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.