Hostile audit — two-reservoir latency theorem

Result audited: QA-RA-S6-T15 and C15.1
Date: 2026-08-28
Outcome: pass after narrowing the relay claim

Adversarial checks

1. The former bounded pattern was not promoted by extrapolation. The all-$q$ argument is an induction over affine run templates. Finite replay appears only as six base cases and as an independent transcription check.
2. The two gap budgets were counted separately. The first is $G_1=9q-33$. It is exhausted by $6q-22$ alternating steps. The second is newly created at the raw seam and is $9q-17$ or $9q-16$, depending on parity. It is not the residue of the first gap.
3. The separated-flank recurrence is never used through overlap. The proof stops the first recurrence at gap zero. It stops the second at gap zero in the even case and gap one in the odd case. Every later update is verified by raw canonical reversal and addition.
4. Parity branches were not merged by analogy. The new $H_q$ state, its gap, its bridge into the alternating $s_r$ family, and the terminal word are certified separately for even and odd $q$.
5. The induction actually closes. The symbolic certificate checks both parity transitions of the first six-step state family, including the parameter shift after an odd cycle. It also links the cycle endpoint to the raw seam start, the seam endpoint to $H_q$, and the final $s_r$ endpoint to the terminal raw word.
6. Complete alternation of the $a_q$ branch is proved. The first seam reaches $U_{6q-2}$, and the exact identity $T^4(U_r)=U_{r+1}$ gives the infinite tail. No long replay is used to justify $(10)^\infty$.
7. The first differing index is not off by one. The responses agree for indices $0,\ldots,12q-13$. At index $12q-12$, $a_q$ outputs zero and $b_q$ outputs one. Therefore the determining horizon lower bound is $12q-11$, one larger than the divergence index.
8. Hamming adjacency is exact. The constants differ by $2^{10}$, and the corresponding bit is zero in 154. No carry changes another source bit.
9. The stronger coinductive relay was removed. The certificate proves a twelve-step increment in first-divergence latency. It does not prove equality of the complete shifted futures of $b_q$ and $b_{q-1}$.
10. No unrelated dynamical claim follows. The result concerns the binary length-growth response. It does not decide palindrome hitting, Lychrel behavior, or the cost of algorithms allowed to inspect raw digits.

Independent implementation

The second verifier represents integers as little-endian bit lists, reverses canonical digit lists, and performs carry addition without Python’s integer bit-reversal operation. It checks the exact first-divergence statement and the complete alternating prefix of the $a_q$ branch on eleven targeted values, including $q=100$. These substitutions guard transcription; the symbolic run identities remain the proof.

Important correction

The research target was initially described as a twelve-step response-pair relay. The attack established a twelve-step latency relay through two serial reservoirs. Those are not synonymous. The former remains unproved; the latter is sufficient for the $4L/3-12$ theorem.