# Multi-scale helix vs radial-accordion test — plan

**Triggered by:** Wolf, 2026-05-02. The harmonic synthesis (Tests 5–9) found that the
same wave geometry recurs at three nested scales — fundamental T₁ (≈ 25–35 Mb on chrX),
T₁/3, T₁/5 — with cross-scale amplitude coupling r = +0.55 (L1↔L2) and r = +0.77 (L2↔L3).
That nested-cycle structure is exactly what would distinguish a helix from a radial
accordion, but the current ED Fig 6 polymer simulation tests the question only at one
scale and only in distance space, where neither model leaves a fingerprint. A proper test
operates per-scale on real Dip-C 3D coordinates.

## Available real data

| Resource | Location | What it gives us |
|---|---|---|
| 14 GM12878 Dip-C 3D structures | `polychrom simulation/dipc_data/extracted/gm12878_*.3dg.gz` (Tan 2018) | Per-bead (chrom_hap, pos, x, y, z) coords at 100-kb resolution |
| 28 mouse cortex P56 Dip-C structures | `polychrom simulation/dipc_data_mouse_cortex/` (if present; needs check) | Per-bead 3D coords as a cross-species replication arm |
| GM12878 phase map at L1 (fundamental) | `go_rerun_v2_2026-04-30/master_annotation_GM12878_hg19.csv` (`cwt_phase` col) | per-bin phase at fundamental scale |
| K562 phase map at all 3 harmonic scales | `harmonic_synthesis_2026-05-01/test5_tsa_harmonic_scale.csv` (`phi_T1`, `phi_T1div3`, `phi_T1div5`) | per-bin phase at L1, L2, L3 in K562 hg38 |
| GM12878 phase map at all 3 scales | **does not exist yet** — needs to be computed by re-running the harmonic decomposition pipeline on GM12878 axis | required for the actual test |

**Decision point 1 (D1):** computing GM12878 multi-scale phase. Two options:
- **D1.A**: rerun the consecutive-peak CWT + multi-scale harmonic extractor on GM12878 (~30 min compute, but follows the same pipeline as K562).
- **D1.B**: use K562 phase as a proxy and lift to hg19. Cheaper but introduces phase-map mismatch on GM12878 cells. Not preferred for a Wolf-grade analysis.

I'll go with D1.A.

## Per-cell pipeline

For each Dip-C cell:

1. **Build cell-specific speckle–lamina axis (ẑ).** Compute centroid c of all beads.
   For each bead, define radial vector r⃗ = (x, y, z) − c. Compute the principal
   axis â of the bead positions weighted by cos(φ_L1) (the fundamental-scale phase):
   â = argmax_{|v|=1} Σ cos(φ_L1)(r⃗ · v). The +â direction is the speckle pole
   (cos(φ_L1) > 0 ↔ A ↔ speckle-proximal under the manuscript convention).
   This avoids circularity: the phase coordinate is from bulk DNase-seq, the axis
   alignment is from per-cell 3D positions; they share no information.

2. **Cylindrical coordinates.** Re-express each bead's 3D position in the cell-specific
   frame as (r, θ, z), where:
   - z = r⃗ · â  (axial, range ≈ −1.5 to +1.5 in normalised nuclear units)
   - r = |r⃗ − z·â|  (radial, range ≈ 0 to 1.5)
   - θ = atan2(r⃗_⊥,y, r⃗_⊥,x) using a perpendicular-frame basis from the second and third
     PCA axes of bead positions

3. **Harmonic-resolved phase per bead.** For each bead at chromosomal position p,
   look up the GM12878 phase map at the three harmonic scales: φ_L1(p), φ_L2(p), φ_L3(p).
   This yields three (r, θ, z, φ) sets per bead.

4. **Per-scale, per-cell statistics.** For each scale L ∈ {L1, L2, L3} and each cell:
   - **Radial-accordion R²**: fit r/r̄ = a + b·cos(φ + ψ) → R²_radial(L)
   - **Slinky R²**: fit z/ẑ = a' + b'·cos(φ + ψ') → R²_slinky(L)
   - **Helix circular-linear correlation r_cl(θ, φ)** (Mardia & Jupp 2000):
     r_cl² = [r²(cos θ, φ) + r²(sin θ, φ) − 2·r(cos θ, φ)·r(sin θ, φ)·r(cos θ, sin θ)]
              / (1 − r²(cos θ, sin θ))
     where r are linear–linear Pearson correlations on the embedded (cos, sin)
     decomposition. r_cl ∈ [0, 1] tests whether θ advances linearly with φ.
   - **Per-cell winding number ω_L = (Σ_chrom dθ_L,chrom) / (2π · n_chrom)**:
     mean number of θ-revolutions per φ-cycle within a chromosome. ω_L ≈ 0
     for pure radial accordion; ω_L ≈ ±1 for a single-turn helix.

5. **Across-scale consistency tests.** A scale-invariant helix predicts ω_L1 ≈ ω_L2 ≈ ω_L3
   (winding propagates to harmonics with the same chirality). A scale-dependent or
   purely radial-accordion structure predicts no such invariance.

6. **Null distribution per scale.** For each cell × scale, permute φ within
   each chromosome (preserves marginal phase distribution and chromosome boundaries
   but breaks position-specific phase association) and recompute the four
   diagnostics. Bootstrap CI for r_cl and ω_L from 1,000 permutations.

7. **Power analysis.** A 250-kb-equivalent per-bead positional uncertainty at L1
   wavelength ≈ 30 Mb gives positional resolution ≈ 1/120 of the wavelength, so
   the test should have power for r_cl > 0.10. At L2 and L3 the per-bead resolution
   becomes harder; we report the power floor honestly.

## Output

**One new figure** (provisional name `extended_data_figure_6b_helix_multiscale.png`):
4 panels at 12×9 inches:
- **a**: per-scale R²_radial (left) vs R²_slinky (centre) vs r_cl (right) for 14 cells,
  three sub-rows for L1, L2, L3. Box plot per panel; expected: R²_radial high at L1
  and (predicted) at L2/L3; r_cl low at all scales if no helix.
- **b**: cross-scale consistency scatter — ω_L1 vs ω_L2 (left), ω_L2 vs ω_L3 (right);
  if helix is real and scale-invariant, points cluster on the y = x line; if absent,
  points cluster near zero with no diagonal structure.
- **c**: per-cell radial accordion fit at L1 + L2 + L3 overlay (one example cell).
- **d**: null distribution of r_cl under per-chromosome phase permutation, with
  observed values overlaid.

**Plus one supporting CSV**:
`per_cell_per_scale_helix_diagnostics.csv` — 14 cells × 3 scales × {R²_radial,
R²_slinky, r_cl, ω_L, n_beads} per chromosome → 14 × 3 × 22 = 924 rows.

## Honest reporting up front

If the data favours the radial-accordion at all three scales: confirms the ED Fig 6
text claim and tightens it across scales (stronger result).
If r_cl emerges at one scale but not others: refines the structural claim and
likely belongs in the main text.
If the diagnostics are too noisy at L2/L3 because of Dip-C resolution: report it
as exploratory at higher harmonics with explicit power floor.

## Time estimate

- D1.A multi-scale GM12878 phase computation: ~45 min (one-off)
- Per-cell pipeline implementation: ~60 min
- Stats + plots: ~30 min
- Caption + figure draft + provenance: ~30 min

**Total: ~3 hours** of focused work, producing a figure that either strengthens or
refines the structural-picture claim. Real data only. No rng outside the
permutation null and bootstrap CI computations (legitimate uses).

## Decision point 2 (D2)

Once results are in: integrate into the manuscript as a new ED figure / Supp Fig,
or restructure ED Fig 6 around this analysis (replacing the polymer-simulation
panels a/b/c with the multi-scale Dip-C diagnostic). Wolf's call when the
analysis is in.

---
*Plan written 2026-05-02 11:15. Awaiting Wolf's go-ahead before execution.*
