Open experiments — running the Kuramoto candidate, chasing the hexasome ΔLk, closing A4¶

Research spike (2026-07-19; concertmaster dispatch). Experiment/derivation only — no rc, no ADR, no production code. FORM-matching only; this does not validate the framework and neither biology nor nonlinear dynamics is superseded ([[user_stance_cascade_matching_substrate_blind_form_not_identity]], [[feedback_no_lineage_claims_in_notebook]]). Provisional throughout. Continues the open experiments of nucleosome_turn_asymmetry_frame_spike.md (fermata F-c, F-a, anomaly A4) and music_discrete_forms_commensuration_shape_spike.md (fermata F-v). Generating script for every number: open_experiments_kuramoto_hexasome_spike.py — driving srmech's own shipped cascade.kuramoto_step (v0.6.0rc14 generalisation), exact Class-N rationals, Class-K pin-slot for sign, no abs(), no numpy / math / fractions ([[feedback_computational_provenance_discipline]]).

0. Bottom line¶

experiment	verdict
E1 — run the Kuramoto / Arnold-tongue candidate (S5, never tested)	The match LIVES on the discriminating question: locking retains the residual, it does not eliminate it. But the dispatch's α hypothesis is partly FALSIFIED — α is not the knob that supplies the asymmetry. Directed (non-symmetric) coupling is.
E2 — hexasome ΔLk, the genuine falsifiable test of S2	Hexasome: NULL again — and now a deep null. But the tetrasome ΔLk was found (CC BY): S2 misses it by 3.5–4.7σ with a sign-flipping residual, and cannot represent its bistability at all. S2 degrades; it does not cleanly die.
E3 — anomaly A4, the Chen 2010 non-closing ledger	RESOLVED: the minus sign is in the PAPER, not the extraction. Three independent readings agree.

Honesty ceiling, carried forward and not relaxed. The music spike established that the shared predicate is generic commensuration-under-closure — calendars, Antikythera gear-trains and floating-point rounding are full members. E1 comes back positive on the discriminating question. That is not evidence of specialness and nothing below should be read as making it so. What E1 buys is a mechanism for clause (d), not a signature.

1. E1 — falsifiability, stated BEFORE the runs¶

Five kill conditions, written with their closed-form predictions before any integration.

How to read this table. Column 2 is the falsifier — the outcome that would have killed the match, stated so it could be tested. It is not a finding. Column 3 is what was measured. Read column 3 for the result. No falsifier fired.

#	falsifier (would have killed the match — NOT a finding)	MEASURED RESULT	verdict
K1	at lock the phase residual φ* → 0, i.e. locking eliminates the residual	Residual RETAINED. φ* ≠ 0 at every locked K, with sin φ* = Δω/K to six decimals; φ* → 0 only as K → ∞. At N = 14 the pin cannot null it at any grip strength.	NOT FIRED — match stands · §2.1
K2	locking requires Δω = 0, i.e. only zero-detuning systems lock	Locks over a finite band, \|Δω\| ≤ K, across the whole swept range.	NOT FIRED — match stands · §2.2
K3	α has no effect on the locked state	α does act: it narrows the tongue as `K_c = Δω/(2 cos α)` and shifts Ω off ω̄ by exactly `−K cos(φ) sin α` (matched to five decimals). But the shift is common-mode, not a re-partition — so α is not* the allocation mechanism (see K5, A6).	NOT FIRED — but α is the wrong knob · §2.3
K4	tongue half-width does not scale linearly in K	Linear. half-width / K = 1.0006–1.0009 for K ≥ 1. Canonical 1:1 Arnold tongue.	NOT FIRED — match stands · §2.2
K5	the allocation is fixed by the closure condition — every locked state splits the residual identically — which would make Kuramoto the "generic beat" the music spike already excluded at clause (d), i.e. nothing new	ALLOCATION-UNDERDETERMINATION CONFIRMED. At fixed A₁₂+A₂₁, φ* is invariant at 0.25268 to five decimals while the split runs 0.05 → 0.95, matching A₁₂/(A₁₂+A₂₁) exactly. The lock threshold is BLIND to the allocation ⇒ coupled asymmetric pairs satisfy clause (d). Mechanism is directed / non-reciprocal coupling, NOT α.	NOT FIRED — clause (d) SATISFIED · §2.4

K5 is the one that mattered, and it came back positive. K1–K4 are textbook and could barely have gone the other way. K5 asked whether running this candidate told us anything the music spike had not already excluded — and it did: an uncoupled beat has nothing to allocate, but a coupled asymmetric pair has a real allocation degree of freedom that the closure condition does not fix.

2. E1 — what the instrument actually did¶

Setup, from the attested periodicities only (script block 0, exact integer arithmetic):

  h_s = 51/5 = 10.2 bp/turn  [PMC6162219]    h_0 = 21/2 = 10.5 bp/turn  [PMC6162219]
  N = 147 bp,  k = 14 contacts               [PMC4512544]

  exact commensuration   N/h_0 = 14/1 = k      ← the ideal the contact lattice wants
  detuning  1/h_s - 1/h_0 = 1/357 turns/bp     ← exact
  ΔØ = 147 × 1/357 = 7/17 = 0.411765 turns     ← reproduces the prior spike exactly

  ω_ideal = 2π/h_0 = 0.598398601 rad/bp
  ω_surf  = 2π/h_s = 0.615998560 rad/bp
  Δω               = 0.017599959 rad/bp        ← the physical detuning driven into the model

2.1 K1 — the discriminating question. The residual is RETAINED, not eliminated.¶

N = 2, α = 0, physical units (time = bp of DNA arc):

K/K_c		φ* (rad)	φ* (turns)	sin φ*	Δω/K
0.50	drift	—	—	—	—
0.90	drift	—	—	—	—
0.99	drift	—	—	—	—
1.00	LOCK	1.566075	0.249249	0.999989	1.000000
1.01	LOCK	1.429960	0.227585	0.990099	0.990099
1.10	LOCK	1.141097	0.181611	0.909091	0.909091
1.50	LOCK	0.729728	0.116140	0.666667	0.666667
2.00	LOCK	0.523599	0.083333	0.500000	0.500000
4.00	LOCK	0.252680	0.040215	0.250000	0.250000
10.0	LOCK	0.100167	0.015942	0.100000	0.100000

sin φ* = Δω/K reproduces to six decimals at every locked row. K_c = Δω exactly (= 0.017599959 rad/bp), and at threshold φ* → π/2.

The answer the dispatch asked for. Two residuals must not be conflated:

the frequency residual Δω is eliminated by locking — that is what locking means;
the phase residual φ* is retained, and it is precisely what carries the eliminated frequency residual. φ* → 0 only as K → ∞.

So locking CONVERTS the residual; it does not remove it. That is structurally the same move the nucleosome makes: the periodicity residual ΔØ is not removed, it is re-expressed as a twist term and absorbed into a ledger that closes (Lk = Tw + Wr). Conversion-with-conservation, not deletion. K1 survives.

2.2 K2 + K4 — the Arnold tongue¶

Normalised (detuning and K in units of Δω_nuc). The scale invariance (Δω, K, t) → (λΔω, λK, t/λ) is exact, so the normalised tongue is the physical tongue.

     K |  -1.50-1.25-1.00-0.75-0.50-0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50
   ----+-------------------------------------------------------------------
   0.00|    .    .    .    .    .    .    #    .    .    .    .    .    .
   0.25|    .    .    .    .    .    .    #    .    .    .    .    .    .
   0.50|    .    .    .    .    .    #    #    #    .    .    .    .    .
   0.75|    .    .    .    .    #    #    #    #    #    .    .    .    .
   1.00|    .    .    #    #    #    #    #    #    #    #    #    .    .
   1.25|    .    #    #    #    #    #    #    #    #    #    #    #    .
   1.50|    #    #    #    #    #    #    #    #    #    #    #    #    #
   1.75|    #    #    #    #    #    #    #    #    #    #    #    #    #
   2.00|    #    #    #    #    #    #    #    #    #    #    #    #    #

      '#' = phase-locked      '.' = drifting      the V opens LINEARLY

K	measured half-width	predicted (= K)	ratio
0.25	0.219833	0.25	0.8793
0.50	0.493996	0.50	0.9880
1.00	1.000557	1.00	1.0006
1.50	1.501396	1.50	1.0009
2.00	2.001503	2.00	1.0008

Canonical 1:1 Arnold tongue, linear in K. K2 and K4 survive. (The two low-K rows undershoot by 1–12%: the relaxation time diverges at the tongue boundary, so a fixed settle window misclassifies marginal cells. A finite-integration artifact, not a deviation from the law — stated rather than smoothed.)

2.3 K3 — what α actually did. Three effects, and none is the one expected.¶

Pure pairwise Sakaguchi (adjacency=[[0,1],[1,0]], zero diagonal), normalised Δω = 1:

α	K_c measured	K_c predicted `Δω/(2 cos α)`	φ* (rad)	Ω − ω̄	residual split
0°	0.499802	0.500000	0.125328	+0.000000	0.500 / 0.500
15°	0.517349	0.517638	0.129773	−1.026571	−0.527 / 1.527
30°	0.577011	0.577350	0.144843	−1.979057	−1.479 / 2.479
45°	0.706711	0.707107	0.177711	−2.783882	−2.284 / 3.284
60°	0.999527	1.000000	0.252680	−3.354102	−2.854 / 3.854
75°	1.930771	1.931852	0.504035	−3.383216	−2.883 / 3.883

Independent cross-check (different angle, converged): the closed form Ω − ω̄ = −K cos(φ*) sin α reproduces the measured column to five decimals at every α.

α does three things:

It narrows the tongue — K_c(α) = Δω/(2 cos α); the tongue closes as α → 90°.
In the mean-field branch it adds a common-mode drift. With adjacency=None the coupling sum runs over j == i, so sin(θ_i − θ_i − α) = −sin α contributes. Verified at n = 1: ω = 0, K = 1, one step of dt = 1 gives θ = −sin α exactly (0 / −0.5 / −1.0 at α = 0° / 30° / 90°). A single oscillator drifts under α. This is the standard Kuramoto–Sakaguchi mean-field convention (the order-parameter form includes the self term), not a defect — but the zero-diagonal adjacency branch does not have it, so the two branches are not comparable without accounting for it. Logged as A7.
It moves the locked frequency off the mean — but as a common-mode drift of the whole pair, not a re-partition between the two members. At α = 0 the split is exactly 0.500/0.500; at α ≠ 0 the split leaves [0,1] entirely (−0.53/1.53, …), i.e. both oscillators are dragged the same way. This is not clean allocation.

K3 survives numerically but is the wrong instrument for the job. See §2.4 and A6.

2.4 K5 — the clause-(d) test. This is the centerpiece.¶

The music spike excluded "generic two-oscillator beat / moiré" at clause (d) — a residual exists but there is nothing to allocate. So the real question is whether a coupled pair has an allocation degree of freedom the closure condition does not fix.

Directed coupling at α = 0, holding the sum A₁₂ + A₂₁ fixed and varying only the ratio. Closed form: dφ/dt = Δω − K(A₁₂+A₂₁) sin φ (threshold depends only on the sum) and Ω = ω₁ + Δω·A₁₂/(A₁₂+A₂₁) (split depends only on the ratio).

A₁₂	A₂₁	sum	Ω	split (osc 0)	φ*	split predicted
1.0	1.0	2.0	0.50000	0.5000	0.25268	0.5000
1.5	0.5	2.0	0.75000	0.7500	0.25268	0.7500
1.9	0.1	2.0	0.95000	0.9500	0.25268	0.9500
0.5	1.5	2.0	0.25000	0.2500	0.25268	0.2500
0.1	1.9	2.0	0.05000	0.0500	0.25268	0.0500

φ* is IDENTICAL to five decimals across every row — the closure state is literally untouched — while the allocation runs from 0.05 to 0.95, matching A₁₂/(A₁₂+A₂₁) exactly. The lock threshold is blind to the allocation.

That is clause (d) verbatim: the closure requirement alone does not fix how the residual distributes; an extra degree of freedom must be fixed from outside. K5 survives — via directed coupling, not via α.

2.5 The N = 14 chain — the residual is retained, but flat; chirality is an AND-gate¶

14 oscillators = the 14 attested contacts, open nearest-neighbour chain (the backbone), each pinned to the exact lattice (pin_anchor ψᵢ = 0), each driven off it by ω = Δω. θᵢ is the DNA's helical phase relative to the lattice demand; the ideal state is θᵢ ≡ 0.

p/Δω	mean θ*	end θ*	centre θ*	end/centre
0.50	43.027647	43.027647	43.027647	1.0000 (drifting — p < Δω, no fixed point)
1.00	1.531641	1.531641	1.531641	1.0000
2.00	0.523599	0.523599	0.523599	1.0000
5.00	0.201358	0.201358	0.201358	1.0000
20.0	0.050021	0.050021	0.050021	1.0000

θ* is non-zero at every grip strength — the arginine-analogue pin cannot drive the residual to zero, it can only trade phase offset against grip, exactly θ* = arcsin(Δω/p) (at p = 2Δω, 0.523599 = arcsin ½). Residual retained, confirming §2.1 at N = 14.

But the profile is perfectly FLAT (end/centre = 1.0000 to four decimals at every p). This is a NULL for "distributed". The residual is retained uniformly, not apportioned into any structure. Nothing here allocates.

The chirality test, 2×2 factorial. End-to-end asymmetry θ₁ − θ₁₄, Class-K signed (no abs()). K = 1, p = 4, chosen so every cell stays LOCKED — at K = 4 / p = 2 the α drive K·deg·sin α overwhelms the pin and the chain drifts, and a drifting profile is not a locked allocation and must not be read as one. All six cells below verified locked = True.

topology	α	ends θ₁ / θ₁₄	centre	asym θ₁ − θ₁₄	Class-K sign
symmetric	0°	+0.25268 / +0.25268	+0.25268	0.000000e+00	0
symmetric	+30°	+0.10634 / +0.10634	+0.00000	0.000000e+00	0
symmetric	−30°	+0.40642 / +0.40642	+0.52360	0.000000e+00	0
directed (f 1.0, b 0.5)	0°	+0.25268 / +0.25268	+0.25268	0.000000e+00	0
directed	+30°	+0.11500 / +0.17834	+0.06254	−6.334172e−02	−1
directed	−30°	+0.39625 / +0.32901	+0.45281	+6.723708e−02	+1

Two things separate cleanly here, and the arc has been conflating them:

α produces STRUCTURE but not CHIRALITY. On the symmetric chain, α = 0 gives a perfectly flat profile, while α = ±30° gives a genuinely shaped one (ends +0.10634 vs centre +0.00000 at +30°; ends +0.40642 vs centre +0.52360 at −30° — the shape inverts with sign α). That structure tracks node degree (interior sites have two neighbours, ends have one), a graph property. But the end-to-end asymmetry stays exactly 0.000000e+00 in all three symmetric cells.
Directed coupling produces CHIRALITY — but only together with α. Directed at α = 0 is still exactly flat and exactly symmetric. Only directed ∧ α ≠ 0 breaks it, and then the sign reverses with sign(α) (−0.0633 at +30°, +0.0672 at −30°).

Chirality is an AND-gate. Neither factor alone breaks the end-to-end symmetry. The symmetric-chain zero is exact, not small — the reflection i → n+1−i maps the α-frustrated symmetric chain to itself, because α enters both neighbour terms identically, so the fixed point must be its own mirror image.

This directly falsifies the dispatch's framing that "α is the load-bearing knob — it breaks the symmetry." It does not break it alone, and on its own it is a distribution knob, not a handedness knob. Logged as A6.

2.6 What this instrument cannot represent — and why that is fair here¶

Result first: no 14:1 tongue exists at any coupling. Two oscillators at frequency ratio ≈ 14:1, swept over coupling. The question under test is the 14:1 column; a 14:1 tongue would show as the winding ratio plateauing at 14.00 over a range of K. It never does — it slides continuously, then collapses to 1:1 once K passes the (large) 1:1 threshold.

K	ω₂/ω₁	winding ratio	14:1 tongue? (the question)	1:1 collapse?
0.1	14.28	14.277260	no — free-running, not locked	no
1.0	14.28	13.992173	no — sliding, no plateau	no
5.0	14.28	9.253016	no — sliding	no
10.0	14.28	3.670906	no — sliding	no
13.0	14.28	1.429954	no — sliding	no
20.0	14.28	1.000000	no	yes — collapsed to 1:1

Sinusoidal coupling has exactly one resonance. Higher-order p:q tongues need harmonics the model does not carry. [NULL — N4.] (The "yes" in the last column is the 1:1 collapse, not a 14:1 lock — it is the model falling into its only resonance, which is the opposite of finding a 14:1 one.)

This is a limitation of the model, but not a defect for this application, and the distinction matters: the nucleosome's commensuration is 1:1 — one DNA helical turn per contact (N/h₀ = 14 turns over 14 contacts). That is the one resonance the model has. Music's comma is 12:7, a high-order commensuration the model cannot express.

So the instrument's own reach tracks the arithmetic-vs-contingent split the music spike found: Kuramoto covers the contingent member (nucleosome, inharmonicity) and not the arithmetic member (the comma). Convergence, not dissonance — an independent line arriving at the prior spike's boundary.

3. E1 verdict¶

The match LIVES, with three corrections to the dispatch hypothesis.

On the discriminating question it survives cleanly. Locking does not remove the residual; it converts a frequency residual into a retained standing phase offset, and at N = 14 the pin cannot null it at any grip strength. Same shape as the nucleosome: measured and carried, not removed.
Kuramoto is NOT the "generic beat" the music spike excluded — a coupled asymmetric pair has a genuine allocation degree of freedom (§2.4) with the lock threshold blind to it. The music spike's row-10 exclusion is correct for an uncoupled beat; a coupled asymmetric pair is a member. (Scope note: this amends a table in another note — flagged as fermata F-α, not executed.)
The dispatch's α hypothesis is partly falsified. α narrows the tongue and adds common-mode drift, but it does not cleanly allocate and it does not create chirality on its own. The two roles separate: α is a distribution knob (it shapes a degree-dependent profile), directed coupling is the handedness knob, and only their conjunction is chiral. So the asymmetry the arc is chasing lives in directed (non-symmetric) coupling — which is also the honest encoding of a handed backbone, and which is why [[user_stance_k2_compare_is_frame_relative_asymmetric_pair]] is better served by the adjacency argument than by α.

And the ceiling holds. All of this is generic coupled-oscillator behaviour. It supplies a mechanism for clause (d); it supplies no evidence of specialness.

4. E2 — the hexasome hunt, and the tetrasome that was already in the notes¶

4.1 Hexasome: NULL again — and this time a deep null¶

No ΔLk for a hexasome exists in any OA source reachable. The null is enumerated, not shallow: PMC full-text hexasome AND "linking number" → 12 hits, all screened, every occurrence definitional or about remodelling; hexasome AND (supercoil* OR topoisomer* OR "DNA topology") → 114 hits; hexasome AND ("change in linking number" OR … OR constrains) → 25 hits, all screened; hexasome AND topoisomerase AND relaxation AND topology → 8 hits. Raw-byte grep for hexasome returned 0 hits in every single-molecule topology paper checked (Vlijm 2015; Ordu 2019; Sheinin 2013 PMC3848035; Kaczmarczyk 2020 PMC6949304; Vlijm 2017 PMC7959483; Recouvreux 2011). The citation trail from Shi 2025 (PMC12041859) resolves to three papers, none of them topology studies.

S2's cleanest test remains untested for the second time. Fermata F-a stays open.

Also NULL: H2A.B / H2A.Bbd ΔLk (Bao 2004 PMC514500 — "linking number" 0 hits, "topoisom" 0 hits; Arimura 2013 PMC3863819 — all zero); hemisome / subnucleosome ΔLk (87 hits screened, not exhausted — Furuyama & Henikoff 2009 PMC2725230 reports supercoiling direction, not a per-particle ΔLk).

4.2 Tetrasome: FOUND — and it was already in our own §0¶

ATTESTED OA, CC BY — Vlijm R, Lee M, Ordu O, Boltengagen A, Lusser A, Dekker NH, Dekker C (2015), PLoS ONE 10(10):e0141267, DOI 10.1371/journal.pone.0141267, PMC4623960, verified CC BY:

"However, subsequently the linking number did not stay constant but was rather observed to change between −0.80 ± 0.05 and +0.86 ± 0.39 turns"

Corroborated (PMC-free but © Biophysical Society, not CC — recorded, not rested on, per the F-h precedent): Ordu, Lusser & Dekker 2019, Biophys J 117(11):2217, PMC6895708 — "Θ_left = −0.31 ± 0.01 turns and Θ_right = +1.38 ± 0.06 turns", with the paper's own drift caveat, and drift-robust ΔΘ_flipping = 1.6 ± 0.2 turns.

The tetrasome's k = 6 is non-circular in the sense the prior spike required — it is fixed by which histone fold is present, not read off the same structure the bp count comes from. [OURS, not attested] the arithmetic that makes 6 the natural count is that the 14 attested contacts sit at SHL −6.5…+6.5 and the (H3–H4)₂ tetramer occupies the central six of them (−2.5…+2.5); the prior spike's table asserts k = 6 without stating that basis, and no source was fetched this spike that enumerates the tetramer's contacts directly. The k = 6 input is therefore the weakest link in §4.3 and is flagged as such — if k differs, N_predicted and hence ΔLk_predicted move with it.

⚠️ A9 — the datum was already in-tree, mis-filed. The nucleosome spike's own §0 already quotes "−0.80 ± 0.05 and +0.86 ± 0.39 turns … at a barrier of only 2.3 ± 0.4 k_BT [PMC4623960]" — but recorded it as evidence that the sign is not fixed (an input to the gate argument), not as a per-particle ΔLk for a k ≠ 14 particle, which is precisely what fermata F-a asked for. A literature hunt was dispatched for a number sitting in the requesting note's first section.

4.3 The S2 test¶

S2's chain is k → hₛ → ΔØ → ΔTw → ΔLk. Attested forms: ΔØ = N(1/hₛ − 1/h₀); Wr = −n(1 − sin δ). Auxiliary and OURS, not attested — and load-bearing for the verdict: that the superhelical turns n and the surface-twist correction ΔSTw both scale linearly with N. Neither is independently pinned for a tetrasome.

N	ΔØ	n	Wr	ΔSTw	ΔTw	ΔLk predicted
147	+0.41176	1.65000	−1.53490	−0.19000	+0.22176	−1.31314
70	+0.19608	0.78571	−0.73091	−0.09048	+0.10560	−0.62530
63	+0.17647	0.70714	−0.65782	−0.08143	+0.09504	−0.56277

σ below is a discrepancy, not a confidence: larger σ = S2 predicting worse. Note especially the sign of the residual column — that it changes between rows is the finding.

N	particle	ΔLk pred	ΔLk measured	residual (meas − pred)	σ discrepancy	what this row says
147	canonical NCP	−1.31314	−1.26 [PMC6162219]	+0.05314	1.06	acceptable; model runs slightly too negative
70	tetrasome (observed N)	−0.62530	−0.80 ± 0.05 [PMC4623960]	−0.17470	3.49	S2 MISSES; model runs too positive — sign of residual has flipped
63	tetrasome (S2-predicted N = 6×10.5)	−0.56277	−0.80 ± 0.05	−0.23723	4.74	S2 MISSES; same direction, larger — but this row inherits the k = 6 assumption

Note the k-dependence, because it bounds how much the k = 6 caveat can rescue: the N = 70 row uses the observed wrap and does not depend on k at all. Its 3.49σ miss therefore stands whatever the true contact count is. Only the N = 63 row (4.74σ) inherits the k = 6 assumption.

Two findings, and the second is the sharper one.

The residual SIGN FLIPS between particles: +0.053 at the canonical particle (model too negative) and −0.175 / −0.237 at the tetrasome (model too positive). No linear-in-N law fits both. The miss is not an offset that a constant could absorb.
The tetrasome ΔLk is BISTABLE and sign-flipping (−0.80 and +0.86, summing to +0.06 — a near-symmetric ± pair). S2 is a single-valued commensuration detuning: it emits one number with one sign. It cannot produce a two-state sign-flipping particle at all. This is a structural problem, independent of any numerical miss.

E2 verdict: S2 DEGRADES but does not cleanly die. The 3.5–4.7σ miss can be absorbed by choosing non-linear auxiliary scalings — but that absorption is anomaly A1 (the beat term is free enough to absorb any mechanism). So this test reinforces A1: S2 cannot be falsified until hₛ(k) is independently pinned, and a hypothesis that cannot be falsified is not passing a test when it survives one. Its prior status ("SURVIVES with a circularity caveat") should be re-read as survives only by invoking free parameters it cannot pin. Fermata F-β.

The ± pair is also a data point for the chirality thread — equal-magnitude opposite-orientation partners is the {θ(αx), θ(α/x)} ±-pair shape of subharmonic_chirality_carrier_findings.md §2. Recorded as a cross-ref, not adjudicated here.

5. E3 — anomaly A4 RESOLVED: the sign is in the PAPER¶

Chen B, Xiao Y, Liu C, Li C, Leng F (2010), NAR 38(11):3643–3654, DOI 10.1093/nar/gkq078, PMC2887952. Three independent readings, all agreeing:

Europe PMC XML, raw characters — ΔSLk = −1.8 … Δϕ = −0.8. U+2212 MINUS SIGN, the same codepoint as the unambiguous −1.8 and −1 in the same sentence — not a hyphen, not an en-dash, not a mangled plus.
Typeset PDF text layer — Áf = À0.8, where the math font maps À → minus and þ → plus, verified against unrelated occurrences in the same document (Lk\xc0Lk°, 10\xc09 M, (a \xfe x \xfe 1/Kapp)). Plus and minus are distinct glyphs; Δϕ gets the minus glyph.
Visual render of the typeset page (journal p. 3653, Discussion, left column) read as an image, not as text — the minus before 0.8 is plainly typeset.

PDF provenance is the key to the verdict. PMC now gates PDF downloads behind a proof-of-work challenge; the retrieved file's MD5 0386437fd351455df59132ebb486c58b matches the checksum PMC declares inside its own full-text XML (<?pdf-md5 …?>, size 5,533,568 also matching). Producer string is the publisher typesetting chain (3B2 Total Publishing System 8.07r/W, 2010-06-16).

PDF and XML do not disagree. The extraction was faithful; the arithmetic failure originates in the published text. No erratum exists (PubMed pubtype plain Journal Article, Europe PMC commentCorrectionList empty).

The internal inconsistency is sharpest within the same paragraph, which uses the opposite-signed convention explicitly:

"The crystal structures of the TBP–DNA complexes showed that the negative Δϕ from DNA unwinding is canceled by the positive ΔSLk gained from 'wrapping' TBP around DNA"

Exact ledger (script block E3): as printed, ΔSLk + Δϕ = −13/5 = −2.60 vs stated ΔLk = −1.00, residual −1.60, does not close. Sign-flipped, −1/1 = −1.00, residual exactly 0, closes.

Reported, not corrected. Per the dispatch: the published arithmetic is not silently fixed. The −0.8 is unusable as an attested datum — not evidence about nucleosome topology in either direction. Anything needing a citable Δϕ must go to the primary (White, Cozzarelli & Bauer 1988, Science 241:323, ref 37 in that paper), which is paywalled and therefore not our basis — derive it instead, per the standing rule.

6. Diagram — where the residual goes¶

graph TD
  subgraph IN["the detuning (contingent, free parameter)"]
    D["Δω = 2π(1/h_s − 1/h_0)<br/>= 2π/357 rad/bp<br/>ΔØ = 7/17 turns over 147 bp"]
  end
  subgraph LOCK["below threshold — K < Δω"]
    U["DRIFT: phase winds forever<br/>no ledger, nothing stored"]
  end
  subgraph RET["above threshold — K ≥ Δω : the residual is CONVERTED"]
    F["frequency residual ELIMINATED<br/>both members at one Ω"]
    P["phase residual RETAINED<br/>sin φ* = Δω/K<br/>→ 0 only as K → ∞"]
  end
  subgraph ALLOC["who bears it — clause (d)"]
    S["symmetric coupling<br/>split fixed at 0.500/0.500<br/>= the EXCLUDED generic beat"]
    A["DIRECTED coupling A₁₂≠A₂₁<br/>split = A₁₂/(A₁₂+A₂₁), free 0.05→0.95<br/>φ* UNCHANGED — threshold is BLIND<br/>= clause (d) SATISFIED"]
  end
  D --> U
  D --> F
  F --> P
  P --> S
  P --> A
  classDef inn fill:#eef,stroke:#446;
  classDef out fill:#fee,stroke:#a44;
  classDef ret fill:#efe,stroke:#464;
  classDef al fill:#ffe,stroke:#a84;
  class D inn;
  class U out;
  class F,P ret;
  class S,A al;

The chirality AND-gate (end-to-end asymmetry of the 14-contact chain):

                        α = 0                     α = ±30°
                 ┌─────────────────────┬─────────────────────────────┐
   SYMMETRIC     │  profile FLAT       │  profile SHAPED (by degree) │
   chain         │  θ ≡ +0.25268       │  ends .10634 / ctr .00000   │
                 │  asym = 0           │  asym = 0.000000e+00        │
                 │                     │  EXACTLY zero — reflection  │
                 │                     │  i→n+1−i maps it to ITSELF  │
                 ├─────────────────────┼─────────────────────────────┤
   DIRECTED      │  profile FLAT       │  profile SHAPED *and* TILTED│
   f=1.0 b=0.5   │  θ ≡ +0.25268       │  asym = −0.0633  (α = +30°) │
                 │  asym = 0           │  asym = +0.0672  (α = −30°) │
                 │                     │  → SIGN REVERSES with α     │
                 └─────────────────────┴─────────────────────────────┘
        all six cells LOCKED (K = 1, p = 4)          ↑
        chirality needs BOTH — it is an AND-gate.
        α alone → distribution, no handedness.  Directed alone → nothing.

7. NULLs (first-class)¶

N1 — hexasome ΔLk. NULL for the second time, now enumerated across 4 search families and 6 raw-grepped single-molecule papers. S2's cleanest test is still untested.
N2 — H2A.B / H2A.Bbd ΔLk. NULL. The prior spike's one genuine open prediction (H2A.B ⇒ k = 10) remains untestable.
N3 — hemisome / subnucleosome ΔLk. NULL (87 hits screened, class not exhausted — scope stated).
N4 — higher-order p:q Arnold tongues. NULL in the sinusoidal Kuramoto model at any coupling. Not a defect here (the nucleosome's commensuration is 1:1), but it means the model cannot express music's 12:7 comma.
N5 — α as a source of chirality. NULL, and exactly zero, not merely small.
N6 — a structured residual profile at α = 0. NULL. The pinned 14-chain is perfectly flat at arcsin(Δω/p) at every grip strength, symmetric and directed. The residual is retained but not distributed into structure. Structure appears only once α ≠ 0, and then it tracks node degree — a graph property, not a commensuration one. "Retained" and "distributed" must not be traded on — the same equivocation warning the music spike raised in a different register, and the reason §2.5 reports them as separate findings.
N7 — chromatosome ΔLk at OA-license tier. Recouvreux 2011 PMC3117191 gives "∼−1.4 turns per particle" but is © Biophysical Society, free-to-read, not CC (Europe PMC fullTextXML 404s, confirming it is outside the OA subset), and the value sits in a figure legend. Per the F-h resolution (no free-to-read tier), recorded, not rested on.
N8 — a non-circular hexasome contact count from an independent structure. Not attempted here.

8. Anomalies¶

A6 — the dispatch's α hypothesis is falsified on the chirality claim. The dispatch stated "α is the load-bearing knob — it breaks the symmetry, which is exactly the asymmetry the whole arc has been chasing." Measured: it does not. End-to-end asymmetry of a mirror-symmetric 14-chain is exactly 0.000000e+00 at α = 0°, +30° and −30°, in locked states. Investigation: 2×2 factorial {symmetric, directed} × {α = 0, α ≠ 0}, all six cells verified locked = True (K = 1, p = 4 chosen so the α drive cannot overwhelm the pin — at K = 4 / p = 2 the chain drifts and a drifting profile must not be read as an allocation), plus the analytic reason: the reflection i → n+1−i maps the α-frustrated symmetric chain to itself because α enters both neighbour terms identically, so its fixed point must be its own mirror image. Verdict: real, and it separates two things the arc had been conflating — α is a DISTRIBUTION knob (it shapes a degree-dependent profile: ends +0.10634 vs centre +0.00000 at +30°) but not a HANDEDNESS knob (asymmetry stays exactly zero). Chirality is an AND-gate on (directed coupling) ∧ (α ≠ 0), with the sign reversing under sign(α) (−0.0633 at +30°, +0.0672 at −30°). Next: the arc's asymmetry should be sought in directed coupling, not in phase frustration alone.

A7 — the mean-field branch drifts a SINGLE oscillator under α. With adjacency=None the coupling sum runs over j == i, contributing −(K/n) sin α to every oscillator. Verified at n = 1: one step gives exactly −sin α. Verdict: real, and convention-correct (the Kuramoto–Sakaguchi mean-field order-parameter form does include the self term) — logged because it is easy to miss and because it makes the adjacency=None and zero-diagonal-adjacency branches non-comparable under α. Next: worth a docstring line if any cascade ships with α ≠ 0.

A8 — S2's residual sign flips between particles. +0.053 canonical, −0.175/−0.237 tetrasome, so no linear-in-N law fits both. Investigation: exact closed form over the attested ΔØ and Wr forms with the two auxiliary scalings flagged as ours. Verdict: real, but not decisive — the auxiliaries are unattested and could be re-chosen. That freedom is anomaly A1, so the test reinforces A1 rather than resolving S2. Next: S2 is untestable until hₛ(k) is pinned with error bars.

A9 — the requested datum was already in-tree, mis-filed (§4.2). The tetrasome ΔLk was quoted in the requesting note's own §0, classified as a handedness anecdote rather than as a per-particle ΔLk for k ≠ 14. Verdict: real; a recognition/filing failure, not a sourcing failure. Next: when a fermata asks for "quantity X for particle class Y", grep the existing notes for Y before dispatching a hunt. Cheap, and it would have saved this one.

A10 — a WebFetch extraction self-contradicted within one response. For PMC3117191 an extraction pass answered "chromatosome removal": ABSENT and then quoted the sentence containing that exact phrase later in the same response. Raw curl + local grep proved the phrase and its −1.4 value genuinely present. Verdict: real; a third instance of the A3 vector (search/extraction summary not surviving fetch), now in a third literature. Next: every number in §4 was re-verified by raw byte grep rather than by trusting an extraction. Keep that as the default.

A4 — CLOSED (§5). Resolution: PAPER, not extraction. Kept as a standing marker that the −0.8 is unusable, and as a methodology exhibit: the MPM chain (declared MD5 inside the XML) is what made a publisher-authentic PDF distinguishable from a mirror.

A11 — PRESENTATION/INDEXING failure is now a pattern, not a one-off — and this note supplied the third instance. Three times this session the fact was correct somewhere in the record while the surface a reader hits first said something else:

#	where	the correct fact	the misleading surface
A5	the source literature	four distinct quantities (superhelical turns, surface-linking number, writhe, linking difference)	all reported as bare numbers near 1–2, reading as one disputed quantity
A9	our own notes	the tetrasome ΔLk for a k ≠ 14 particle	filed under "handedness anecdote", so a hunt was dispatched for a number already in §0
A11	this note's own §1 summary table	§2.4 states clause (d) is satisfied, with φ* invariant at 0.25268 while allocation runs 0.05 → 0.95	the K5 row stated the hypothesis in the negative ("the allocation is fixed … the match dies") with the reversal carried only by a "survives" verdict cell — read cold, it asserted the opposite of the finding

Investigation: caught by the music spike, which went to this note's body, cited §2.4 correctly, and declined to edit another agent's file — logging the discrepancy as a source note instead. That was the right call and it is why the defect surfaced at all. Verdict: real, and structural rather than accidental — a verdict column reading "survives" attached to a claim phrased as "the match dies" is ambiguous by construction, so the defect was latent in every row of that table, not just K5. Fixed: §1 rebuilt so the measured RESULT is its own column and the falsifier column is explicitly labelled "NOT a finding"; §2.6 and §4.3 tables given result-bearing columns for the same reason. Next (methodology, cheap and general): a summary row must state the RESULT, never the hypothesis-under-test with the reversal delegated to a verdict cell. Falsifiers may be recorded — pre-registering them is the discipline — but they must be visibly marked as the thing that did not happen. The failure mode is distinct from citation hallucination and from [[feedback_fluent_domain_vocabulary_failure_mode]]: nothing here was unattested or wrongly worded, the indexing was wrong. Worth a memory line if the pattern recurs a fourth time.

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

F-α (cross-note table amendment). — EXECUTED by the music spike, 2026-07-19. §2.4's result was carried across correctly: the clause-(d) exclusion is narrowed, not deleted — an uncoupled beat stays excluded, a coupled asymmetric (directed) pair is now a member on a measured result. That note's count moves to 15 systems / 6 excluded at four clauses, its headline is unchanged (discriminating but not special), and the comma ↔ nucleosome verdict is untouched. It also records the §2.6 convergence independently. Nothing outstanding.
F-β (S2's status). — EXECUTED, 2026-07-19. S2 is now recorded as DEGRADED, not as surviving a test: k-independent 3.49σ miss, sign-flipping residual, and — stated more sharply there than here — a decisive arity failure against an observed bistable particle (a single-valued model cannot emit a two-state ± pair). A1 reinforced; A9 logged against ourselves; F-a (hexasome) still OPEN; k = 3 untouched. Nothing outstanding.
F-γ (recording a published arithmetic error). A4 is resolved as an error in the published text. Does the project record that observation anywhere durable, or simply drop the source? It must not be "corrected" on the author's behalf either way.
F-δ (the chromatosome tier). The −1.4 value is free-to-read but not CC-licensed and lives in a figure legend. The F-h precedent says no free-to-read tier. Confirm the exclusion stands?
F-ε (does E1 warrant anything shipped?). Recommendation: NO. cascade.kuramoto_step already does everything this spike needed; nothing here is a missing op. The one surface observation is A7 (the α self-term convention), which is a docstring line at most, not an rc.
F-ζ (directed coupling as the arc's asymmetry carrier). A6 relocates the asymmetry from α to non-symmetric adjacency. That touches [[user_stance_k2_compare_is_frame_relative_asymmetric_pair]] and the chirality thread. Worth a follow-up, or fold into the existing chirality note?
F-η (does A11 warrant a memory line?). Three presentation/indexing failures in one session (A5 / A9 / A11), each a correct fact behind a misleading surface, and distinct from both citation hallucination and fluent-domain-vocabulary. The candidate rule is one line — a summary row states the RESULT, never the hypothesis-under-test with the reversal delegated to a verdict cell; a pre-registered falsifier must be visibly marked as the thing that did not happen. Bank it as feedback memory, or leave it in-note until a fourth instance? Not decided here — three instances inside one session is a weak base rate, and the agent that tripped the defect is not the one who should judge whether it generalises.

10. Sources (attestation status)¶

Inherited attested (prior spikes, re-used not re-derived): hₛ ≈ 10.2 / h₀ ≈ 10.5, ΔØ ≈ +0.4, Wr = −1.53, ΔTw ≈ +0.2, ΔLk = −1.26 [Segura et al. 2018, Nat Commun 9:3989, PMC6162219, CC BY] · 14 minor-groove contacts at SHL −6.5…+6.5 [Hodges et al. 2015, Genetics, PMC4512544].

Newly attested OA this spike: Vlijm et al. 2015, PLoS ONE 10(10):e0141267, DOI 10.1371/journal.pone.0141267, PMC4623960, CC BY verified verbatim — tetrasome ΔLk −0.80 ± 0.05 / +0.86 ± 0.39 turns (full text fetched twice, raw-HTML grepped).

Re-verified this spike (E3): Chen et al. 2010, NAR 38(11):3643, DOI 10.1093/nar/gkq078, PMC2887952, CC BY-NC — typeset PDF retrieved and authenticated against PMC's own declared pdf-md5; XML raw codepoints inspected; page rendered visually. ⚠ Its printed Δϕ sign does not close and is unusable — see §5.

Free-to-read but NOT OA-licensed (weaker tier — recorded, not used): Ordu, Lusser & Dekker 2019, Biophys J 117(11):2217, PMC6895708 (© Biophysical Society) · Recouvreux et al. 2011, Biophys J 100(11):2726, PMC3117191 (© Biophysical Society; the −1.4 chromatosome value, in a figure legend) · Furuyama & Henikoff 2009, Cell 138(1):104, PMC2725230 (NIH author manuscript).

REJECTED (paywalled-only ⇒ not attestation): White, Cozzarelli & Bauer 1988, Science 241:323 (the SLk primary; ref 37 in Chen 2010) · Read, Baldwin & Crane-Robinson 1985, Biochemistry 24:4435 (ACS) · the classic Prunell / Hamiche / Alilat tetrasome papers, JMB 1998–99 and PNAS 1996 (Elsevier / not PMC-deposited) — not used; the Dekker-lab OA papers carry the same physics with better provenance.

Checked, returned nothing useful: Bao et al. 2004 PMC514500 · Arimura et al. 2013 PMC3863819 · Sheinin et al. 2013 PMC3848035 · Kaczmarczyk et al. 2020 PMC6949304 · Vlijm et al. 2017 PMC7959483 · Prunell 1998 PMC1299595 · Bharath et al. 2003 PMC167642 · Wu & Travers 2019 PMC6765122 · Zlatanova & Victor 2009 PMC2839809 · Shi et al. 2025 PMC12041859 · Ordu, Lusser & Dekker 2016 PMC5167136.

Method note ([[feedback_fluent_domain_vocabulary_failure_mode]]). Every numeric claim in §4 was re-verified by raw-byte grep of downloaded source rather than by an extraction summary, after A10 showed an extraction self-contradicting inside one response. Where the field's phrasing and the source's phrasing differ, the source's word is used; the tetrasome quote is given verbatim rather than paraphrased into topology idiom.

Cross-links: nucleosome_turn_asymmetry_frame_spike.md (S2 §3.2, S5 §3.1, fermatas F-a / F-c, anomalies A1 / A4) · music_discrete_forms_commensuration_shape_spike.md (clause (d) §1, fermata F-v, the arithmetic-vs-contingent split §2) · subharmonic_chirality_carrier_findings.md §2 (the ±-pair as equal partners — the tetrasome ± pair) · chromatin_histone_structural_machinery_findings.md (row 1 / G3) · srmech.amsc.cascade.kuramoto_step + srmech_cascade_kuramoto_step_general_f64 · [[user_stance_k2_compare_is_frame_relative_asymmetric_pair]] · [[feedback_computational_provenance_discipline]] · [[feedback_paywall_is_about_open_quotability_derive_instead]].