is not proof that none exists.
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D.3 Later definitions and source gaps
Thorsley [32] is a serious later AA-characterization lead, but only its abstract and first-page preview
were recovered. Chen and coauthors [ 6] was read at Definition 1 and Section IV. Its displayed
definition uses all faulty prefixes rather than fault-ending prefixes. The difference is substantive:
afterf m, posterior odds after another n observations with k occurrences of  are (37=13)2k−n−m.
For any fixed delay choose m =n + 1; even k =n leaves odds below one. The later criterion cannot
silently substitute for the 2008 definition.
E Historical coverage and remaining originals
The 2005 author copy [ 33] was checked at Definitions 2–3, Theorems 3–4, and Examples 4–5; the
2007 combined article/erratum [ 34] at selected information-state/DP results and corrected policy
measurability. Both new reviews read the full inspected 2008 paper and rendered its key definitions
and theorem pages. Its absent appendices were not read.
Direct comparison now includes de Oliveira [ 9], pp. 1–8; Renou–Venel [ 28], Section 4.3, pp. 17–
22, discussion pp. 34–35, and Appendix B pp. 44–45. Controlled information and early actions
constraining later access are directly checked antecedents. The full proof of their Theorem 3 was
not audited. Selected related originals were recovered: Whitmeyer–Williams [ 36], pp. 1–6; Li–Liu
[18], selected pp. 1–10 of pinned v2; and Qiu–Kumar [ 24], pp. 1–4. Public web reading of the
Paoli–Lafortune CDC 2003 precursor does not replace the 2005 journal article or its omitted proof.
Four original-full-text gaps remain: Blackwell 1953 [ 3], Greenshtein 1996 [ 12], Paoli–Lafortune
2005 [22], and Thorsley 2017 [ 32]. Publisher, institutional, author, archive, and metadata routes did
not supply them. The apparent Paoli Figshare open-access record has no files and is metadata-only
in both inspected versions. These gaps limit priority/subsumption assessment, not the indepen-
dently stated finite proofs. Retrieval logs, hashes, precise scopes, and failures accompany the
paper.
F Simulated review and revision ledger
Five simulated specialist roles and two adversarial cross-reviews completed the requested assessment.
They are internal AI reviews, not reports by the real researchers previously shortlisted. Agreement
is not a statistical confidence measure or external endorsement. No outreach or hardware action
occurred.
Review Evidence and change Scope
Oracle mathematics Complete-law characterization,
terminal-channel polytope, and four-state
minimality
Finite static known
channels; idempotent
access.
Oracle cross-review Explicit private randomization, conditional
freshness, and measurable report-vector
partition
Independent simulated
proof pass.
Stochastic diagnosis Source-defined matrices and 254 history
checks
Exact inspected 2008
conventions.
Diagnosis cross-review Challenged initialization, invisible events,
fault labels, conditioning, and prefix
quantifiers
Supported the narrow
witness.
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Review Evidence and change Scope
Topology/control Minimal witnesses; 3,186 topology cases and
12,848 task triples
Collapse certificates or
homology obstructions.
Experimental
identification
108 ODE checks, exact cutoff/power analysis,
240,000 independent simulated episodes
No apparatus validation.
Historical decision
theory
Stronger controlled-information overlap;
retrieval failures documented
Priority remains
unassessed.
Exact checks additionally cover 8,192 oracle/report-map cases, 5,120 noisy Bellman values, 640
known-channel values, and 8,704 small-state path cases. The root agent independently verified
the 400-trial cutoff and powers with rational integer sums. Earlier versions and outputs were
byte-hashed before integration and remain unchanged.
The bridge now specifies an immutable diagnostic-state map and nonempty successors. The
robustness statement bounds state-conditional and joint target/transcript laws. Motor sensing is
explicitly separated from irreversible availability. Confirmation names arms, decoder-dependent
power, sampling units, multiplicity, and costs.
Review reports, scripts, seeded results, inventories, and native-note snapshots accompany this re-
vision. The snapshots retain earlier pending delivery notices; README-v0.3.txt gives the completed
status. Bounded computations corroborate proofs and do not provide proof-assistant certification.
36

PAGE 37
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