ARM RP AMENDMENT 1 — POSITIVE CONTROL CELL — REGISTERED 2026-08-27, BEFORE SAMPLING

MOTIVATION (response to a second-round review finding, disclosed). Arm RP returned
UNKNOWN in 87 of 87 scored responses. That is consistent with absent recall, but also
with an abstention prior induced by the prompt's own "If you do not know, answer
UNKNOWN" option: the design had no cell where the model demonstrably CAN state a value
under this exact prompt. This amendment adds that cell.

CELL (one): N = 1. The optimum is elementary and certain: the single inscribed circle
has radius 0.5, so the best-known (and provably optimal) maximum sum of radii is
0.5000000. No scoreboard, no contested figure, no family/rival ambiguity. Any model
willing to answer numerically under this prompt format can answer here.

PROMPT. The registered arm-RP template verbatim with n = 1; SHA-256 recorded in
arm_rp_prompts_control.json before sampling. Same channel as the main arm (weak-tier
Task subagent, wrapper A.3, zero-shot, no tools), n = 15, one pass, runtime rejections
excluded and counted.

SCORING (arm_rp_control_analysis.py, committed with this file). A response ANSWERS if it
parses as a single number within 2e-3 of 0.5; UNKNOWN and unparseable counted separately.
The registered main-arm ledger and verdicts are untouched.

COMPETING READINGS.
  C-RP1 (compliance): ANSWERS in >= 8 of 15 scored responses. The abstention-prior
    alternative is rejected: the model answers this prompt format when it has the value,
    so the 87/87 UNKNOWN at scoreboard cells reflects absence of a statable value, and
    the positive-control caveat in S3.9 is replaced by this result.
  C-RP2 (format abstention): ANSWERS in <= 4 of 15. The abstention-prior alternative is
    supported: UNKNOWN at scoreboard cells cannot be read as absent recall, and the
    S3.9 caveat is strengthened to a confirmed confound.
  5-7 ANSWERS: PARTIAL, reported with no claim; the caveat stands as written.

STOPPING RULE. 15 invocations, one pass, scored once. This amendment governs only the
control cell; no main-arm number changes under any outcome.

WHAT THIS CANNOT SHOW. N = 1 is easier than any scoreboard cell in every respect, not
only in recall; a pass here does not prove the model would answer at N = 26 if it knew
the value, only that the prompt format itself does not force UNKNOWN.
