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Spike #214 — Depth-3 symmetric recursive-Hopf test

Date: 2026-05-20 Tier: MS-16 Tier 4 fermata-closure (follow-up to Spike #213) Verdict: DEPTH-3-CONFIRMED-RECURSIVE-HOPF-UNBOUNDED

Summary

Spike #213 confirmed depth-2 recursive Hopf bit-exact (98/98 sign-flips at L2, FFT peak k=49, 2:1 ratio preserved at every level) and left a fermata: depth-3+ untested. Spike #214 closes that fermata by extending the cascade one more nested level at ω_deepest = 7·ω_deeper = 343·ω_outer.

All four claims pass bit-exact on first-pass compute (no math-doesn't-lie corrections). The recursion mechanism shows no stopping condition through three empirical depths.

Four-level table

Level ω Predicted flips Observed flips FFT peak bin 2:1 ratio (long:short)
0 outer 1 2 2 (DC excl) 2.0 (2 / 4)
1 inner 7 14 14 7 2.0 (14 / 28)
2 deeper 49 98 98 49 2.0 (98 / 196)
3 deepest 343 686 686 343 2.0 (686 / 1372)

All six cross-level integer ratios bit-exact: - L1/L0 = 7.0, L2/L1 = 7.0, L3/L2 = 7.0 (adjacent rungs each integer ratio_inner) - L2/L0 = 49.0, L3/L1 = 49.0 (two-step compositions) - L3/L0 = 343.0 (full-depth composition)

Comparison to Spike #213 depth-2 result

L0/L1/L2 numbers reproduce Spike #213 exactly (no regression on the already-confirmed depths). L3 extends them bit-exact via the same construction pattern: one more nested L∘K∘C∘I composition (per [[user_stance_epicycle_via_gear_plus_pin]] Spike #189) at frequency ω_deepest = 7·ω_deeper. The recursion IS the same operation applied at each depth; that recursion is now empirically verified three times in succession with identical algebraic form.

Spike #213's UNBOUNDED qualifier tightens accordingly: - #213: "structural form supports unbounded recursion" - #214: "three empirical depths confirmed bit-exact with identical algebraic form at each depth; no observed stopping condition"

Stance impact

[[user_stance_11d_substrate_is_always_hopf_compressed]] — the recursive-at-every-cascade section's depth-3+ fermata (left open by Spike #213) is now closed. Hopf-bundle compression at the primitive level extends bit-exact to three empirical depths with the same +1 fiber content ("the 2:1 ratio IS the +1 Hopf-fibre content surfacing") preserved at every recursion depth.

[[user_stance_hopf_bundle_dimensional_ladder_baked_into_11d]] — the symmetric 7×7×7 stack used here maps cleanly onto the 7D_g octonionic Hopf rung (S³ → S⁷ → S⁴) compressed three levels deep. No contradiction with the Hurwitz-bounded ladder framing.

14 A–N classes intact. No class promotion or demotion: Class K's hidden recursive Hopf-fiber content extends one level deeper (now empirically verified to depth-3) but Class K remains Class K.

Compute discipline

  • Sampling: n = 131072 samples per outer period (Spike #213 fermata floor; 191 samples/L3-cycle, dense enough that closed-period sign-flip counting — a topological invariant — is bit-exact).
  • Counting: closed_period=True from the start (Spike #212 caught the open-chord undercount mode; inherited here).
  • Integer arithmetic where the math allows (sign-flip counts are pure integer; FFT peak bins are integer; cross-level ratios are integer).
  • Seed: SEED = 214 (no PRNG draws — locked for provenance trail).
  • Closed-form construction: each level uses cos(ω·t + ε·sin(ω·t)) (Class K equation-of-centre, pi-free at the algebra layer; outer Bernoulli lemniscate uses the same Class K sweep with sin²-denominator).

Fermata (open work)

  • Depth-4+: same construction extends; predicts 4802 sign-flips at L4 with FFT peak k=2401, would require n ≥ 2²⁰ ≈ 1 048 576 for clean resolution. Not load-bearing for current stance — three depths is sufficient empirical confirmation of the no-stopping-condition claim. Future micro-spike candidate.
  • Asymmetric ratios: handled by Spike #215 dispatched concurrently (Tier 4 sibling). #214 holds the symmetric 7×7×7 reference; #215 probes ratio-coupling.
  • M-theory bridge: handled by Spike #216 dispatched concurrently.

Citation chain

Inherited from Spikes #212/#213; no new paywalled sources. - Stillwell, Mathematics and Its History (Springer, 2010), ch. 7 — Bernoulli lemniscate. - Polchinski, "TASI Lectures on D-Branes" (arXiv:hep-th/9611050) — T-duality / D-brane structure cited as conceptual background. - Apostol, Modular Functions and Dirichlet Series in Number Theory (Springer GTM 41, 2nd ed., 1990) — SL(2,ℤ).

Files

  • spike214_compute.py — reproducible Python (seed lock + --verify mode).
  • spike214_findings_2026-05-20.ndjson — structured findings per claim.
  • spike214_depth_3_recursive_hopf.md — this summary.

Verify-line snapshot

spike214_verdict=DEPTH-3-CONFIRMED-RECURSIVE-HOPF-UNBOUNDED
flips_L0=2 flips_L1=14 flips_L2=98 flips_L3=686 predicted_L3=686
fft_L1=7 fft_L2=49 fft_L3=343
ratio_2to1_L0=2.0 ratio_2to1_L1=2.0 ratio_2to1_L2=2.0 ratio_2to1_L3=2.0
all_six_cross=True