Ampere: arbiter's report (Step 3)

Project: Assis and Chaib, Ampere's Electrodynamics (Apeiron 2015), the only document (PDF with text layer).

Source gap note (repeated from claims.md): there is no unreadable-text gap, but the book was read selectively. The 1826 Theorie translation (Part VII) and the translations of Ampere's early papers (Part VI) were not read by Step 1 or by any reviewer. Statements about their content rest on the book's own commentary and table of contents. The book is by two partisans of Ampere/Weber. Nothing here rests on [memory] alone for a verdict, but several verdict-relevant outside sources (Moyssides 2014, Graneau, Wesley, Peoglos) were seen only as abstracts or search snippets; no reviewer or I read those papers in full.

Reading guide for the reader

Ampere's 1820s force law gives the force between two small pieces ("elements") of current-carrying wire. Mainstream physics instead uses the Biot-Savart law plus the Lorentz force. For complete loops the two give the same net force; they differ only in how they split it among elements. Most of the dispute is therefore about what can be measured on pieces of a circuit, not on whole circuits.

Disagreements recomputed

D1. Do all values of k give the same force on closed circuits? The three reviewers disagreed. The physicist said the k-dependent part integrates to zero on closed circuits. The contrarian said the whole family indexed by k gives identical closed-circuit forces. The mathematician said this is false and that only k = -1/2 matches the field law.

I recomputed with my own script (scratch file, not part of the book): two skew circular loops, 600 elements each, central force (1/r^2)(cos eps + h cos theta cos theta') r-hat with h = k - 1, compared with the Grassmann (Biot-Savart-Lorentz) force ds2 x (ds1 x r-hat)/r^2 [computed]. Two different loop geometries gave the same pattern:

So in the book's parametrisation, with its prefactor held fixed, the net force on a closed loop changes with k, but only as an overall factor (2+k)/3, along the same direction as the Grassmann force. Two conclusions: - The mathematician is right that the force is not k-independent for this form of the law. - All three are partly right. The direction and geometry dependence of closed-circuit forces is the same for every k, so closed-circuit experiments cannot tell the shape of the law apart; k enters only as a scale that is degenerate with the overall constant. The sign (-1/2 times Grassmann) shows my normalisation differs from the book's by a convention; I did not check the book's prefactor. Maxwell's own statement is that closed-circuit experiments cannot fix the element law without an extra assumption, and that Ampere's (third law) and Grassmann's (zero collinear force) are two such assumptions [web: https://en.wikipedia.org/wiki/Amp%C3%A8re's_force_law, retrieved by search summary]. That page also says that integrating around the second circuit removes k in its form of the law. That is a different parametrisation from the book's, and I did not reconcile the two. Treat "which k-family leaves closed-circuit forces unchanged" as open at the level of normalisation, but settled at the level of the physics: closed-circuit data do not distinguish Ampere from Grassmann.

The physicist's separate numerical check (force along an element from a closed loop vanishes only at h = -3/2) is consistent with the mathematician's exact-differential argument: only k = -1/2 gives the "Lorentz-like" property that the force from a closed circuit on a test element is perpendicular to it.

D2. Is the "bridge self-force" a defect of the field law? The physicist said refuted by calculation. The mathematician said "tested, not supported". The contrarian said "not supported" for the defect framing but "never properly tested" for whether Lorentz quantitatively matches the bridge. Resolution: in the Lorentz picture the force on an element is i dl x B with B from all currents in the circuit, including the bridge. That is the definition of the law, not an addition [read: the book's own Sec. 21.2 concedes the field explanation works only with this term, per the physicist and contrarian]. The framing "needs an extra self-force" is therefore a definitional point, settled by the law's definition. The separate question, whether the quantitative bridge forces agree, rests on Moyssides (2014): he reports that the theoretical velocity of a floating wire in a mercury trough, computed with both the Biot-Savart-Grassmann-Lorentz and Ampere laws, agrees within error with the measured velocity [web: https://link.springer.com/article/10.1140/epjp/i2014-14034-2 via search abstract; the page itself needs a login redirect and I could not read the full text]. That is one author's report, not a neutral replication, and not a computation by any reviewer or me.

D3. How to label the bridge result. The physicist and mathematician split it into "observation: Established" and "discrimination: Not testable as stated". The contrarian split it into "observation: Established (replicated)" and "discrimination: Never properly tested by a neutral control". Both agree on facts. I use "Not testable as stated" for discrimination between Ampere and Lorentz on net force of a movable open piece in a complete circuit, because both laws predict the same net force there (Moyssides above; Maxwell above). I keep the contrarian's point visible: that is about this class of experiment. Other claimed discriminating tests (Graneau's wire-fragmentation and longitudinal-force experiments) are "Never properly tested by a neutral party", not dismissed without testing, since mainstream authors did publish rebuttals.

D4. Where k = -1/2 comes from. The contrarian and mathematician corrected Step 1 in compatible ways. The null experiment (no continuous rotation) gives k = (1-n)/2 only for a given n. Ampere assumed n = 2 in 1822 and said there was "no proof deduced directly from experiment" [read, Sec. 7.5, via physicist and mathematician]. An astatic-coil experiment (performed March 1822, analysed and first presented to the Academy June 1822; Round 2 correction, [read: deep read, PDF pp.132-135]) showed no continuous rotation, which with the exponent n gives k = (1-n)/2, so n > 1 means k < 0, and the bridge test (Sept. 1822, Geneva, de la Rive) then confirmed the surprising consequence (collinear repulsion) [read, Sec. 8.3, via contrarian]. Ampere never performed his 1825 "similarity" equilibrium (he said he had not built the instruments); only Ettingshausen (1878), Felici (1882) and Tricker did, and they found n = 2 [read, PDF p.191-192, via QE]. CORRECTED in the QE revision below.

D5. Weber verdict. The physicist gave "Tested, not supported", the other two "Historically superseded but instructive". Resolution below under Claim 5.

Verdicts per claim

Claim 1: central 1/r^2 force with k = -1/2, strong third law

• Agree: math is consistent (all three; mathematician re-derived k = (1-n)/2 from exactness of the torque form; contrarian and physicist numerics agree). Closed-circuit forces are consistent with all data.
• Disagree: whether "unique element law" is untestable, never tested, or superseded.
• Stronger argument: the physicist's and mathematician's. Element forces are not separately measurable, and the strong third law for elements is replaced in mainstream by field momentum (Poynting; radiation pressure) [memory, high; consistent with the contrarian's note]. The contrarian's added point stands: the discriminating tests were run mostly by advocates.
• Final verdict (split): closed-circuit forces from the law: Established. Unique per-element law: Not testable as stated for steady closed-circuit data; for the open-element tests (Graneau and others) Never properly tested by a neutral party, not dismissed without testing. As a complete electrodynamics (no radiation, no retardation): Historically superseded but instructive.

Claim 2: k from experiment; bridge confirms collinear repulsion

• Agree: the bridge moves away from the current source and this has been reproduced; the sign chain n > 1, k < 0 gives repulsion for collinear elements (1 - 3/2 = -1/2) [computed by all three].
• Disagree: only the label for the discrimination point (D3).
• Final verdict (split): the derivation and the observed bridge motion: Established. That the bridge confirms Ampere over Lorentz: Not testable as stated by this experiment, since the Lorentz law with the full-circuit field gives the same net force (Moyssides, one author; Maxwell). k = -1/2 came from the null experiment plus an assumed n = 2 (D4), not from the bridge.

Claim 3: field law gives zero collinear force and breaks the strong third law; needs a bridge self-force

• Agree: the element-level Grassmann force is zero for collinear elements and not equal-and-opposite (ds' parallel to r-hat kills the cross product) [calculation, mathematician and contrarian].
• Disagree: how to label the "needs self-force" part.
• Stronger argument: all three agree the self-term is part of the definition; I side with the physicist that this is settled by definition.
• Final verdict (split): zero collinear force and third-law violation at element level: Established (mathematics, and a different element law, not a flaw). "Needs an ad hoc self-force so the field law is defective": Refuted by calculation as a defect claim (the term is part of the standard field calculation). Whether the field law quantitatively reproduces the bridge force: Plausible, untested by neutral parties (one author reports agreement; no independent replication seen).

Claim 4: all magnetism reduces to currents

• Agree: weak form fine; electron spin and nuclear moments are intrinsic, so "literal molecular currents" is not how modern physics describes it. Bohr-van Leeuwen: classical physics gives no magnetisation [memory, high; not web-checked, because no verdict turns on it].
• Final verdict (split): dipole = loop, solenoid = magnet, magnets described by bound currents for fields and forces: Established. All magnetism is classical circulating currents: Historically superseded but instructive.

Claim 5: Weber's law is a better foundation than the field concept

• Agree: Weber's law reproduces Coulomb, Ampere and induction in slow regimes; it has no retardation, no waves, no Lorentz invariance; Hertz's 1888 radio waves and Michelson-Morley-type results favour Maxwell [web: https://en.wikipedia.org/wiki/Weber_electrodynamics: "only applicable for electrostatics, magnetostatics and the quasistatic approximation", "not suitable for describing electromagnetic waves"; Helmholtz's negative inertial mass objection is disputed; neon-lamp tests showed no deviation from Maxwell]. The book gives Weber only about two pages and no derivation (mathematician, [read]).
• Disagree: the label.
• Stronger argument: the physicist's, using the scale as written. "Tested, not supported" applies where experiments were run and went against it (waves, relativistic effects). "Historically superseded" describes the same fact in historical terms. I keep the contrarian's distinction: no head-to-head controlled test of Weber against Maxwell was run; Maxwell simply explained more.
• Final verdict (split): unification in the slow regime (Coulomb, Ampere, induction): Plausible, untested against Maxwell in any regime where they differ. "Better foundation / replacement for the field concept": Tested, not supported (radiation and relativistic phenomena, tested by Hertz and later), and Historically superseded but instructive. Modern retarded variants of Weber exist [web, found via contrarian: https://www.researchgate.net/publication/370275906_On_the_Modernisation_of_Weber's_Electrodynamics] and were not assessed here.

Tested / never tested / dismissed (kept visible)

Corrected claim list

Numbering below follows the revised claims.md (1-6). The sections above use the older numbering (their Claim 4 = new claim 5, their Claim 5 = new claim 6; the magnet-rotation claim is new claim 4). Changes from QE are marked (QE).

1. The element force is central, 1/r^2, with angle factor sin(alpha) sin(beta) cos(gamma) + k cos(alpha) cos(beta) and k = -1/2; it obeys the strong third law and, with the book's prefactor (eq. 2.16), gives exactly the Biot-Savart-Lorentz (Grassmann) force on any element from a closed source circuit (QE). Closed-circuit data cannot fix the element law.
2. (QE) The constants came from null results: k = -1/2 follows from the 1821 Ampere-Despretz non-rotation (in hindsight), the 1822 null rotation and Savary's 1823 combination with the ring-magnet null. n = 2 was asserted in 1822, derived by Savary in 1823, and measured directly only in 1878-1882 (Ettingshausen, Felici, Tricker). Ampere never ran his own similarity experiment, so "deduced uniquely from experience" is partly rhetorical. (Round 2) Ampere's own 1820 text says the element law "could not be found by experiment" and is a hypothesis (PDF pp.305, 307), and in Sec. 29.1 he claims to rely "solely on experimentation" (PDF p.346); the null-rotation date is March 1822 (performed) and June 1822 (analysis, k = (1-n)/2).
3. (QE) The 1822 bridge moves, as Ampere predicted from k < 0 (collinear repulsion). The field law gives zero collinear force but, using the field of the whole circuit including the bridge, gives the same net force on the bridge; for a closed source the field law with the bridge's self-term equals Ampere's law without it, element by element. So the bridge cannot discriminate the laws, and the claim that the field law fails quantitatively is refuted by calculation. Moyssides 2014 is no longer needed to support agreement.
4. (QE, new) A magnet spins about its own axis (Ampere 1821-22; Faraday had thought it impossible); an opposite reaction torque acts on the external wire or mercury. The book's charge that the field explanation needs a self-torque violating Newton's third law is a bookkeeping point: net torque on the whole closed system is zero in both pictures.
5. Dipole = small loop, solenoid = magnet is standard (Established). Spin magnetism is intrinsic, not classical circulating charge. (Round 2) Ampere himself conceded in 1820 that a bare helix acts as a magnet only if the axial return wire cancels its straight-wire component (Part VI fn 32, PDF p.319). A 1914 Nature note called Onnes's persistent current an "Ampere molecular current" (digest only).
6. Weber's force law covers slow electrostatic, Ampere and induction phenomena but has no waves and is not Lorentz invariant; the strong claim that it is the better foundation is not supported. (Round 2) Ch. 22 was deep-read: it has no Weber equation, derivation, experiment or regime of validity and never compares Weber with fields, so "better foundation than fields" is the reviewers' and Assis's wider argument, not something the book demonstrates.

What would settle it

• (Superseded by the QE revision: the quantitative bridge question is settled by calculation.) Whether the quantitative bridge force agrees with the field law: a numerical Biot-Savart-Lorentz calculation of the full hairpin-in-mercury geometry (including the current spread in the mercury), plus a force measurement on a bridge with a pre-agreed protocol, run by someone other than Ampere/Weber partisans or Lorentz partisans, with data and code shared. Convention: a simple version of the calculation is a good exercise for the reader.
• Whether the Ampere element law is distinguishable at all: find any observable (not net force on a closed circuit, not net force on a piece in a complete circuit) on which k = -1/2 and Grassmann predict different measurable results. If none exists, the claim is Not testable. Exploding-wire fragmentation is the only candidate cited, and it needs a controlled analysis of thermal and magnetic-pressure effects.
• Reconciling normalisations: a short symbolic calculation showing exactly which family of k and overall constants leaves closed-circuit forces equal to Grassmann's (my numerics show (2+k)/3 scaling in the book's parametrisation).
• Weber: a retarded-Weber model that reproduces radiation and relativistic data, or a regime in which Weber and Maxwell predict different, measurable results.

Open questions

1. Is there any measurable quantity on which the Ampere and Grassmann element laws differ?
2. How strong is the Moyssides result? I saw the abstract and summary only.
3. Do the Graneau wire-fragmentation and Wesley/Peoglos experiments hold up under a neutral replication?
4. CLOSED (QE): Sec. 21.3 was deep-read (see QE revision). Still open: whether Davy's 1823 mercury experiment tests the reaction-torque prediction, and the cited counter-rotation papers (not read).
5. PARTLY CLOSED (Round 2): Part VI (PDF pp.290-323), Sec. 29.1 (PDF pp.345-349), Ch. 6.3-7.5, Ch. 22 and PDF pp.420-465 were deep-read. Still not deep-read: the 1826 Theorie Sec. 29.2-29.11 (the four equilibrium cases in Ampere's own words) and Note 31.1.
6. CLOSED (QE): prefactor confirmed as eq. 2.16 with ii' <-> (mu0/2pi)II' [read: PDF p.32]; with it k = -1/2 matches Grassmann.

Final verdict summary

(New claims.md numbering. Rows changed by QE are marked.)

Web sources: https://link.springer.com/article/10.1140/epjp/i2014-14034-2 (abstract via search); https://en.wikipedia.org/wiki/Weber_electrodynamics; https://en.wikipedia.org/wiki/Amp%C3%A8re's_force_law (via search).

QE revision

QE verdict was MINOR. Round 1 used deep read 1 (Ch. 21, PDF pp.266-281, Opus, deep/Amperes-Electrodynamics-28a544.md). Round 2 (below) adds the other deep reads appended to the same file (Ch. 22; Ch. 6.3-7.5; ether and vis viva, PDF pp.420-465; Part VI, PDF pp.290-323; Parts VII-VIII opening, PDF pp.324-349). The text below this paragraph up to "Round 2" is the Round 1 record. I did no new web checks; the one verdict-relevant [memory, medium] item (Davy 1823) is left "unknown" rather than asserted.

New evidence

• Book concession [read, PDF p.271]: "Grassmann's force can only explain quantitatively the measured values ... if one includes the force exerted by the bridge on itself." The book gives no numbers and names no specific calculation (footnotes 11-12 list Assis and Bueno papers and books only).
• Identity check [computed, deep read: 200,000-segment closed tilted-ellipse source on one element]: Ampere force (1.28957, -0.89818, -0.66334)e-3 equals Grassmann force to every printed digit. Hence Grassmann(whole circuit incl. bridge) = Ampere(rest of circuit), because Ampere's bridge self-force is zero. QE's example.py agrees (to 3e-14, eq. 2.16 prefactor).
• Magnet self-rotation (Sec. 21.3): Faraday (1822) thought it impossible; Ampere did it Nov-Dec 1821, reported 7 Jan 1822. Faraday and Biot explain it by the magnet's pole acting on current inside the magnet. Ampere says the driving torque comes from current in the mercury or external circuit, with an equal opposite reaction. Order-of-magnitude [computed, deep read]: with i = 1 A, B about 0.5 T, a = 6 mm, driving torque about 9e-6 N m versus self-terms about 6e-10 N m (ratio 1e-4). The "self-torque" is a split of forces, not a net torque on an isolated system. Ampere's insulating-coat control (Theorie Sec. 29.22) and his energy argument against distance-only forces are further points. Note: the heading of Sec. 21.3.6 puts "Ampere" in quotes; its third-law argument is the authors' reconstruction, not Ampere's text.
• Davy 1823 is cited as confirming mercury counter-rotation, but Ch. 21 gives no details, and the confirming quote is third-person text from Ampere's own bulletin. Not confirmed.
• Spot checks from QE: Ampere said n = 2 had "no proof deduced directly from experiment" because the experiments used magnets (PDF p.141); the "elongate the current" quotation is a contemporary journal report, not Ampere's words; the Grassmann critical angle is arccos(sqrt(2/3)) = 35.26 degrees.
• Missed material now recorded (QE section 3): the Ampere-Despretz null of 20 Jan 1821 (torque proportional to (1-k) + (1+2k) sin^2, zero net torque gives k = -1/2), Savary 1823 (n - 1 + 2k = 0 with kn + 1 = 0, giving (k, n) = (1, -1) or (-1/2, 2)), Biot-Savart bent-wire tan(rho/2) error cancellation, Ampere's energy argument, his hedging on whether the force is truly elementary (ether or fluid origin; digest contradiction at PDF p.424 vs p.461 unresolved), and Grassmann and Clausius independently finding the same element force.

Changed verdicts

1. Claim 3, quantitative bridge agreement: "Plausible, untested neutrally" becomes Refuted by calculation (for the claim that the field law fails quantitatively). Reason: the identity above, plus the book's own concession, make the two laws' net bridge force equal for any closed source; no neutral experiment is needed, and Moyssides 2014 (abstract only) is no longer load-bearing. The calculation is shown in the deep read and in primer Part B. What remains is interpretation, not a measurable difference.
2. Claim 2 / D4 wording: Ampere did not perform the similarity experiment; "repeated by Ettingshausen, Felici and Tricker" is wrong. n = 2 was asserted (1822), derived (Savary 1823), measured (1878-1882). The k = -1/2 route also has the 1821 Despretz null.
3. Claim 4 (magnet self-rotation, previously "Not rated"): rated as split in the verdict table. The "self-torque violates Newton's third law" charge is Refuted by calculation as a claim about the whole system; at element level the strong third law does fail for Grassmann (claim 3), which is a property of the force law, not an observable net torque.
4. D1 prefactor question closed: with the book's prefactor k = -1/2 reproduces Grassmann exactly; the (2+k)/3 scaling in my earlier numerics reflected my normalisation. The "one-parameter family indexed by k gives the same closed-circuit force" in facts.json is wrong at fixed prefactor.

Unchanged: claim 1, 5 (old 4) and 6 (old 5) verdicts. Claim 6 (Weber) still rests on a two-page summary; deep read 6 is outstanding. Unchanged also: the weak citation of researchopinion.in by the physicist should be ignored in favour of the book (PDF p.271); "[computed]" is used as a tag and is defined in the primer.

Round 2

Round 2 used the additional deep reads appended to deep/Amperes-Electrodynamics-28a544.md: Ch. 22 (PDF pp.282-289), Ch. 6.3-7.5 (PDF pp.117-142), ether, fluids and vis viva (PDF pp.420-465), Part VI Ch. 23-25 (PDF pp.290-323) and Parts VII-VIII opening (PDF pp.324-349). Page tags are PDF pages (printed = PDF - 3). I did no new web checks. Result: no verdict changes. Several wordings and facts are corrected or sharpened.

New findings: - Despretz null (20 Jan 1821) confirmed [read: pp.118-119]. Two perpendicular wires at equal distance from a small needle: Biot-Savart predicts no net torque; Ampere with k = 0 predicts torque (1 + sin^2 b)/c, i.e. horizontal 1 versus vertical 2, net 1 unit, general k giving (1-k) versus (2+k). The null result refuted k = 0 and led Ampere to stop for about two years [read: pp.120, 122]. This supports claim 2's k = -1/2 route and leaves claim 1 untouched. - Date correction (D4). The astatic-coil null was performed in March 1822 and analysed between March and June 1822; first Academy presentation 10 June 1822, with no mention of the failed prediction (chemise 206 bis predicted rotation) [read: pp.132-135]. The 27 March 1822 positive wires-only rotation is a separate experiment [read: p.128]. My D4 text said May 1822; corrected above. The "no proof" quote on n = 2 is confirmed in full [read: p.141, Ampere 1822o p.418]. Until mid-1822 Ampere held k = 0 (hesitating because of Gay-Lussac and Thenard's unpublished attraction before a spark, which would imply k > 0), and the k = -1/2 collinear repulsion was so unexpected that he tested it with de la Rive in Geneva, Sept 1822 [read: pp.91, 148]. - "Uniquely from experience" (claim 2). In 1820 Ampere wrote that the law "could not be found by experiment" and adopted a hypothesis of "sufficient probability" [read: pp.305, 307]. In 1826 (Sec. 29.1) he says he relied "solely on experimentation" and that the formula holds whatever the physical cause [read: pp.346-347]. His method B is equilibrium (null) experiments on finite conductors, which he admits is of "much more restricted application". The "four cases of equilibrium" in PDF pp.326-336 are the translators' summary, not Ampere's words, and Ampere's own statement of them (Sec. 29.2 onward) is still unread. The claim is a methodological program resting on assumptions (central, equal-and-opposite forces), not a pure experimental output. - Part VI is qualitative only [read: pp.290-323]. No k, no n, no measured value; the torsion balance is described but reports no data; failures are recorded (Earth acting on a current, helix anomaly, mercury contact), and the vacuum experiment seems unperformed (p.296 fn 26). So Part VI neither supports nor contradicts claim 1, and "Ampere measured k" is wrong. - Ether and fluids; the digest contradiction (PDF p.424 vs p.461-462) is resolved. Ampere proposes an electric-fluid mechanism (two fluids in motion, "ether" being his name for their neutral combination, fn 218), declines to rule an ether in or out, and says the formula stands either way [read: pp.425-428, 462]. So the book's "pure action at a distance" framing is too strong for Ampere's views, though his formula is a direct central two-element force and he has no Maxwell-type field. This does not touch any verdict. - Vis viva argument [read: pp.426, 459-460]: forces that are functions of distances between fixed points are conservative, so a rigid system cannot gain speed per revolution. That is correct physics. It is aimed at Biot and Faraday-style fixed-pole models; Faraday's continuous rotation is consistent because the mercury contact makes the circuit non-rigid in his sense. - Weber (claim 6) [read: pp.285-286]. Ch. 22 contains no equation, derivation, experiment, number or regime of validity. Weber is described in words (distance, radial velocity, radial acceleration), the book says it will not go into detail, and the support is a bibliography of mostly sympathetic authors (some fringe, [memory, medium]). The claim "better foundation than the field concept" is not made in Ch. 22, which never mentions fields. The Weber verdict was already based on outside sources and is unchanged. - Unresolved: Davy 1823 (unread, still "unknown"); the p.119 sentence that Ampere "did not doubt" k, n/m should be non-zero seems to contradict "n/m = k = 0" on the same page (cannot be resolved from the text); eq. 6.7 is the authors' reconstruction.

Verdict table after Round 2 (changes against the Round 1 table: none):

Corrected claim list changes in Round 2: claim 2 (March 1822 null date; Ampere's own 1820 and 1826 statements on "from experience"), claim 5 (helix caveat) and claim 6 (Ch. 22 contains no derivation and no field comparison). Still outstanding: Sec. 29.2-29.11 and Note 31.1 of the 1826 Theorie, and the Davy 1823 and counter-rotation papers.