Skip to content

Round 10 entry-point A — is ℓ=7 individually a Mersenne node? No. The {3,7} signal is the octupole.

Dispatched 2026-05-25 (sequential, no subagents). Picks up parking-lot thread 1: Round 6.A's one remaining (open) verdict — the Mersenne {1,3,7} prediction, where ℓ=1 (dipole) and ℓ=3 (octupole) were present and ℓ=7 was the open testable. Run on attested, already-committed data — no new download, no healpy, no fabrication.

Generating code + provenance: verify_ell7_mersenne_specificity.py + .ndjson. Reuses the attested per-ℓ C_ℓ table committed by Spike #190 (docs/srmech/notes/spike190_findings_2026-05-19.ndjson; SMICA-nosz anafast; Planck 2018 IV, doi 10.1051/0004-6361/201833881).

The distinction the round forces

Spike #190 found a {3,7} aggregate concentration: 6.19× over uniform null, p = 0.006, on the low-ℓ TT spectrum. But {3,7} is the pair together. Round 6.A's open prediction was about ℓ=7 specifically. Does ℓ=7 individually carry a Mersenne signature, or does ℓ=3 carry the whole {3,7} result?

Three ℓ=7-specific tests (committed code)

1. Split of the {3,7} concentration.

mode share of the {3,7} pair-power
ℓ=3 (octupole) 80.4%
ℓ=7 19.6%

The {3,7} signal is ℓ=3-dominated. ℓ=7 contributes under a fifth.

2. Per-mode ranking in the ℓ=2–8 band (ratio of each mode's power-share to the uniform 1/39 null):

ranked descending: ℓ=3 (9.97×) > ℓ=5 (6.29×) > ℓ=4 (4.39×) > ℓ=2 (3.75×) > ℓ=7 (2.42×) > ℓ=6 > ℓ=8

ℓ=7 ranks #5 of 7. It is outranked by three non-Mersenne modes — ℓ=5, ℓ=4, and even the famously suppressed quadrupole ℓ=2. A distinguished Mersenne node would not sit below ℓ=4 and ℓ=5.

3. Local-maximum parity test. ℓ=7 is a local maximum (> ℓ=6, > ℓ=8) — but so is the non-Mersenne odd ℓ=5. The local maxima in the band are {3, 5, 7}: that is odd-ℓ parity alternation, not a Mersenne fingerprint. ℓ=5 sitting in the same pattern, and ranking above ℓ=7, kills the Mersenne-specific reading.

Verdict per Spike #229 tiers

🟠 (open) → WEAKENED on the ℓ=7-specific claim.

  • There is no ℓ=7-specific Mersenne signature in the attested Planck low-ℓ TT data. ℓ=7 is unremarkable: 2.42× uniform, 5th of 7, below three non-Mersenne modes, local-max-by-parity not by Mersenne.
  • This satisfies Round 6.A's own falsifier criterion ("robust ℓ=7 absence would weaken the Mersenne prediction"). The ℓ=7 element of the {1,3,7} prediction is not supported and should be withdrawn as a per-ℓ claim.
  • The {3,7} aggregate result (Spike #190) is untouched — it stands at 6.19×, p=0.006, but it is now correctly understood as ℓ=3 (octupole)-driven, not evidence for ℓ=7.

What this does to §11.9.6 (future promotion-PR)

The §11.9.6 amendment (already queued by Rounds 8.A + 9.A) gains a third correction. The Mersenne reading should be stated honestly:

  1. ℓ=1 = the observer fiber-coordinate (kinematic dipole; removed from the anisotropy spectrum by convention — consistent with the fiber-leak reading of Round 8.A).
  2. ℓ=3 (octupole) = the real low-ℓ concentration carrier; part of the quad-oct alignment.
  3. ℓ=7 = withdraw the per-ℓ Mersenne claim. Not distinguished in the data. The {3,7} structural pairing (parallelizable-sphere ladder 1+3+7; Hopf-fiber dims) remains valid as an algebra-layer identity, but its CMB-multipole projection does not single out ℓ=7.

This is the layered discipline ([[user_stance_identity_not_implementation_discipline]]): the structural 1+3+7 Hopf-fiber identity is unchanged; only its empirical CMB-projection fingerprint contracts (ℓ=7 drops out).

Honest scope

  • Per [[feedback_dont_pre_commit_spike_query_operators]]: this is a negative result on a framework prediction, reported plainly. The {3,7} aggregate was not allowed to stand in for ℓ=7.
  • Per [[feedback_computational_provenance_discipline]]: committed code; reuses attested Spike #190 input; no hand-entered spectrum.
  • Pseudo-C_ℓ caveat inherited from Spike #190 (no MASTER mode-coupling correction; for the relative ℓ-distribution test this is immaterial, as Spike #190 noted).
  • Cosmic variance at low ℓ is large; "WEAKENED" not "FALSIFIED" — but the direction is unambiguous: ℓ=7 carries no excess over its non-Mersenne neighbours.
  • PR #679 stays open; §11 SSoT frozen until a promotion-PR.

Disposition

  • §11.9.6 amendment now three-fold: boosting=β (8.A) + alignment=geometric (9.A) + ℓ=7 withdrawn, {3,7}=ℓ=3-driven (10.A).
  • Round 6.A's (open) ℓ=7 verdict → resolved: WEAKENED / withdrawn as a per-ℓ claim.
  • The {1+3+7} Hopf-fiber algebra identity is preserved; only the CMB-multipole projection of it loses ℓ=7.