Biased-representative finite-policy experiment
Scope and frozen instance
For each pair, draw A_i ~ Bernoulli(a) independently of a mismatch vector
E, and set B_i = A_i xor E_i. The teacher has iid mismatches with rate
tau = 1/4. The oracle mismatch law is the fixed two-component mixture
R_E = (11681887 / 16777216) Bern(1/4)^n
+ ( 5095329 / 16777216) Bern(1/2)^n.
These rational parameters are frozen from the phase-2 selected
n8_B1_10_single_d1_4 model. They are not retuned at a new dimension, bias, or
round cap. Required cases are n=4, a in {1/2,2/5,1/4}, K in {2,3,4}.
The extension is n=6, the same three biases, and K in {3,4}, and is run
only if the prespecified resource gate succeeds. The gate is independent of
the numerical scientific outcome.
At every round, the policy may select any subset of currently masked coordinates, using all past information and private randomization. Selected coordinates are sampled independently from the oracle's exact singleton conditionals at the pre-round context. All coordinates must be completed by round K; empty batches allow earlier completion. This is a fixed-head, fixed-oracle experiment, not optimization over oracle parameters.
Nine-type state and exact singleton conditionals
A state is the histogram of these pair types, in order:
empty, A0, A1, B0, B1, 00, 01, 10, 11.
Unlike in the fair case, A and B are not interchangeable, B-only observations carry information about the latent component, and bit flips are not a valid symmetry. The fair three-count state is therefore not used.
For component mismatch rate p, put q_p = a + (1-2a)p. The likelihoods of
the nine types are
1, 1-a, a, 1-q_p, q_p, (1-a)(1-p), (1-a)p, ap, a(1-p).
For a labeled partial context with histogram h, its component posterior is proportional to its component weight times the product of these likelihoods raised to the histogram counts. The multinomial coefficient cancels between components if one conditions on the histogram instead of a labeled context.
Within component p, the probabilities of value 1 for the six possible head types are
empty A: a empty B: q_p
B | A=0: p B | A=1: 1-p
A | B=0: ap/(1-q_p) A | B=1: a(1-p)/q_p.
The oracle heads average these six quantities against the full current component posterior. In particular, the nonlinear reverse heads must not be obtained by substituting the posterior mean p into the displayed fractions. That substitution would double-condition incorrectly.
For example, a=2/5, component weights (7/10,3/10), rates (1/4,1/2), and a
context observing only B=1 give R(A=1 | B=1)=18/31. This is an independent
reverse-head regression fixture.
Actions and transitions
An aggregate action is the seven-tuple
(empty_A, empty_B, empty_both, fill_A0, fill_A1, fill_B0, fill_B1).
The first three numbers partition a subset of empty pairs; the last four select a subset from each singleton type. At the final round the unique legal action completes all masked coordinates. In other rounds every legal tuple, including the empty action, is enumerated.
For an empty pair selected in both coordinates, the two sampled bits are independent Bernoulli draws with the frozen empty-A and empty-B heads. Their joint law is not the oracle's pair law. All other selected-coordinate draws are also conditionally independent in the round. Exact multinomial laws within each selected type, followed by histogram convolution, give the rational transition kernel.
For n pairs there are C(n+8,8) states and C(n+3,3) terminal histograms.
Strict positivity makes every histogram reachable after one round. A full
nonfinal layer has C(n+15,15) state-action pairs. Thus for K>=2 the layered
occupation LP has
C(n+3,3) + (K-2) C(n+15,15) + C(n+8,8)
action variables, before the additional terminal absolute-value variables. The required n=4 cases have 530, 4406, and 8282 action variables for K=2,3,4. The n=6 K=3,4 extension has 57351 and 111615 action variables. The extension gate explicitly constructs all n=6 exact kernels at a=2/5, checks the K=4 edge count, and enforces 120000 action variables, 3000000 transition entries, and 120 seconds of kernel-building time. Each LP has a 90-second solve cap.
Why the terminal objective is full TV
The four completed pair types (00,01,10,11), not merely mismatch count, are the terminal observable. Both the teacher and the uniformly lifted policies are invariant under permutations of pairs. A terminal four-type histogram is exactly an orbit of that permutation group. Both distributions are constant on each orbit, so summing the labeled-coordinate L1 distance over an orbit gives the L1 distance of its histogram masses. Consequently, four-type histogram TV equals the full 2n-coordinate TV for the constructed policy. A mismatch-count-only TV would in general be insufficient.
Every aggregate action can be lifted by uniformly choosing disjoint pair subsets within each current type. This gives a legal, permutation-equivariant policy with exactly the aggregate transition law. Conversely, after any full labeled history, an arbitrary labeled action has an aggregate tuple and the displayed aggregate kernel. This fact justifies the Bellman lower bound for all history-dependent policies, without assuming their final output is symmetric. Equivalently, pair-permutation symmetrization cannot increase TV to the symmetric teacher, and occupation-measure Markovization then yields the finite aggregate LP optimum.
Rational certificates
The LP minimizes half the terminal L1 distance using action occupation measures. Mixtures are optimized before TV is evaluated; this is not an average of deterministic-policy TVs. The numerical solver proposes a bounded terminal test and a feasible policy candidate. Its floating-point optimality report is not taken as a rigorous certificate.
For the lower certificate, clip the LP-derived test into [0,1], interpret
its printed decimals exactly, and round down to the dyadic grid 2^-40.
Starting from this test, perform backward Bellman minimization over every
legal aggregate action with the exact rational transition kernel, rounding
each expectation down to the same grid. The result is a lower enclosure
of the minimum expected test value over all policies. Subtract the exact
teacher mean of the dyadic test and truncate below at zero. Since TV
dominates E_Q f - E_P f for every f in [0,1], the result L is an all-policy
lower bound. Bellman rounding contributes at most K * 2^-40.
For the upper certificate, interpret nonnegative recovered action weights
as exact decimal rationals and normalize each state. Missing states use a
legal empty action, or the forced completion action in the last round.
Propagate state mass on a 2^-80 grid, rounding each edge contribution down.
This produces a componentwise lower terminal mass vector q_lo. Its missing
mass e=1-sum(q_lo) bounds the total omitted probability. Therefore
U = min(1, (sum_h |P(h)-q_lo(h)| + e)/2)
is an upper bound on the full TV of this explicitly legal lifted policy.
Every case reports exact rational L and U as authoritative fields, with
floating-point values for display. The interval L <= inf_policy TV <= U
is not a proof-assistant formalization.
The exact kernel is shared by LP assembly and rational certification; LP assembly converts it to floating point. Independence comes from the separate literal-coordinate kernel checks and literal Bellman/LP comparison, not from claiming two independently written reduced kernels.
KL, model TV, and head errors
The teacher and oracle use the same A law. Under the invertible map
(A,E) -> (A,B), their joint likelihood ratio depends only on mismatch
count. Thus their joint KL and TV are exactly the mismatch-count KL and TV,
independent of a at fixed n. The code additionally verifies model TV against
the full four-type histogram distribution. Natural logarithms are enclosed
using rational range reduction, a finite atanh series, and a rigorous tail
bound. The exact KL upper bound is checked against 1/10.
The n=8 calibration is not transferred as an equality to n=4 or n=6. Actual
KL is reported at each dimension; the oracle is not adjusted to use the
remaining budget. Similarly, bias changes can change the uniform head
error. Every valid partial histogram and every available head are enumerated
exactly. The maximum error and a witnessing context are reported, including
whether it exceeds 1/4. No point is excluded on that basis, and this is not
described as a common-head-budget comparison. In particular, a component
shift by 1/4 need not bound reverse-head error: a=2/5, tau=1/4, p=1/2 gives
an A|B=1 head difference of 4/15.
Positive L-TV(P,R) would certify finite-depth cost beyond the serial
oracle-law baseline. Nonpositive values do not refute the theory. Positive
L_K-U_(K+1) certifies a finite-dimensional depth separation. Neither metric
is assumed positive when choosing the grid or deciding the resource gate.
Validation and command line
--self-test uses only the standard library and exhaustively compares n=2
literal heads and all selected-coordinate subset kernels at all three
biases. It also checks exact-teacher two-round controls and the reverse-head
counterexample. The default scientific run additionally compares all n=2,
K=2,3, three-bias histogram LPs with independently assembled full-coordinate
LPs. Every a=1/2 scientific case is compared to the frozen phase-2 fair
three-count LP implementation in vendor/policy_lp.py.
python biased_policy.py --self-test
python biased_policy.py --config phase3_config.json --profile-only
python biased_policy.py --config phase3_config.json --gate-profile --output gate.json
python biased_policy.py --config phase3_config.json --lp-self-test --output lp_checks.json
python biased_policy.py --config phase3_config.json --output biased_results.json
The main result is checkpointed after every case. Failed cases are retained with their errors; the resource-gated extension is explicitly marked run or skipped. No failure or negative scientific outcome is silently removed.