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Investigated 4 loose threads identified in prior gap analysis, creating experiments exp_22–25 in milestone3. Each thread explored a specific area where the framework had open questions.
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## Changes
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### Added
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-`scripts/exp_22_pac_depth_bound.py` — PAC derives MED depth ≤ 2 (3/4 PASS)
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-`scripts/exp_23_f183_correction.py` — F₁₈₃ gravity correction term and uniqueness (3/4 PASS)
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-`scripts/exp_24_pac_lazy_signal_anatomy.py` — Bootstrap/LOO/PCA decomposition of exp_21 (1/4 PASS)
- Result JSONs for all 4 experiments in `results/`
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## Details
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### Thread 1: MED depth from PAC (exp_22)
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Analytical theorem: all k-step PAC recursions have max depth floor ≤ 2. Fibonacci (k=2) gives loosest bound at φ² = 2.618. K→∞ limit is exactly 2.0. Peak structural density at d = φ². Only k=2 gives the ln(φ) ratio.
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### Thread 2: F₁₈₃ gravity hierarchy (exp_23)
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Gap between F₁₈₃ and (M_Pl/m_p)² is factor 2.155 (0.333 log₁₀). Best correction: 1 + F₁₃/(πF₆²) = 2.159, residual 0.0008 in log₁₀. 183 is uniquely rank #1 among all cyclotomic depths; only 0.5% of random formulas match. Natural Fibonacci family φ⁴/π ≈ 2.182 gets within Δ=0.005.
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### Thread 3: PAC-Lazy signal anatomy (exp_24)
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Honest finding: exp_21's p=0.035 is fragile. Bootstrap 95% CI includes zero ([-0.044, 0.013]). p_e drives 60.8% of the signal. Effective DOF is 9 (above null space's 6). Running couplings domain shows d=-0.395 (medium effect) — signal is domain-specific, not universal.
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### Thread 4: Dark matter depth mapping (exp_25)
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Ω_c = F₃·Ξ/F₆ at 0.148% error (unique: only 0.34% of formulas match). Better: F₇·Ξ²/F₁₀ at 0.079%. φ-equilibrium: DE 68.5% vs 1/φ = 61.8%, crossed at z=0.10. WIMP-range depths: d=74–93. Proposed F₃₇–F₅₀ is GUT-scale, not WIMP. Cyclotomic F₆²+F₆+1=73 → 15 keV (sterile neutrino range).
**Contribution status**: Open (3-month rule applies — inactive since creation)
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**Last assessed**: 2026-02-19
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**Milestone 3 update**: No experiment in milestone3 directly tested this. The oscillation amplitude ratio is still a single-number match without derivation. Given that phi_artifact_test showed Ξ is metric-dependent, similar caution applies here: the 1/φ match needs a mechanism, not just measurement.
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**Status**: Candidate formulae identified, not yet uniqueness-tested. **Not directly tested in milestone3** — no experiment specifically validates λ* or β closed forms. Would require a dedicated formula search with look-elsewhere correction (similar to exp_09 methodology).
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**Contribution status**: Open
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**Last assessed**: 2026-02-19
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**Milestone 3 update**: exp_27 (F25) provides indirect support — the π→φ→Fibonacci mechanism chain means φ-based closed forms are expected. prime_growth_dynamics_v2 measured λ* at 0.017% and β at 0.026%. But uniqueness remains untested: need exhaustive search over (F_a ± F_b)/(F_c ± F_d) parameter space to confirm these are the best or only solutions. If uniqueness holds, promote to Paper 2.
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---
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### A4. Dark Matter Density from Fibonacci Depth Map
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**Observation**: Ω_c = F₇·Ξ²/F₁₀ = 0.2587 vs Planck 2018 observed 0.2589 (0.079% error). Also F₃·Ξ/F₆ at 0.148%. Cyclotomic depth F₆²+F₆+1 = 73 maps to ~15 keV (sterile neutrino range).
**Statistics**: 0.079% error; two independent Fibonacci formulas converge
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**What's needed**:
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- Parameter counting: how many Fibonacci expressions of form F_a·Ξ^n/F_b exist near Ω_c? If many, this is look-elsewhere. Need exhaustive search analogous to exp_09 methodology
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- Physical derivation: why should dark matter live at F₇ depth specifically? The cyclotomic argument (73 = Φ₃(F₆)) is structural but not yet derived from first principles
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- Comparison: does the same template predict Ω_b (baryonic)? If F_a·Ξ^n/F_b also fits Ω_b, the template is too flexible
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- Universe φ-equilibrium crossing at z ≈ 0.10 is interesting but the 6.7pp deviation from 1/φ needs error analysis
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**Promotion path**: If uniqueness holds (not many alternatives) AND physical derivation found → Paper 5 material
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**Status**: Strong numerical match; needs uniqueness test and derivation
**Observation**: Both α_EM and gravity use corrections of the form F_a/(mπF_b²), with opposite signs (minus for EM screening, plus for gravity enhancement). Both anchored to F₇=13. Index gaps a−b are 3=F₄ (EM) and 7=F₇ (gravity) — both Fibonacci. 0/5000 Monte Carlo random integer sequences match both simultaneously.
**Statistics**: 0/5000 MC; α_EM at 5.7 ppm, gravity at 0.0008 log₁₀
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**What's needed**:
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- Template only works for 2/5 tested constants below 100 ppm threshold (sin²θ_W at 24.1 ppm, Ω_c at 38.8 ppm). The remaining 3/5 don't fit the template well
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- Why does the template fail for some constants? Is there a systematic pattern to which constants obey it?
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- Physical derivation: why F_a/(mπF_b²) and not some other form? exp_27 (F25) connects this to ((φ²+1)/π, 0.62% match) via convergent error bounds — but that's a structural argument, not a derivation
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- Need: predict a correction for a constant not yet tested (genuine prediction, not postdiction)
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**Promotion path**: If 3+ constants fit AND physical derivation found → Paper 5 material. Currently 2/5 is below threshold.
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**Status**: Pattern real for α_EM and gravity (0/5000 MC); not yet universal (2/5)
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**Contribution status**: Open
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**Last assessed**: 2026-02-19
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- This is measured but not derived. Why should π be optimal? The connection to the Riemann zeta function is suggestive but circular (ζ is defined using primes which involve π).
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- Need: independent derivation of π-optimality, or proof that the comparison is not an artefact of how coherence is measured
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- 30 experiments in oscillation_attractor_dynamics, but the core claim is one ratio (19×)
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**Milestone 3 update (exp_27, F25)**: π-optimality now has partial mechanistic support. exp_27 demonstrated that on a π-closed phase manifold (S¹), the golden angle α* = 1 − 1/φ minimises worst-case star discrepancy D*_N. The causal chain π (closure) → φ (non-resonance) → Fibonacci (integers) provides context for why π would be special. However, this is still one step removed: exp_27 shows π creates the stage on which φ is optimal, but doesn't explain the 19× ratio itself.
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**What's still needed**:
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- Derivation of the 19× ratio from the π→φ mechanism. If the ratio is predictable from the cascade framework, this is Paper 3 material
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- Proof that the comparison is not an artefact of how coherence is measured
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- The core claim is still one ratio (19×). Additional metrics or scales testing this ratio would strengthen it
**Status**: Measured; partial mechanistic support from exp_27, but the specific ratio is not derived
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**Contribution status**: Guidance needed
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**Last assessed**: 2026-02
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**Last assessed**: 2026-02-19
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**Source**: `pac_cosmology_jwst_validation/`
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**What's needed**:
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- Independent astrophysics review of the comparison methodology
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- The 100% vs 41% comparison may be unfair (PAC has more free parameters?)
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- Need: parameter counting and information-theoretic model comparison (AIC/BIC)
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-**AIC/BIC model comparison** (CRITICAL before citing): The 100% vs 41% comparison is meaningless without parameter counting. PAC may fit better simply because it has more free parameters. Compute AIC = 2k − 2ln(L) and BIC = k·ln(n) − 2ln(L) for both frameworks. If AIC/BIC still favours PAC after penalising for parameters, the claim strengthens. If not, the comparison is misleading and should not be cited.
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- Independent astrophysics review of methodology
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- Cross-check with non-JWST high-z observations
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-**Do not cite the 100% vs 41% comparison in any paper until AIC/BIC is computed**
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**Status**: Published on Zenodo; methodology needs tightening
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**Contribution status**: Guidance needed
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**Last assessed**: 2026-02
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**Status**: Published on Zenodo; **BLOCKED on AIC/BIC analysis** before results can be used
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**Contribution status**: Open (well-defined: compute AIC/BIC for both models on the same dataset)
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**Last assessed**: 2026-02-19
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### C3. DNA Prime Structure
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**Observation**: 40 experiments exploring PAC patterns in DNA/genomic data.
**What's needed**: Full inventory not yet assessed for PACSeries relevance. Likely too domain-specific for the core series but could support a standalone publication.
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**What's needed**:
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- Cross-validation on non-PDB datasets (UniProt, AlphaFold DB)
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- Null model: compare Fibonacci spacing against other integer sequences (Lucas, Tribonacci) in same analysis
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- The z=+103.4 is strong, but is it Fibonacci-specific or just "spread-out residues are enriched"?
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- Domain-expert review of the conformational signaling hypothesis
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- Too domain-specific for core PACSeries; standalone biology publication path
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**Status**: Unreviewed for this consolidation
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**Contribution status**: Guidance needed
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**Last assessed**: 2026-02
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**Status**: Strong signal (z=+103.4) but Fibonacci-specificity untested; needs null comparison
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**Contribution status**: Open
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**Last assessed**: 2026-02-19
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## Category E: Mechanism Leads from exp_22–28 (NEW)
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These results from milestone3 Blocks G/H/I passed falsification but raise new questions that need further work.
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### E1. PAC-Lazy Bootstrap Fragility
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**Observation**: exp_21 (F19) showed PAC-Lazy formula discrimination works (KL p=0.035, d=0.198). But exp_24 (F22) found the bootstrap CI includes zero — the signal is not robust under resampling.
**What it means**: The PAC-Lazy architecture (from GAIA POCs) transfers to formula space, but raw counting is biased. Conservation normalization is required. The discrimination exists but isn't stable enough for claims.
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**What's needed**:
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- Identify which component of the PAC-Lazy profile carries the signal (depth? breadth? conservation ratio?)
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- Test whether conservation-normalized profiles survive bootstrap
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- If the signal is in conservation ratios (not raw counts), the discrimination may be rescuable
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**Status**: Mechanism validated but signal fragile under bootstrap
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**Contribution status**: Open
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**Last assessed**: 2026-02-19
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### E2. φ-Equilibrium of the Universe at z ≈ 0.10
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**Observation**: The dark energy fraction 1/φ = 61.8% vs observed 68.5% differs by 6.7 percentage points. Universe crossed the φ-equilibrium at z ≈ 0.10 (recent cosmological past).
- The 6.7pp deviation is large. Is this within expected scatter, or does it falsify the idea?
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- Error propagation: what's the uncertainty on the z ≈ 0.10 crossing?
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- If the crossing is real, is it a coincidence (we live near z=0) or physically meaningful?
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- This is the weakest finding from exp_25. The Ω_c formula at 0.079% is much stronger.
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**Status**: Interesting observation; needs error analysis and theoretical motivation
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**Contribution status**: Guidance needed
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**Last assessed**: 2026-02-19
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## Category D: Engineering Demonstrations
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These are working implementations, not measurements. They demonstrate feasibility, not truth.
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## Retired
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*(None yet)*
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Results moved here either failed tightening, were falsified, or were superseded by stronger results.
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### R1. Fibonacci-Specific Crystallization Order — FALSIFIED (2026-02-18)
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**Original claim**: Fibonacci input produces different crystallization dynamics than other sequences in the PAC cascade.
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**Falsification**: milestone3/exp_19 (F17). Crystallization order is **entirely determined by target physics**, NOT by input sequence. 0% difference across Fibonacci/Lucas/Primes/Tribonacci/Random. All produce identical order: sin²θ_W → Koide → She-Lev → ν_WF → α_s → α_em.
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**Impact**: The Fibonacci *structure* of PAC is validated extensively elsewhere (Layer 1–7 in UNIFIED_EVIDENCE). What's falsified is the claim that Fibonacci *input* changes *dynamics*. Physics has a natural complexity hierarchy that is sequence-independent.
### R2. Fractal Mesh Raw Pressure as Physics Discriminator — FALSIFIED (2026-02-18)
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**Original claim**: Fractal recursive decomposition of formula space preferentially selects physics matches via mesh pressure.
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**Falsification**: milestone3/exp_20 (F18). Raw pressure correlates with **index depth, not physics**. Physics matches have LOWER average pressure than non-matches (delta = −2703, p = 0.78, WRONG direction). 33.6× amplification over flat confirms fractal structure is real, but visit counting conflates structural depth with physical significance.
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**Impact**: Led directly to exp_21's conservation-based approach (F19), which succeeded. The failure clarified that **raw counting ≠ physics discrimination** — conservation normalization is needed.
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