Discrete mathematical forms of music — do they share the nucleosome's commensuration shape?¶

Research spike (2026-07-19; concertmaster dispatch). Scoping/derivation only — no code shipped, no rc, no ADR. FORM-matching only; this does not validate the framework and music theory is not superseded ([[user_stance_cascade_matching_substrate_blind_form_not_identity]], [[feedback_no_lineage_claims_in_notebook]]). Provisional throughout. Follows nucleosome_turn_asymmetry_frame_spike.md; companion to subharmonic_chirality_carrier_findings.md and ../../antikythera-maths/subharmonic_chirality_collapse_stub.md. Generating script for every number: music_discrete_forms_commensuration_shape_spike.py — exact integer/rational arithmetic, Class-N logs via srmech.amsc.rational.log1p_series_truncate, Class-K pin-slot for sign, no abs(), no float in any load-bearing result ([[feedback_computational_provenance_discipline]]).

0. Bottom line¶

The dispatch's central test resolves AGAINST the analogy as posed — and then finds a different one that survives.

The Pythagorean comma is an arithmetic non-closure (a theorem). The nucleosome's ~10.2-vs-10.5 detuning is a contingent one (a free physical parameter). They differ in kind, so comma ↔ nucleosome is decorative.

But music contains two structurally different non-closures, and the dispatch pointed at the wrong one. Music's second non-closure — string inharmonicity — is contingent in exactly the nucleosome's way. That match survives on structure, not on "both involve ratios."

pairing	verdict
Pythagorean comma ↔ nucleosome periodicity detuning	FAILS — arithmetic vs contingent
string inharmonicity ↔ nucleosome periodicity detuning	SURVIVES — both contingent detunings off an exact integer ideal
comma ↔ nucleosome dyad parity (146 bp)	SURVIVES — both arithmetic; but the DNA instance is a construct artifact
any of the above ↔ MFO's asymmetric-resonator comb	FAILS — §5; MFO's own exemplar is the one the predicate excludes

1. The predicate — and what it EXCLUDES¶

P(a) a discrete lattice generated by iterating a ratio · P(b) a continuous domain it must close within · P© an irreducible residual (closure fails) · P(d) the residual must be allocated by a policy underdetermined by (a)–©

Clause (d) needed sharpening. In its naive form ("a human chooses") it is anthropocentric and excludes DNA by definition. The load-bearing form is allocation-underdetermination: the closure requirement alone does not fix how the residual distributes, so an extra degree of freedom must be fixed from outside the arithmetic. Under that reading the agent may be a tuner, a calendar committee, or a histone fold.

15 systems tested. 6 excluded, at four different clauses — the signature of a predicate doing real work rather than one clause carrying it. (Was 14/6 on first pass; the beat row split in two under the measured amendment of §1.1 — the exclusion count is unchanged, the member count went up by one.)

system	a	b	c	d	verdict
Pythagorean / syntonic comma	1	1	1	1	MEMBER — © arithmetic
maximal evenness / Euclidean rhythm	1	1	1	1	MEMBER — © arithmetic
calendar leap-year rules	1	1	1	1	MEMBER — © contingent
Antikythera gear-train ratios	1	1	1	1	MEMBER — © contingent
floating-point rounding modes	1	1	1	1	MEMBER — © arithmetic
string inharmonicity / Railsback	1	1	1	1	MEMBER — © contingent
coupled asymmetric pair (directed coupling)	1	1	1	1	MEMBER — © contingent; measured, see §1.1
nucleosome periodicity detuning	1	1	1	?	PARTIAL — (d) weak-form only
nucleosome dyad parity (146 bp)	1	1	1	1	MEMBER — construct artifact only
polyrhythm / tuplets	1	1	0	0	EXCLUDED at © — closes exactly at LCM
UNCOUPLED two-oscillator beat / moiré	1	1	1	0	EXCLUDED at (d) — nothing to allocate
bell / drum inharmonic spectrum	0	1	1	0	EXCLUDED at (a) — no ratio-lattice ideal
thermal equilibration	0	1	1	0	EXCLUDED at (d) — allocation is determined
statistical variance across particle classes	0	0	0	0	EXCLUDED — not a residual at all
MFO subharmonic comb (period-doubling)	0	1	0	0	EXCLUDED at (a),© — bifurcation

Verdict: DISCRIMINATING, not vacuous. Clause (d) is what kills "any system with two periodicities" — but the boundary sits one level in from where this note first drew it (§1.1): an uncoupled pair has a residual and nothing to allocate; a coupled one does. Clause (a) is second: it kills bells and drums, whose inharmonic spectra are not perturbations of any ratio-generated ideal.

1.1 Amendment — the beat row splits, on a measured result (fermata F-α)¶

The first pass excluded "generic two-oscillator beat / moiré" wholesale. That was too broad, and the correction comes from a measurement, not a preference.

Attribution: this result is not ours. It is K5 of the open-experiments spike — open_experiments_kuramoto_hexasome_spike.md §2.4 + its committed provenance script — run on srmech's own cascade.kuramoto_step. Verified against that note's §2.4 body and data table before citing.

With directed (non-reciprocal) coupling at α = 0, holding the sum A₁₂ + A₂₁ fixed at 2.0 and varying only the ratio, the locked phase offset is φ* = 0.25268 — identical to five decimals across every row — while the allocation runs 0.05 → 0.95, tracking A₁₂/(A₁₂+A₂₁) exactly. Closed form: the threshold depends only on the sum, the split only on the ratio. The lock threshold is blind to the allocation.

That is clause (d) verbatim in the strong sense: the residual is real, it closes a ledger, and how it splits between the two members is not determined by the closure. So:

case	status	why
uncoupled two-oscillator beat / moiré	still EXCLUDED at (d)	a residual exists, but there is nothing to allocate — the original reasoning stands for this case
coupled asymmetric pair (directed / non-reciprocal)	MEMBER	coupling creates the allocation degree of freedom, and directedness leaves it underdetermined by the closure

Coupling is what creates something to allocate. That is the whole content of the split — an uncoupled pair has two independent phases and no shared ledger; a coupled pair has one shared locked state whose allocation between the members is a free direction. Note the mechanism is directed coupling specifically: the source records the effect as arriving via non-reciprocity, not via the Sakaguchi α frustration parameter.

This does not inflate the verdict. Adding coupled asymmetric pairs makes the member class larger, not more exclusive — the headline stands unchanged below. This is generic commensuration-under-closure with a mechanism attached, not a signature.

But the predicate does not single out music + DNA. Calendars, gear-trains and floating-point rounding are full members. The shape is a general commensuration-under-closure shape, not a signature of anything cosmic. Reported plainly, because the dispatch asked what the predicate rules out and the honest answer includes ruling out its own specialness.

⚠️ Equivocation caught. The nucleosome spike's "the honest object is a DISTRIBUTION, not a constant" uses distribution in the statistical sense (variance across particle classes). Clause (d) uses it in the allocation sense (spreading one residual over several slots). These are different objects and the word bridges them illegitimately. Row 13 above excludes the statistical reading at every clause. The two senses must not be traded on.

2. Q2 — the discriminator. ARITHMETIC vs CONTINGENT: they differ in kind.¶

2.1 Music's non-closure is a theorem¶

Lemma (rational-closure exclusion). Let r = p/q in lowest terms, r > 1. If rⁿ = 2ᵐ for positive integers n, m, then q = 1 and p is a power of 2.

Proof. rⁿ = 2ᵐ ⇒ pⁿ = 2ᵐqⁿ. Since gcd(p,q) = 1, gcd(pⁿ,qⁿ) = 1; yet qⁿ divides pⁿ, forcing q = 1. Then pⁿ = 2ᵐ, so p is a power of 2 by unique factorisation. ∎

Corollary 1. The just fifth 3/2 never closes the octave — not for 12 fifths, not for any n.
Corollary 2. Any equal division of the octave into n > 1 parts uses an irrational step 2^(1/n). Rational intervals and octave closure are mutually exclusive.

The failure is prohibited by unique factorisation. No parameter can be adjusted to remove it. Computational witness: no 3^a is a power of 2 for a ∈ 1..399 (and the lemma says none ever is).

Attribution — read before citing. The fact is attested: Scholtz 1998 endnote 8 states that (3/2)ⁿ "can never be an exact multiple of 2 since every power of 3 is an odd number" [E]. That is a parity argument, and it covers the fifth only. The UFD/coprimality framing above, and the generalisation to arbitrary r = p/q with its two corollaries, are OURS — Scholtz does not frame it that way and himself falls back on an "intuitive musical proof" for the fourths case. No number-theory-grade OA source for the general statement was located [NULL]. Flagged so the lemma is not mistaken for a cited result.

2.2 The nucleosome's is not¶

The coherency condition is N/h = k. Solving for h gives h = N/k = 147/14 = 21/2 = 10.5 — and this is solvable for any integers N, k, because h is a free real parameter. There is no coprimality argument, no unique-factorisation argument, nothing that forbids hₛ = 10.5. The measured surface periodicity simply is not 10.5.

⇒ The two non-closures differ IN KIND. The comma ↔ nucleosome analogy is decorative.

2.3 …but the nucleosome DOES have an arithmetic non-closure — on a different member¶

A 2-fold dyad axis passing through a base pair requires an odd bp count. That is a ℤ/2 parity obstruction — arithmetic, and canonically Class K (pin-slot / phase boundary).

Luger 1997 used a 146 bp (even) palindrome; the particle could not satisfy the parity condition and absorbed the 1 bp deficit by stretching — i.e. an arithmetic impossibility forced a residual that had to be distributed through the structure [attested prior spike, PMC4378457].

That is a genuine clause-(a)–(d) instance and it is type-1 (arithmetic), matching the comma. But it is a crystallography construct artifact, not the native particle. Flagged; not rested on.

So the arithmetic non-closure in the nucleosome exists — it just isn't the one the dispatch was looking at.

2.4 Convergence — the instrument's reach reproduces the same split, by a different route¶

Worth recording because it arrives independently. The open-experiments spike reports that sinusoidal Kuramoto has exactly one resonance — 1:1 — with no higher-order p:q Arnold tongue at any coupling, because higher tongues need harmonics the model does not carry [open_experiments_kuramoto_hexasome_spike.md §2.6 / N4].

Line that up with §§2.1–2.2:

	commensuration	inside the instrument's reach?
nucleosome	1:1 — one helical turn per contact (N/h₀ = 14 turns over 14 contacts)	yes — it is the one resonance the model has
music's comma	12:7 — twelve fifths against seven octaves	no — needs a higher-order tongue the model cannot reach

So a reach limit of the instrument, derived from the model's harmonic content, lands on exactly the same boundary as the §2.1 unique-factorisation argument — the contingent 1:1 detuning on one side, the arithmetic high-order comma on the other. Two independent routes to one split.

Stated with its limit: this is a convergence, not a proof. A sinusoidal model's reach is a fact about the model, and the p:q structure of the comma is a fact about the integers; that they partition the same way is corroboration that the partition is real, not a second derivation of it. It does mean the framework's own instrument would have found the boundary had the arithmetic not.

3. The enumeration — formalism → lattice → domain → residual → policy¶

Comma ratios and cents are exact (theorems; no citation needed). Historical/empirical rows carry attestation status per §6.

formalism	discrete lattice	continuous domain	residual	distribution policy	non-closure kind	verdict
Pythagorean comma	⟨3/2⟩ iterated 12×	pitch (log-freq circle)	531441/524288 = 23.4600 ¢	— (untempered: dumped in one wolf fifth)	arithmetic	MEMBER
syntonic comma	⟨3/2⟩⁴ vs 5/4	pitch	81/80 = 21.5063 ¢	meantone: ¼ each over 4 fifths	arithmetic	MEMBER
schisma	PC ÷ SC	pitch	32805/32768 = 1.9537 ¢	absorbed / ignored	arithmetic	MEMBER (degenerate)
diaschisma	—	pitch	2048/2025 = 19.5526 ¢	—	arithmetic	MEMBER
lesser diesis	—	pitch	128/125 = 41.0589 ¢	—	arithmetic	MEMBER
greater diesis	—	pitch	648/625 = 62.5651 ¢	—	arithmetic	MEMBER
equal temperament (12-EDO)	ℤ/12	pitch	PC spread evenly	uniform, 1.9550 ¢/fifth = PC/12 [C]	resolves (a) by leaving ℚ	MEMBER
¼-comma meantone	4-fifth chain	pitch	SC over 4 fifths	concentrated, 5.3766 ¢/fifth; consonant major thirds [C]	"	MEMBER
well-temperament	12 unequal fifths	pitch	PC unevenly	deliberately unequal, commas "divided and dispersed" [C]	"	MEMBER
just intonation	small-d rational lattice	pitch	non-closure accepted	none — residual kept visible	arithmetic	MEMBER
Tonnetz / prime lattice	ℤᵏ over primes (monzo)	pitch	comma vectors = lattice kernel	tempering = quotient by the kernel	arithmetic	MEMBER
overtone vs tempered scale	ℕ·f₀ vs 2^(k/12)	frequency	per-degree error	absorbed by ear/context	mixed	MEMBER
string inharmonicity	ℕ·f₀ (ideal)	frequency	f_n = n·f₀√(1+B n²), B ≈ 10⁻³ [A]	— (physical)	contingent (B→0 removes it)	MEMBER
Railsback stretch	tempered keyboard	frequency	inharmonicity accumulated	measured & applied: "stretching" [B]; octaves larger than nominal [A]	contingent	MEMBER
beat frequencies (uncoupled)	two partials	frequency	\|f₁−f₂\|	none — nothing to allocate	n/a	EXCLUDED at (d)
coupled asymmetric pair	two phases, directed A₁₂ ≠ A₂₁	phase circle	locked φ* with a free split	allocation = A₁₂/(A₁₂+A₂₁), threshold blind to it	contingent	MEMBER — §1.1
polyrhythm / tuplets	p:q pulses	time	zero — closes at LCM	none needed	n/a	NULL at ©
maximal evenness / Euclidean	k onsets in n pulses, k∤n	time (cyclic)	gaps cannot all be equal	maximally even (Bjorklund)	arithmetic	MEMBER
nucleosome periodicity	14 contacts	bp / turns	ΔØ = 7/17 turns	weak-form only	contingent	PARTIAL
nucleosome dyad parity	ℤ/2 on bp count	bp	1 bp	stretched through structure	arithmetic	MEMBER (artifact)

3.1 The sharpest structural point: JI and ET are dual, not competing approximations¶

By the §2.1 lemma you may have rational intervals or octave closure, never both:

just intonation = keep ratios rational, accept non-closure
equal temperament = accept irrational ratios, achieve exact closure

Temperament is therefore not "a better best_rational" — it is the deliberate move off the rational lattice that best_rational exists to stay on. Same object, opposite direction. This is the cleanest thing the spike found and it is a theorem, not a reading.

3.2 Numerology hazard, checked and defused¶

PC/12 = 1.955001 ¢ vs schisma = 1.953721 ¢ — differ by 0.001280 ¢. Near-equal to ~10⁻³ cents and not the same number: the schisma is the rational 32805/32768 (monzo 2⁻¹⁵3⁺⁸5⁺¹); PC/12 is not a rational interval at all. Recorded as a coincidence, explicitly NOT used as evidence.

4. Q3 — does srmech already express it? The JI half: yes, verbatim. The temperament half: no, and it should not ship as music.¶

Exact continued fraction of log₂(3/2), computed by integer power comparison only (no floats):

[0, 1, 1, 2, 2, 3, 1, 5, 2, 23, 2, 2]

Its convergents are the historically-used equal divisions of the octave:

convergent	EDO	tempered fifth	error vs just 3/2
⅗	5	720.0000 ¢	+18.0450 ¢
7/12	12	700.0000 ¢	−1.9550 ¢
24/41	41	702.4390 ¢	+0.4840 ¢
31/53	53	701.8868 ¢	−0.0682 ¢
179/306	306	701.9608 ¢	+0.0058 ¢

And srmech's shipped Class-N op reproduces them directly:

best_rational(log2(3/2), max_d=  5) = 3/5     ->   5-EDO
best_rational(log2(3/2), max_d= 12) = 7/12    ->  12-EDO
best_rational(log2(3/2), max_d= 60) = 31/53   ->  53-EDO
best_rational(log2(3/2), max_d=400) = 179/306 -> 306-EDO

Choosing a temperament IS choosing max_denominator. That is Class N verbatim. Class I (ℤ/12 pitch-class / octave group) covers the cyclic side, consistent with the Z₁₂/D₁₂ reading already in audio-scoping-2026-05-09.md.

Recommendation — do NOT ship a music/tuning catalog.

The JI half is redundant. best_rational + continued_fraction already are the just-intonation problem. A tuning catalog would re-wrap a shipped op.
The temperament half is genuinely absent but is not music. Tempering out a comma = quotienting the prime-exponent lattice ℤᵏ by the sublattice generated by the comma vectors, then choosing a section. That is integer linear algebra (Smith normal form / kernel), squarely in srmech's integer-ALU idiom and adjacent to Class L. If it ships it should ship as the general lattice-quotient + residual-allocation op, with music as one worked instance — not as a music op-family. Shipping it as "tuning" would privilege a substrate, against [[feedback_no_privileged_primitive_classes]].
Surface note (observed this run, not a defect): best_rational is uint64-bounded on both inputs, so the exact Class-N rational for log₂(3/2) (a ~90-digit numerator out of log1p_series_truncate) is rejected by _ensure_uint64. Any use of Class-N best_rational on a transcendental needs an explicit pre-truncation step. Worth documenting if a worked instance ever ships.

Fermata F-i — conductor decides whether the lattice-quotient op is worth a tracker item at all.

5. Q4 — the MFO tie. Checked, not assumed — and it does NOT triangulate.¶

MFO's actual claim (notebook head, and §XIV.8):

"All matter and force fields are inharmonic and subharmonic excitations of a single metric field. An asymmetric resonator."

"a body resonator is asymmetric and inharmonic / subharmonic (a cymbal's overtones are not an integer harmonic ladder…)"

Three findings, and they cut against the bridge:

MFO has already decoupled the two words itself. §XIV.8 measured the comb from a logistic map and recorded the honest correction: "the comb comes from nonlinearity, not from spatial asymmetry." So "asymmetric universal resonator" bundles two signatures MFO's own notebook has separated. The bridge must be checked per-signature; bundled, it is a phrase, not a claim.
MFO's measured signature is a bifurcation cascade, not a detuning. The comb is period-doubling (f/2, f/4, f/8, f/16) with Feigenbaum δ ≈ 4.669. That is not a commensuration residual at all — there is no ratio-generated lattice failing to close. It fails predicate clauses (a) and ©.
MFO's chosen exemplar is precisely the one the predicate excludes. A cymbal/bell spectrum is inharmonic in the no-nearby-integer-ideal sense; a piano string and the nucleosome are detuned in the small-perturbation-off-an-exact-integer-ideal sense. Opposite ends of the same word. In-tree corroboration already exists: Spike #40 measured bell as geometric-decay and drum as Bessel-zero frequencies — neither is a perturbation of ℕ·f₀.

This answers the open fermata F-g. The nucleosome spike's fermata-execution pass logged the user framing — "DNA carries both continuous and discrete domain mathematics; read as a physical instantiation of MFO's asymmetric universal resonator" — MFO-side, recorded but not adjudicated, with two guard-rails: it must not retro-justify S1/S2, and it does not soften anomaly A1. This spike adjudicates it: the reading does not go through by commensuration (§§2, 5). Both guard-rails are observed — nothing here retro-justifies S1/S2, and A1 (the beat term under-determined to the width of what it explains) is untouched and if anything reinforced, since §2.2 shows the detuning is a free parameter.

Q4 VERDICT: it is NOT a three-way triangulation. It is a genuine two-way form-match (nucleosome ↔ string inharmonicity) plus a third thing wearing the same word. The dispatch asked whether DNA ↔ music ↔ cosmos genuinely triangulates or is "three different things wearing one word" — the answer is two of the three match; the cosmos member does not join via commensuration.

This does not refute MFO. It says the bridge the user proposed does not go through by this route, and names the route that does (§2.3, §5 above). Per [[feedback_mfo_vs_srmech_notebook_split_rule]], the ontology stays MFO-side; §§2–4 (lemma, lattice, Class-N) are srmech-side. Fermata F-ii.

6. Attestation ledger¶

Exact / no citation required (arithmetic theorems, reproduced in the script): all comma ratios and monzos · all cents values · the §2.1 lemma and both corollaries · the continued fraction of log₂(3/2) and its convergents · the best_rational outputs · maximal-evenness onset sets · LCM closure of polyrhythms · the nucleosome parity argument.

Inherited attested (prior spike, re-used not re-derived): 14 minor-groove contacts [PMC4512544] · hₛ ≈ 10.2 / h₀ ≈ 10.5 and ΔØ ≈ +0.4 [PMC6162219] · the 146-vs-147 resolution and the stretch [PMC4378457] · MFO §XIV.8 Feigenbaum comb [in-tree].

ATTESTED OA, full text fetched this spike:

tag	claim	source
[A]	stiff-string law f_n = n·f₀√(1+Bn²); partials run sharp; B ≈ 10⁻³ typical for piano; "octaves slightly larger than should"	Gràcia & Sanz-Perela, The wave equation for stiff strings and piano tuning, arXiv:1603.05516v2 (UPC; Reports@SCM 3, 2017) — verbatim: "the frequency spectrum is no longer harmonic, but of the form fn = n f √(1 + Bn²)"; "For piano strings the value of the inharmonicity parameter B is about 10⁻³." Corroborated: Roy, Edinburgh Student J. Sci., DOI 10.2218/esjs.9815 (OA, 2024)
[B]	Railsback 1938; tuners deviate from ET ("stretching"); caused by inharmonicity	Hinrichsen, Entropy-based Tuning of Musical Instruments, arXiv:1203.5101v1 — verbatim: "first explained by O. L. Railsback in 1938, who showed that this perception is caused by inharmonic corrections in the overtone spectrum… Professional aural tuners compensate this inharmonicity by small deviations, a technique known as stretching."
[C]	ET flattens fifths by 1/12 Pythagorean comma; meantone divides the syntonic comma over four links; well-temperament distributes deliberately unequally	Scholtz, Algorithms for Mapping Diatonic Keyboard Tunings and Temperaments, Music Theory Online 4.4 (1998) (peer-reviewed OA musicology) — verbatim: "Equal temperament flattens the fifths… by 1/12 of a Pythagorean comma"; "equally dividing the syntonic comma over the four links of each major third"; "The more that the commas were divided and dispersed, the more that well temperament approached equal temperament."
[D]	maximal evenness; Bjorklund ≡ Euclidean algorithm; k∤n is the hard case; 7-of-12 is the diatonic pattern	Toussaint, The Euclidean Algorithm Generates Traditional Musical Rhythms, BRIDGES 2005 extended version (McGill-hosted, author's own copy) — verbatim: "If k divides evenly… the solution is obvious… The solution is less obvious when k and n are relatively prime"; "it is the same pattern as the pitch pattern of the major diatonic scale."
[E]	no sequence of just fifths ever closes an octave	Scholtz, MTO 4.4 (1998) endnote 8 — verbatim: "which can never be an exact multiple of 2 since every power of 3 is an odd number."

REJECTED (paywalled-only ⇒ not attestation): Railsback, Scale Temperament as Applied to Piano Tuning, JASA 1938 (the primary) · Explaining the Railsback stretch…, JASA 138(4):2359 (2015) — HTTP 403 · Clough & Douthett, Maximally even sets, J. Music Theory 35 (1991) — reference-only inside [D], not fetched; do not quote their words · Fletcher, Blackham & Stratton, JASA 1962.

NULL — could not attest (do NOT write these): - The Railsback stretch magnitude in cents. Every trail terminated at a Wikipedia image file (Hinrichsen's own Fig. 1 credit line is en.wikipedia.org/wiki/File:Railsback2.png). The "~30 cents" figure in circulation has no fetchable source. ⚠️ This is the same failure mode as the prior spike's A3 — a number that looks sourced but dissolves on fetch. - The phrase "treble sharp / bass flat." Not present in any fetched OA text. Only the aggregate "octaves slightly larger" is attested. The note's §3 row was corrected accordingly. - "Key colour" / "key character" as a sourced term — absent from [C]. The unequal distribution is attested; the name for its consequence is not. Corrected in §3. - "Meantone gives PURE major thirds" — [C] says "consonant", and reserves "pure" for just intonation. Corrected in §3. - Per-string measured B values — Roy's results are locked in a figure, not a table. - A UFD-grade source for the general closure lemma (§2.1 attribution note).

⚠️ Attested-source conflict, flagged not resolved: [A] gives typical piano B ≈ 10⁻³; Hinrichsen [B] states B ranges "between 0.0002 for bass strings up to 0.4 for treble strings" — but his own figure axis tops out at 10⁻¹, below his quoted 0.4. Treat 0.4 as suspect; do not cite it. The script brackets B ∈ {10⁻⁴, 5×10⁻⁴, 10⁻³}, consistent with [A].

7. Diagram — the lattice-vs-continuum closure failure¶

  CONTINUOUS DOMAIN: the octave circle, one full turn = 1200 cents
  DISCRETE LATTICE:  iterate the just fifth 3/2 (= 701.955 cents), fold into the octave

   0¢ ┌────────────────────────────────────────────────────────────────┐ 1200¢
      │  the lattice generated by ⟨3/2⟩ is DENSE in this circle and     │
      │  NEVER lands on 0 again — by the §2.1 lemma, not by bad luck    │
      └────────────────────────────────────────────────────────────────┘
        ↑                                                            ↑
      start                                                    after 12 fifths
        │                                                            │
        └───────────────── 12 fifths = 7 octaves + 23.4600¢ ─────────┘
                                                    │
                                      PYTHAGOREAN COMMA — the residual
                                                    │
              ┌─────────────────────────────────────┼─────────────────────────────┐
              │                                     │                             │
        ┌─────▼─────┐                        ┌──────▼──────┐              ┌───────▼───────┐
        │  EQUAL    │                        │  MEANTONE   │              │     WELL-     │
        │ spread    │                        │ concentrate │              │  TEMPERAMENT  │
        │ evenly    │                        │ on 4 fifths │              │  deliberately │
        │ 1.955¢    │                        │  5.377¢     │              │   UNEQUAL     │
        │ ×12       │                        │  ×4         │              │  = key colour │
        └───────────┘                        └─────────────┘              └───────────────┘
              └────────────── clause (d): SAME residual, ALLOCATED differently ──────────┘

  ─────────────────────────────────────────────────────────────────────────────────────
  CONTRAST — the nucleosome (why the shape does NOT carry over):

     147 bp / (21/2 bp per turn) = 14.000 turns  ← EXACT. The lattice DOES close.
     147 bp / (51/5 bp per turn) = 14.412 turns  ← detuned, residual ΔØ = 7/17

     the residual exists only because the MEASURED h_s is 10.2 rather than 10.5.
     Nothing forbids h_s = 10.5.  h is a free real parameter.
              ⇒ CONTINGENT, not arithmetic. Different kind of failure.

graph TD
  subgraph ARITH["ARITHMETIC non-closure — no parameter can remove it"]
    A1["Pythagorean / syntonic comma<br/>3^12 ≠ 2^19 by unique factorisation"]
    A2["maximal evenness: k ∤ n<br/>gaps cannot all be equal"]
    A3["nucleosome DYAD PARITY<br/>Z/2 — odd bp required<br/>(construct artifact only)"]
  end
  subgraph CONT["CONTINGENT non-closure — a parameter that happens to differ"]
    C1["string inharmonicity<br/>f_n = n·f₀√(1+Bn²); B→0 removes it"]
    C2["nucleosome PERIODICITY<br/>h_s ≈ 10.2 vs h₀ = 10.5; h free"]
    C3["calendar / gear ratios"]
  end
  subgraph OUT["EXCLUDED by the predicate"]
    X1["bell / cymbal / drum<br/>no ratio-lattice ideal — clause (a)"]
    X2["MFO subharmonic comb<br/>bifurcation, not detuning — (a),(c)"]
    X3["generic beat / moiré<br/>nothing to allocate — clause (d)"]
  end
  A1 -. "SAME KIND" .-> A3
  C1 == "SAME KIND — the surviving match" ==> C2
  A1 -. "DIFFER IN KIND — analogy fails" .-x C2
  X2 -. "MFO's exemplar is here, not with C1/C2" .-> X1
  classDef ar fill:#eef,stroke:#446;
  classDef co fill:#efe,stroke:#464;
  classDef ou fill:#fee,stroke:#a44;
  class A1,A2,A3 ar;
  class C1,C2,C3 co;
  class X1,X2,X3 ou;

8. NULLs (first-class)¶

N1 — polyrhythm / tuplets. Every rational polyrhythm p:q closes exactly at LCM(p,q). Residual identically zero. There is no rhythmic comma. The dispatch's "same problem in rhythm" framing does not survive. (The time domain does have a member — maximal evenness — but by a divisibility obstruction, not a commensuration one.)
N2 — beat frequencies, uncoupled only. Excluded at clause (d): a residual exists but there is nothing to allocate and no policy. Amended — the coupled asymmetric case is a member, on a measured result (§1.1). The null now applies strictly to the uncoupled case.
N3 — "both involve ratios." The stated null hypothesis. Confirmed vacuous: it is satisfied by every row of §3 including the excluded ones.
N4 — the predicate's specialness. Excluded. Calendars, gear-trains and floating-point rounding are full members; the shape is generic commensuration-under-closure.
N5 — MFO three-way triangulation. Does not hold (§5).
N6 — PC/12 ≡ schisma. Coincidence to ~10⁻³ ¢, not an identity (§3.2).
N7 — the "distribution" equivocation. Statistical variance ≠ residual allocation (§1).
N8 — the Railsback stretch magnitude. The most-quoted number in this whole area ("~30 cents at the extremes") has no fetchable source; it traces to a Wikipedia image file. The curve is attested [B]; its magnitude is not. See §9 A3.
N9 — "key colour", "pure meantone thirds", "treble sharp / bass flat". Three phrases that read as standard and are not in any fetched OA source. All three were in this note's first draft and were removed. See §6.

9. Anomalies¶

A1 — the surviving match is with the member music theory treats as an error, not as structure. Temperament (the comma side) is the celebrated, theorised, notated part of music's commensuration problem. Inharmonicity is the part treated as a defect of real strings — yet it is inharmonicity, not temperament, that shares the nucleosome's shape. Investigation: §2 lemma vs §2.2 free-parameter argument, both exact. Verdict: real, and it inverts the dispatch's expected mapping. Next: if the framework wants the DNA↔music reading, it must be built on the Railsback/inharmonicity member and explicitly not on the comma — the opposite of the intuitive route.

A2 — the nucleosome's arithmetic non-closure survives only in an artificial construct. The one genuinely arithmetic obstruction found on the DNA side (ℤ/2 dyad parity) manifests as a distributed residual only in Luger's 146 bp palindrome — a crystallographer's construct. The native 147 bp particle satisfies the parity condition exactly and has no residual to distribute. Investigation: parity arithmetic + the attested 146/147 resolution [PMC4378457]. Verdict: real; means the type-1 match is with an experimental artifact, not with biology. Next: conductor decision on whether an artifact-only match may be cited at all.

A3 — the prior spike's A3 failure mode recurred, in a new domain. The nucleosome spike logged a search summary that did not survive fetch. Here the same vector appeared as a number: the Railsback stretch magnitude is quoted everywhere, is cited by an arXiv paper — and that paper's own figure credit is a Wikipedia image file, not a measurement. Investigation: direct fetch of every candidate source; the JASA primary (1938) and the 2015 JASA explanatory paper are both paywalled (the latter returned HTTP 403). Verdict: confirmed — an apparently well-sourced quantity with no attestable origin. Next: the Railsback curve may be cited [B]; its magnitude may not. Worth a methodology exhibit line alongside the nucleosome A3 — two independent instances now, in unrelated literatures.

A4 — three "standard" phrases failed attestation. "Key colour", "pure thirds in meantone", and "treble sharp / bass flat" all read as textbook and none is in any fetched OA source; "pure" is actively contradicted by [C], which says consonant. All three were in this note's first draft. Verdict: real; the failure mode is fluent-domain-vocabulary, distinct from citation hallucination — the phrases are plausible register, not invented facts, which makes them harder to catch. Next: none required; logged as a pattern.

10. Fermatas (conductor decisions — this pass is NOT authorized to decide)¶

F-i (srmech surface). Ship a general lattice-quotient + residual-allocation op (integer Smith-normal-form over a prime-exponent lattice), with tuning as one worked instance — or park it? Recommendation in §4 is not as a music catalog. Also: document the best_rational uint64 bound?
F-ii (notebook placement). §5 is an MFO-side ontological finding (the resonator bridge does not go through by commensuration); §§2–4 are srmech-side tooling/math. Split per [[feedback_mfo_vs_srmech_notebook_split_rule]], or hold both here pending the F-d decision still open from the nucleosome spike?
F-iii (A2). May an artifact-only (146 bp construct) match be cited in framework prose, or is "arithmetic non-closure on the DNA side" to be recorded as native-absent?
F-iv (the inversion). A1 says the DNA↔music reading must be built on inharmonicity and not on the comma — inverting the intuitive route. Worth a follow-up spike on the Railsback member specifically (it is the one with a measured, applied, systematic deviation curve), or park?
F-v (Kuramoto, still open). The nucleosome spike's F-c (Arnold-tongue / mode-locking, untested; cascade.kuramoto_step ships with Sakaguchi-α) is now more motivated: §3 shows two-periodicity detuning is the contingent-class mechanism shared by strings and nucleosomes. Fold F-c into a follow-up, or keep parked?

Cross-links: nucleosome_turn_asymmetry_frame_spike.md (Q1/Q3, ΔØ = 7/17, fermata F-c) · chromatin_histone_structural_machinery_findings.md (row 1 / G3) · subharmonic_chirality_carrier_findings.md (Class-K chirality, overtone/undertone ±-pair) · ../../antikythera-maths/subharmonic_chirality_collapse_stub.md · spike_40_musical_wave_epicycle_shape_2026-05-17.md (bell/drum spectra — the clause-(a) exclusions) · audio-scoping-2026-05-09.md (Z₁₂/D₁₂ pitch-class group) · MFO notebook head + §XIV.8 (F1179–F1186) · srmech.amsc.rational (Class N) · [[feedback_no_privileged_primitive_classes]] · [[feedback_mfo_vs_srmech_notebook_split_rule]].