Three documents and the dimension sweep, in order: bond angles (pressing and tilt), bending frequencies, and the one-at-a-time and grouped four-direction substitutions.

Bond Angles from the Four-Direction Orbit Picture: Pressing and Tilt

Document 1 of 3. 9 November 2026, revised after independent verification the same day. Author: the author. For assembly into the case. Scripts and outputs: bond_angles.py/.out/.json, bond_angles_followup.py/.out, projection_bend.py/.out, projection_survey.py/.out, tilt_dimensions.py/.out in agents/claude/periodic-table-10d-20261109/. Verification report: verification-doc-1.md. Measured angles were quoted from memory when the work was done; the verifier sourced every one (NIST CCCBDB, Kisiel's microwave tables, primary papers). Values that changed or need a phase label are marked. Measured values are test data, not inputs to any fit except where stated.

1. Setting and premises

Electrons in an atom follow orbits in four directions (the three observed and the fourth) around the nucleus. In a molecule a bonding pair orbits both nuclei (electron sharing is literal); a lone pair orbits the central nucleus only. Electrons do not change orbit size within a bond, so multiple bonds are electrons on equal orbits kept apart by phase. These are the stated premises, recorded in chemistry-note.md.

Two consequences of the orbit picture are used:

(a) Pressing. The pairs on the central atom's shell repel. A lone pair stays near the centre and counts fully; a shared pair spends part of its orbit near the other nucleus and counts for a fraction f near the centre.

(b) Tilt. The four-direction distance from nucleus to electron is what the attraction fixes, so motion into the fourth direction forces the three observed directions to compensate. A bond that is a direction in four dimensions is seen as its projection into three. Lone pairs drive the tilt; by construction of the rule, bonds without lone pairs do not tilt.

2. Pressing: the calibration on water and ammonia

Model: four pairs as points on the unit sphere, Coulomb repulsion, weights (f, f, 1, 1) for two bonds and two lone pairs, (f, f, f, 1) for three bonds and one lone pair; energy minimised over all directions with 24 random starts. The bond angle falls smoothly and monotonically with f: 109.47 at f = 1 (tetrahedral), 104.4 at 0.80, 94.5 at 0.50, 85.2 at 0.30 for AX2E2; 109.47, 106.7, 99.6, 90.5 for AX3E. A variant with the shared pair placed further out instead of lighter is not monotonic and was dropped.

Result: water (104.4776) needs f = 0.803; ammonia (106.67) needs f = 0.799. Two molecules with different lone-pair counts fix the same number to half a percent. One shared-pair weight, 0.80, accounts for both.

Extrapolation by electronegativity fails: O-H and N-H have the same f despite different electronegativity gaps, so every other hydride is predicted near 104 to 106 degrees against measured values near 90 to 93 for the third period and beyond. The weight each heavy hydride would need (H2S 0.443, PH3 0.352, H2Se 0.415, AsH3 0.326, SbH3 0.316, H2Te 0.402) falls with bond length, loosely (f x R^2 ranges from 0.71 to 1.11). Document 2 shows those small weights are inconsistent with the measured bending stiffness (relative to water they would make the heavy bends a factor two too soft); the heavy angles are not made by weaker repulsion. Halides (NF3, OF2, PF3, PCl3, SF2, SCl2, NCl3, Cl2O) need weights from 0.45 to above 1: larger ligands open the angle, which a model without ligand size cannot give.

3. Tilt: the compensation geometry

For unit bond directions with fourth components w1, w2 and four-direction angle Theta:

cos(theta_observed) = (cos Theta - w1 w2) / sqrt((1 - w1^2)(1 - w2^2)).

Opposite tilts (w1 = -w2 = s) close the observed angle; equal tilts open it.

Two bonds at the tetrahedral four-direction angle, tilted oppositely:

Water, 104.4776, is reproduced at s = 1/3 (cos theta = -1/4). This is a one-parameter calibration that lands on a simple number. Two other exact routes give the same number, five equidistant directions in four dimensions and the proton model's push/pull ratio 1/4; water alone does not distinguish them, and Document 3 argues against the push/pull route.

4. Survey of two-bond molecules at s = 1/3

Base angles: 120 with one lone pair, 109.47 with two. Measured values are the verifier's sourced gas-phase values.

The one-lone-pair value 115.94 follows from the 120 base with water's tilt and was not fitted. With the gas-phase nitrite value, all four one-lone-pair molecules are under-predicted, by 0.9 to 3.4 degrees: the 1/9 tilt is systematically a little too large for that class. The two-lone-pair non-hydrides are over-predicted by 1.1 to 1.8.

No lone pairs means no tilt by construction, so CO2 at 180, BF3 at 120 and CH4 at 109.47 are consistency checks, not evidence; three bonds tilted by 1/3 without a lone pair would put BF3 at 133.4.

Heavy and carbene-type two-bond molecules need a larger tilt:

The heavy hydrides cluster at s = 0.555 to 0.574, near 1/sqrt(3) = 0.577: at s^2 = 1/3 the two-lone-pair base projects to exactly 90 degrees. Singlet CF2 at the same s shows the water angle from the 120 base.

What the tilt values do and do not show. For the hydrides there are two values: s^2 = 1/9 for the oxygen and nitrogen hydrides and s^2 near 1/3 for the third-period and heavier hydrides. This is not a rule by period: singlet CF2 and CH2 are second-period centres and need the heavy value, while SCl2, SF2, SO2 and ClO2 are third-period centres and sit near or below 1/9. 1/9 is a fit to water and 1/3 was chosen to match the H2S to H2Te cluster. The reading is restricted to hydrides, and every heavy-hydride prediction at 1/3 (90 degrees) lies below the measured angle, by 0.3 to 2.1 degrees for the two-bond molecules: a systematic offset, not scatter.

5. Pyramidal molecules: tilt into two hidden directions

Three equivalent bonds cannot tilt in mutually opposite senses in one hidden direction; equal tilts open the angle (s = 1/3 gives 120 against ammonia's 106.67). They can tilt into a hidden plane with tilt vectors 120 degrees apart, w_i . w_j = -|w|^2/2. Two natural normalisations exist: the same tilt magnitude as the two-bond case, |w| = s, which gives 108.21 for ammonia at s = 1/3; or the same tilt per hidden direction, |w|^2 = 2 s^2, which gives 106.60. The second was chosen after ammonia failed the one-direction rule, because it reproduces ammonia; that is disclosed here. With it:

Ammonia is reproduced to 0.07 degrees from water's number under the chosen normalisation. The exact values 104.48 (cos -1/4), 106.60 (cos -2/7) and 90 all come from 1/9 and 1/3. All heavy pyramids are under-predicted, by 1.6 to 3.3 degrees, the same sign as the two-bond hydrides.

Pyramidal halides, s^2 per hidden direction that each would need (not s): NCl3 0.072 (sourced angle 107.78; below 1/9), NF3 0.208 (r0 102.37) or 0.217 (r_e 101.85), XeO3 0.197 (crystal value 103; no gas-phase value exists), PCl3 0.243, PBr3 0.230, PF3 0.272, AsCl3 0.262 (rs 98.9: 0.259), AsF3 0.287 (r_e 95.88: 0.290). H3O+ (about 113, cited second-hand) needs none. The halides sit between 1/9 and 1/3 except NCl3, which is below 1/9.

The base has one fourth direction; a second hidden direction must be a pair direction, along which a nucleus's quarks are spread and so have no position. the stated readings: the electron, bound in six directions, herds the quarks enough to give the nucleus a position along a pair direction; or the electron moves into the pair directions on its own. Energy that moves out of the observed directions on one part of the orbit returns on another part or along another direction; the total is conserved. Not computed.

6. How the two consequences divide the work

Document 2 (bending frequencies) shows that the small weights fitted to the heavy angles are inconsistent with the measured stiffness, and that one pressing strength, f = 0.80, is consistent with the heavy hydrides' bends (within 8% relative to water for the two-bond hydrides, within 3% among the heavy pyramids), under the assumption that whatever closes the heavy angles adds no stiffness. Combined with this document the reading is: pressing with f = 0.80 sets the angle for second-period centres; for heavier centres the pressing is the same and the further closure is the tilt, which changes the observed angle without changing the force. The frequencies exclude the alternative (weaker repulsion); they do not establish the tilt on their own. Ammonia's umbrella frequency is a separate, unexplained failure of the repulsion model (Document 2, section 6).

7. Status

Established arithmetic: the projection formula and every predicted or needed value in the tables (verified independently); the repulsion model's calibration values. Measured values: sourced by the verifier; six were corrected or labelled (NO2-, NCl3, H3O+, NF3 r_e, XeO3 phase, SnCl2 unverified). Supported: one pressing weight for water and ammonia; the heavy angles are not from weaker repulsion (with Document 2). Suggestive: the two-bond tilt at s^2 = 1/9 for oxygen and nitrogen hydrides and near 1/3 for heavier hydrides, with a systematic under-prediction of 0.3 to 3.3 degrees at 1/3; the pyramids' tilt into a hidden plane with the same tilt per direction, which reproduces ammonia under a normalisation chosen for that purpose. Open: what fixes 1/9 and 1/3; what gives a nucleus a position along a pair direction; the one-lone-pair class's 0.9 to 3.4 degree under-prediction; halide ligands (ligand size); the carbene-type scatter.

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Bending Frequencies from the Pair-Repulsion Model

Document 2 of 3. 9 November 2026, revised after independent verification the same day. Author: the author. For assembly into the case. Scripts and outputs: bend_frequencies.py/.out, bend_frequencies_fixed_f.py/.out in agents/claude/periodic-table-10d-20261109/ (both rerun after the correction in section 2). Verification report: verification-doc-2.md in the same directory.

1. Purpose

Molecular bending frequencies are among the most precisely measured quantities in chemistry. Document 1 gives a model for bond angles with one parameter, the shared-pair weight f. The same model has a stiffness, the curvature of its energy in the bond angle. This document converts that stiffness to a bending frequency with physical constants and compares it with measurement. No parameter is fitted to a frequency.

2. Model

Energy: four pairs on the central atom's shell, Coulomb repulsion, weights f for shared pairs and 1 for lone pairs, energy unit e^2/R with R the bond length (the shell radius is set equal to the bond length; this choice fixes the absolute scale of every prediction by the same factor, so ratios between molecules are the firmer result). Stiffness k: second derivative of the energy in the bond angle at the measured angle, lone pairs relaxed at each angle (central difference, step 0.5 degrees). Kinetic term: Wilson G-matrix element for the symmetric bend of XH2, G = 2/(m_H R^2) + 2(1 - cos theta)/(m_X R^2), omega^2 = k G (verified against a full B-matrix computation: rigid-bond and full treatments differ by 0.3% at most); for XH3 the symmetric umbrella motion with the bond angle converted exactly to the polar angle and the central atom's recoil along the axis included, inertia 3 m_H R^2 (1 - 3 m_H sin^2 beta / (m_X + 3 m_H)). The first version of this document had the recoil with the wrong sign (inertia raised instead of lowered); the verifier caught it, and the pyramid numbers below are the corrected ones (verified against a full B-matrix computation to 0.3%). Constants: e^2/(4 pi eps0) = 2.307077e-28 J m, u = 1.66053907e-27 kg, m_H = 1.00783 u.

Inputs (angle, bond length in angstrom, bend in cm^-1) with the verifier's sourced values (NIST WebBook / Shimanouchi, CCCBDB) where they differ: H2O 104.48, 0.958, 1594.7 (harmonic 1649); H2S 92.1, 1.336, 1182.6 (harmonic 1215); H2Se 90.9, 1.460, 1034.2; H2Te 90.3, 1.658 (sourced r_e 1.651), 861.0; NH3 106.67, 1.012, 950.2 (centre of 932.4/968.1; sourced 932.5/968.3); PH3 93.5 (sourced 93.3), 1.420 (1.421), 992.1; AsH3 92.1, 1.511, 906.8 (sourced 906.0); SbH3 91.5, 1.704 (sourced 1.700), 782.2 (sourced 781.5). None of the differences exceeds 0.5% in frequency or 0.01 angstrom; the largest effect on a prediction is 0.6% (H2Te, R^-1.5 scaling).

3. Run 1: weight as fitted to each molecule's angle (Document 1, section 2)

Relative to water, the six heavy hydrides are predicted a factor of about two softer than they are (ratios 0.52 to 0.59 against water's 1.075). This comparison does not depend on the absolute scale. The small weights that the heavy angles seemed to need (Document 1, section 2) are inconsistent with the measured stiffness: whatever closes the heavy angles, it is not weaker repulsion.

4. Run 2: weight held at the second-period value, angle as measured

Within-family ratios, which do not depend on the scale choice: H2S : H2Se : H2Te relative to water predicted 0.684 : 0.602 : 0.500 against measured 0.742 : 0.649 : 0.540, about 8% low throughout; AsH3 : SbH3 relative to PH3 predicted 0.909 : 0.761 against measured 0.914 : 0.788. The near-exact absolute agreement for the heavy XH2 molecules in the table therefore rests on the shell-radius choice happening to land near water's 7.5% excess; the robust statement is the 8% within-family agreement.

Run 2 evaluates the curvature at the measured angle, which for the heavy molecules is not the model's own minimum (at f = 0.80 the minimum is near 104 degrees). This presumes that whatever holds the heavy angle at 90 to 93 degrees does so without adding stiffness of its own. That is an assumption about the tilt of Document 1, not something these frequencies demonstrate.

5. Interpretation

• Robust: the fitted small weights are inconsistent with the stiffness (section 3). The heavy angles are not made by weaker repulsion.
• Suggestive: with one shared-pair weight, 0.80, for every hydride, the heavy two-bond hydrides are predicted within 8% relative to water and within 1% absolutely at the scale used; the heavy pyramids within 8 to 12% absolutely and within 3% relative to each other.
• Not demonstrated: that the heavy angles are geometric (the tilt). The frequencies are consistent with it, under the assumption in section 4, and they rule out the alternative of weaker repulsion. They do not establish the tilt on their own.

6. Ammonia: an unexplained model failure

Ammonia's umbrella mode is predicted at 1832 cm^-1 against 950, a factor 1.93. The measured value is the centre of the inversion doublet (932.4 and 968.1), and the nitrogen does pass through the plane of the hydrogens over a barrier of about 2000 cm^-1 (not sourced here). A double well of that depth lowers the observed fundamental below the harmonic value by roughly 7 to 10%; the harmonic umbrella frequency is about 1020 to 1060 cm^-1 (verifier's figure, from memory, not sourced). That accounts for a small part of the discrepancy and leaves a factor of about 1.75. Ammonia is a failure of this model, not an inversion effect. The pyramids of the third period and beyond, whose barriers are high, are described within 8 to 12%.

7. Caveats

The shell radius is taken as the bond length. The model gives a harmonic curvature and should be compared with harmonic frequencies where known (water and H2S above). Central masses are standard atomic weights (effect below 0.1%). Measured inputs are from memory, verified within the thresholds stated in section 2.

8. Status

Established arithmetic: the stiffnesses, the kinetic terms (XH2 exact to 0.3%; XH3 corrected and verified to 0.3%), the conversions. Supported: the fitted small weights are excluded by the stiffness. Suggestive: one pressing weight across both periods. Open: the ammonia factor 1.75; the shell radius; whether the tilt adds stiffness; a full G-matrix treatment.

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Switching the three-dimensional formulas to four directions, one at a time and together

9 November 2026. Scripts: force_law_test.py (.out/.json), dimension_sweep.py (.out/.json), hidden_separation.py (.out/.json), hidden_separation_abs.py (.out). Calculation: repeat the calculation changing one formula at a time to four dimensions, then in groupings, until the solutions appear. Score: bending frequency ratio to water against measured (scale-free), rms of log ratios; the ammonia value separately.

Switches

law: pair potential 1/d (three-direction Coulomb) or 1/d^2 (four-direction Gauss form). length: bond length as measured, or the four-direction length R/sqrt(1 - |w|^2) from the tilt. angle: pairs at the observed angle with a three-direction kinetic term, or at the four-direction angle (tilt undone) with the four-direction kinetic term (the coordinate Jacobians cancel, so this is the four-direction frequency). separation (fourth switch): pair distances measured with the bonds' hidden components included (two bonds at +s, -s in the fourth direction; three bonds with tilt vectors 120 degrees apart in a hidden plane), lone pairs free in the full space.

Results

Per-molecule fitted weights score worse than one weight in every configuration (heavy rms 0.68 to 1.68), the factor-three softness of the fitted small weights persisting under every law.

Fourth switch, hidden separations with the 1/d law, three-direction length and kinetic term:

• Weights: water 1.000, ammonia 1.005. With the bonds' hidden components in the separations, the shared-pair asymmetry disappears: equal pairs, tilted as Document 1 describes, give both angles. The heavy hydrides need 1.01 to 1.11 at their tilts (H2S 1.067, H2Se 1.028, H2Te 1.008, PH3 1.111, AsH3 1.068, SbH3 1.049).
• Frequencies, equal weights, absolute: H2O 1825 (1.145), NH3 1887 (1.99), H2S 839 (0.71), H2Se 728 (0.70), H2Te 604 (0.70), PH3 647 (0.65), AsH3 571 (0.63), SbH3 473 (0.61). Relative to water the heavy six are 0.53 to 0.62 (stiffness about one third), ammonia 1.74.

What was found and what was not

• No configuration of the three switches improves on the original. The original with one weight reproduces the six heavy bends to 6% relative to water; every four-direction substitution makes them worse, the bond length worst. The heavy hydrides' bending is three-dimensional in force law, length and motion.
• Ammonia's factor stays between 1.46 and 1.99 in every configuration, including the fourth switch. It is not an artefact of any of the four three-dimensional assumptions identified. Not solved.
• The fourth switch gives one new result for the angles: with hidden components included in the separations, no shared-pair weight is needed; equal pairs tilted by 1/9 (one direction) and 2/9 (two directions) give water and ammonia, and the heavy hydrides are within 1 to 11% of equal weight at their tilts. So the 0.80 of the pressing model may be the three-direction shadow of equal pairs separated in hidden directions. The same picture fails the frequencies: heavy stiffness one third of measured, ammonia twice to three times.
• The two pictures are complementary and inconsistent with each other: the three-direction pressing with f = 0.80 gets the frequencies and needs the tilt for the heavy angles; the tilted equal-pair picture gets all the angles and fails the frequencies by factors near sqrt(3) in both directions (heavy low, ammonia high).

Remaining candidates (not formula substitutions)

• Ammonia's umbrella as a motion of the tilt vectors in the hidden plane rather than of the hydrogens in three directions; a rigid rotation of the tilt vectors preserves all separations and so has no restoring force in these models, and a change of tilt magnitude needs a law for what fixes the tilt, which is not known.
• A force law for the pair-pair tail from the field equation itself rather than any power law.
• The observed bend of the heavy hydrides treated as the three-direction projection of a mixed motion, with the tilt oscillating; needs the same unknown law.

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Independent verification of the bond-angle document

Independent verification of Document 1 (bond angles: pressing and tilt)

9 November 2026. Verifier: a separate Opus agent launched by the author at the stated instruction, read access to the packet and the web, no write access to the packet. Report summarised faithfully below (the full text is retained in the session archived record); the author's response follows.

Verifier's findings

Formulas: all correct. The projection formula was derived independently and checked on random four-vectors to 1e-15; opposite tilts close and equal tilts open the angle; the hidden-plane case gives cos theta = (c + |w|^2/2)/(1 - |w|^2), so |w|^2 = 2/9 gives -2/7 (106.6015) and 2/3 gives 90; five equidistant directions in four dimensions have cosine -1/4 (104.4775); CF2 at s^2 = 1/3 on the 120 base gives 104.4775; BF3 with an equal 1/3 tilt would be 133.43.

Arithmetic: every number in the tilt table, the one-lone-pair prediction 115.94, every "s needed" value and every pyramidal per-direction value reproduces; the pressing-curve values and calibrations 0.803 and 0.799 match bond_angles.json; the three fast scripts rerun byte-identical through the compute runner; bond_angles_followup.out is consistent with bond_angles.json.

Labelling errors: (a) the pyramidal halide list in section 5 gives s^2 per direction but the text calls them tilts next to tables that list s; (b) "between the two values" is false for NCl3, which is below 1/9.

Measured values against sources (NIST CCCBDB experimental geometry pages; Kisiel's microwave tables quoting primary papers; primary papers where named). Hold within 0.3 degrees and 0.01 angstrom: H2O 104.4776, NH3 106.67, H2S 92.11, PH3 93.3 (r0; r_e about 93.36), H2Se 90.9, AsH3 92.083, SbH3 91.557 (R 1.700), H2Te 90.26 (R 1.651), O3 116.8, SO2 119.5 (r0; r_e 119.33), ClO2 117.40, CF2 104.86, SiF2 100.77, GeF2 97.15, CH2 singlet 102.4 (from memory), SiH2 92, OF2 103.1, Cl2O 110.89, NH2 103.4, SCl2 102.71, SF2 98.05, PH2 91.5, NF3 102.37 (r0; r_e 101.85), PF3 97.7 (R 1.561), PCl3 100.1, PBr3 101, AsF3 96.2 (rg; r_e 95.88), AsCl3 98.6 (rs 98.9). Wrong or wrong phase: NO2- 117.5 +- 2 gas phase (Ervin, Ho and Lineberger 1988; 115.4 is the NaNO2 crystal value); NCl3 107.78 (Cazzoli, Favero and Dal Borgo 1974); H3O+ about 113 (Tang and Oka 1999, cited second-hand); XeO3 103 is crystal-only (Templeton et al. 1963); SnCl2 95 unverified (no authoritative source found). No bond length off by more than 0.01 angstrom (largest PF3 0.009, H2Te 0.007, NF3 0.006), so the f x R^2 statements do not change.

Recomputed: NO2- error -1.6 (s needed 0.267); NCl3 s^2 per direction 0.072; NF3 at r_e 0.217; AsF3 at r_e 0.290; AsCl3 at rs 0.259; PH3 error -3.3 to -3.4; PH2 s 0.562; H3O+ still none. With the gas-phase NO2- all four one-lone-pair molecules are under-predicted (O3 -0.9, NO2- -1.6, ClO2 -1.5, SO2 -3.4): the 1/9 tilt is systematically too large for that class and the NO2- match disappears.

Overclaims: (1) "s^2 = 1/9 for the second period and 1/3 for heavier" is contradicted by the document's own tables: CF2 and CH2 (second period) need the heavy value; SCl2, SF2, SO2, ClO2 (third-period centres) sit near or below 1/9. The rule holds for hydrides only. 1/9 is a fit to water; 1/3 was chosen to match the H2S to H2Te cluster. (2) Every 1/3 prediction (90) is below the measured angle (by 0.3 to 2.1 for two-bond, 1.5 to 3.5 for pyramids): a systematic offset, not scatter; "suggestive" fits better than "supported". (3) NH3 to 0.07 rests on choosing |w|^2 = 2 s^2 after ammonia failed the one-direction rule; the other natural normalisation |w| = s gives 108.21; the document should say so. (4) The measured column was unverified and contained crystal values. (5) Water at s = 1/3 is a one-parameter calibration landing on a simple number, correctly marked "sets s". (6) "Roughly as 1/R^2" is loose (f x R^2 from 0.71 to 1.11); "no lone pairs means no tilt" is built into the rule, so CO2, BF3, CH4 are consistency checks, not evidence; the Open list is honest.

Verdict: geometry and arithmetic correct; scripts reproduce; most measured values hold to 0.3 degrees; six values wrong or wrong phase (NO2-, NCl3, H3O+, XeO3, NF3 at r_e, SnCl2 unverified); these do not affect water, ammonia or the heavy-hydride 1/3 cluster but remove the NO2- agreement and push NCl3 outside the halide band; main weakness interpretive: narrow the 1/9-1/3 reading to hydrides, disclose the systematic under-prediction and the post-hoc normalisation choice; fix the section 5 label and sentence; replace or label the six values.

Author's response

Accepted in full. Document 1 was revised: measured values replaced (NO2- 117.5 gas phase, NCl3 107.78, H3O+ about 113, NF3 r_e 101.85 alongside r0) or labelled by phase (XeO3 crystal, SnCl2 unverified); the one-lone-pair class now reads as systematically under-predicted by 0.9 to 3.4 degrees; the two-value reading is restricted to hydrides with the contradicting non-hydrides named; the systematic sign of the heavy-hydride errors is stated and the status lowered to suggestive; the |w|^2 = 2 s^2 choice is disclosed as one of two natural normalisations chosen after ammonia failed the one-direction rule, with the alternative's value given; the section 5 label and sentence are fixed; CO2, BF3 and CH4 are described as consistency checks. The earlier notes (bond-angles-note.md, projection-survey-note.md, tilt-dimensions-note.md) are superseded by the document on these points and are left unchanged as frozen records.

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Independent verification of the bending-frequency document

Independent verification of Document 2 (bending frequencies)

9 November 2026. Verifier: a separate Opus agent launched by the author at the stated instruction, with read access to the packet and the web, no write access to the packet. Report reproduced verbatim below; the author's response follows.

Verifier's report

Reruns: both scripts ran through scripts/agent-compute.py; outputs match the stored .out files exactly; the tables in sections 3 and 4 match.

Physics: constants correct (e^2/4 pi eps0 = 2.307077e-28 J m, u = 1.66053907e-27 kg, m_H = 1.00783 u, nu = omega/(2 pi c)). The XH2 formula G = 2/(m_H R^2) + 2(1 - cos theta)/(m_X R^2), omega^2 = kG, is the standard Wilson element; a full numerical A1 G-matrix from the B-matrix agrees, rigid-bond elimination of the stretch changes the result by at most 0.3% (H2O 1714 -> 1711). The XH3 umbrella kinetic term had the recoil sign wrong: the script used I = 3 m_H R^2 (1 + 3 m_H sin^2 beta / m_X); recoil must lower the inertia; the correct value is I_beta = 3 m_Y R^2 (1 - 3 m_Y sin^2 beta / (m_X + 3 m_Y)) = 3 m_Y R^2 (m_X + 3 m_Y cos^2 beta)/(m_X + 3 m_Y), equivalently G_theta = (2/3)(1 + 2c)(m_X + 3 m_Y)/[(1 + c) m_Y R^2 (m_X + m_Y (1 + 2c))] with c = cos theta; a full B-matrix check agrees to 0.3%. The document's G was too small by 0.715 (NH3), 0.877 (PH3), 0.947 (AsH3), 0.967 (SbH3). Corrected pyramid predictions at f = 0.799: NH3 1832, PH3 1111, AsH3 1010, SbH3 845 (ratios 1.93, 1.12, 1.11, 1.08); run-1 values PH3 584, AsH3 495, SbH3 404, NH3 1832. The bond-angle to polar-angle conversion is exact.

Sourced values (NIST WebBook/Shimanouchi, CCCBDB): H2O bend 1594.59 (harmonic 1649), R 0.958, angle 104.4776; H2S 1182.7 (harmonic 1215), 1.336, 92.11; H2Se 1034.2, 1.460 (r0), 90.9; H2Te 861, R 1.651 (r_e; document 1.658), 90.26; NH3 950 (932.5/968.3), 1.012, 106.67; PH3 992.1, 1.421, 93.3 (document 93.5); AsH3 906.0, 1.511, 92.083; SbH3 781.5, 1.700 (document 1.704), 91.557. No value exceeds the flag thresholds (0.5% frequency, 0.01 angstrom). Rescaled ratios with the corrected kinetic term: NH3 1.93, PH3 1.12, AsH3 1.115, SbH3 1.085. Central masses are standard atomic weights (effect below 0.1%). The model gives a harmonic curvature and should be compared with harmonic frequencies where known: H2O 1714/1649 = 1.039, H2S 1172/1215 = 0.965.

Interpretation: "heavy pyramids within 5 to 8%" is wrong; corrected, 8 to 12% high, ammonia 93% high. The relative ratios do not support "one universal strength" as stated: the absolute scale is arbitrary (shell radius = R); within-family ratios miss by about 8% (XH2) and PH3/NH3 is predicted 0.606 against 1.044 measured. Run 2 evaluates curvature away from the model's minimum and assumes the tilt holds the angle without adding stiffness; that is an assumption, not demonstrated. Run 1's constant factor shows the small fitted weights are inconsistent with the bends but does not by itself discriminate in favour of the tilt. Ammonia facts correct (doublet 932.4/968.1; barrier about 2000 cm^-1, about 2020 in the literature, not sourced; nitrogen passes through the plane), but a double well of that depth lowers the fundamental by roughly 7 to 10% (harmonic omega2 about 1020 to 1060 cm^-1, from memory); it cannot account for a factor 1.6 to 1.9. The excess is a model failure, not an inversion effect.

Verdict: arithmetic reproduces exactly; XH2 treatment and constants correct; measured data hold within thresholds (largest differences 0.007 angstrom for H2Te R, 0.09% for AsH3 and SbH3 frequencies); XH3 recoil sign error must be fixed; the step to "one universal pressing strength, heavy angles are geometric" is not supported as stated; present run 2 as suggestive, compare against harmonic frequencies where available, treat ammonia as an unexplained model failure.

Author's response

Accepted in full. The recoil sign is corrected in both scripts and both were rerun; the document's tables now carry the corrected pyramid values. Sections 4 to 8 of the document were rewritten: the within-family ratios are given as the robust comparison, the absolute agreement is attributed to the scale choice, the non-stationary-curvature assumption is stated, the interpretation is reduced to "the fitted small weights are excluded; one weight is consistent; the tilt is not established by the frequencies", harmonic values are quoted for water and H2S, and ammonia is recorded as an unexplained model failure. The sourced input values are listed in section 2 of the document; the predictions were not rerun for them since the largest effect is 0.6%. Document 1's references to Document 2 were softened to match.

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