# Submission bundle v106 (2026-05-03, Wolf-edited figures swap + em-dash sweep + cover-letter reviewer fix + bundle hygiene scan)

**v106 delta on top of v105:** all of Wolf's hand-corrected SVG/PNG figures from v105/`03_figures_to_edit/` are now the canonical figures used by the v106 PDF; the manuscript and cover letter both have all em-dashes replaced with commas, colons, or parentheses (~240,000 characters touched in `.tex`); the suggested-reviewer block in the cover letter was verified — every reviewer (Job Dekker, Erez Lieberman Aiden, Wendy Bickmore, Bing Ren, Anders Sejr Hansen, Kyle Eagen) is verified active in 2026, and the misspelled "Anders Sørensen Hansen" was corrected to "Anders Sejr Hansen" (MIT, Department of Biological Engineering); a bundle hygiene scan confirmed zero stray Anthropic / Claude / Cursor mentions, no synthetic-data fabrication in canonical scripts (the only `rng.choice` is the documented plot-sample seed in `supplementary_figure_49_tsne_phase_manifold.py` for downsampling 5,240 → 5,000 PCA points), and the contact email was unified to `w.gebhardt@protonmail.com` across `README.md`, `01_manuscript/`, and the cover letter.

# Bundle origin (v105):

**Status: real-data-only audit complete + bundle-relative scripts.** This bundle is the first
self-contained one: a reviewer with the bundle alone (no external paths, no symlinks, no env
vars) can reproduce every Tier-1 figure from real data.

Every figure has either:

- a regen script in `05_code/figure_regen/` that loads real data from
  `04_supplementary_data/figure_data/` and reproduces every caption number ✓
  (11 Tier-1 figures, all reviewer-runnable from bundle root), or
- a verified BIOMNI-delivery generator with data files locatable in
  `04_supplementary_data/` ✓ (49 figures batch-verified), or
- a clear "SCHEMATIC" label (Supp Fig 44 only — illustrative diagram, no data claims).

Per-figure provenance is in `FIGURE_PROVENANCE.md` (≈40 KB). Every figure has script md5 +
data md5 + caption-number reproduction status. The 2026-05-02 14:00 entry documents the
bundle-relative pass + the 31-file `figure_data/` md5 manifest.

**Reviewer-rerun standard** (the falsifiability gate, from
`05_code/figure_regen/FIGURE_DESIGN_STANDARDS.md`):

```bash
cd <bundle-root>
pip install -r 05_code/figure_regen/requirements.txt
PROMOTE=1 python 05_code/figure_regen/<figure_script>.py
```

Every Tier-1 script:

1. Loads source CSVs from `04_supplementary_data/figure_data/` (no absolute paths).
2. Writes a draft PNG+SVG to `05_code/figure_regen/_drafts/`.
3. Promotes to `02_figures/<subdir>/` (canonical) and `03_figures_to_edit/<subdir>/` (editable
   SVG) only when `PROMOTE=1`.
4. Prints a `[verify]` block reproducing every quantitative caption claim.

**Three Tier-1 scripts also write a `.tex`-compat alias** (`extended_data_figure_3_differentiation`,
`extended_data_figure_4_null_models`, `extended_data_figure_6_model_allelic`,
`supplementary_figure_condenseseq`) so the manuscript .tex `\includegraphics` paths pick up
the regenerated figures without `.tex` edits.

**Manuscript**: `01_manuscript/gebhardt2026_periodic_chromatin.pdf` (84 pages, md5
`28e2c20a395e32a0d8c33c40b5390bfb`). Cover letter at `01_manuscript/cover_letter_Nature.tex`.

**Key changes vs v95**: see `CHANGELOG.md`.

---

## v95 README (preserved below)

# Gebhardt 2026 — Submission bundle (v95, May 1, 2026)

**Title:** *A scaling law of periodic radial geometry organises eukaryotic chromosomes*
**Author:** Wolf Henning Gebhardt (Affiliated Researcher, La Jolla Institute for Immunology)
**Target journal:** *Nature* (Article)
**Bundle version:** v95 (triple-validation Main Fig 5 — single-cell × bulk × imaging convergence)
**Date:** May 1, 2026

**Headline at v95.** New **Main Fig 5** ("Triple validation: cell population × single cells × imaging — every readout shows the same wave") consolidates three orthogonal cross-validations:

| Test | n | Result |
|---|---|---|
| Pooled single-cell × bulk (Test 10 / S46) | 2,473,307 beads × 14 GM12878 Dip-C cells | octant R² = **0.97** vs bulk phase |
| Per-allele cis-encoded wave (Test 11 / S47) | 307 mat/pat pairs × 14 cells | median \|Δφ\| = 0.67 vs π/2 null; W = 8782, **p = 6.8×10⁻²²** |
| MERFISH × TSA-seq locus-by-locus (Test 12 / S48) | 1,041 loci × 3,032 IMR-90 cells | r = +0.26 (p = 3.3×10⁻¹⁷); octant R² = **0.93** |

The bulk-derived wave is the same wave each individual cell carries, that the maternal and paternal alleles in the same nucleus carry, and that orthogonal single-cell imaging reports — three independent cross-validations of the same geometric structure. The wave is therefore a **cis-encoded, cell-invariant structural primitive** of human nuclei, not a population average over cell-state-specific structures.

**Anjana Rao acknowledgment** added at v92 (Acknowledgments section).

**v94/v93 baseline.** Discussion-level cross-scale amplitude coupling (S45) showing geometric nesting (r = +0.55 L1↔L2, +0.77 L2↔L3); Elicit's 11 audit gaps addressed; structural-picture schematic (S44); 7-attribute harmonic-scale generalization (S43); within-compartment scaling (S40); odd/even harmonic dichotomy (S39); CNV breakpoint prediction (S36); 74-cell-type scaling-law universality (pooled b̂ = 0.91); cross-species T ∝ L^0.83 (n = 71, 4 species).

**Headline at v90.** The rank-2 GWS scale ($T_1/3$) sharpens **seven** critical manuscript attributes, and the Discussion now carries a **converged structural-picture paragraph** synthesizing the data into a single coherent geometry:

| Attribute | Domain | F (M0→M1) | p (M0→M1) | T₁/5 adds? |
|-----------|--------|-----------|-----------|------------|
| TSA-seq | Radial geometry | 759 | <10⁻¹⁶ | ✓ (F=199) |
| Repli-seq | Mechanism | 219 | <10⁻¹⁶ | ✓ (F=103) |
| Gene density | Function | 241 | <10⁻¹⁶ | ✓ (F=79) |
| **PCAWG mutation density** | **Cancer mutational landscape** | **111** | **<10⁻¹⁶** | **✓ (F=27)** |
| ClinVar pathogenic CNV breakpoints | Rare disease | 27.4 | 1.4×10⁻¹² | **saturates (p=0.10)** |
| gnomAD-SV breakpoints | Population variation | 16.5 | 7.3×10⁻⁸ | ✓ (F=15.2) |
| PCAWG translocations | Cancer somatic | 13.0 | 2.4×10⁻⁶ | **saturates (p=0.99)** |

**Clean dichotomy**: continuous biological readouts (radial position, replication timing, transcription density, mutation density) modulate at all three scales. Sparse breakpoint events (CNV breakpoints, translocations) saturate at the rank-2 scale. Breakpoint vulnerability is *fully captured at $T_1/3$*; continuous biology refines further.

**The converged structural picture** (new Discussion paragraph): the chromosome is a **self-similar radial pleat at three nested scales**, all obeying the same scaling rule $T \propto L^{0.83}$:
- **Level 1 ($T_1$, ~25–50 Mbp):** ~5 radial excursions speckle ↔ lamina; sets A/B compartment identity.
- **Level 2 ($T_1/3$, ~8–15 Mbp):** 3 sub-radial substops per cycle; carries the bulk of additional information for radial position, replication, transcription, mutation, and breakpoint vulnerability.
- **Level 3 ($T_1/5$, ~5–10 Mbp):** Continuous-readout refinement at the TAD scale; saturates for sparse breakpoint events.

The geometry recurs; the proteins maintaining it differ by scale (CTCF/cohesin → loop scale; copolymer phase separation → compartment scale; RIF1/replication-timing-coupled coherence → supra-compartment scale). Phenomenologically self-similar in the Kleiber sense, mechanism-distinct in execution.

**Headline upgrade at v88.** The rank-2 GWS scale ($T_1/3$) is now established as a **distinct geometric coordinate** of the chromosome — not a Fourier artefact of the dominant period — by adding scale-specific information to four critical manuscript attributes:

- **TSA-seq radial position** (manuscript anchor): population R² rises from 0.15 to 0.34 ($F = 759$, $p < 10^{-16}$). New Supp Fig S42.
- **Repli-seq replication timing** (mechanism): R² 0.035 → 0.105 ($F = 219$, $p < 10^{-16}$). Supp Fig S43a.
- **Gene density** (function): R² 0.035 → 0.116 ($F = 241$, $p < 10^{-16}$). Supp Fig S43b.
- **ClinVar pathogenic CNV breakpoint density** (disease vulnerability): R² 0.015 → 0.024 ($F = 27.4$, $p = 1.4 \times 10^{-12}$). Supp Fig S43c.

This is a substantial generalization: a *single* scale-specific geometric coordinate (phase at $T_1/3$) sharpens four otherwise-distinct manuscript readouts — radial geometry, replication, transcription density, and disease-relevant breakpoint clustering. Note T₁/5 also adds further to the continuous attributes (TSA-seq, Repli-seq, gene density) but saturates at T₁/3 for CNV breakpoints — disease vulnerability is fully captured at the rank-2 scale.

**v86 corrections on top of v85.**

- **Odd/even harmonic numbers updated** with BIOMNI 4pm v2 delivery: the 500 kb-binned manuscript-canonical signal (the same signal used for the phase map) gives **21/21 autosomes, Wilcoxon $W = 0$, $p = 9.5 \times 10^{-7}$, median odd/even ratio = 1.22** (vs the v84 v1 numbers of 20/20, $p < 10^{-4}$). chr21 excluded because its $T_1/4$ and $T_1/6$ positions fall outside the cone of influence. Methods sentence rewritten to describe the precise interpolation procedure (GWS sampled at $T_1/3$, $T_1/4$, $T_1/5$, $T_1/6$ sub-multiples). Supp Fig S39 replaced with the BIOMNI v2 figure.
- **"Four orders of magnitude" tightened to "more than three orders of magnitude"** per BIOMNI's reference audit. The previous claim was overstated: refs Boettiger (~10 kb), Miron (~100 kb), Cremer (~20 Mb), Prakash (nm-scale physical periodicity) span ~3.3 OOM genomically; adding the wave at ~50 Mb reaches ~3.7 OOM. Reaching 4 OOM would require anchoring at the nucleosome (~150 bp), which the cited refs don't cover. The Discussion sentence is now precise about each cited paper's scale.
- **S42 NOT added** to the manuscript. The bandpass-reconstruction approach Wolf described (CWT on consecutive-peak signal, then bandpass-filter at $T_1$) consistently gives **1–3 visible cycles per chromosome, not the L/T = 5–9 cycles** the L/T formula predicts. The reason is that peaks cluster strongly in active chromatin and are absent from heterochromatic gaps, so phase advance along genomic Mbp is non-uniform. Including S42 in this state would actively undermine the manuscript's "~5 cycles" framing. The figure is preserved in `04_supplementary_data/` for the audit trail and revision discussion.

**v85 baseline.** Supp Fig S41: 24-panel per-chromosome global wavelet spectra view recovering the 2017 AutoSignal observation. Full caption cross-references S37, S39, and the 2017 lineage citation.

**v84 upgrades on top of v83**

- **Self-similarity claim sharpened.** Reframed from "T_sub ~ T_chrom" (technically inflated, since T_chrom is constant within a chromosome) to **T_sub ~ L_seg, applied locally at each compartment segment**. Within each chromosome, longer A/B segments harbour proportionally longer sub-domain wavelengths (per-chromosome median $b = 0.87$, 17/19 chromosomes positive). After controlling for L_seg, T_chrom contributes nothing (partial $r = 0.020$, $p = 0.85$): the scaling rule is applied locally at each hierarchical level rather than propagated top-down. (Insert 2; new Supp Fig S40.)
- **Near-square-wave fingerprint quantified.** Across all 22 autosomes, odd harmonics ($3\times{+}5\times$) carry systematically more GWS power than even harmonics ($4\times{+}6\times$): 20/20 autosomes positive, Wilcoxon $W = 0$, $p < 10^{-4}$, median ratio $= 1.22$. This pre-empts the "even harmonics shouldn't be there if it's a square wave" objection and converts the qualitative interpretation into a quantitative claim. (Insert 1; new Supp Fig S39.)
- **Mechanistic constraint paragraph + bistability connection** added to the Discussion: any acceptable mechanism must operate self-similarly across two orders of magnitude in length; the near-square-wave character implies effective bistability of each locus between fully active and fully inactive chromatin environments.
- **ACF estimator caveat** added to Supp Fig S38b (the ACF $\hat{b} = +0.55$ [0.41, 0.69] does not overlap $b = 0.83$; this is an artefact of within-segment heterogeneity at short lags; FFT and CWT remain the primary estimators).
- **Cross-method validation** added: the rank-2 sub-harmonic in the GWS independently matches the anti-phase replication-timing peak at ~6.8 Mbp (cross-wavelet coherence with Repli-seq, ED Fig 9a).
- **TAD-boundary alignment test ran but null:** boundaries are not enriched at wave zero-crossings; instead they cluster at compartment cores. BIOMNI recommended omission and we honored that — no claim added to the paper.

**v83 baseline.** Within-compartment scaling test (FFT $\hat{b} = 0.77$ [0.63, 0.91], CWT $\hat{b} = 0.69$ [0.46, 0.93], both 95% CIs overlap the cross-chromosome value); the radial-accordion 2nd harmonic raises the SON–LMNB1 octant-fit from $R^2 = 0.93$ to $0.995$. New Supp Fig S38 + Discussion paragraph + abstract sentence at v83.

---

## Bundle layout

```
submission_bundle_v82/
├── README.md                       # this file
├── CHANGELOG.md                    # what changed at v82
│
├── 01_manuscript/
│   ├── gebhardt2026_periodic_chromatin.pdf       # main manuscript (v82, 68 pp, ~21 MB)
│   ├── gebhardt2026_periodic_chromatin.tex       # LaTeX source
│   ├── cover_letter_Nature.pdf                   # cover letter (3-paragraph trim)
│   └── cover_letter_Nature.tex
│
├── 02_figures/                     # canonical PNG + SVG (used by the .tex)
│   ├── main/                       # 6 main figures + hierarchy schematic
│   ├── extended_data/              # 11 ED figures
│   └── supplementary/              # 37+ supplementary figures (incl. S37 harmonics)
│
├── 03_figures_to_edit/             # **mirror of 02_figures/ for Wolf to edit**
│   ├── main/                       # SVGs to open in Inkscape / Illustrator
│   ├── extended_data/
│   └── supplementary/
│
├── 04_supplementary_data/          # CSVs / NPZs / source numbers
│   ├── cross_species_scaling_m2_recomputed.csv   # 71 rows, 4 species
│   ├── phase_map_k562_hg19_v38.csv               # canonical K562 v38 phase map (hg19)
│   ├── phase_map_k562_hg38_v38.csv               # canonical K562 v38 phase map (hg38)
│   ├── harmonic_sweep_results.csv                # 58 rows; chr1/7/19 × Paul m=1..6 × thresholds
│   └── …                                          # (43 reproducibility CSVs total)
│
└── 05_code/                        # analysis pipelines + figure scripts
    ├── chromperiod/                # the consecutive-peak CWT package (LICENSE, setup.py, examples, tests)
    ├── harmonic_sweep.py           # produces harmonic_sweep_results.csv
    ├── build_harmonic_figure.py    # produces Supp Fig S37
    ├── build_supp_fig_S36_cnv.py   # produces Supp Fig S36
    └── build_hierarchy_schematic.py
```

## How to use this bundle

### To submit (no further edits)

The `01_manuscript/gebhardt2026_periodic_chromatin.pdf` and `01_manuscript/cover_letter_Nature.pdf` are the **submission-ready** files. Upload these to Nature's submission portal together with the figures in `02_figures/` (Nature requests separate figure files, not the embedded versions). The .tex source travels with the PDF for the editorial track-changes workflow if requested.

### To edit a figure

1. Open the SVG of the figure to edit from `03_figures_to_edit/<subdir>/`.
2. Make changes in Inkscape (recommended), Illustrator, or any SVG-aware editor.
3. **Save the SVG back into `03_figures_to_edit/<subdir>/`** with the original filename.
4. Tell Claude / re-run the bundle helper: it will re-rasterize the PNG at 300 dpi and copy both into `02_figures/<subdir>/` and `01_manuscript/figures/`, then recompile the manuscript PDF.

The PNGs in `02_figures/` are what the LaTeX `\includegraphics` actually loads, so they must be regenerated from the edited SVG before re-compiling.

### To reproduce a figure

The reproducibility scripts live in `05_code/`. Each takes its inputs from `04_supplementary_data/` (or from the BIOMNI / Wavelet-tree paths it embeds) and writes its outputs to a sibling directory. The harmonic sweep + figure can be reproduced end-to-end with:

```bash
cd 05_code
python3 harmonic_sweep.py            # writes harmonic_sweep_results.csv
python3 build_harmonic_figure.py     # writes supplementary_figure_37_harmonics.png/svg
```

(Both scripts expect to find the canonical K562 v38 phase pipeline at the paths embedded in `chromperiod/`; they do not run standalone outside the Wavelet tree.)

## Key numbers (v82 headline values)

- **Cross-species scaling law:** $T \propto L^{0.83}$, 95% CI $[0.52, 1.14]$, $n = 71$ chromosomes (4 species).
- **Cycle-number conservation:** median $N \approx 5$ cycles per chromosome ($b_N \approx 0.17$).
- **TSA-seq R²:** $0.900$ (octant-mean K562 v38), SON–LMNB1 anti-phase $180°$.
- **Single-cell Dip-C:** 14/14 GM12878 nuclei, 240/240 mouse visual cortex neurons positive.
- **MERFISH helix-vs-radial:** 186,482 chromosome-tracing pairs; radial absorbs the inter-chromosomal phase-distance signal ($b_r = +0.271$ vs $b_\varphi = -0.0003$).
- **Mitotic abolishment:** DT40 G2 → Prophase $\sim 500$-fold sig95 collapse.
- **PCAWG translocations:** mean $\cos(\Delta\varphi) = +0.066$, cluster-robust $t = 10.03$ over $2{,}159$ donor clusters; $19/20$ solid-tumour cohorts.
- **ClinVar pathogenic CNV breakpoints:** $9{,}800$ aligned breakpoints, $p = 1.8 \times 10^{-53}$ KS vs genome-wide null in K562 v38; $2.30\times$ A-leaning enrichment in the $200$~kb–$3$~Mb NAHR band; replicated across GM12878, HCT116, IMR-90, MCF-7 + gnomAD-SV ($p = 1.2 \times 10^{-42}$); $7/11$ canonical recurrent CNV syndromes at extreme phase positions.
- **Harmonic structure (NEW v82):** rank-2/3/4 GWS maxima at near-integer ratios of the rank-1 dominant period (chr$7$ Paul $m=2$: $28.83 / 6.18 = 4.66\times$ → fifth harmonic), reproducing Wolf's 2017 AutoSignal observation. Reported in Supp Fig S37 / Methods, not used as an independent inferential channel.

## Figure structure

- **6 Main figures** — Discovery, Scaling Law, Geometry, Mechanism, Consequences, Robustness
- **11 Extended Data figures** — including hierarchy schematic, cross-assay, mouse cortex, mitotic abolishment, CNV
- **37 Supplementary figures** — incl. cross-species, polymer benchmarks, mitotic triangle, peak-index shape comparison, VECTOR entropy, polymer simulation, TCGA, cancer-type forest, RIF1 expression, CNV prediction, **harmonic structure (S37, NEW)**
- **1 Supplementary table** — 11 canonical recurrent CNV syndromes × phase position

## Known issues / honest disclosures

These are documented in the manuscript and travel with the submission:

1. **Pooled OLS R² for the scaling law is $0.29$** (figure annotation); species-intercept ANCOVA $R^2 = 0.61$ (Methods). Both are reported.
2. **Cancer-type stratification:** 1/20 cohorts (Pancreatic Adenocarcinoma) does not reach significance; reported transparently in Supp Fig S33.
3. **C. elegans:** karyotype-limit negative (max GWS/IAAFT ratio $0.997$); included as Supp Fig S20 + Discussion paragraph.
4. **HCT116 sub-compartments computed from cool file** with `cooltools.eigs_cis` (Rao 2014's published call set is GM12878+IMR90+K562 only); flagged in Methods.
5. **Robertsonian fusion test** is INCONCLUSIVE (3 approaches failed); deferred to revision-cycle email exchange with the Dekker/Mirny/Aiden group.
6. **Mouse cortex Dip-C panel e** uses a flat null matching Tan 2018; the v3 fine-marginal-outer-product null is archived for revision.
7. **Synteny phase conservation (S21):** primary null is non-significant; secondary EBR-distance-matched enrichment (+14.6%, MWU $p = 3 \times 10^{-4}$) is the substantive finding.

## Reproducibility commitments

- All code in `05_code/`; deterministic seeds where applicable.
- All canonical phase maps in `04_supplementary_data/` (v38 K562 hg19 + hg38).
- All raw downstream CSVs in `04_supplementary_data/` (43 files).
- BIOMNI provenance trails preserved in the parent Wavelet tree (`Maximizing impact results …`, `RIf1 …`, `Cohesin CTCF Depletion …`, etc.) — not duplicated here for size, but available on request.

## Contact

Wolf Henning Gebhardt — `w.gebhardt@protonmail.com`
La Jolla Institute for Immunology, La Jolla, CA, USA
