# HIPG Formal Toy Model v0.6 ## Meta-Operator Learning, Learned Dynamics, Belief Quotients, Joint Identifiability, Arithmetic Solver Bridges, and Learned Failure Diagnosis **Framework**: HIPG — Heterogeneous Intelligence Protocol Generation **Phase**: A — Formal Toy Model **Version**: v0.6 **Date**: 2026-08-14 **Status**: Executable finite research scaffold --- # 0. Purpose v0.5 crossed: $$ \boxed{ \text{Hypothesis Selection} \rightarrow \text{Hypothesis Construction}. } $$ But it still relied on several supplied structures: - the Boolean operator grammar; - the transition table in the trajectory quotient; - canonical-field anchors for partner learning; - hand-coded impossibility routing classes; - a propositional-only solver bridge. v0.6 attacks these assumptions one by one without claiming to remove them all. The new scaffold instantiates: $$ \boxed{ \begin{aligned} &\text{Meta-Operator Selection / Library Learning}\\ &+\text{Learned Transition Model}\\ &+\text{Belief-State Task Quotient}\\ &+\text{Joint Partner–Task Identifiability Analysis}\\ &+\text{Causal Symmetry Breaking}\\ &+\text{Arithmetic Constraint Solver Bridge}\\ &+\text{Learned Failure Diagnosis}\\ &+\text{Cost-Aware Protocol Regret}. \end{aligned} } $$ --- # 1. Meta-operator learning v0.6 does **not** yet invent an arbitrary new formal language. Instead, it receives a finite operator superset: $$ \mathcal O_0=\{\land,\lor,\oplus\} $$ and multiple tasks. It searches operator subsets and minimizes: $$ \boxed{ L(\mathcal G) = \sum_{T\in\mathfrak T} \operatorname{MDL}(h_T\mid\mathcal G) + \lambda_G|\mathcal G|. } $$ Canonical training tasks: $$ E\oplus J, $$ $$ (E\oplus J)\land O, $$ $$ (E\oplus J)\oplus O. $$ The learned operator subset is: $$ \boxed{ \{\land,\oplus\} } $$ while $\lor$ is rejected as unnecessary for this task family. This is **operator-family selection**, not unbounded grammar invention. --- # 2. Reusable subexpression discovery Across the learned task solutions, v0.6 mines repeated subexpressions. It discovers: $$ \boxed{ M_0:=E\oplus J } $$ appearing three times. Description length changes from: $$ 13\rightarrow10. $$ For a novel task: $$ (E\oplus J)\land\neg O, $$ the plain expression has MDL: $$ 6, $$ while the macro representation: $$ M_0\land\neg O $$ has MDL: $$ \boxed{4}. $$ Thus v0.6 contains the first executable form of: $$ \boxed{ \text{cross-task operator library learning}. } $$ --- # 3. Transition structure is learned from intervention traces v0.5 computed trajectory quotients from a known transition table: $$ P(x'\mid x,a). $$ v0.6 hides that table from the learner. The runtime collects: ```text state_before intervention observed_state_after reward_observed ``` and estimates: $$ \widehat P(x'\mid x,a). $$ In the canonical finite system, each state-action pair is sampled 7 times with transition-observation noise: $$ p_{\mathrm{noise}}=0.08. $$ The minimum majority confidence is: $$ \boxed{0.7142857}. $$ The learned deterministic transition model exactly recovers the hidden evaluator transition table, then reconstructs the horizon-2 task quotient: $$ \boxed{ \{\text{goal}\}, \{s_0\}, \{s_1\}, \{\text{trap}\}. } $$ The hidden transition table is used only for evaluation. --- # 4. Partial observability: belief-task quotient Two hidden states may produce the same observation: $$ O(h_0)=O(h_1)=\text{ambiguous}. $$ Then a task representation cannot always be a quotient over directly observed state IDs. v0.6 introduces belief states: $$ b=P(X\mid H_t). $$ Canonical beliefs: $$ b_{h0}=(0.9,0.1), $$ $$ b_{mid}=(0.5,0.5), $$ $$ b_{h1}=(0.1,0.9). $$ For actions $A,B$ with success tied to $h_0,h_1$, their task signatures are: $$ Q(b,A)=P(h_0\mid b), $$ $$ Q(b,B)=P(h_1\mid b). $$ The belief-task quotient separates all three beliefs even though the raw observation aliases the hidden states. This is the first HIPG toy step toward: $$ \boxed{ \text{belief-space task semantics}. } $$ --- # 5. Joint partner/task semantics can be behaviorally known but ontologically non-identifiable v0.6 removes the direct canonical-field anchor in one benchmark. It jointly enumerates: - 48 generated partner adapters; - the bounded Boolean task-expression library. The entire surface behavior can be learned exactly: $$ \text{behavioral task accuracy}=1. $$ Yet the same surface truth table admits: $$ \boxed{12} $$ distinct exact adapter/task decompositions. Therefore the runtime returns: ```text UNKNOWN_NONIDENTIFIABLE ``` rather than promoting behavioral success to semantic identification. This executable result instantiates: $$ \boxed{ \text{Operational Equivalence} \not\Rightarrow \text{Unique Semantic Decomposition}. } $$ --- # 6. Causal interventions break the symmetry A second joint benchmark adds **known intervention semantics**, not direct partner-field labels. From a baseline state, the runtime performs: ```text TOGGLE_E TOGGLE_J TOGGLE_O ``` and observes which surface bits change. This identifies the unique adapter: $$ \boxed{ p_0=\neg O,\qquad p_1=E,\qquad p_2=J. } $$ Then binary reward over surface states identifies a unique task expression: $$ \boxed{ (J\land O)\oplus E. } $$ Hence: $$ \boxed{ \text{passive behavioral data} \rightarrow \text{non-identifiable}, } $$ while: $$ \boxed{ \text{semantically known interventions} \rightarrow \text{identifiable in this finite model}. } $$ This is a concrete HIPG demonstration of causal symmetry breaking. --- # 7. Arithmetic solver bridge v0.5 introduced a real propositional solver via SymPy SAT. v0.6 adds a continuous linear constraint bridge using: ```text scipy.optimize.linprog(method="highs") ``` Contract: $$ \text{confidence}\ge0.70, $$ $$ \text{latency}\le5, $$ $$ \text{error}\le0.10, $$ $$ \text{confidence}-\text{error}\ge0.65. $$ Injected candidate: $$ (0.60,8.0,0.15) $$ is invalid. The solver minimizes L1 repair distance and returns one feasible repair: $$ \boxed{ (0.70,5.0,0.05) } $$ with repair objective: $$ \boxed{3.2}. $$ The runtime now contains both: - discrete SAT-style verification; - continuous LP-style constraint repair. It is still not a general SMT/proof assistant. --- # 8. Failure diagnosis is learned from examples Earlier versions routed failure modes by explicit runtime logic. v0.6 adds a separate benchmark in which a decision tree learns from synthetic examples with measurable diagnostic features: $$ (C,D,F,A,G,\Tau,R, \text{bit margin}, \text{mapping unknown}, \text{budget low}). $$ Target classes: ```text STRUCTURAL INFORMATION NONIDENTIFIABLE RESOURCE REPAIRABLE ``` On a held-out 1000-example synthetic test set: $$ \boxed{ \text{accuracy}=0.977. } $$ Five canonical probes are all classified correctly. The classifier does not receive a benchmark `mode` label at inference. This does **not** replace theorem-based impossibility checks. It only demonstrates that diagnostic routing can itself become a learned object. --- # 9. Cost-aware protocol regret v0.5 measured task-regret only. v0.6 uses: $$ \boxed{ R_T = \sum_t \left[ \Delta V_t + \lambda_E C_{\mathrm{experiment},t} + \lambda_M C_{\mathrm{model},t} \right]. } $$ Canonical mean total regret over 100 runs: $$ \boxed{ R_{\mathrm{COST\text{-}AWARE}}=4.1150 } $$ $$ R_{\mathrm{ACTIVE\ EIG}}=4.7644, $$ $$ R_{\mathrm{RANDOM}}=4.7586, $$ $$ R_{\mathrm{BLIND}}=4.7586. $$ Thus the cost-aware constructor wins this finite benchmark. Again, this is an empirical toy result, not a universal regret theorem. --- # 10. Certificate v0.6 Every canonical benchmark emits: ```text HIPG-CERT-0.6 ``` validated on: $$ \boxed{ \text{Schema} + \text{SHA-256 Lineage} + \text{Semantic Consistency}. } $$ A new semantic invariant is added: > If behavior is perfectly predicted but the adapter/task decomposition is non-identifiable, the certificate must remain `UNKNOWN`, not `SUCCESS`. This prevents an operationally correct bridge from being mislabeled as a uniquely understood semantic bridge. --- # 11. Canonical validation v0.6 preserves all v0.5 cases and adds eight new benchmarks. Canonical result: $$ \boxed{ 34/34\ \text{benchmarks PASS} } $$ $$ \boxed{ 34/34\ \text{certificates PASS schema+hash+semantic validation} } $$ $$ \boxed{ 24/24\ \text{unit tests PASS}. } $$ --- # 12. What v0.6 actually establishes The justified engineering claim is: $$ \boxed{ \begin{aligned} &\text{A finite HIPG constructor can learn}\ &\text{operator subsets, reusable semantic macros,}\ &\text{transition structure, belief-task distinctions,}\ &\text{and diagnostic classifiers,}\ &\text{while explicitly preserving semantic non-identifiability.} \end{aligned} } $$ It also demonstrates that causal interventions can turn one finite joint partner/task problem from non-identifiable to identifiable. --- # 13. What v0.6 does not establish v0.6 does **not** prove: - B-TSDPC; - universal grammar discovery; - universal causal representation learning; - general POMDP quotient convergence; - unique latent ontology recovery; - general impossibility diagnosis; - general SMT equivalence; - cross-domain regret superiority. The learned grammar still starts from a supplied operator superset. The transition learner still knows the finite state/action names. The causal-identification benchmark knows intervention semantics. These are now the next exposed boundaries.