Hacking the Fine Structure Constant in Leptons Geometry

On the basis of a single field “below” the SM fields we first compute the leptons magnetic moment anomaly from the leptons masses without the help of quantum electrodynamics. Next, we compute the fine structure constant from the leptons masses using neither the anomaly nor QED. Here we begin to understand what it is.


Introduction
Particles physics has been dictated by field theories for many decades.The result is the standard model and its extensions.But those theories are based on generic mathematical frameworks that can accommodate an infinite number of free parameter-dependent universes.Is that physical?On the other hand, field theories are essentially based on couplings (and symmetries) between which little or no coherence is required; the current status is not surprising.Once this is understood, huge progress might come from understanding the origin of free parameters and symmetries.A simple angle of attack that seems available at present is to pay attention to the data itself and assume that there is now enough of it with sufficient precision for a pertinent analysis to be made.But then the "problem" must be hidden in mathematical abstractions from the very beginning of quantum physics.
Here we are not only ignorant; the risk is to being blinded by preconceptions because it is fairly easy to disregards the possibility that a theory is emergent.Then we restarted from the Bohr model and de Broglie's thesis (De Broglie, 1925) and tried a different route; using modern data we reached coherent results (Consiglio, 2013(Consiglio, , 2014a(Consiglio, , 2014b) ) (repeated briefly in addendum).In short, we showed that the mass spectrum is entirely coherent in numbers and geometry and only three types of resonances come naturally; for this we rely on a single equation, which was found reasoning close to the Poincaré stress (Poincaré, 1906): Where all entities have a classical origin except D: 1) X is the reverse of a density.2) K, P, and K are integral numbers.3) D is a distance specific of the particle group (charged leptons, quarks, or massive bosons).4) μ is a small self-energy that we compute for leptons.But now the equation has no less than 6 free parameters and we only have 12 masses and three particles groups to test it.Its relevance must be proven using additional criteria which can be of several kinds: 1) Small resonance numbers, 2) coherence of those within or between all particles groups, 3) geometrical coherence between and within particle groups, 4) coherent parameters X, D, and μ between groups, and finally 5) agreement with existing knowledge that is not only to compute particles mass.
Under the first four criteria, the formula was proven relevant and we showed (Consiglio, 2014a) that it suggests the existence of a unique field "below" the SM fields to which resonances "give" symmetry.In second analysis (Consiglio, 2014b), we show that the parameter X is constant and that the particle-group dependence of D is physically coherent; it leads to computing all masses coherently and a trivial approach to the bosons and top quark widths also gives perfect results.This last paper concludes that the known fields are emergent and it ends with the following conjecture.
A physical action is a product of currents of the field below.Conservation laws imply alternative algebras.
This theorem is of high importance in physics since the SM is built on three and only three symmetries as U(1) Y ×SU(2) L ×SU(3) C and it is known that the following are isomorphic: C to U(1), H to SU(2) and O to SU(3).
The conjecture is appropriate as long as no other symmetry is discovered, but we just assume mathematically coherent limitations that explain why nothing else is observed.
In this paper, we use this conjecture to address the fifth criterion: We compute the electron and muon magnetic moment anomaly using resonance numbers, α, and geometry.Then, we find how to compute α from the leptons resonances given by the equation, and (of course) independently of the anomaly.
We must say, however, that our hack is incomplete in geometry, in particular where 4D and 3+1D rotations are addressed.We use 3D slices to picture 4D rotations and the 3+1D or 4D geometry is not clear (even though the two 3D pictures are simple enough) and it looks like the 4D geometry that comes is new.The method can be criticized but the results stand; firstly by precision, and secondly by simplicity in reasoning and equations.Our first purpose in this paper is to recover precision data to find new directions.

The Field Below
According to de Broglie, each particle of energy is associated to a frequency given by the Plank-Einstein relation; then using the minimal Dirac (1993) G D = e/2 α we write: It suggests that a current G D interfering with a charge e is an action which repetition is energy.The idea is indirectly suggested by Lochak (1995Lochak ( , 2007) ) who "recognizes" the symmetry of magnetism in the Dirac equation and finds a massless monopole equation -a magnetic current.Then, we assume that the electron wave is also a magnetic current (Consiglio, 2013).Comparing phases and using the de Broglie's analysis of the Bohr model led to the following equation: where V is the de Broglie wave phase velocity, and g a charge associated to a constant magnetic current carried by the de Broglie wave.This result was shown coherent with a total magnetic current (de Broglie's standing wave) obeying the Dirac condition and with Bohr's energy levels.Now, we interpret (2) as the fundamental quantum that can also be oriented in the time direction, in which case we name it a time-current.Then it is trivial to get fractional charges from a breach in the time direction and 4D space (V = c ± v), or 3+1D space (V = c 2 /v) as it gives the same results.The up and down type quarks electric charges come from currents velocities "thru time" which are: A time breach of this form is of interest when it comes to SU(3) in 4D space; since octonions can be generated as = ⨁ , it suggests two "superposed" 4D spaces giving the degrees of freedom of SU(3).
The elementary field constituent is given directly by equations (2 -3) augmented with space-currents.Here the massive particles constituent is a time-current ±e c/2 which manifests an electric charge dependent on its sign and its time direction, associated to or interfering with space-currents; merging two time-currents of opposite charges and directions gives an electron and the Dirac condition.A toy model based on this idea was instrumental to understand the field and compute the bosons masses; it is complete in the sense that no new particle appears (Consiglio, 2014b).Here we only need to picture leptons, for which we deduced differences in time-currents, two or four: Electron: where notations (up, down and charge) are trivial.

Method
The method relies on the use of equation ( 1) to analyze the mass spectrum structure the resonances.In this paper we present our calculus and equations in the same intuitive manner as the ideas and the logic came.This is the natural way but it is important first to outline the minimal ideas and principles that guide the reasoning: -Geometry.Our reasoning and equations take place in a broken 4D space.It is explicit in (3) and results in a relative rotation of the symmetry of currents depending in their direction in time.
-Rotations in 4D are around a 2D plane which, for leptons, is defined by the time and the magnetic moment axis.We use two 3D slices in our calculus because we do not know what 4D rotation group is valid.
-Single current quantum.We assume a single quantum of current obeying equation (2) that applies in the time direction and possibly in space.We do not need to assume a different space current quantum.
-Action.This is the bottom line; we do not discuss energy/momentum that we consider emergent.
All the formulas below correspond to putting two orthogonal Poincaré cones in 4 dimensions; it gives cylinders where the currents trajectories are helicoids and action comes as products of currents plus some basic trigonometry.The logic will become clearer when used in the first part of the paper.So the principle here is minimalism.
-Measurable action.Measurable quantities are only differences which are related to action.The difference can be in rates of action (e.g.mass ratios) or a multiple of the quantum -which here corresponds to an integral number of rotations.It gives in particular the ratios tan(x)/x, or y sin(x/y) that we shall use, where x is a helicoid angle and y = π/2 -x is the complimentary angle.When comparing action it gives a general formula where tan(x) compares in action to tan(n x)/n where n is a number of turns.
-Inter-action.The systems we discuss are analyzed symmetrically along privileged orthogonal axis.
o Reciprocal actions relates to currents translations along privileged rotation axis; it gives formulas like sin(x)/sin(π/2 -x), tangents, where x is the translation angle of the helicoid.
This part of the geometry is also explained Figure 2. Using (4.1 -4.3) we find an important coincidence that will be central to the discussion in this paper:

The Dirac Condition and Leptons Parameters, a Semi-Empirical Search
In a celebrated paper, Dirac (1993) analyzes the possibility of existence of magnetic monopoles using quantum mechanics.Based on the mathematical properties of the electron wave function interpreted as a density of probability of presence, he shows that a monopole is compatible with the existence of quantum mechanics in Hamiltonian form if and only if the so called Dirac condition is respected: It results in the elegant idea that the existence of magnetic poles fixes the electric charge and conversely.But here we assume that the electron wave is a physical magnetic current; since Dirac's demonstration is based on the "fields of force" acting on the electron wave it comes that magnetic currents acting on apparent electric charges (or conversely) must obey the same condition.
But in our model e is an apparent electric charge and simultaneously a sum of magnetic currents; the latter must be taken into account in the condition as part of the total current; for any charged lepton it should be: Now compare with our data.The fundamental resonance in equation (4.3) corresponds to a theoretical half electron, that is N = P = 1, probably K = 0, and a self-energy μ/2 that we shall first ignore.It gives, as per (1 -4.3): This mass is purely theoretical but fundamental and then it should be compared to μ which, in our model, comes from the interaction of the time-currents (not the apparent charges) and then, for an electron, from the product e 2 /4.The rest of the electron mass (N = P = K = 2) is given by space currents and, according to our conjecture, it should also correspond to a product; then in (8) the numbers (N = P = 1) correspond to a hypothetical particle where a current G is interacting with e/2 which mass is given by an action corresponding to the product G e/2.  9) by ( 10), and in light of (7) we add e/2, which is the μ/2 that we first ignored in (8); we find: We recognize the modified Dirac condition in (7).The fine structure constant appears straightforwardly from the numbers in a form that gives or confirms the nature of the field.At first sight, the relative discrepancy (-1.65 10 -3 ) seems acceptable since we analyze a hypothetical particle but we shall see how this numerical value holds.
There is a second aspect related to the Dirac condition which comes from the toy model and the charges e/3 and 2e/3 going respectively down and up the time and supposedly merged as the electron time-current; assume their individual self-interactions are squared charges: ( /3) (2 /3) → (1/3) (2/3) = 5 /9 (12) Now compute from (9 -12): 4( 5 /9)/ = 137.03247151/ (13) The relative discrepancy with respect to α is ≈ 2.26 10 -5 .The coincidence can be seen redundant with equation ( 11) as it is almost identical, but it comes from a different interaction and we shall see that this value also holds.

The Electron Mass and Spin
According to our conjecture, the electron mass comes from a product that we first write in complex form: where e/2 represents the currents, not the apparent charges.Now we write (15) in quaternion form: The rationale for this equation is that in (3) the time-velocities are on each side of the light speed singularity.
According to relativistic tachyon theory (Recami & Migneni, 1976) such currents in space will see the other rotated on a hyper-complex plane.Here we assume the same for time-currents (or O = H⊕H).We recognize in (15) the masses of ( 14) and then we multiply this equation by -j to get real numbers and the observable parts: We get the real squared charges of ( 14) and the rest is angular momentum that splits into two components; one is a real number like the mass and then observable -it is the magnetic moment.
The other is imaginary and unobservable; it is found along the time axis, like e/2 in (15); then it is a 4D-gyroscope which rate can be interpreted as the origin of inertia using special relativity.For convenience, we will name this component spin -even though it is not its usual form.Now we have e G = h c → G = 2 G D ; this is still coherent with (11).
Then we identify the squared charges in ( 16) with the masses in (5) as it should help understanding; it gives: Substituting G = G D , we get 1 = √2/4π which is ridiculous; hence the coincidence (5) does not relate to energy but to a relation between two physical actions.Multiplying each side of ( 17) by the Planck constant and using also ( 16) we get the following correspondence: which interpretation is obvious: An action h at each period of a lepton pulsation makes its spin and its magnetic moment; but (5) gives an approximate equality which seems incompatible in precision with the leptons masses.
Now in ( 16), the mass comes from a 4D rotation around a 2D plane defined by the spin and the magnetic moment.
Since leptons have a so called magnetic moment anomaly those are not pure rotations, they must include a translation corresponding to a helicoid angle α around the time axis.Then around the magnetic moment axis it covers an angle (π/2 -α) (assuming Euclidian 4D space or SO(4)).Hence retroaction implies a second translation angle α/(π/2 -α).Now we can replace α 2 in ( 5) by two coefficients corresponding to those two angles: where sin(α)/sin(π/2 -α) corresponds to a ratio of action per volume element which must then be multiplied by the angle (π/2 -α) to get the full action.Using (4.1) we can compute the absolute error in ( 19); it reduces to 0.007 meV.
It shows that the ratio (m -μ)/μ holds with a relative precision of 2.9 10 -8 , which will be important, and agrees with current knowledge (precision) of the electron mass.(The case of other leptons is discussed section 6.4,where we take into account their resonance numbers and obtain the same precision.)The coefficient 4π in (19) shows that the phenomenon giving the angle α/(π/2 -α) is isotropic in 3D space.It is then the wave where the rotation is that of a magnetic current obeying J m = -∇ × E in standard notations.
Finally, the interactions of each apparent electric charge with its space and time-current lead to two hyper-complex planes which probably requires a formal treatment using octonions; but since we compute a physical action we shall use real numbers.Considering that each is a half-electron, and then a current G/2 interacting with two currents e/2, and self-interacting electric charges (e/3, 2e/3) it gives: which of course is identical to (13).Then we get: This equation also gives an approximate equality with the fine structure constant.Together with (11), it suggests equilibrium where space currents interfere with the time currents and the apparent charges under distinct angles.

Wave Geometry
The change in phase of the de Broglie wave over the first Bohr orbit of a hydrogen atom is 2π, while the Compton wavelength change in phase over this orbit is 2π/α.Then over one Compton wavelength, we have: ∆ = ∆ (22.1)where φ D and φ C are the de Broglie and Compton (standing wave) phases, and ∆φ D and ∆φ C are their changes in phases over any length.On the n th orbit we find:   Charged currents interact and the equation (1) was found assuming the existence of a pressure field: Space-currents (horizontal) receive a pressure dependent on cos(π/2 -α); time-currents (vertical) receive a pressure dependent on cos(α).Then cos(π/2 -α)/cos(α) = tan(α) is the ratio of action between space and time currents.In space, the pressure depends on cos(π/2 -α) = sin(α) and implies a second translation angle α/(π/2 -α) that applies to the solid angle 4π, "thru" the angle (π/2 -α), which is the amplitude of the space-currents.It gives (19), where the ratio of mass (m -μ)/μ is pressure-dependent.Hence, time-currents are μ ↔ e 2 /4 and space-currents are (m -μ) ↔ G 2 .

Other Resonance Coefficients
In Figure 1, we depict the main characteristics of the geometry of the resonances found in Table 1: Two space-currents (corresponding to N = P) of opposite direction interfere and give the product NP in (1).Two or four time-currents corresponding to K interfere together and with the space currents and add the KD in (1).Logically, the main resonance NP corresponds to G 2 in equation ( 16) while K corresponds to e 2 /4.Then the product NP makes the spin and the true space-resonance cycle is (NP -2) K which is a product G 2 e 2 .The power available depends on the number of currents C while the mass μ is constant; then we divide this coefficient by the number of currents.
But the spin corresponds to a product G e, (the square root of G 2 e 2 ,) and we get a spin-dependent coefficient where the spin relates to the apparent electric charges giving equation ( 22).It is: In the direction of time (K in Figure 1), the same reasoning gives NK 2 for a product e 2 /4.We get a spin-independent coefficient which then relates to equation ( 11) and the time-currents; it is: Now for the muon and tau the coefficient corresponding to the time current rotation is not α like in ( 23), but it depends on the resonance numbers.The electron is the special case because all resonance numbers are identical and even (N = P = K = 2) and then all phases are identical.For the muon and the tau, N = P and K are odd and prime with each other, and then the cycle is NK.Now using (24) for an electron, the cycle uses N = K = 2 and its angle should be written 2 α/2.Then for a muon and a tau the corresponding coefficient is: 3) The correction mixes angles and resonance and fits with the interaction of current where action is angle-dependent; it is similar to (19) and it will be the general form used in this section.We introduce the angle α/2 which we now consider as the physical angle of each time-current -it gives α for two currents of opposite directions.

The Calculus for the Electron
Now we want to compute the anomaly but the resonance geometry of is not understood.Then we can only rely on our conjecture, on currents geometry, on the anomalous values in (11 -21), and on minimal reasoning.We define: -From (11): -From (21): -a T =a 0 a 1 a 2 where a 0 is in (24), a 1 depends on β 2 , a 2 on β 1 and a T = (g -2)/2 is the full correction.
According to our conjecture a T is a product giving a measurable quantity where a 0 corresponds to the angle α in (23) or φ in (24.3), a 1 to the anomalous apparent electric charges (21), and a 2 to the anomalous currents interactions (11).
Since β 1 and β 2 are deduced from the leptons masses, they are related to the tangent of some angles part of the resonance geometry (possibly to the rotation of space currents (β 2 ), with respect to time-currents (β 1 ) that must be bent by velocity in the same manner as in Figure 1).The anomaly is angular and differential and then a 1 and a 2 must be computed as the ratios involving an arctangent respectively of β 2 and β 1 and resonance numbers.
Therefore for the electron the first correction term a 1 e is given by an expression of following form:

tan( ) tan ( ) →
It links an action given by α in the standard theory and β 2 thru the resonance numbers.Now β 2 relates to the apparent electric charges giving the spin; then Y = E as defined in (24.1) and we get: This is still incomplete because the translation angle α/2 of the time-currents also impacts the coefficient and subtracts from K. It might imply a tangent, or it might simply be an amplitude, but it will not impact the results precision significantly.Then, to simplify notations we write: Now β 1 comes from the time-currents of the electron; we must make a similar correction but our reasoning must involve F defined in (24.2).Naturally, this correction will be similar in form to the equation above.The logic is: -Firstly, the first order effect is null; it is a second order correction where the cross-products cancel.
-Secondly the angle must be α instead of α/2 since the two angles α/2 on the axis of K sum up.
It gives, for an electron: Note that in the equations (25 -26) the angle α/2 affects K and -2α 2 affects K 2 ; it is actually the same geometry where only K is impacted.Now from (23 -25 -26) we find: The CODATA experimental value of the electron g/2 is: /2 = 1.00115965218076 (27) (28) From ( 19), the relative precision of the ratio (m -μ)/μ is 3 10 -8 , and it applies to μ/X and then to β 1 and β 2 ; the relative error in a T with respect to CODATA (29) is 2.6 10 -9 .This is one order of magnitude better that we should expect from (19).We see that our hack in reasoning and coefficients is effective for the electron.

Muon and Tau
We get the equations needed to compute the muon anomaly using (24.1 -24.2 -24.3 -25 -26), and including the four currents given by the toy model and the resonance numbers in Table 1.We get: /2 = 1.00116592081 (29) while the CODATA experimental value of g/2 for a muon is: /2 = 1.00116592091 (54) (33) (30) The result is within uncertainty due to the lesser precision of experimental data.Our reasoning and coefficients are also effective for the muon, but with no additional hadronic corrections, as expected with a single field.Note that the SM prediction includes those but disagrees with experiment and results in a 2-4σ discrepancy.Typically: − = (2.8 0.8) 10 (31) The very short lifetime of the tau makes impossible at present to measure its anomaly.The SM prediction is: /2 = 1.00117721(5) (32) Using the same equations and the resonance numbers in Table 1 we get: /2 = 1.001257893 (33.1)But on the other hand, in the tau resonance, N = P = 9 is not a prime number and then, perhaps, we should use 3 instead of 9 in the equations to compute its anomaly (see also section 6.2).It gives: /2 = 1.001170374 (33.2) where the difference with the SM prediction is more coherent with that of muons.

Second View on Leptons Resonances
Our analysis of the resonances in Table 1 fits with the spin and magnetic moment, and two translation angles α/2 where (π/2 -α) is complimentary; the time axis and the magnetic moment define a 2D plane which is the axis of a 4D rotation augmented with translations.
But now we get a quasi-symmetrical picture that suggests the existence of a second view on the leptons resonances where a different mass μ' can be associated to an angle (π/2 -α); in rough approximation and using angular ratios, we should have: μ' = μ(π/2 -α) ≈ 378 eV/c 2 .Of course geometry is not so simple since the symmetry between space and time is broken and this is what we want to check.
Starting with this approximate value and using equation ( 1), an empirical search targeting the same masses as in Table 1 (to all shown decimals) gives Table 2  3) The resonance numbers are small and their logic is new as we get P' = K' instead of N = P, and N' = 2 P' except for electrons: N' = 2 p'-1 = 2; this difference agrees with the definition of φ in (24.3) as compared to (23), and we also find N' = 2N -2 which should hide the spin.Importantly, we get P' = K', which can only correspond to the spin and the magnetic moment axis, and those are consecutive numbers.Two coherent patterns together with small resonance numbers seem to confirm the existence of a second view on the same phenomenon -which is quite stunning.Now a few more verifications of coherence; first assume a modified Dirac condition is there, we empirically find: 3( ′/ ′ 2) = 68.396648611/2 (35) which is reminiscent of (11) with a different symmetry.Then searching a simple relation between D and D' gives: √2 3 √3 (36) which compares the diagonal D of a cube, to that of its face which is 3D' with a discrepancy of 3%, about twice the difference on our prediction of μ'.
Finally, we get a better match with μ' = μ (π/2 + π α + α/π -4 α 2 ) where the relative error is 5.4 10 -7 ; the formal difference with the initial idea is almost equivalent to adding 2 rotations in μ' and then removing the translation; it might not be a known rotation group -or the two views model the rotations in different spaces.On the other hand, this is expected since we instinctively analyze Table 1 using two 3D slices and space-time does not seem to work like that -from the expression of μ', there seem to be a double inversion.But now, one step back, the significance of the result is important: The translation and the rotation are absorbed in μ'.They must also split in the resonance numbers and the consequences are of high interest as we shall see.

Alpha
With a single field, α defines the "field of forces" at work in Table 1 in the space directions (N, P) while D corresponds to the time direction (K).But in Table 2, we find K' = P' and it puts on equal ground the time and the magnetic moment axis.The N's are orthogonal to those, but now they are pure harmonics and they depend on K'.It implies that α only influences N' and creates a unique resonance path (for all leptons) that depends only on integral numbers -including for currents and then it is necessarily 137.
But then α depends only on Pi, on 137, and on lengths defined by 1/N'.where the relative errors with respect to (4.1) and (34.2) are 9.6 10 -10 and 8.3 10 -10 respectively.(Note that the decomposition just works like a division; the left term is the closest square to D -2 from which it is subtracted; the middle term is the division of the rest by π 2 that gives a small residual term.Then we search known numbers.) Note that a few other interesting solutions exist for the negative terms in an acceptable precision range.
In those expressions, the proof of a single field is fivefold: 1) The identity in form of the three terms of the two expressions (including 7 = 3+3+1).
3) Finding 274 here is a direct and strong confirmation of 137 and 1/137 in equation ( 37).
5) In equation ( 1), D and D' have the dimension of 1/NP, and from the equation ( 37) it comes that the squared pseudo-norm giving α -2 also has the dimension of 1/NP.Then D (or D') and 1/NP are of opposite dimensions -which is coherent with the equation ( 1) and with the split in space and time currents.
It is now crystal clear that the two views address the same geometry and agree with the existence of a single field.

Questioning and Empirical Coincidences
In this section, we try to find some order in the integral numbers as they seem at first arbitrary.
Moreover the equations ( 41) and ( 42) show a peculiar symmetry between path lengths and resonance numbers, where: 1) All elementary oscillation length sum-up below the minimal resonance number.2) All known resonance numbers coexist in the (same) universe and sum as 137; as though we can add no other.Now 137 suddenly looks like a very fundamental number which is obviously related to bounded complexity, cutoff, and conservation; and it looks at first like the answer to 137 is (42).

Only 137?
In light of (41 -42), of the existence of Tables 1 and 2, and of the inversion between (40.1) and (40.2), let us be curious and compute a pseudo-norm converse to (37), that is an inversion between space and time: The coincidence is shocking as it gives a prime number γ -1 ≈ 37 that immediately compares to α -1 ≈ 137 and it suggests a converse geometry based on dimensional inversion -similar to the inversion between ( 41) and ( 42).
Now compute the sums for each generation (excluding bosons); that is taking leptons N, P, K, and quarks N, P.
So it is similar to Σ' and Σ in (41 -42) since each Σ'(n) is bounded by 1 and each Σ(n) relates to an integral number based on 37 and 10 (note that there are several other coincidences of the same kind).Hence it is not only 137 but also apparently 37, and it suggests that simple arithmetic is at work.Then we should easily find a direct correspondence between the numbers in Tables 1 and 2 since firstly α must be there in both cases and secondly we find the same integral numbers in D and D'.The resonance numbers are 2, 3, 4, 8, 16 in Table 2, and 2, 3, 5 and 9 in Table 1; the 2 and 3 are identical, we trivially find 5 + 3 = 8, and 4 = 2 + 2 = 2 2 can be a sum or a product of currents; similarly, 16 = 4 2 = 5 2 -3 2 for the tau.Hence the correspondence is obvious and ( 37) is coherent with Table 1.It is interesting that only the tau uses squares in the two tables and that the link relates to the first primitive Pythagorean triple.It also justifies our lack of confidence for the tau magnetic moment as computed in (33.1).

And the Wave?
It is stated that ( 19) is the wave, but this equation is only valid with the electron mass.We must rewrite it for the muon and the tau in a manner similar to φ in (23.3), taking resonances into account.We write: where R and Q are unknown coefficients which depend on the resonance numbers, possibly 137 and other numbers, and m the associated lepton mass.
It is easy to find R from the relations between the resonance numbers in Tables 1 and 2 as found section 5.1.Since N and N' hold the spin it absorbs an angle α/2 (and α in the electron case); the spin is 1/2 and the non absorbed part is α/2 (and for the muon and the tau we have N+1 = 2K in Table 1 while N = K for the electron).We get: But it is another issue to get Q as it should not be related to the resonance numbers but rather to 137 as it deals with the other angle.Using (44 -45), an empirical search for similar expressions involving "known" numbers gives: = (1/4)(137 1 − (274 − 3 − 37/2) ) (46.1) = 2(137 37 (137 − 137/74) ) (46.2) Using those, the equation ( 44) holds with precision better than 3 10 -8 , which is coherent with (19).The integral parts of those expressions are doubtless but we cannot use CODATA recommended values to get such precision.We use Table 1 as it gives leptons masses coherent with μ; the non-integral parts can be doubtful.

Discussion
In this section, we discuss two questions connected to the theory in this paper.
7.1 QED Calculus of Alpha at Order 10, Hacking β 1 and β 2 Aoyama, Hayakawa, Kinoshita, and Nio (2012) have computed the 10 th order QED contribution to the electron anomaly: Now, re-computing the electron anomaly with this new value comes in very good agreement with CODATA, which is not very significant as it exceeds by three orders of magnitude the expected precision on β 1 and β 2 .We do not understand why we should have such precision as it comes from the leptons masses which uncertainty is much larger.We need a better theory to conclude, but without theory we can still hack β 1 and β 2 which are close to α -1 .They should be pseudo norms and we can use the same method as for D and D'; by definition, they depend on each other thru X/μ and it adds one more constraint that, in principle, enables to computing more decimals.Relying on "known" numbers, an empirical search gives: ( where the relative error with respect to the empirical values of β 1 and β 2 is 1.6 10 -11 .When X/μ is computed twice from those expressions the relative difference is ≈ 1.6 10 -16 , and the relative difference with (4) is 5.4 10 -11 .It is better than expected and coherent with the precision in (27) -which seems to confirm the validity of (50).
We see that β 2 holds the spin (+π 2 like α) as inferred, and β 1 seems to absorb it (-5π 2 ), which can also read (2 -7)π 2 since we get 7π 2 in D. Recall that β 2 is associated to the coefficient E in (24.1) which depends on (NP -2)K where the term -2 corresponds to the spin absorption.The 7 has a double interest as we get 1/14 in front of the next term.Hence, this part seems coherent in geometry.At the opposite the negative terms must be incomplete.
-With respect to a e (theory) in (47), we find a difference of -1.06 ×10 -12 which can be seen compatible; but interestingly, we now have a problem similar to the muon where the SM gives an overestimate.
Then, perhaps, QED is reaching a limit where (or just before) hadronic corrections are significant.

On the Proton Charge Radius Conundrum
The muonic hydrogen Lamb shift was measured by Pohl et al. (2011) The fundamental difference between our analysis of the anomalous magnetic moment and QED is that we use resonances as the de Broglie wave coefficients of action.But we get odd resonance numbers for the muon and even for the electron.If those are physical the Dirac equation is not applicable as-is to muonic orbitals.By definition, the phase of a lepton wave depends on the space currents giving N P (while its action is given by the coefficients in equations (24.x)).For muons, it gives 25 and compares to 4 for electrons.Then, at first order, we must replace α by the following expression in the calculus of muons energy levels: ε = tan tan(6.25 ) 6.25 = 0.0073024 (52) which makes a huge difference since we must use its square ε 2 to compute energy levels (just replace α 2 in the Bohr model or α in the Dirac equation).It gives lower energy levels and higher transition energies in proportions of: which is in good agreement with measurement.The next issue is to understand why the same effect does not appear with helium.In this case the energy loss given by standard equations is multiplied by 4 and then we have to multiply by 4 the phase coefficient 6.25 which becomes an integral number: Since 4NP compares to 4 for an electron, the wave connection is equivalent to that of the electron and the Dirac equation is valid.Finally a discrepancy will come with any atom nucleus of odd charge but not with even charges.Hence this theory of leptons resonances can be tested further, for instance with lithium.

Conclusions
We first compute the electron and muon magnetic moment anomalies out of the leptons masses using only resonances, special relativity and the heretic deduction of a single field "below".It requires no additional hadronic corrections for the muon in agreement with the existence of a unique field.The toy model also passes the test since it is used in several manners in this calculus -which is important as it relates to the nature of the field.
Then we compute the fine structure constant from the leptons resonances and geometry.It looks as though the long standing puzzle of its origin has a solution, incomplete at present, related to the dissymmetry between space and time since a pseudo-norm gives it a geometrical status complimentary to the velocity of light; say α and D are geometrical keys to pierce the light cone in the direction of time -quite a basic definition of mass.
Now the logical bet is that there is no free parameter at all and that the integral number based world that we deduce is the natural one in a broken 4D space.The next problem is to link in concepts to field theories but the way to do so does not appear straightforwardly and the field geometry is not well understood at present.A dimensional inversion seems to be there and, as far as we know, this is unheard of -maybe it comes from the true geometry of space-time.
We must mention, however, that 7, 19, and 37 are centered hexagonal numbers or cube differences; the next one is 61 and it is also the full SM particles count (including all types, charges, generations, and colors).It can hardly be a coincidence and it provides with a simple approach to studying the field as a whole; for instance, it is straightforward to put all particles on hexagonal centered spots where charges, generations, stability, oscillations and resonance numbers are organized in a coherent manner.In this picture, the field is a single system where the integral numbers 2, 7, and 19 are natural.Then, perhaps, this is the direction that we search.
Note: All numerical data in this paper were computed in an Excel spreadsheet provided as supplementary material; Microsoft guarantees 15 decimals which is sufficient for all our results.Still, we made a few verifications where the calculus requires high precision.
The Table 3 gives the quarks computed masses (natural scheme).The top quark resonance width is currently estimated to 2 ± 0.5 GeV/c 2 .With respect to the equation ( 1), the width is related to K, since this coefficient addresses the time-currents, because those are transformed or separated in any decay.
We showed that the quarks N are sub-harmonics of a fundamental circular resonance given by 266 = 2×7×19, where N is a circular resonance while P is radial.
Taking into account P = 3, orthogonal to N, leads to computing a mass using (K ± (3√2)/266) in (1) which gives a difference +1.97 GeV with respect to the pole mass in Table 3 (or -1.94 to depending on the sign of the correction).This mass is, with respect to the pole, the delta-energy at which the resonance breaks.It is then the top quark width, and the logic and calculus are in perfect agreement known principles.

A.2 Massive Bosons
The equation ( 1) is modified for massive bosons, as shown in equation (A2).The modification is initially empirical, but we later find its significance as related firstly (Consiglio, 2014a) to the difference between the Dirac and Klein-Gordon equations, and secondly (Consiglio, 2014b) to the phase lock between two resonances paths (as explained below): Due to the large bosons mass, we neglect μ.We also use the same value of X as in Table 1.The massive bosons parameter D x are deduced from the field analysis and depend on the time-currents, they are: W ± and Z 0 → D WZ = α 2 /(1 + α 2 ) + α D/2(1 -α 2 ) -D 2 /6(1 + α 2 ) = 5.62404904 10 -5 (A3.1) where D is that of leptons (4.2).Then we find the equation to compute the parameter k in (A2) that corresponds to synchronizing two resonances; it is: where D x addresses a specific bosons type.Finally we compute k using (A3.1 -A3.2); it gives: It leads to the masses predicted in Table 4, shown together with the SM prediction where relevant.Note that no adjustment is possible; the coherence of geometry and coefficients makes the approach predictive and falsifiable.The resonance widths are computed in a manner similar to the top quark, but using the equation (A4) to understand the resonances.Let us simplify this equation using k = 1 and take its cube: The left-hand side is twice the volume of a 4-sphere of radius 1/144 divided by half its circumference: π/144 3 = π 2 (1/144) 4 / (π/144) The right-hand-side is twice the volume of a 4-sphere of radius 266 D divided by its radius: Figure 2. Curr Figure 3. Reso of α is also derived from theory and from existing measurement of the anomaly: α -1 (Order 10) = 137.035999 173 (35)(48) The resonances, the spin and the electric field are taken into account in (37); then we can only miss two currents, each corresponding to 1/274.Those are not part of the resonance, implying a positive coefficient which gives: Now using D = D x /π gives:π/144 3 = 266 3 π 2 D 3