# R12 Counterfactual Conjugate Commit hypothesis **Status:** theory only. No implementation or H100 launch is authorized before the preregistered carry-only motor is scored. This document records a possible successor if writer repair alone leaves a measured reader/cycle bottleneck. ## 1. Empirical premise The post-DRS evidence localizes a transaction failure rather than a missing local arithmetic rule: - first transition after DRS SFT: 497/500; - native frozen-cycle first state: 38/50; - all 12 native failures first diverge at serialized carry; - target result digit under teacher forcing: 50/50; - final-layer carry linear-probe test accuracy: 694/800 = 86.75%; - paired next-call active-digit switch: 40/50; - autonomous integrated two-call cycle: 9/50. For a fixed-width local transition, the operands, operation, cursor, and result prefix are already serialized. The only persistent arithmetic state is one bit: \[ F_x(c_p) = (r_p, c_{p+1}). \] The open failure is whether one semantic bit survives the complete map \[ q_{in}(E(A(h_p))) = q_{out}(h_p), \] where `h_p` is the late residual, `A` emits the carry token, `E` embeds that token on the next call, and `q_in/q_out` recover its semantic value. ## 2. Minimality and the C3-1 hypothesis For each local symbol `x=(op,a_p,b_p)`, decimal execution is a two-state Mealy transducer: \[ F_x(c_p)=(d_p,c_{p+1}). \] Carry zero and one are behaviorally distinguishable by a one-symbol continuation, while one bit realizes every local transition. The minimal arithmetic quotient therefore has exactly two states. The cursor and immutable tape remain visibly serialized; C3-1 is not allowed to add a hidden result tape, retry loop, solver, or extra transformer pass. Conditional on a successful frozen writer, learn one nonzero consumer direction `u in R^576` at one fixed result-digit site. Define \[ P_u={uu^T\over u^Tu},\qquad K_u(h,c)=(I-P_u)h+(2c-1)u. \] This is a hard one-bit clamp rather than an additive hint: \[ u^T K_u(h,c)=(2c-1)u^Tu. \] The carry-flip map \[ G_u=I-2P_u \] is an exact involution and conjugates the committed token labels: \[ G_u^2=I,\qquad K_u(h,1-c)=G_uK_u(h,c). \] The complete transaction is therefore \[ h_p^{write}\to W_8\to token(c_{p+1})\to E(c_{p+1}) \to K_u\to h_{p+1}^{consume}. \] Training must still establish all 400 `(op,a,b,c)` local cells. Flip equivariance alone is not evidence of arithmetic execution. For a tied optimization geometry, freeze a writer-side carry axis `v_o` from fit-only data and parameterize the consumer chart as a Householder transport of that axis. A single Householder reflection can map `v_o` to any consumer axis, so this tied arm and a freely learned `u` have the same hypothesis class. Any advantage is optimization geometry, not expressivity. ## 3. Equivalence boundary C3 is not yet a new computational primitive. A transformer can represent it already, and its state machine is computationally equivalent to a two-state finite transducer. The writer alone is an output adapter. The only potentially useful contribution is the tied output/input group action, hard quotient projection, and counterfactual commutation objective. It changes optimization geometry, not expressivity. The frozen rank-8 writer has 4,634 parameters. C3-1 adds exactly 576, for 5,210 total. With `(u^Tu)^-1` precomputed, the consumer costs one dot product, one scalar projection, and one vector update: 1,153 multiplies and 1,152 adds/subtracts. It retains exactly one bit and adds no token, KV slot, hidden persistent state, or sequential step. ## 4. Required matched arms Only after a carry-writer result localizes a remaining reader failure: 1. frozen base; 2. equal-parameter additive consumer `h+(2c-1)u` with ordinary paired-row CE; 3. untied hard clamp with freely parameterized `u`; 4. tied C3-1 Householder transport from the frozen writer axis; 5. C3-1 with counterfactual pairs shuffled inside identical nuisance strata; 6. C3-1 without the `(I-P_u)` erase term. All learned arms must share base, data, initialization scale, optimizer update count, forward positions, and decoding. Primary endpoints are the frozen 50-case cycle, autonomous episodes, unseen widths, counterfactual selectivity, and direct transcripts. Teacher-forced carry is secondary. ## 5. Discriminating predictions 1. Writer-only repair improves carry serialization; a true tied transaction gives additional integrated-cycle and full-episode gain. 2. Literal carry-token flips and `G_u` interventions agree on the next active digit. Donor carry interchange changes only the source case's local transition; double reflection restores baseline. Random orthogonal and irrelevant-result shams do neither. 3. Removing hard projection may preserve one-step behavior but loses accuracy as chain length grows. 4. A genuine one-bit mechanism transfers to widths 8 and 10. Fit-width-only gain falsifies the state claim. 5. All 400 local cells must pass. A persistent low cycle score after local success proves another cursor, tape, or control map is missing. ## 6. Kill criteria Reject C3-1 mechanism support if any condition holds: - the frozen writer is below 99% carry accuracy on a fresh board; - local `(digit,next-carry)` exactness is below 99% overall or below 98% in any width/style/operation/carry stratum; - the frozen cycle is below 45/50, fresh two-call exactness is below 90%, or autonomous full-trace exactness is below 90% separately at widths 4/6/8/10; - tied C3-1 is less than +10 points over both additive and untied controls on cycles or less than +15 points on width-8/10 full traces on any frozen seed; - shuffled pairing retains at least 50% of treatment gain; - carry-flip interventions are below 95% selective; - any non-DWS router fire or gate-off logit change occurs; - removing hard projection changes long-chain accuracy by less than 2 points. At 95% per-step reliability, a ten-step trace succeeds only `0.95^10 = 59.87%` under independent errors; width-10 reliability of 90% requires at least 98.952% per step. Local accuracy alone is never the primary claim. If the additive or untied adapter matches C3-1, reject the coupling claim and retain the simpler interface repair. If every local gate passes but full traces fail, reject one-bit carry as sufficient and localize the remaining cursor/tape/control state instead of enlarging the packet speculatively.