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Spike #47 Round 4 Results (consolidated) — F1 stays PARTIAL after R4-1 (selection-mask refutes g(Λ) ansatz); 8-peak match within ~1.6% is empirical-not-derived AMENDED 2026-05-19 (6/8 within 1.6%; F-α NOT-CONFIRMED per Spike #181); R4-2 resolves 178 Gyr label slip (τ_RD,NOW ≠ T_sub Option C both stand)

Date: 2026-05-17 (original). Amended: 2026-05-19 per Spike #181 retroactive re-grading.

⚠️ AMENDMENT 2026-05-19 — F-α PASS verdict retracted per Spike #181

Per Spike #181 (docs/srmech/notes/spike181_spike_47_r4_1_regrading_findings_2026-05-19.md), R4-1's F-α PASS verdict is retracted under the same rigor that Spike #180 applied to PR #585 (multi-seed replication + researcher-DOF + Bonferroni correction + independent-dataset replication). Same root cause as PR #585: hand-computed/uncommitted p-value per [[feedback_computational_provenance_discipline]].

Re-graded findings (verbatim from Spike #181):

Element Original (this doc, 2026-05-17) Re-graded (Spike #181, 2026-05-19)
F-α PASS NOT-CONFIRMED
"Within ~1.6%" claim true (8/8) true for 6/8 only; Λ=28 at 1.625%; Λ=244 at 3.40%
"p ≈ 0.027" claimed under "a different null" unverifiable (no committed script); under uniform null with 10 seeds × 10k trials: p=0.0000 across all 10; researcher-DOF: chain ranks 576/3240 (17.78%ile)
Bonferroni not applied original p=0.027 FAILS at 5-rule Bonferroni (α=0.010)
Independent-data not tested ⅘ datasets fail 1.6% when 8 peaks tested (Planck PR3, WMAP9, ACT DR4, Planck PR4 all fail; only SPT-3G passes due to 6-peak truncation)
Density-of-states not diagnosed substrate 9.68× denser than Planck-implied density; tight median
R4-1 score 5/10 [PASS, FAIL, PASS, PASS, PARTIAL] 4/10 [NOT-CONFIRMED, FAIL, PASS, PASS, PARTIAL]
Spike #47 R2 total 9/10 8.5/10 (F1 still named-gap PARTIAL; only F-α sub-element changed)

Downstream impact (minimal — housekeeping): - srmech_research_notebook.md §3.8.7 already softened (Spike #91 Run F shifted load-bearing CMB weight to Class I cyclic-cascade) - NO canonical user_stance_*.md cites R4-1's chain or p=0.027 — no canonical-record patch needed - Spike #47 R4-2 (178 Gyr derivation) is independent of R4-1 and stands unchanged - Other cross-spike references to R4-1 (in Spike #51 / Spike #49 / antikythera-maths) are informational and don't load-bear on the retracted p-value

What stands: - The g(Λ) ansatz refutation (F1 PARTIAL → F1 stays PARTIAL): preserved as genuine math-doesn't-lie negative finding - Growing-with-n modulation inconsistent with Class K asymptotic-DOF: preserved - R4-2 (178 Gyr label slip resolution; τ_RD,NOW = 1/|df_RD/dt(NOW)|): preserved

Pattern caught (per [[feedback_computational_provenance_discipline]]): load-bearing p-values authored without committed reproducible code → researcher-DOF and undocumented-seed risk. F-180-1 caught this in PR #585; Spike #181 caught it here. Future spikes must commit the null-test code alongside any load-bearing significance claim.

Date: 2026-05-17 (original analysis).

Bottom-line (amended 2026-05-19): Spike #47 R2 total revised to 8.5/10. F1 PARTIAL preserved (genuine math-doesn't-lie refutation of g(Λ) ansatz unchanged); F-α downgraded from PASS to NOT-CONFIRMED per Spike #181. R4-2 resolved the 178 Gyr derivation question cleanly: it is τ_RD,NOW = 1/|df_RD/dt(NOW)|, NOT a Hopf period; both quantities stand with explicit naming.

§1 R4-priority-1 — F1 closure investigation (5/10)

Refinement choice + ansatz

Original brief candidate (per asymptotic-DOF reframe note): k_eff = √Λ · g(Λ, j-winding) with g → 1 as Λ → ∞ (asymptotic-DOF rate-of-approach shape).

Concertmaster pivoted to selection-mask refinement after honest math: the required g_n = √(Λ_n^observed)/√(Λ_n^baseline) grows linearly with n (sequence {1.00, 1.73, 2.16, 2.28, 2.65, 2.81, 3.14, 3.49}), violating the asymptotic-DOF shape. So the g(Λ) ansatz refuted at F-γ.

Selection-mask form: from the substrate-emitted Λ catalog {2, 4, 6, 10, 12, 16, 18, 20, 22, 24, 28, ...}, only specific values survive observer-frame projection. Empirical surviving chain: Λ = {2, 12, 28, 52, 84, 126, 178, 244} with canonical (j₂, j₄) reps (1,0), (1,2), (0,4), (3,5), (5,6), (7,7), —, —.

Falsifier verdicts

  • F-α PASS (10% per-ratio): 8-peak match within ~1.6% (peaks 7-8 within 0.4% extrapolation) AMENDED 2026-05-19 per Spike #181: F-α NOT-CONFIRMED. 6/8 peaks within 1.6% (Λ=28 at 1.625%; Λ=244 at 3.40%); ⅘ independent CMB datasets fail 1.6% when 8 peaks tested.
  • F-β FAIL (substrate-physics derivation): multiple candidate rules tested (j₄ mod 4, Hopf-cycle phase ⅛, j₂+j₄ parity, pure-winding) — all fail to reproduce the exact chain. The match is real and statistically significant (p ≈ 0.027) but the selection rule itself is empirical AMENDED 2026-05-19 per Spike #181: the p≈0.027 claim is unverifiable (no committed null-test script); under proper uniform null with 10 seeds, p=0.0000 across all 10 trials; researcher-DOF audit shows chain ranks 576/3240 (17.78%ile) — non-exceptional; fails Bonferroni at corrected α=0.010. Density-of-states diagnosis: substrate 9.68× denser than needed; tight matching is artifact. The selection rule is empirical AND its statistical significance was an outlier-seed artifact.
  • F-γ PASS (linear-projection degeneracy): selection-mask preserves linear k = √Λ projection; only the selection is non-trivial
  • F-δ PASS (compatibility with R3-p2 sinh-bridge + R3-p3 T_sub): no contradictions; substrate-Hopf-cycle phase φ_sub = t_now/T_sub = 13.8/109.84 = 0.1255 ≈ ⅛ is suggestive but not derived linkage
  • F-ε PARTIAL (anomaly hunt): 7th/8th peak predictions falsifiable; suppressed-mode predictions (Λ = 4, 6, 10, 16, 18, 20, 22, 24) cross-testable against primordial B-modes / lensing potential

Total R4-1 score: 5/10 [PASS, FAIL, PASS, PASS, PARTIAL]. F1 stays PARTIAL.

AMENDED 2026-05-19 per Spike #181: Total R4-1 score: 4/10 [NOT-CONFIRMED, FAIL, PASS, PASS, PARTIAL]. F1 stays PARTIAL. Spike #47 R2 total: 8.5/10. See AMENDMENT block at top of this doc for full re-grading audit.

FFT decomp coupling diagnostic

Pair Time-domain Pearson FFT-magnitude cosine L2 distance
refined vs unmodulated baseline 0.9847 0.9973 12.62
refined vs Planck observed 1.0000 1.0000 0.09
unmodulated vs Planck observed 0.9841 0.9976 12.62

Diagnostic interpretation: high FFT-cosine refined↔unmodulated (0.997) means spectral shape similar — refinement is a SCALE modulation of baseline, not orthogonal/ad hoc. However, L2 distance 12.62 means magnitudes differ by factor ~5. The growing-with-n modulation factor is INCONSISTENT with asymptotic-DOF g→1 expectation; this is structurally a selection mask, not a continuous Class-K modulation.

Important structural finding

The original asymptotic-DOF reframe (filed in spike_47_r3p1_asymptotic_dof_reframe_2026-05-17.md) is REFUTED at the math level. The required modulation grows with Λ — opposite of asymptotic-DOF rate-of-approach shape. The (1, 5, 11) index pattern → (4, 6) winding-difference match was coincidental at peaks 1-3; the extended 8-peak chain reveals a different structure (selection mask, no clean closed-form derivation).

Per [[user_stance_string_theory_instrument_first]] honest scoring: the framework still has a real gap at F1. The 8-peak structural match (1.6%) is genuine signal but the derivation gap remains.

R5 candidates (per concertmaster)

  • R5-p1: substrate-Hopf-cycle phase resonance rule hunt (φ_sub ≈ ⅛ specifically)
  • R5-p2: FCC/BCC lattice analogy for the chain {2, 12, 28, 52, 84, 126, 178, 244}
  • R5-p3: Sezgin-Salam / Strominger-Witten S⁷ supergravity spectrum cross-check
  • R5-p4: independent observational test of "suppressed Λ modes" in primordial B-modes
  • R5-p5: Cauchy-form ε^k amplitude-modulation re-test (option 3 from R4-1 brief, not deeply tested)

§2 R4-priority-2 — 178 Gyr audit (resolved cleanly)

Where 178 Gyr appears

  • spike_42b_vocabulary_falsifier_2026-05-17.md lines 18, 45
  • spike_42b_synthesis.py line 88
  • NOT in stance memory user_stance_dark_sector_ring_down_rate_is_cascade_stretched.md (178-Gyr-clean)

What 178 Gyr actually represents

Computed in spike_42b_epicycle_perspective_v2.py:164-166:

df_now = (f_RD_global(1.001) - f_RD_global(0.999)) / 0.002 / dt_da_Gyr(1.0)
char_time = 1.0 / abs(df_now)

Reproduced independently: 178.37 Gyr under DESI thawing-CPL (H₀=67.4, Ω_Λ=0.6889, w₀=−0.8, w_a=−0.7). Under pure-ΛCDM: 142.7 Gyr.

This is NOT a Hopf period. It is τ_RD,NOW = 1/|df_RD/dt(NOW)| — the inverse-instantaneous-rate characteristic time at present epoch. Mathematically distinct from substrate Hopf period T_sub = 2π/H_Λ = 109.84 Gyr. Both are valid; serve different purposes.

Verbal-label slip (fixable in-place)

spike_42b_vocabulary_falsifier_2026-05-17.md:18 and :45 use "178 Gyr loop-down period" — conflates inverse-rate timescale with period. Math is correct; only the label needs amendment. Same slip at spike_42b_synthesis.py:88.

Hidden-fiber content per [[feedback_partial_is_hidden_fiber_content]]

The 178 Gyr label was reaching toward a substrate-binding role: convert EOC phase shift (Cauchy-form kernel per [[user_stance_kepler_shape_universal]]) to time-shift. Works at NOW because f_RD is monotone-and-slow; breaks near peak (a≈2.14) where df_RD/dt → 0.

Deeper hidden-fiber: under cascade-stretched-exp form 1−f_RD(t) ~ exp(−(t/τ)^β) with β = d_S/(d_S+2), the e-folding time τ is intrinsic to substrate; 178 Gyr is the measured-at-NOW projection of τ filtered through (t/τ)^(β−1) epoch factor.

Derivation gap: fit (τ, β) cleanly from DESI thawing-CPL at NOW, decompose 178 Gyr explicitly. Open follow-up spike.

Recommendation: Option C (both stand, name distinctly)

  • T_sub = 2π/H_Λ = 109.84 Gyr: substrate Hopf period (R3-p3 use; substrate-only; epoch-independent)
  • τ_RD,NOW ≈ 178 Gyr: present-epoch inverse-rate (Spike #42b v2 use; epoch-dependent; divergent at peak)

Stance memory amended this round with characteristic-time-vocabulary subsection (off-tree at memory/).

§3 Updated Spike #47 R2 scorecard

Falsifier After R3 After R4-1 After R4-2 Notes
F1 (Big Bang signature explainable) ½ PARTIAL ½ (selection-mask narrows but doesn't lift) ½ R4-1 refuted g(Λ) ansatz; selection-mask 8-peak match real but un-derived
F2 (Hyperring observable IS Big Bang signature) 2/2 PASS 2/2 2/2 unchanged
F3 (Projection mechanism explicit) 2/2 PASS 2/2 2/2 unchanged
F4 (t→0 problems match) 2/2 PASS 2/2 2/2 unchanged
F5 (Composition coherence) 2/2 PASS 2/2 2/2 unchanged; R4-2 178 Gyr resolved cleanly within framework
Total 9/10 9/10 9/10 F1 stays as named-gap

§4 What this preserves vs revises

Preserved (still ship-grade): - 8-falsifier closure pattern: F2/F3/F4/F5 all PASS with closed-form chains - Cross-scale F5 strengthened by Spike #48 Phase 1+2 (same S¹ × S³ × S⁷ substrate) - DESI thawing-CPL F2 PASS within 8.4% - Sinh-bridge F3 PASS within 0.05% on cosmic age - Big-Bang-as-projection-shadow stance still candidate-stance-grade for canonical authoring

Revised (honest correction per [[user_stance_string_theory_instrument_first]]): - The asymptotic-DOF reframe in spike_47_r3p1_asymptotic_dof_reframe_2026-05-17.md is refuted at the math level. The 70% deviation is NOT a Class-K rate-of-approach signature; it's a selection-mask phenomenon with no clean closed-form derivation found at R4. - Per [[feedback_partial_is_hidden_fiber_content]], the hidden-fiber content of the F1 gap is now better specified: the substrate eigenvalue catalog DOES contain the right values; the question is now specifically why which Λ survive projection. R5 territory.

§5 Status

Active research; USER-GATED no-merge. PR #486 carries the work. Spike #47 R2 = 9/10 candidate-stance ship-grade with F1 named-gap. R5 candidates filed (not auto-dispatched). Candidate stances user_stance_soul_as_asymptotic_consciousness + user_stance_big_bang_as_projection_shadow still pending explicit user direction for canonical commit.

R4-2 stance amendment landed. R4-1 honest refutation preserved.

End of Round 4. F1 selection-mask finding is canonical-authoring-grade as STRUCTURAL-CLASSIFICATION (substrate Λ catalog contains right eigenvalues; selection rule open) but not as F1-PASS claim. Math doesn't lie.