E35 to E40: does the flush-to-zero rule ever matter? (after two wrong explanations)

01Why it matters

One obscure rule about tiny numbers is pinned in the spec. If it never matters, is it needed?

02Method, step by step

1. Count how often tiny values arise (E35, E38).
2. Analyse exp exhaustively over 2^32 inputs (E39).
3. Classify every reduction and RMSNorm intermediate (E40).

03Results, exact and sourced

1. R1E35: zero of 21,248,280 exp evaluations reached the band it studied (a band later shown not to be where the flush rule acts).

   Zero of 21,248,280 `exp_pinned` evaluations across six prompts

2. R2E38's wider sweep found 2 arguments in that band on one long decode (Qwen2.5-0.5B, 256 tokens); the digest did not move.

   with 2 of that cell's 22,626,240 arguments inside

3. R3E39: the mechanism we had given in four places was wrong; the conclusions survive, the reason did not.

   the mechanism claim in E29, E35, E38 and erratum E-10 is wrong

4. R4The claim that a denormal arises in a real decode was withdrawn.

   is withdrawn.

5. R5E40 classified 562,531,070,920 intermediates across both pinned checkpoints; none was subnormal.

   **562,531,070,920** classified

6. R6No weight operand in either model is subnormal.

   **0 of 628,547,776 weight operands across both §0 models are subnormal**

7. R7Conclusion: on the normative vector the rule does not affect the digest.

   On the normative vector, §1.3's FTZ/DAZ is not digest-relevant.

05Scope and caveats

• Read the four documents in order; two of our own mechanism claims were retracted, while the measurements stood.
• Holds for the two pinned checkpoints and the stated prompts only.
• Run by us; no unaffiliated reproduction.
• E39 shows the flush rule can change bits in the softmax division step on constructed inputs; the finding is only that it did not on the normative vector.
• E40 reports a defect in E39's own counting instrument, found by a control and fixed; the source states it could not bite at that call site.
• The 2-argument E38 finding was in a band that E39 later showed is independent of the flush rule.

06Outcome

07What we learned

We never found a case where the flush rule changes the digest. The explanations we first gave for why were wrong, and we published the corrections next to the exhaustive count that replaced them.

08Check it yourself

• cis2-spec / E35_DENORMAL_REACH.mdhttps://github.com/Aefinity-AI/cis2-spec/blob/main/docs/E35_DENORMAL_REACH.md
• cis2-spec / E38_REACH_SWEEP.mdhttps://github.com/Aefinity-AI/cis2-spec/blob/main/docs/E38_REACH_SWEEP.md
• cis2-spec / E39_FTZ_IS_NOT_WHERE_WE_SAID.mdhttps://github.com/Aefinity-AI/cis2-spec/blob/main/docs/E39_FTZ_IS_NOT_WHERE_WE_SAID.md
• cis2-spec / E40_MATVEC_RMSNORM_REACH.mdhttps://github.com/Aefinity-AI/cis2-spec/blob/main/docs/E40_MATVEC_RMSNORM_REACH.md
• cis2-spec / CHANGELOG.mdhttps://github.com/Aefinity-AI/cis2-spec/blob/main/CHANGELOG.md

Read the four pages in order and the CHANGELOG (links above).