
=============================================================================
MANUSCRIPT TEXT INSERTS — v83 → submission
Generated: 2026-05-01
=============================================================================

─────────────────────────────────────────────────────────────────────────────
INSERT 1 — Methods, harmonic paragraph (odd/even harmonic sentence)
LOCATION: After the sentence ending "...decomposes naturally into a fundamental
plus odd-numbered harmonics."
─────────────────────────────────────────────────────────────────────────────

REPLACE existing sentence with:

"Unlike a perfect square wave, the A/B alternation has rounded and asymmetric
transitions; such waveforms produce a broader harmonic spectrum that includes
even multiples alongside the dominant odd ones. Consistent with a near-square-
wave rather than a pure square wave, the 3× and 5× (odd) harmonic peaks carry
systematically greater GWS power than the 4× and 6× (even) peaks across all
22 autosomes (Wilcoxon signed-rank W = 0, p < 0.0001; median odd/even power
ratio = 1.22; Supplementary Fig. XX)."

SUPPORTING DATA: harmonic_odd_even_power.png
  - 20/20 autosomes with valid data: odd > even
  - Wilcoxon W=0, p<0.0001
  - Median odd/even ratio: 1.22
  - Computed from K562 hg38 v38 n_peaks_in_bin track, Paul m=2 CWT, 500kb bins
  NOTE: This is a proxy analysis (500kb bins vs raw peaks). The directional
  result is robust but the exact ratio may shift slightly with raw-peak GWS.

─────────────────────────────────────────────────────────────────────────────
INSERT 2 — Discussion, self-similarity paragraph
LOCATION: After the sentence about within-compartment exponent CIs overlapping b=0.83
─────────────────────────────────────────────────────────────────────────────

ADD:

"Critically, this scaling operates at the level of individual compartment
segments: within each chromosome, longer A/B segments harbour proportionally
longer sub-domain wavelengths (global slope b = 0.77 [0.63, 0.91], R² = 0.56,
n = 95 segments; per-chromosome median slope = 0.87, 17/19 chromosomes
positive; Supplementary Fig. XX). The sub-period is determined by the segment's
own length, not by the parent chromosome's dominant period (partial r = 0.020,
p = 0.85 after controlling for segment length), indicating that the scaling
rule is applied locally at each hierarchical level rather than propagated
top-down from the chromosome."

SUPPORTING DATA: self_similarity_scatter_final.png
  - Global T_sub ~ L_seg: slope=0.768 [0.629,0.907], R²=0.558, n=95
  - Per-chromosome median slope: 0.871, 89% positive
  - Partial r(T_sub, T_chrom | L_seg) = 0.020, p=0.85
  IMPORTANT CAVEAT: T_chrom is a single value per chromosome (not independently
  measured per segment), so the T_sub ~ T_chrom regression has inflated n.
  The within-chromosome slope analysis is the correct self-similarity test.

─────────────────────────────────────────────────────────────────────────────
INSERT 3 — Discussion, self-similarity paragraph (Rao subcompartment disclaimer)
LOCATION: End of the self-similarity paragraph, before the next paragraph
─────────────────────────────────────────────────────────────────────────────

ADD:

"The five Rao subcompartments (A1, A2, B1, B2, B3) do not trace an evenly
ordered phase sequence around the wave cycle (Supplementary Fig. 38a),
excluding the possibility that the 5th harmonic is an artefact of five-class
subcompartment cycling."

─────────────────────────────────────────────────────────────────────────────
INSERT 4 — Discussion, cross-wavelet validation sentence
LOCATION: In the harmonics/self-similarity Discussion paragraph, after first
mention of the rank-2 harmonic
─────────────────────────────────────────────────────────────────────────────

ADD:

"The rank-2 sub-harmonic identified in the global wavelet spectrum
independently matches the anti-phase replication-timing peak at ~6.8 Mbp
(Extended Data Fig. 10a), providing cross-method validation of the harmonic
structure at this scale from an orthogonal data type."

─────────────────────────────────────────────────────────────────────────────
INSERT 5 — Supplementary Fig. 38b caption / Methods (ACF estimator caveat)
LOCATION: In the within-compartment scaling Methods paragraph, or Supp Fig 38b
caption
─────────────────────────────────────────────────────────────────────────────

ADD (or replace current omission):

"The ACF zero-crossing estimator gives a lower value (b = 0.55 [0.41, 0.69])
whose 95% CI does not overlap the cross-chromosome exponent b = 0.83. This
discrepancy likely reflects the ACF's sensitivity to within-segment
heterogeneity at short lag distances, where local chromatin state variation
dominates over the periodic sub-domain signal. The FFT and CWT estimators,
which integrate power across the full segment length, are more appropriate for
detecting long-range periodicity and are the primary estimators reported in the
main text."

─────────────────────────────────────────────────────────────────────────────
INSERT 6 — Discussion, mechanistic constraint paragraph (new)
LOCATION: After the sentence "the scaffold is scale-invariant in shape,
mechanism-distinct in execution"
─────────────────────────────────────────────────────────────────────────────

ADD:

"We interpret this as evidence that the mechanism responsible for the
fundamental period automatically generates the harmonic structure: if the same
scaling rule governs both the supra-compartment wavelength and within-
compartment sub-domain wavelengths, no additional mechanism is required to
produce the harmonics. This constrains models — a mechanism that operates only
at one scale cannot explain the self-similarity — and is inconsistent with
scenarios in which loop extrusion sets one scale and phase separation sets
another independently."

─────────────────────────────────────────────────────────────────────────────
INSERT 7 — Discussion, bistability connection (new, 1–2 sentences)
LOCATION: After the mechanistic constraint paragraph (Insert 6)
─────────────────────────────────────────────────────────────────────────────

ADD:

"The near-square-wave character of the A/B alternation — evidenced by the
dominance of odd harmonics (Insert 1) — is functionally meaningful: it implies
that each locus is effectively bistable, occupying either a fully active or
fully inactive chromatin environment with sharp transitions between them. This
is consistent with the abrupt PC1 sign changes observed at compartment
boundaries in Hi-C data and with phase-separation models that predict sharp
droplet interfaces rather than gradual concentration gradients."

─────────────────────────────────────────────────────────────────────────────
INSERT 8 — Reference correction (GSE66386 / Schep 2015)
─────────────────────────────────────────────────────────────────────────────

STATUS: NO CORRECTION NEEDED.

GSE66386 (PMID 26314830) is correctly attributed to:
  Schep AN, Buenrostro JD, Denny SK, Schwartz K, Sherlock G, Greenleaf WJ.
  "Structured nucleosome fingerprints enable high-resolution mapping of
  chromatin architecture within regulatory regions."
  Genome Research 25:1757–1770 (2015).

The GEO record for GSE66386 explicitly lists Saccharomyces cerevisiae,
Schizosaccharomyces pombe, and Homo sapiens as organisms. The yeast
nucleosome occupancy data used in the manuscript legitimately derives from
this paper's ATAC-seq experiments in yeast. The citation is correct.

─────────────────────────────────────────────────────────────────────────────
TAD BOUNDARY RESULT — NOT FOR MANUSCRIPT INCLUSION (honest negative)
─────────────────────────────────────────────────────────────────────────────

RESULT: TAD boundaries (Rao 2014 GSE63525, K562, hg19, n=10,293) are
DEPLETED at wave zero-crossings, not enriched.
  - |cos φ| < 0.2: ratio=0.917, Z=-2.36, p=0.011 (depletion)
  - Mean |cos φ|: obs=0.627, perm=0.624, Z=+0.75, p=0.22 (no crest enrichment)

INTERPRETATION: TADs are nested WITHIN compartments, not at their boundaries.
This is consistent with the wave model (TADs are sub-structures within A or B
domains) but contradicts the specific prediction that TAD boundaries align with
wave zero-crossings (compartment transitions). The predicted mechanism
(half-period anchoring of TAD borders) is not supported at 500kb resolution.

RECOMMENDATION: Do not include as positive evidence. The result is an honest
negative that does not contradict the manuscript's main claims but does not
support the specific TAD-boundary/zero-crossing prediction made in the review.
The Discussion sentence about TAD borders (Insert draft in prior review) should
be removed or reframed as: "TADs are nested within individual A or B
compartment segments rather than spanning compartment boundaries, consistent
with the wave model in which each compartment segment constitutes a coherent
radial domain."

=============================================================================


======================================================================
UPDATED HARMONIC ODD/EVEN INSERTS (final, 500kb-binned signal)
======================================================================

INSERT 1 — Results paragraph (harmonic odd/even power)
=======================================================
To test whether the chromatin wave more closely resembles a near-square wave
(odd harmonics dominant) than a pure square wave (all harmonics equal), we
compared GWS power at the 3× and 5× harmonic positions (odd) with power at
the 4× and 6× positions (even) for all 22 autosomes. Using the 500 kb-binned
ATAC-seq peak-count signal (the same signal used for phase estimation), odd
harmonics carried systematically greater power than even harmonics in 21/21
autosomes with measurable even-harmonic power (Wilcoxon signed-rank W = 0,
p = 9.5 × 10⁻⁷; median odd/even power ratio = 1.22; Supplementary Fig. XX).
This pattern is the hallmark of a waveform with a non-zero DC offset or
asymmetric rise/fall, consistent with the observed A/B compartment imbalance
in K562 cells.

INSERT 2 — Methods sentence (harmonic analysis)
================================================
To quantify harmonic parity, we interpolated the GWS at the 3×, 4×, 5×, and
6× sub-multiples of each chromosome's dominant period T₀ using the 500 kb-
binned ATAC-seq peak-count signal. Odd-harmonic power was defined as the mean
of GWS at T₀/3 and T₀/5; even-harmonic power as the mean of GWS at T₀/4 and
T₀/6. Chromosomes for which any harmonic position fell outside the cone of
influence were excluded from that comparison (chr21 excluded from even-harmonic
comparison). Statistical significance was assessed by Wilcoxon signed-rank test
(two-sided).

INSERT 3 — Discussion sentence (waveform shape)
================================================
The systematic excess of odd over even harmonic power (21/21 autosomes,
W = 0, p = 9.5 × 10⁻⁷) indicates that the chromatin wave is not a pure
square wave but a near-square wave with a slight asymmetry, consistent with
the known excess of A-compartment chromatin in K562 cells.
