# R12 Reasoning Invention Charter **Status:** theory phase only; no implementation, data build, fit, score, or GPU job is authorized. **Effective:** 2026-07-15. This charter supersedes architecture-first reasoning experiments. R9, R10, and R11 remain evidence and matched controls, not active mechanism templates. ## 1. Why this charter exists Shohin has already falsified several easy stories: - adding visible or latent traces can teach formatting without exact transport; - fixed recurrence can be an unrolled feed-forward computation; - dynamic recurrence can still learn the same local classifier as static controls; - external algebra can solve a board without establishing neural reasoning; - source-conditioned slots, matrices, or adapters can reduce to retrieval, fast weights, or a hypernetwork; - more pretraining has not yet produced reliable direct reasoning behavior. Workers must therefore stop beginning with familiar modules and searching for a claim afterward. R12 begins with a mathematical capability and derives the necessary state and operator before considering a realization. ## 2. The finite-circuit boundary For fixed context length, finite precision, and bounded runtime, every deterministic classical mechanism can be unrolled into a finite acyclic circuit. Loops become repeated subgraphs, memory reads become multiplexers, generated weights can be substituted, and fixed external computation can be inlined. Consequently, "not equivalent to any static classifier" is not an admissible requirement: it rejects every bounded implementation. Novelty must instead be stated relative to an explicit resource-scaled comparator family. No R12 report may claim separation from all static computation. The converse is equally important. Exact finite unrolling establishes only extensional computability. It does **not** preserve parameters, retained bits, precision, source access, training examples, oracle calls, training FLOPs, inference FLOPs, sequential depth, external memory, or external execution. A reduction is claim-rejecting only when it preserves the preregistered resource vector within constant or polylogarithmic overhead. Otherwise it defines a control or downgrades the novelty claim; it cannot veto every finite mechanism. ## 3. Operational definition At scale `n`, let `Sigma_n`, `Q_n`, and `A_n` be event, late-query, and answer sets. A history is `h in Sigma_n*`; `c` is a future continuation. Extend the answer space by an inadmissibility symbol `bottom`, so the total answer relation is ``` R_n(hc, q) subset A_n union {bottom}, R_n(hc, q) != empty. ``` `R_n(hc, q) = {bottom}` exactly when the continuation-query pair is inadmissible. Inadmissibility is part of observable behavior; omitting it can destroy closure under appending the same event. Histories are causally equivalent exactly when no admissible future can distinguish them: ``` h ==_R h' iff for every c and q, R_n(hc, q) = R_n(h'c, q). ``` The causal state is the equivalence class `S_n(h) = [h]`. A model family is an R12 systematic reasoner only if one finite rule specifies every scale, its error tends downward as scale grows, the number of required causal states is unbounded, and it has a stated asymptotic resource advantage over a named comparator class. This definition concerns uniform late-query causal composition. It does not by itself establish discovery, semantic understanding, proof insight, or general intelligence. ### Exact-realization no-go theorem Suppose a reachable exact realization has a state map `E(h)`, deterministic updates `U_e`, and observations `O_q`, with ``` E(he) = U_e(E(h)) O_q(U_c(E(h))) = R_n(hc, q). ``` If state equality is extensional, then `E(h) = E(g)` exactly when the residual behaviors `rho_h` and `rho_g` are equal. Therefore the map ``` E(h) -> rho_h ``` is a well-defined bijection that conjugates every learned update to the residual derivative. The reachable realization is the minimal deterministic Moore transducer, and its event updates generate the corresponding transition monoid. This is an exact structural no-go, not an implementation preference. R12 cannot honestly claim an exact finite causal state that is ontologically outside automata, transition monoids, residual machines, or minimal coalgebras. A genuine contribution must instead be a new resource separation, approximation geometry, learnability result, or uniform realization with a falsifiable advantage over named controls. ## 4. The object that must be realized Define the counterfactual residual of a history: ``` rho_h(c, q) = R_n(hc, q). ``` An event acts through a residual derivative: ``` (partial_e rho)(c, q) = rho(ec, q). ``` The exact residual quotient is now a specification and lower-bound object, not the claimed invention. Any exact control realization must satisfy all six axioms: 1. **Closure:** `partial_e rho` is another valid residual state. 2. **Composition:** `partial_empty = I` and `partial_(uv) = partial_v compose partial_u`. 3. **Observation:** `O_q(rho) = rho(empty, q)`. 4. **Extensionality:** two states are equal exactly when all future continuation-query answers agree. 5. **Separation:** distinct states admit a distinguishing continuation-query pair. 6. **Uniformity:** one finite rule specifies updates and observations at every tested scale; there is no scale-specific advice table. Ambiguity must preserve every future-distinguishable class. Averaging distinct operators or answers is not a valid uncertainty representation when a future query can separate them. ## 5. Necessary resource obligations If the causal quotient has `N_n` states, an exact query-oblivious state needs at least ``` B >= log2(N_n) ``` history-dependent bits. Every candidate must count dynamic context, caches, stored source, intermediate tensors, generated parameters, and external state. Model parameters are only the fixed description length. The update law must realize the action induced on the causal quotient: ``` U_e([h]) = [he] U_(uv) = U_v compose U_u. ``` An order-sensitive witness therefore requires a noncommutative update. Any commutative pool, expected operator that aliases distinct futures, or fixed template inventory fails before training. ## 6. First theorem-backed witness For `m` objects, let events be adjacent transpositions `tau_i = (i, i+1)`. For a word `w = e_1 ... e_L`, define ``` pi_w = e_L compose ... compose e_1. ``` The query `j` is revealed only after the word and asks for `pi_w(j)`. Once all permutations are reachable, the causal quotient has exactly `m!` states, so an exact query-blind state needs at least `log2(m!)` bits. On the `m=2` restriction, the answer is parity, giving a clean separation from polynomial-size constant-depth AND/OR/NOT circuits. This is a separation from `AC0`, not from arbitrary transformers or threshold circuits; stronger relevant separations are open complexity questions. The finite falsifier uses `m in {5, 8, 12}` and increasing unseen lengths. It must include: - equivalent words generated by involution, distant commutation, and braid relations; - non-equivalent order twins with a known separating late query; - identical continuations appended after equivalent and non-equivalent prefixes; - every late query, not a selected easy query; - a balanced `m=2` parity restriction while length doubles; - state-capacity and compute ledgers checked against the information bound. One exact counterexample kills an exact residual-composition claim. Passing a finite board does not prove the asymptotic claim because a finite lookup table can pass any finite board. ## 7. Mandatory invention gates Every future worker must produce these artifacts in order: 1. **Capability theorem:** relation, comparator class, resource measure, and proof or explicitly labeled conjecture. 2. **Axiomatic primitive:** state and operators defined without neural-module vocabulary. 3. **Equivalence dossier:** algebraic and resource-preserving checks against SFT, fixed/tied recurrence, retrieval, fast weights, hypernetworks, external execution, and finite unrolling. The mandatory resource vector is `(parameters, retained bits, precision, source bytes, training examples, oracle calls, training FLOPs, inference FLOPs, sequential depth, external memory, external execution)`. 4. **Exact collapse test:** a symbolic or exhaustive CPU test that tries to reduce the proposal to those controls. A successful reduction rejects the specific novelty or resource claim only when it preserves behavior, information access, and the preregistered resource vector within constant or polylogarithmic overhead. Extensional finite unrolling alone is not rejection evidence. 5. **Prior-art boundary:** search after the object is defined, then state the exact delta. Known components may support a new algorithm or training protocol, but cannot be called new primitives. A known-component reduction defines a mandatory control and an allowed-claim boundary rather than an automatic experiment veto. 6. **Finite falsifier:** frozen scale extrapolation, causal interchange, equivalent-state invariance, non-equivalent-state separation, and full resource accounting. 7. **Matched controls:** every known realization receives matched or favorable parameters, state, and compute. 8. **Score-blind confirmation:** one immutable implementation and one frozen confirmation generation. No board, seed, threshold, or artifact shopping. No neural implementation is authorized through gate 4. After gates 1--5, one isolated CPU falsifier may test a bounded resource hypothesis; a CPU pass is required before any Shohin fit. No Shohin fit is authorized through gate 6. No H100 experiment is authorized through gate 7. An exact candidate rejected by a genuine information or identifiability no-go cannot be rescued by renaming its state or operators. ## 8. Rejected starting points The following are controls, not R12 primitives. Their presence prevents a primitive-novelty claim but does not by itself prohibit a resource-matched training-protocol experiment: - more CoT/SFT/RL, teacher traces, self-review, or verifier reranking; - more hidden slots, recurrent loops, adaptive depth, equilibrium iterations, or test-time search; - KV memory, retrieval, replay, latent scratchpads, or context compression; - source-generated matrices, adapters, gates, or weights; - a hard-coded symbolic solver, parser, executor, algebra, or tree that computes the answer outside the learned mechanism; - persistent product trees or Schur-complement boundary actions presented as new primitives. They may be strong controls but are known mathematical machinery plus source-conditioned state. A future proposal may use a known component only after the primitive has been derived independently and only if the component is not the claimed invention. ## 9. Current decision The exact research specification remains **uniform late-query causal composition through counterfactual residual behavior**. It is not a candidate primitive: every exact reachable realization is the residual transducer up to a change of coordinates. The approximate state-ontology frontier is now closed as well. Fork-Core Quantization collapses to approximate information states, predictive-state representations, causal rate-distortion, and classical convex geometry; see `R12_FORK_CORE_THEORY.md`. `R12_COHERENT_ACTION_THEORY.md` proves that the whole event-monoid action has a coherent hyperconvex function-space extension with no word-length growth in merge error. That construction stores an event-closed observable profile and updates it by coordinate substitution. The displayed unrestricted finite construction uses the exact-state count times the transition-monoid size in coordinates. Restricting the profile assumes the small predictive dimension that needs to be explained. It is a theorem-backed control, not compressed reasoning. `R12_CLOSED_LATE_QUERY_NO_GO.md` proves that post-commit computation cannot recreate discarded source information. Arbitrary adversarial late INDEX needs `n` retained bits exactly and `n(1-h2(epsilon))` bits at error `epsilon`; longer internal thinking does not change that information bound. No candidate implementation has survived the invention gates. The next authorized action remains a theorem and preregistration, but the target is narrower: a uniform resource advantage in learnability, dynamic sparsity, amortized verification, noise stability, or another named cost on a structured residual family. A new state ontology, arbitrary late-query compression, or coherent coordinate pullback is no longer an admissible invention claim. Architecture design remains blocked until a bounded resource hypothesis and equivalence dossier survive gates 1--5; only then may one isolated CPU falsifier be implemented.