Document 3 of 3. 9 November 2026, revised after independent verification the same day. Author: the author. For assembly into the case. Scripts and outputs: precession_bend.py/.out, projection_bend.py/.out in agents/claude/periodic-table-10d-20261109/. Verification report: verification-doc-3.md.
1. The non-central term in the orbit model
In the four-direction orbit model (agents/claude/nucleus-lock-20261107/lock_size.py) the up quarks pull with charge 4/3 through four-direction distance and the down quark pushes with charge 1/3 through three-direction distance. Near the three-space, with w the fourth coordinate and r the three-direction radius, for a point nucleus,
V = -1/r + (2/3) w^2 / r^3 (model)
V = -1/r + (1/2) w^2 / r^3 (a central four-direction pull -1/rho)
so the non-central part is (1/6) w^2 / r^3. The radial force is -1/r^2 (orbital rate Omega^2 = 1/r^3) and the fourth-direction restoring force is -(4/3) w / r^3, so the fourth-direction oscillation runs at sqrt(4/3) times the orbital rate in the small-tilt, point-nucleus limit, a drift of sqrt(4/3) - 1 = 0.1547 turns per orbit (verified independently; a numerical check at softening s = 0.01 gives a ratio 1.15470054).
Correction from verification. The first version of this document matched 0.1547005 to the stored line 0.1546669 of the hydrogen runs. That was a coincidence of digits: the stored number is the angular frequency in atomic units of the (x1, x2) motion in the s = 100, 1 a0, 10 degree reference run, essentially the orbital frequency itself, not a drift per orbit; in that run the fourth-direction line is 0.4693522, a ratio near 3, because the nucleus is softened to atom size. The point-nucleus run that does bear on the drift is findings item 5 of hydrogen-tests-quick-20261109 (s = 0.01, tilt 10 degrees): lines 0.99998 and 1.15758, a drift of 0.1576 per orbit, consistent with sqrt(4/3) - 1 at finite tilt. The drift depends on tilt and on softening; 0.1547 is its small-tilt point-nucleus limit.
The split into central and non-central parts is a choice of reference. The fourth-direction frequency comes entirely from the 4/3 pull; the push weakens the in-plane binding. The hydrogen findings (item 3) show no second line and no precession without the down-quark term, which is why the non-central part is called the precession term here.
2. Scale of the non-central term against what bending needs
In the pair-repulsion model with all four pairs of equal weight, holding two bonds at the water angle instead of the tetrahedral one costs 0.00165 in units of e^2 over the shell radius; ammonia's angle 0.00051; H2S's 92.1 degrees 0.0218 (verified by an independent minimisation, which also finds the same energies in four dimensions: the lone pairs gain nothing by entering the fourth direction, supporting section 5). The non-central energy of one pair at tilt alpha is (1/6) sin^2(alpha)/r in the same units; equating it to half the bending cost gives tilts of 4.0, 2.2 and 14.8 degrees. The stored hydrogen runs use 10 degrees.
What this comparison is and is not. It is a comparison of scales: the non-central term at hydrogen-run tilts is of the same order as the bending energies. It is not a demonstration that the term drives the bend: (a) the term is positive, it raises the energy of a tilted pair and pulls it back toward w = 0 (that is why the fourth-direction rate exceeds the orbital rate), so it resists tilt rather than paying for a bend; (b) the coefficient 1/6 is the proton's; by section 4(c) a nucleus with equal protons and neutrons has non-central part (Z/2) w^2/r^3, three times the proton's per unit charge, so oxygen's value differs; (c) screening by inner electrons and the pair charge 2e are ignored, and section 3 tilts the bonds while this section tilts the lone pairs. The first version of this document said "strength is not the obstacle"; the defensible statement is that the magnitudes are comparable, status reading.
3. Direction: compensation geometry
Two bonds at the tetrahedral four-direction angle, tilted into the fourth direction by s and -s, show cos(theta) = (s^2 - 1/3)/(1 - s^2); water's 104.48 is exact at s = 1/3 (Document 1, section 3). Three equivalent bonds tilted equally open the angle (s = 1/3 gives 120 against ammonia's 106.67), so the compensation geometry in one hidden direction fails on the pyramidal class; Document 1, section 5, treats a second hidden direction.
4. Three routes to cos(theta) = -1/4 and their standing
(a) Compensation geometry with s = 1/3 (above). (b) Five equidistant directions in four dimensions: n + 1 equidistant unit vectors in R^n sum to zero when centred, giving mutual cosine -1/n; for five in four dimensions, -1/4 (arccos = 104.4775). Water has four valence pairs; a fifth repelling point (the core pair) would give this angle to water but also to ammonia and H2S, which do not show it. Not favoured. (c) The proton model's push/pull ratio (1/3)/(4/3) = 1/4 read as -cos(theta). A proton (uud) pulls 4/3 and pushes 1/3; a neutron (udd) pulls 2/3 and pushes 2/3, net zero; a nucleus with Z protons and N neutrons pulls (2/3)(2Z + N) and pushes (1/3)(Z + 2N); for N = Z the ratio is 1/2 (arccos(-1/2) = 120). Oxygen-16 has N = Z, so this route would not give water 104.48. Least favoured.
5. Repulsion cannot supply the tilt
For any four unit vectors in any dimension, 0 <= |sum v_i|^2 = 4 + 2 sum of the six pair cosines, so the cosines average at least -1/3 and the largest is at least -1/3, with equality only for the regular tetrahedron, which spans three dimensions. Four mutually repelling pairs therefore gain nothing by leaving the observed three directions. The tilt of Document 1 needs something other than pair repulsion: the nucleus's non-central pull (section 2 shows only that its scale is comparable), or something the shared orbit does because it is anchored to a second nucleus. Repulsion does fix which molecules can tilt in one hidden direction: two bonds can take opposite senses, three cannot, which is the split Document 1 finds.
6. On the tilt value
The orbit model has no intrinsic tilt: the 10 degrees of the stored runs is a launch condition. The value the water fit asks for, s = 1/3 (19.5 degrees), and the heavy value s^2 = 1/3 (35.3 degrees), are not derivable from the model as it stands. Whether a lumpy, rotating nucleus drives the orbit to a definite tilt is the dynamical question. The full test is a two-centre orbit simulation in the four-direction model with the nuclei free to move in the fourth direction: does a bound molecule bend into the fourth direction on its own, and by how much. Not started.
7. Status
Established: the expansions and the sqrt(4/3) ratio (section 1); the bending energies (section 2); the tetrahedral bound (section 5); the simplex cosine and the nucleus push/pull ratios (section 4). Readings: the non-central term as "the precession force"; the scale comparison of section 2; which route gives -1/4; the orbit anchoring as the tilt's source. Withdrawn: the match of 0.1547 to the stored line 0.1546669; the statement that strength is not the obstacle. Open: the tilt's dynamical origin.
Beats between two electrons on one shell
The excess fourth-direction drift over the flat orbit was integrated for tilts from 0.5 to 45 degrees (point nucleus, both nuclear types) and fitted as about 0.10 sin^2(tilt) for the proton and 0.25 sin^2(tilt) for an equal-protons-and-neutrons nucleus. Each measured fine-structure splitting divided by the valence electron's hydrogen-like orbit frequency gives a beat fraction; the tilt difference that produces it on the curve is read off. The tilt differences are read from the lines, not derived.
r=1/4: excess ~ k sin^2(tilt), k = 0.122, 0.120, 0.100 at 2, 3, 5 degrees
r=1/2: excess ~ k sin^2(tilt), k = 0.321, 0.264, 0.251 at 2, 3, 5 degrees
line cm-1 Zeff fraction tilt r=1/4 tilt r=1/2
C 3P 1-0 16.42 1.82 1.81e-04 2.2 1.0
C 3P 2-1 27.00 1.82 2.97e-04 2.8 1.6
C+ 2P 63.42 2.68 3.23e-04 3.0 1.7
N+ 3P 1-0 48.74 2.95 2.04e-04 2.4 1.2
N+ 3P 2-1 82.64 2.95 3.46e-04 3.1 1.8
O 3P 2-1 158.27 2.00 1.44e-03 6.9 4.3
O 3P 1-0 68.72 2.00 6.26e-04 4.5 2.8
O++ 3P 1-0 113.18 4.02 2.55e-04 2.6 1.4
O++ 3P 2-1 193.00 4.02 4.36e-04 3.5 2.2
Ne+ 2P 780.42 3.47 2.36e-03 8.9 5.6
Si+ 2P 287.24 3.29 3.27e-03 10.4 6.5
Independent verification of the precession-force document
Independent verification of Document 3 (the precession force)
9 November 2026. Verifier: a separate Opus agent launched by the author at the stated instruction, read access to the packet, no write access. Report summarised faithfully; the author's response follows.
Verifier's findings
Section 1: the expansion of the model potential to -1/r + (2/3) w^2/r^3, the central-pull expansion -1/r + (1/2) w^2/r^3, the non-central part (1/6) w^2/r^3, the frequencies Omega^2 = 1/r^3 and omega_w^2 = (4/3)/r^3, and sqrt(4/3) - 1 = 0.1547005 are all correct; a numerical check at s = 0.01 gives a ratio 1.15470054. Discrepancy: the match to the stored line 0.1546669 is wrong. In findings.md item 6 and the zeeman outputs, 0.1546669 is the angular frequency in atomic units of the (x1, x2) motion in the s = 100, r = 1 a0, 10 degree reference run (orbital frequency 0.156489975 in zeeman.out), not a drift per orbit; the w line there is 0.4693522, a ratio near 3. The agreement of digits is a coincidence. The point-nucleus run (findings item 5, s = 0.01, tilt 10 degrees: lines 0.99998 and 1.15758, drift 0.1576) is the correct comparison and is consistent with sqrt(4/3) - 1 at finite tilt. The statement that the rate does not depend on tilt holds only in the small-w point-nucleus limit.
Section 2: both scripts reran byte-identical through the runner. Independent minimisation (vectorised, Powell, six starts, three and four dimensions): tetrahedral energy 3.6742346 (6/sqrt(8/3)); bending costs 0.00165, 0.00051, 0.02182; tilts 4.03, 2.24, 14.82 degrees; the four-dimensional results are identical to the three-dimensional ones, supporting section 5. The arithmetic is right; the physical reading is not: (a) the non-central term is positive, raising the energy of a tilted pair and pulling it back, so it is a cost of tilting, not energy available to pay for a bend; (b) 1/6 is the proton's coefficient, and by section 4(c) an N = Z nucleus has non-central part (Z/2) w^2/r^3, three times the proton's per unit charge, with omega_w/Omega = sqrt(2); (c) screening, pair charge 2e and the lone-pair versus bond geometry are ignored.
Sections 4 and 5: arccos(-1/4) = 104.4775; the projection values at s = 1/3 (-1/4 opposite, -1/2 equal); the simplex argument (n + 1 equidistant vectors in R^n, c = -1/n); the Gram argument for four vectors (sum of six cosines at least -2, equality only for the regular tetrahedron); the push/pull formulas including the neutron's net zero; all correct.
Statuses: "the identification of the non-central term and its drift" as established is overstated because of the false match to the stored line; "the force that causes the precession" is defensible with care (no second line or precession without the down-quark term, but the central/non-central split is a choice of reference and the w frequency comes from the 4/3 pull); "strength is not the obstacle" and "suffices in strength" are not established and should be a scale comparison with status reading; the tetrahedral bound, the 4(c) ratios, the "not favoured" and "least favoured" readings and the open dynamical question are honest.
Verdict: arithmetic and geometry check out; two inferences need repair: remove or replace the 0.1546669 match (cite the s = 0.01 run, drift 0.1576 at 10 degrees); reduce the strength argument to a scale comparison.
Author's response
Accepted in full. Section 1 now states the coincidence, cites the point-nucleus run and gives the drift's dependence on tilt and softening; section 2 is rewritten as a scale comparison with the three objections stated; section 7 lists the two withdrawn statements. The author also told the research programme directly that the earlier claim of a match to the stored line was wrong.