| # Independently checked finite-dimensional certificates |
|
|
| This verifier uses only Python's standard library. Its bounds are computed with |
| integers and `Fraction`; JSON floating-point fields are display values. It does |
| not treat solver tolerances or an LP status as a proof certificate. |
|
|
| ## Scope and model |
|
|
| The teacher has independent fair representative bits A_i and independent |
| mismatches E_i ~ Ber(tau), with B_i = A_i xor E_i. The model uses the same fair |
| representative bits and a rational finite mixture of iid Bernoulli mismatch laws. |
| All coordinates are sampled from the original, coherent conditional heads of |
| this model. A round selects its subset before its independent fresh outcomes; |
| heads are frozen throughout a round. Decisions may depend on the full history |
| and private randomness. Every coordinate must be committed by round K. |
| |
| This is a certificate for this interface, not for rejection, remasking, |
| head-correction, postselection, joint sampling, or arbitrary neural decoders. |
| |
| ## Why three integers suffice for a lower certificate |
| |
| Let s=(c,m,o): c completed pairs, m mismatches among them, and o pairs with one |
| visible endpoint. The remaining f=n-c-o pairs are fresh. Fair singleton values |
| and their endpoint orientations give no information about E. The model's next |
| completion mismatch probability p(c,m) is its exact mixture posterior mean. |
| |
| Any original action is characterized, as far as the next aggregate state is |
| concerned, by a=(u,v,b): complete u open pairs, cosample v fresh pairs, and reveal |
| one endpoint of b fresh pairs. Every legal action satisfies 0<=u<=o and v+b<=f. |
| Its transition is |
| |
| c' = c+u+v, o' = o-u+b, m' = m+X+Y, |
| X ~ Bin(u,p(c,m)), Y ~ Bin(v,1/2), independently. |
| |
| The transition is valid conditional on the entire original history, not merely |
| after averaging under the teacher. Therefore, for a terminal test depending |
| only on mismatch count, a Bellman minimum over every legal (u,v,b) lower-bounds |
| the expected test under **every** adaptive, privately randomized original |
| policy. Random mixtures cannot beat the smallest action expectation. In the |
| last round the only allowed count action is (o,f,0). |
| |
| For the feasible-policy upper certificate, lift each chosen count action by |
| uniformly selecting the concrete eligible pairs and uniformly choosing the |
| orientation of every fresh singleton. The target, model, and this lifted policy |
| are invariant under pair permutations and independent simultaneous flips of |
| both endpoints. These symmetries act transitively on full outputs with the same |
| mismatch count. Consequently the count TV equals full-output TV for the lifted |
| policy. The lower bound itself requires no assumption that an arbitrary policy |
| has symmetric outputs: a terminal count test is already a legitimate full-output |
| test. |
| |
| ## All-policy lower certificate |
| |
| Convert the supplied test to a fixed dyadic function h(m) in [0,1] by rounding |
| down on grid S=2^40. This defines a new legitimate test; there is no need to |
| assume an unquantized floating test is exact or optimal. |
| |
| Set terminal integer values V_0(n,m,0)=S h(m). For remaining rounds t, compute |
|
|
| V_t(s) = min_legal a floor(sum_j W_j V_{t-1}(s_j) / D_a), |
| |
| where W_j and D_a are exact integers for the transition law. If p=A/D in lowest |
| terms, convolving Bin(u,p) and Bin(v,1/2) gives denominator D^u 2^v. Induction |
| shows V_t/S is a lower bound on the exact Bellman value. Its absolute downward |
| error is at most t/S. Completed states are absorbing. |
| |
| Compute E_P h exactly using rational binomial probabilities. Then |
|
|
| L = max(0, V_K(0,0,0)/S - E_P h) |
| |
| is an all-policy full-output TV lower bound. The test orientation is Q-P: |
| h should be large on counts overrepresented under generated Q. The verifier |
| remains valid for any test, even a poor or sign-reversed one, but may return 0. |
|
|
| ## Rigorous upper certificate for a recovered policy |
|
|
| Decimal action weights are interpreted as exact rational numbers and normalized |
| separately at each provided state. Missing states use an empty action until the |
| last round, when all remaining coordinates are sampled. This completely specifies a feasible |
| count policy, even at histories whose occupancy was numerically zero in the LP. |
| Thus the certified policy can differ negligibly or materially from the LP's |
| intended policy; the reported certified upper, not its original objective, is |
| authoritative. |
|
|
| Propagate masses on grid F=2^80. For each directed action/outcome edge, round its |
| exact contribution down. Inductively, every stored state mass is no greater |
| than that state's true probability under the reconstructed policy. Let q^- be |
| the terminal subprobability and delta=1-sum_m q^-(m). Then Q(m)>=q^-(m) and |
| sum_m[Q(m)-q^-(m)]=delta, hence |
|
|
| max(0, TV(P,q^-)-delta/2) <= TV(P,Q) |
| <= min(1, TV(P,q^-)+delta/2). |
| |
| Both endpoints are exact rational numbers. The right endpoint is a feasible |
| policy upper bound U for the optimum. The reported interval [L,U] is valid even |
| when numerical solver diagnostics are poor. A large gap means certification |
| has not established near-optimality. |
|
|
| ## Model TV, all-context heads, and KL budget |
|
|
| The model count probabilities are computed exactly. Fairness and exchangeability |
| make count TV exactly equal to joint model TV, rather than merely a projection |
| lower bound. The verifier checks every (c,m), 0<=m<=c<n, to find the largest |
| absolute completion-head error |p(c,m)-tau|. Fresh-coordinate heads coincide with |
| the teacher's fair heads, so this covers all partial-coordinate contexts. |
| |
| KL(P||R) is an expectation of log(P_count/R_count). For each positive rational x, |
| write x=2^k y with 1<=y<2 and use |
| |
| log y = 2 sum_{j=0}^{N-1} z^(2j+1)/(2j+1) + remainder, |
| z=(y-1)/(y+1), |
| 0 <= remainder <= 2 z^(2N+1)/[(2N+1)(1-z^2)]. |
| |
| Here z<=1/3, N=48, and each rational term is bounded on a common 2^-120 grid. |
| The same enclosure is used for log 2. Negative k reverses interval endpoints |
| when multiplying. Summing with exact teacher probabilities gives rigorous KL |
| bounds. A witness meets a supplied budget only if the **upper** bound is <= that |
| budget. An interval straddling the budget is not accepted as certified feasible. |
| |
| ## Interpretation of the execution gap |
| |
| The reported signed quantity G=L-TV(P,R) is a certified lower bound on the |
| excess of optimal K-round output TV over model joint TV. A positive G establishes |
| an unavoidable extra execution cost for this fixed rational witness. A negative |
| G is not evidence of a negative cost; it means this lower certificate does not |
| establish positive excess. For a witness satisfying all constraints, L is also |
| a valid lower bound on the outer minimax problem, without any claim that the |
| search globally optimized over models. |
| |
| ## Self-tests and limits |
| |
| `--self-test` checks six n=2 cases by comparing an exact count Bellman recursion |
| against an independently implemented, literal 2n-coordinate recursion. The |
| latter enumerates all hidden-coordinate subsets and independent outcomes using |
| the full rational joint law. Tests also check Bellman rounding enclosures, a |
| one-pair privately randomized mixture with zero true TV, and logarithm interval |
| consistency. These are implementation regression tests, not a formal proof in |
| Lean or a guarantee against all programming errors. |
| |
| Example: |
| |
| python src/certify_witness.py --self-test |
| python src/certify_witness.py --input results/witnesses.json \ |
| --output results/certificates.json |
| |
| Input may be one witness, a list of witnesses, or an object with a `witnesses` |
| array. Witnesses specify n, tau, rational weights and component probabilities, |
| optional rational budget/head_bound, and cases containing K, terminal_test, and |
| optionally the recovered policy. Round numbers in policies start at one. |
| |
