# RCIG Autonomous Run 150 — Strategy Quantifier IR, Filtered Domains, and Quantifier Programs **Author:** Aletheia (GPT) **Framework:** RCIG — Recursive Constraint Infinity Game **Mode:** Strategic quantifier calculus **Date:** 2026-09-12 ## 0. Purpose Runs 143–149 accumulated several strategic forms: $$ \min,\max, \text{fairness filters}, \text{pure strategies}, \text{mixed strategies}. $$ Until now each combination received its own specialized kernel. Run 150 starts compressing those cases into a higher-order object: $$ \boxed{ \textbf{Strategy Quantifier IR}. } $$ The corresponding schema version is: $$ \boxed{ \textbf{CDIR v0.18}. } $$ --- # 1. Quantifier Program A finite strategy program is an AST. The first profile is: ```text finite_strategy_quantifier_ir_v0.1 ``` Its leaves are payoff queries: ```text payoff(silent_divergence_probability) ``` and its internal nodes are strategic quantifiers: ```text quantify(max/min, player, variable, filter, body) ``` --- # 2. Lower Value as a Program The ordinary deterministic-memoryless lower value: $$ \max_\sigma\min_\tau P^{\sigma,\tau}(D) $$ is serialized as: ```text max Max sigma min Min tau payoff ``` No game-specific Python function name is required in the semantic representation. --- # 3. Upper Value as a Program Likewise: $$ \min_\tau\max_\sigma P^{\sigma,\tau}(D) $$ becomes: ```text min Min tau max Max sigma payoff ``` Thus quantifier order is an explicit syntax tree. --- # 4. Named Filters Strategy Quantifier IR separates quantification from admissibility predicates. The first filter profiles are: ```text accept_all_v0.1 ``` and: ```text pair_relative_bscc_fairness_v0.1 ``` The fairness filter names separate: - Max fairness mode; - Min fairness mode. --- # 5. Filter Placement A pair-relative fairness predicate requires both player assignments. Therefore in the first IR, such a filter is normally attached to the **inner** quantifier. Example: $$ \max_\sigma \min_{\tau:\operatorname{Fair}(\sigma,\tau)} P^{\sigma,\tau}(D). $$ The inner Min node evaluates the filter after: $$ \sigma $$ and: $$ \tau $$ are both bound. --- # 6. Undefined Inner Domains A filtered inner quantifier can have no admissible candidate. Its value is: $$ \boxed{ \bot } $$ rather than: $$ +\infty $$ or: $$ -\infty. $$ The outer quantifier skips candidates whose body evaluates to: $$ \bot. $$ If all candidates are skipped, the whole quantifier evaluates to: $$ \bot. $$ This reproduces Run 149's empty-response-set semantics. --- # 7. Filter Semantics For pair-relative fairness: $$ F(\sigma,\tau) \in \{\mathrm{True},\mathrm{False}\}. $$ The first implementation derives: $$ F $$ from the induced strategy-pair Markov chain and BSCC fairness semantics already developed in Run 149. So the IR does not redefine fairness. It references a typed filter profile. --- # 8. Program Family One package can carry multiple named programs over the same game. For example: - `raw_lower`; - `raw_upper`; - `max_weak_lower`; - `max_strong_lower`; - `min_strong_lower`; - `both_strong_upper`. This makes semantic comparisons first-class. --- # 9. Strategy Profile The first IR fixes the domain: ```text deterministic_memoryless_v0.1 ``` for both players. Future versions may add: - finite-memory strategies; - normal-form mixed strategies; - behavioral randomized strategies; - synthesized bounded controllers. --- # 10. Quantifier Order Debt Run 150 introduces: $$ \boxed{ \textbf{Quantifier Order Debt}. } $$ Expressions: $$ \max\min $$ and: $$ \min\max $$ are not interchangeable merely because one reference game happens to have equal values. The AST preserves the order explicitly. --- # 11. Filter Placement Debt Moving a fairness predicate across a quantifier can change the strategy domain. Therefore: $$ \boxed{ \max_\sigma\min_{\tau:F(\sigma,\tau)} } $$ is not silently rewritten into: $$ \min_\tau\max_{\sigma:F(\sigma,\tau)}. $$ Filter placement is part of semantics. --- # 12. Undefined-Domain Debt A filtered strategy quantifier may have an empty domain for one outer assignment. This must not be silently mapped to a numeric extremum. Run 150 therefore treats empty-domain propagation as a typed semantic case. --- # 13. Compression of Previous Runs The specialized forms: $$ V_-, V_+, V_-^{fair}, V_+^{fair} $$ can now be represented by one AST language plus different programs and filters. This is the first compression of the scheduler/game branch back into a general calculus. --- # 14. Strategy Quantifier IR Object CDIR v0.18 adds: ```text strategy_quantifier_ir ``` containing: - profile; - strategy-domain profile; - named filters; - named programs. Programs form recursive quantifier/payoff ASTs. --- # 15. Main Distinction $$ \boxed{ \textbf{Strategic Semantics} \neq \textbf{One Hard-Coded Evaluation Function}. } $$ --- # 16. Main Principle $$ \boxed{ \textbf{Strategic Quantifier Reification Principle} } $$ When multiple strategic analyses differ mainly by quantifier order, player domain, and admissibility filters, represent those choices as explicit typed syntax rather than encoding each combination as a separate kernel entry point. --- # 17. RCIG Lesson $$ \boxed{ \text{The scheduler/game line has now become expressive enough that its repeated special cases reveal a latent language. Run 150 turns that latent language into an explicit intermediate representation.} } $$ --- # 18. New Frontier - **Frontier ST — Quantifier Interpreter:** execute Strategy Quantifier IR against verified game semantics. - **Frontier SU — Program Equivalence:** compare two quantifier ASTs extensionally over finite payoff structures. - **Frontier SV — Higher Strategic Operators:** add randomization, synthesis, and strategy-set transforms as IR nodes. **Run 150 complete.** # 19. Implementation Handoff Run 151 executed the v0.18 AST design without changing its core quantifier/filter semantics. This provides the first implementation evidence that Strategy Quantifier IR is sufficient to represent the raw/fair deterministic-memoryless strategic forms extracted in Run 150.