B-THEORY SENSOR RELIABILITY: FINAL THREE-ITERATION RESEARCH REPORT
Iteration03 of03 | 4 November2026 | Internal AI-assisted research

MAIN RESULT

The robust two-reference boundary is solved for worst-current-mode expected
sensing cost. Let each reference have known marginal error1/5, and let the
unknown joint-error probability t=Pr(F1=F2=1) lie in [0,T]. A single policy
must have accepted error<=1/10 and coverage>=gamma in every current sensor
mode, for every law in that interval. Define
  K=3/5-8T,  M=1+K,  L=3/5+K.
For 0<=T<=3/40, the sharp frontier is
  gamma_max=L/M=(6/5-8T)/(8/5-8T),
  min sup_t max_m E[N_reference | current mode=m] = ((2+K)/L)*gamma,
  sensing cost = 3/100+(2/25)*((2+K)/L)*gamma,
for 0<=gamma<=gamma_max. Above T=3/40, no positive common coverage is possible.
The result covers adaptive randomized policies in the declared interface.

At T=1/25, an interval whose upper endpoint is the independent-error law,
maximum guaranteed coverage is11/16, not the known-law value3/4. The reference
count slope is57/22. At the threshold T=3/40, maximum guaranteed coverage is
3/5, and both references are required on every started episode. This is a
strict uncertainty penalty even though marginal reference quality is fixed.

This is a finite specialization with analytic proofs, exact computational
certificates, and fresh internal AI review. Historical novelty and external
peer review are not established. The prior-average cost frontier under this
ambiguity class is left open; no worst-mode result is relabeled as an average.

1. COMPLETE CONTRACT AND WHAT THE GUARANTEE CONDITIONS ON

An episode begins with fair actuator polarity theta in{0,1}; it stays fixed
through that episode. Command a produces x=a XOR theta. A terminal estimate d
implements desired target g using command a=g XOR d, with loss1{d!=theta}.

The primary sensor has modes H (healthy), R (reversed), U (uninformative).
H observes its input with fresh BSC error epsilon=1/10; R reverses the input
before the same noise; U returns a fresh fair bit. Initial mode L0 has law
pi=(2/5,2/5,1/5), independent of theta.

Timing and charged interface inherited from iteration02:
  1. Mandatory C observes a trusted injected physical zero through L0, cost1/50.
  2. Exactly one transition opportunity occurs before the routine reading:
       Pr(L1=m|L0=l)=(1-h)1{m=l}+h*pi(m), with main case h=1/4.
  3. Mandatory routine residual P observes theta through L1, cost1/100.
  4. Up to two optional reference readings Q_i=theta XOR F_i, cost2/25 each.
  5. Act or abstain. L1 does not change after P. The actuator never changes.

h is refresh probability, not necessarily a different-mode probability.
Actual Pr(L1!=L0)=16h/25. Thus it equals4/25 in the main case. C/P order and
their mandatory combined cost3/100 are fixed; these results do not optimize
omission or reordering of the pair. The reference orientation is stipulated,
as are its marginal error rates. Independence of modalities would not by itself
identify orientation. No added modality is needed for the solved boundary.

New ambiguity contract: choose one unknown t in[0,T], T in[0,1/5]. Before each
episode sample a potential error pair independently of theta,L0,L1, all primary
noise, and policy coins. Its joint law is
    (F1,F2)   00       01       10       11
    mass      3/5+t    1/5-t    1/5-t    t.
Both marginal error rates are exactly1/5. Independence corresponds to t=1/25;
the interval also permits negative dependence. The second error exists even
if the policy never buys its reading. The first reading cannot influence it.
The reference pair is independent across episodes; t itself is shared and fixed
across episodes. Fresh theta and primary episode variables are sampled each
episode. Equivalently, the result is a single-episode result with the specified
fair prior and no inherited actuator evidence. Keeping the same physical theta
while accumulating prior episodes' reference evidence would change the problem.

Policies know T, not t. They may use C,P, purchased reference answers, and fresh
private randomization, but have no calibration dataset or cross-episode memory.
They do not get free adaptation to the shared unknown law. This restriction is
explicit; amortized learning of a common law is a different future problem.

For acceptance A, write at a particular fixed t
  C_m(t)=Pr_t(A | L1=m),
  E_m(t)=Pr_t(A and d!=theta | L1=m).
The uniform guarantee is, for every t in[0,T] and every current m,
  C_m(t)>=gamma,  E_m(t)<=C_m(t)/10.
For positive coverage, this is risk Pr_t(d!=theta|A,L1=m)<=1/10. It averages
over the fair theta, old mode, refresh, noise, and policy coins. It does not
condition on L0, a particular theta, whether a change happened, or an individual
history. At zero coverage the conditional risk is undefined. Objective is worst
over t and current mode of expected reference count, with mandatory cost added.
It is neither reference count conditional on acceptance nor prior-average cost.

2. SHARP ROBUST FRONTIER: LOWER BOUND AND CAPACITY PROOF

Condition on current U. C,P is independent of theta, and the pair's law does
not depend on t. Before buying the second reference, all observable prefix laws
are independent of t because the first reference marginal is fixed. Therefore
one policy determines the same realization weights for every t.

By averaging over bit-label symmetry, and using the least-error terminal action
for the observed references, a lower-bound description uses nonnegative weights:
  b: accepted mass with no references;
  c: accepted mass after exactly one reference;
  q1,q2: first/second query probabilities under U;
  u,v: realization weights of accepting respectively after agreeing/disagreeing
       reference pairs (the outcome probability is not included in u,v).
They satisfy q1+b<=1, q2+c<=q1, and u,v<=q2. CP-dependent coins and stopping
are included. Replacing a wrong-way report by the better polarity can only
decrease error in this U relaxation; imposing other modes could only restrict
the full policy class further.

Set A_t=3/5+2t and B_t=2/5-2t. Then
  C_U(t)=b+c+A_t*u+B_t*v,
  E_U(t)>=b/2+c/5+t*u+(1/5-t)*v,
  N_U=q1+q2>=c+2u.
The agreement posterior error is t/A_t, disagreement error is1/2, and a single
reference has error1/5. At t=T the risk condition implies
  4b+c+M*v <= K*u.
Let z=K*u-c-4b-M*v>=0. Coverage at the other endpoint is
  C_U(0)=L(u-v)-3b-z.
Let s=(2+K)/L. Since N_U>=c+2u, substitution gives
  N_U-s*C_U(0) >= v+(3s-4)b+(s-1)z >=0.
Here s>=13/6, so every coefficient is nonnegative. Any policy satisfying common
coverage gamma therefore has worst-mode reference count at least s*gamma.

For capacity, b+c+u<=1 follows from the query constraints. Substitution yields
  M(u-v)<=1+3b+z,
and hence
  C_U(0)<=L/M+(L/M-1)(3b+z)<=L/M.
This proves gamma<=L/M. The proof deliberately uses risk at T and coverage at0.
Protecting only one endpoint would miss the uncertainty penalty.

These are lower bounds on the full adaptive policy class because every policy
induces feasible U weights. A reference-only policy below attains them in all
three modes, making the bounds sharp for the full problem rather than merely
for the relaxed U calculation.

3. ATTAINING POLICY AND THE DEPENDENCE THRESHOLD

For 0<=T<=3/40 and gamma<=L/M:
  - After mandatory C,P, buy Q1 with probability q1=M*gamma/L; otherwise abstain.
  - After Q1, accept d=Q1 with probability K/M.
  - Otherwise buy Q2, and accept iff Q2=Q1, using their common answer.
All coins are independent. This policy ignores the primary pair for its
decisions, so its conditional performance is identical in H,R,U.

The second-query probability is q2=gamma/L, and the early accepted mass is
c=K*gamma/L. At an actual t,
  C_m(t)=gamma*(K+3/5+2t)/L,
  E_m(t)=gamma*(K/5+t)/L,
  E[N_Q|m]=(2+K)*gamma/L.
Coverage is minimized at0 and equals gamma there. The difference
E_m(t)-C_m(t)/10 is affine increasing in t and is zero atT. Thus the guarantee
holds throughout the ambiguity interval; it is not a policy that secretly
changes with the unknown parameter.

When T>3/40, choose the admissible endpoint t=T. In U the smallest attainable
posterior error with two references is t/(3/5+2t)>1/10. One-reference and
uninformative leaves have still larger errors. Mixing or selective acceptance
cannot lower every accepted posterior below that bound. Positive U coverage
is impossible, so positive common coverage is impossible. At T=3/40 equality
is allowed: accept only two-reference agreement. Robust coverage can be3/5.
Thus the feasible maximum drops discontinuously to zero immediately beyond
the threshold. This concerns exact uniform guarantees, not a continuity claim
about average performance under an estimated law.

Representative exact results at maximum guaranteed coverage:
  T         gamma_max     reference-count slope    total sensing cost
  0         3/4           13/6                     4/25=.16
  1/25      11/16         57/22                    69/400=.1725
  3/50      9/14          53/18                    127/700
  3/40      3/5           10/3                     19/100=.19
  >3/40     0             no positive frontier     .03 if mandatory pair paid

At T=1/25 and maximum robust coverage, actual coverage ranges from11/16 at
t=0 to3/4 at t=1/25. Risk ranges from7/110 to1/10. These distinguish guaranteed
minimum coverage, actual coverage, and accepted risk.

4. KNOWN LAW, UNKNOWN LAW, AND IDENTIFICATION

If the policy is told the exact t<=3/40, replace L by
  L_known=K+A_t=6/5-6t, with K=3/5-8t.
The sharp worst-mode reference slope becomes (13/5-8t)/(6/5-6t), and maximum
coverage is L_known/(1+K)=3/4. The same weight proof applies, now imposing risk
and coverage at the same known law. For t>3/40 positive coverage is impossible
even when the law is known. At t=1/25 this recovers iteration02's19/8 slope.

Choosing the known-T policy does satisfy the error bound for every t<=T,
but its coverage is lower at t=0. It cannot claim the known-law coverage3/4
uniformly. The robust policy rescales its initial query probability to protect
this lower endpoint; beyond gamma_max that probability would exceed one.

The dependence parameter is population-identifiable from paired references:
  Pr(Q1!=Q2)=2/5-2t.
This does not require knowing theta. The same t shared across iid episodes
therefore can be estimated with a paid paired-reference training program.
Neither its observation cost nor estimation uncertainty is included in this
single-episode frontier. Finite training cannot simply certify an arbitrary
exact bound with certainty when competing laws have overlapping support.
Prior evidence about a persistent actuator would also alter the fair-theta
assumption. No such free information is supplied here.

Iteration01's separate identifiability results remain useful: routine readings
alone leave actuator/sensor orientation pairs observationally equivalent; a
trusted reference and routine channel separate all six fixed states. One RP
block has rank4 for arbitrary population mixture weights despite six distinct
columns; RPP has rank6 with determinant magnitude
|1-2delta|^3|1-2epsilon|^6/4. Pooled PP,PC,R designs identify all six weights
with at most two readings per episode. Thus individual-state discrimination,
mixture-law identification, and parameter learning are different questions.
Their earlier proofs depend on their own persistent-mode/fresh-noise contracts.

Iteration02 showed CP alone is uniform for every refresh h, while the triple
sign moment E[(-1)^(C+P+Q1)]=(192/625)(1-h) identifies h in replicated episodes
with known other parameters. Changing dependence between Q1,Q2 does not change
this single-reference moment under the independence-from-primary contract.

5. CONSOLIDATED CONTROL, CALIBRATION, AND ARCHITECTURE FINDINGS

Iteration01 solved a different, freely chosen three-test fixed-mode interface.
Its exact policy takes P,P, then buys C only on agreement; otherwise it abstains.
Loss including sensing and abstention is5453/25000=.21812 versus.25 for every
policy restricted to at most two tests. Its overall coverage is.756, accepted
error61/378, but U-conditional accepted error is.5. These numbers must not be
compared directly against the later mandatory-pair cost frontiers.

Iteration02 added the timed refresh. Under independent references, known h,
and ordinary Bayes error plus abstention cost, the optimal incremental value is
min{1/4,13/100+(7/20)e,22/125+(26/125)e},
where e=61/250+(32/125)h. The second-reference branch begins at h=1419/4544;
immediate abstention becomes optimal at h=1453/3328. These are refresh values;
actual mode-change probabilities multiply by16/25.

At h=1/4, stale h=0 and updated policies have identical actions and total loss
.2678, but stale forecasts are miscalibrated. At h=1/2, updated total loss is
.28 from abstaining after the mandatory pair; stale loss is.2902. The exact
mode-constrained mean-cost frontier at h=1/4 under independence is
.03+(11539/62500)gamma for gamma<=.75. That is an inherited PRIOR-AVERAGE
result for a known reference law, not this iteration's robust worst-mode result.

For probability scoring, freeze Y=1{theta=1} across all episodes, including
abstentions. For forecast Q and r(Q)=E[Y|Q],
  Brier = E[(Q-r(Q))^2] - Var(r(Q)) + Var(Y).
Reliability error, resolution, and irreducible uncertainty are distinct.
Iteration01's true posterior and constant.5 forecast are both calibrated,
yet only the former has resolution. A calibrated-score ordering also need
not order task risk: forecasts.24/.76 equally often yield Brier.1824 and
optimal terminal loss.24 at abstention cost.25; forecasts0,.5,1 with masses
.1,.8,.1 yield Brier.2 and terminal loss.2. Both target the same balanced bit.

In iteration02, correct posterior reporting has reliability error0 while the
ordinary Bayes policy still has U-conditional accepted error.2. Calibration
under an assumed law cannot certify an unknown-law subgroup guarantee. A
robust interval for risk does not automatically supply a single calibrated
point forecast for every t. This report makes no such universal forecast claim.

All policies can be compiled. The Bayes comparator in iteration02 stores an
action and probability report for each of28 possible histories. The new robust
policy stores T-dependent coin probabilities, the first reference bit, and the
stage. If an explicit joint estimator and a compiled controller use the same
information and random seed with the same rule, they match path by path.
No explicit-self-model advantage follows from a label, and equal engineering
cost is not assumed. These are functional reliability/control examples, not
evidence about consciousness or human metacognitive sensitivity.

6. CLAIM LEDGER: INHERITED, ESTABLISHED, NEW, AND OPEN

Inherited B-Theory/report material:
  R4's known-noise actuator tracking and explicit/compiled/external null;
  M5's calibration, memory and action-conditioned evidence;
  the acquisition report's distinction between sufficiency and acquirability;
  the original review's latent-sensor proposal and proper-score caution.

Established mathematical antecedents:
  behavioral-mode diagnosis; Bayesian filtering across transitions; controlled
  sensing; reject-option decisions; full-support impossibility; linear mixture
  identification; proper scoring; minimax uncertainty classes; LP duality.
  These general ideas are not novel contributions claimed here.

Derived in iterations01-02, with explicit proofs/exact checks:
  the six-state constructions, mixture-block/pooling comparison, nonmyopic
  costed example, hidden-U risk gap, refresh thresholds, and independent-
  reference conditional-risk frontiers. Their specialized publication priority
  remains unverified. Internal AI review is not external peer review.

New in iteration03:
  the ambiguity class with explicit episode semantics;
  sharp robust worst-mode capacity and cost slope over [0,T];
  the exact threshold3/40 and matching policy;
  known-law versus one-unknown-law policy comparison;
  analytic lower bounds plus rational certificates and direct pair enumeration.
  These are proved for this model. No claim of a new general robust-testing
  method, unique historical priority, or empirical sensor validation is made.

Open or deliberately outside scope:
  robust PRIOR-AVERAGE optimal cost using primary information;
  paid multi-episode learning and confidence sequences for t;
  persistent theta across episodes; state-dependent reference dependence;
  unknown orientation or marginal accuracy; more than two references;
  arbitrary observation ordering, physical hazards, resource-matched learning;
  external review and a broader novelty audit. No fourth iteration is begun.

7. PRIMARY SOURCES AND ACCESS LIMITS

Access dates:2026-11-04. Earlier full snapshots and manifests remain in
iteration01/sources and iteration02/sources, preserved byte-for-byte.

B-Theory R4, actual manuscript:
https://www.b-theory.com/manuscripts-html/reflective-self-models/
MIND full manuscript, including M5:
https://www.b-theory.com/manuscripts/mind-five-finite-constructions-r1-2026-10-05.pdf
Acquirable Predictive Models:
https://www.b-theory.com/manuscripts-html/acquirable-predictive-models/
Relevant model, scoring, calibration and scope passages were inspected in prior
iterations. These are working manuscripts, not external referee certification.

de Kleer and Williams(1989), Diagnosis with Behavioral Modes:
https://ijcai.org/proceedings/1989/212
Primary proceedings mode/measurement passages inspected. Joint fault-mode
inference and discriminating measurements are close established antecedents.

Fleming and Daw(2017), Self-evaluation of decision-making:
https://www.princeton.edu/~ndaw/fd17.pdf
Author PDF obtained; relevant model/discussion passages inspected. Second-order
evaluation already relates self-performance to inference about another actor;
our functional controller comparison does not identify a distinct mental faculty.

Gneiting and Raftery(2007), Strictly Proper Scoring Rules, Prediction, and Estimation:
https://sites.stat.washington.edu/people/raftery/Research/PDF/Gneiting2007jasa.pdf
Author PDF definitions/scoring passages inspected. The Brier decomposition is
classical Murphy(1973), whose publisher full page was unavailable in iteration01.
https://doi.org/10.1175/1520-0450(1973)012%3C0595:ANVPOT%3E2.0.CO;2

Adams and MacKay(2007), Bayesian Online Changepoint Detection:
https://arxiv.org/abs/0710.3742
https://www.cs.princeton.edu/~rpa/pubs/adams2007changepoint.pdf
Run-length/hazard recursion inspected; our scheduled one-transition model does
not constitute a new quickest-change detection theory.

Nitinawarat and Veeravalli(2015), controlled Markovian sensing with non-uniform cost:
https://arxiv.org/abs/1310.1844
https://arxiv.org/pdf/1310.1844
Primary preprint obtained; observation, cost and risk scope inspected. General
adaptive sensing and distinct risk constraints precede this finite application.

Gul and Zoubir(2015), Minimax Robust Hypothesis Testing:
https://arxiv.org/abs/1502.00647
Primary abstract inspected this iteration: it addresses uncertainty classes,
robust tests, and fixed/sequential extensions. Its entropy/contamination classes
differ from our fixed-marginal two-bit dependence class. No full theorem or
priority comparison was completed. A search also located Huber(1965), DOI
10.1214/aoms/1177699803, but its primary page could not be opened; no detailed
claim here relies on the secondary search descriptions.

Other bounded comparisons and their limitations remain recorded in earlier
reports: Allman-Matias-Rhodes latent identification, Hui-Walter no-perfect-
standard identification, Chow rejection, Maniscalco-Lau metacognitive sensitivity,
and Shiryaev change detection. This is targeted literature research, not an
exhaustive novelty review. Source access failures are not treated as full reads.

8. VERIFICATION AND DELIVERY

exact_frontier.py implements a seven-variable, eight-inequality U relaxation
with endpoint risk/coverage constraints. SciPy proposes solutions; rational
arithmetic checks primal feasibility, dual signs, reduced costs, dimensions,
and zero duality gap. Fourteen certificates cover capacities and cost points
including the threshold and impossible regimes. The analytic proof supplies
the continuum result; grid points alone would not do so.

A second calculation directly enumerates all four reference-error pairs and
the policy's random branches using Fraction arithmetic. A fresh adversarial
AI reviewer independently constructed a separate six-variable LP and checked
ten rational primal-dual solutions, and independently verified the lower-bound
algebra. The reviewer emphasized episode resets and the prohibition on free
learned-law adaptation; both are explicit above. This remains internal AI work.

Reproduction:
  python -B outputs\iteration_03\exact_frontier.py --verify
uses only the standard library to verify saved certificates. Omit --verify to
regenerate with installed SciPy. No network, GPU, paid API or randomness is
needed for these calculations. prior_hashes.json records both previous folders;
verification.json records final preservation checks. No Git commands were used.

The companion video_script.txt provides a reviewable three-minute explanation
with timing, diagrams to draw, exact numerical examples, and interpretation
limits. It is a script/storyboard, not a rendered or published video.

All three requested research iterations are complete. Stop here. If later
publication is desired, external mathematical review and a targeted novelty
audit should precede it; they require a separate request. No additional model
run or new chat is needed now. A later proof review belongs in this chat if it
reuses these contracts; empirical implementation would merit a separate scoped
chat once a physical sensor and cost model exist.
