According to the ref. No. [15] the mass of the only other stable fermion in free space besides the electron, the proton (antiproton), results from the rotation of the S³ sphere (assumed to be at rest in R⁴), so that within the sphere the centrifugal energy

E P = m 0 2 [ ω P P ] 2 = 1 2 m 0 c 2 (1)

is generated. In association with the de Broglie-formula delivers the former equation the four-dimensional height λ 4 ≡ λ w = h = 0.814013 × 10 − 13     cm such that reversely applies

m 0 = 2 λ w ( 1 − sin A ) c ≡ 2 h ( 1 − sin A ) c = 2 τ w ( 1 − sin A ) (2)

angle Α (capital ALPHA) = 43.788˚ being the Galaxy’s latitude in the “northern hemisphere” (arbitrary definition) of the spinning Universe, h Planck’s constant, m₀ rest mass of proton, c_Α speed of light at cosmic latitude Α and λ w / c ≡ λ 4 / c = τ w = h / c quantum of (rest) time (see Chapter 5.). “This implies, contrary to common view, all fermionic material spatially to be build up four-dimensionally, although with the extremely tiny fourth spatial dimension 2λ_w”. If the preceding is true, then the instability of all fermion masses (resonances) > m_0(Α) (proton) = 2τ_w(Α) , of the meson masses < m_0(Α) and of the free neutron must be due to the fact that their four-dimensional height ≠ 2λ_w. This requires the mass of the only other stable particle in free space, the electron (positron), for reasons of elementary topology to be a fixed stable fractional amount of m₀ (see Chapter 3.2). In accord with the preceding must the Euclidian quantum of time be of value

τ w = h c = 2.715256 × 10 − 24     s (3)

Equations (2) and (4) are in accord with the CGS-system such that especially is valid (referred to R⁴): m (g) = λ_w (cm)/c = τ_w (s) and h really results in λ_w (cm) [15].

Now, how does τ_w, or duration on quantum scale, come into existence physically? I.e., if we demand the cosmos really to be built-up as an Einsteinian hypersphere (hyper-ellipsoid) and material particles to exist as smallest possible geometrically four-dimensional objects rooting in the geometrically three-dimensional “surface” of the hypersphere. In the following is proposed that rest time principally is representable as the duration of the physical mapping of four-dimensional λ_w into the geometrically three-dimensional rest frame of the respective particle λ_w → τ_wc. This also can be written as the difference

− λ x 2 + λ y 2 + λ z 2 − λ w 2 = 0 (4)

− λ x 2 + λ y 2 + λ z 2 − τ w 2 c 2 = 0

λ w = τ w c

because for four-dimensional λ_w no equivalent exists in the geometrically three-dimensional space of our world, whereby λ_w and λ_x belong together in a special way, as we shall show immediately.

Other as the three spatial dimensions of space-time is λ_w linked to the latter dimensions only in the dimensionless tie point − λ x 2 + λ y 2 + λ z 2 − λ w 2 = 0 . This implies that the rigid four-dimensional quantum of length can be mapped into R³ at the latter tie point firstly as the quantic continuum τ_w of points dτ_w with highest possible velocity ω P ≡ c = const . such that be valid d λ w = ω P d τ w and thus:

τ w = 1 ω P ∫ 0 λ w d λ w = 1 c ∫ 0 λ w d λ w = ∫ 0 τ w d τ w (5)

Secondly this physical-topological notion of the quantum of time evidently is associated with the proton’s four-dimensional height 2λ_w and the associated inertial mass according to Equation (2) in the form

m 0 = 2 ∫ 0 τ w d τ w = 2 τ w (6)

Thirdly, the four-dimensional 2λ_w is considered to rotate in four-dimensional space around the tie point 0 (zero) perpendicular to its three-dimensional embedding, with the zero point between the two λ_w. This four-dimensional rotation of the fermions is perceived as spin of their hexagonal diameter in physical measurements (see Chapter 3.3 below).

The presented quantum-geometric notion of the origin of time as well as the protonic mass also leads directly to the physical basis of the relativistic time dilation and the energy-momentum relation. To this end, abstract from the dimensions λ_y and λ_z so that the tie point 0 (zero) is formed by the mutually perpendicular dimensionsλ_x and λ_w. Consider the four-dimensional λ_w and thus the tie point now moving along the three-dimensional axis 2λ_x with velocity v. The direction of λ_x is given by the velocity vector v such is valid: v 2 τ ′ x 2 = v λ ′ w / c = v τ ′ w until the tie point meets the movement of ω P = c along λ_w which becomes relative to the moving tie point to c τ ′ w = λ ′ w . With this we have a right-angled triangle with the catheters cτ_w and v τ ′ w and the hypothenuse c τ ′ w , so that the Pythagorean expression holds:

c 2 τ ′ w 2 − τ ′ w 2 v 2 = c 2 τ w 2 (7)

τ ′ w 2 ( c 2 − v 2 ) = c 2 τ w 2

τ ′ w = τ w 1 − v 2 c 2

From Equation (7) the relativistic energy-momentum relation of the proton, E ′ 2 − E ′ 2 β 0 2 = E 2 directly follows through the demand that 2λ_w is always conserved and, thus, by multiplying both sides of the second line of Equation (7) by 4c² follows directly:

4 λ w 2 ( c 2 − v 2 ) 1 − v 2 c 2 = 4 λ w 2 c 2 = m 0 2 c 4 = E 0 2 (8)

Factor (1 − sinΑ) is neglected. I.e., the fundamental conservation of four-dimensional elementary length λ w = m 0 c / [ 2 ( 1 − sin A ) ] = E 0 / [ 2 c ( 1 − sin A ) ] in the special relativistic energy conservation law (first left-hand sided ratio of Equation (8)) seems to be the very origin of the association of rest energy E₀ and rest mass m₀ of the proton but of the electron as well and, therewith, of all stable matter. This demand is justified, since beyond the tie point time does not exist, so that the four-dimensional centrifugal energy of cτ_w + λ_w= 2λ_w of (1) is mapped to the quantum geometric environment (metric) of the tie point according to the Einstein general relativity theory as a deviation from the undisturbed flat Euclidian metric η_nm:

ϰ T m n = 8 π G c 2 T m n = 2 U = 2 E 0 = 2 τ w [ ω P P ] 2 = 2 λ w c = m 0 c 2 = R m n − 1 2 R g m n (9)

The energy-momentum tensor T_mn (cosmic latitude Α neglected) represents the distribution of protonic matter (energy), and the Ricci tensor R_mn the respective symmetric warping of space. In Newtonian approximation, constant ϰ of the left-hand side of the Einsteinian field equation can be replaced through the respective value ϰ = 8πG/c², T_mn through T₄₄ = ϱc². If now the Poisson potential U = 4πGϱ of classical gravity is introduced, then eventually ϰT_mn = 2U follows. If we replace the Poisson potential U by the centrifugal energy E₀ of the resting proton according to Equation (1), then the left-hand side of the Einsteinian field equation takes the form of Equation (9). Thus in the result, firstly the quantum geometric basis of the relativistic time dilation and secondly of the relativistic energy-momentum relation is revealed, third the already longer assumed connection between energy and time.

One also should notice the numerical value λ_w = 0.814013 × 10^‒13 cm of the proton’s four-dimensional geometric height to be very near to the recent experimental finding that the former’s charge radius ≈ 0.8335 (95) × 10^‒13 cm. It is clear that the apparent rest time continuum of any observation every observer in the three-sphere will experience in reality is the finite sum

∑ 0 Δ t τ w = Δ t (10)

of the finite time quanta τ w = h / c associated with λ_w and that the physical presence really lasts only τ_w second with reference to the respective rest-frame in the sphere. This explains why also in quantum physics time always takes a “sharp” (Feynman) value. It is clear that according to the preceding this must be valid for mass and energy as well. Furthermore, a speculative time travel is not possible since existing time quanta can be relativistic delayed though, but, other than assumed by Gödel, past and future time quanta have no physical existence.

Above has been shown that the rest mass of the only stable fermion next to the electron (positron) in the Universe, the proton (antiproton), is directly related to the four-dimensional time quantum τ w = λ w / c = h / c in dependence on the degree of latitude in the order of magnitude (1 − sinΑ). Because of this very basic nature of the proton’s mass it is obvious, that the only other stable fermion with defined rest mass, the electron (positron) should be derivable from the former basically. As shown above, follows the relativistic association of the proton’s mass and energy from the preservation of the four-dimensional height λ_w of a double four-simplex. Now, it is suggested that the electron be build up on the same elementary four-dimensional scale as the proton, but with the variation that the basic three-dimensional quantum volumes of these basic particles, proton and electron (and their antiparticles), are not equal, such that one can write

m e = x × 2 τ w ( 1 − sin A ) ≠ m p (11)

where x stands for the still to be found reduction of the three-dimensional volume fraction of m_p, which obviously has the value λ₁ × λ₂ × λ₃ = 1, to the smallest volume possible in nature. From the preceding it is clear that on this elementary scale between the individual Euclidic grid points λ₁ − λ₃ other points do not exist so that on this smallest possible spatial scale a circle 2λ₁π cannot be constructed within the microscopic λ₁-grid system. The same is true for the construction of the area of an elementary circle π(λ₂)² within the λ₂-grid and of a sphere 4π(λ₃)² within the λ₃-grid, respectively, with the consequence that radius r_e must be other than the former quantum lengths.

Because, thus, in neither spatial dimension on quantum scale radius r_e can be constructed from rational λ₁ nor irrational λ₂ and λ₃, respectively, this implies the former to be transcendental and of the order of magnitude of 1/π. I.e. besides the dimension-dependent quantum lengths λ₁ − λ₃ a transcendental grid with π in the denominator must exist to render construction of circular bodies on quantum scale in all three spatial dimensions. It is clear that within the π-grid, higher π-values can only be generated by multiplying once or more by π itself. So the quantum π-units have to be derived for each spatial dimension individually, depending on the respective smallest length of the three spatial dimensions:

λ 1 2 π 1 1 ⇒ λ 1 2 π 1 = λ 1 π , λ 2 2 π 1 1 ⋅ π 2 2 ⇒ λ 2 2 π 3 = λ 2 π π , λ 3 2 π 1 1 ⋅ π 2 2 ⋅ π 3 3 ⇒ λ 3 2 π 6 = λ 3 π 3 . (12)

Of course, be valid π 1 ≡ π 2 ≡ π 3 . Thus, the mass ratio m_p/m_e takes the form

m p m e = 2 τ w ( 1 − sin A ) λ 1 λ 2 λ 3 ⋅ ( 2 τ w ( 1 − sin A ) λ 1 λ 2 λ 3 3 ! ⋅ π ⋅ π π ⋅ π 3 ) − 1 = 6 π 5 (13)

where 3! in the denominator results from the multiplication of λ₁, λ₂/2 and λ₃/3 to receive in connection with Equation (11) the smallest possible volume of the electron. Therefore is given:

2 τ w ( 1 − sin A ) 6 π 5 ≡ m p 6 π 5 = 9.10956 × 10 − 28     g ,       m p m e = 1836.118109 (14)

One should notice the latter value to be not quite, though very near to the tabular one m_e = 9.109387015 × 10⁻²⁸ and m_p/m_e = 1836.152668. The reason is that both, m_p and m_e, are considered as masses resting in their respective rest frame. However, this does not apply to the electron, because its mass is so tiny that it can only be measured in the orbit about a resting atomic nucleus with known rest mass. But according to the symmetric extension of special relativity, every earthly observer resting in her or his frame of reference considers her or his rest frame to be resting relative to the CMB, whereas the respective atom, consists of the nucleus at rest and the orbiting electron. Therewith the latter translationally is moving with v_0(CMB) ≈ 370 km∙s⁻¹ relative to the CMB. It is thus to be expected that for every observer, who is at rest relative to the CMB, the volume and therewith the mass of the respective electron as well as its orbit will appear relativistically stretched due to this one-way propagation of the atom relative to the CMB by the amount of

Δ x ( 1 -way ) = 370 × Δ T ( 1 -way ) = 370 × Δ x 01 ( v ⊤ ) 1 -way − Δ x 01 ( 1-way ) 2 × π × 370 × Δ t 01 (15)

With unit 1 (one) for one electron’s mass this takes the form:

Δ m e ( 1 -way ) = [ ( 1 − 370 2 c 2 ) − 1 − 1 ] × 370 2 2 × π × 1 = 0.03318362127 (16)

In view of this result, Equation (14) becomes to

m p 6 π 5 = 9.10956 × 10 − 28     g ,     m p m e = 1836.151293 (17)

Notice that the latter value comes very near to the experimental one m_p/m_e = 1836.152668. The slight deviation is most probably due to a slightly higher speed relative to the CMB than the above accepted 370 km/s. E.g. calculation with v_CMB = 373 km/s results in m_p/m_e = 1836.152386 and with v_CMB = 374 km/s in 1836.152749. Finally, it should be mentioned that Lenz pointed out in a short article as early as 1951 that the ratio m_p/m_e comes very close to the product 6π⁵ [16].

The proposed four-dimensionality of the fermions proton (antiproton) and electron (positron) also explains two strange properties that contradict the view and the known physics. On the one hand, this is the experimental fact that a rotation of the axis of rotation by 360 degrees does not correspond to the initial value, e.g. + (plus), but − (minus) results and only after a further rotation by 360 degrees results again + (plus). I.e. two full rotations of 360 degrees each are required to reach the initial value again, while in the geometrically three-dimensional world of our view a rotation of 360 degrees is sufficient for this. Another physical curiosity is the half-integer spin of all fermions and their antiparticles, including the neutron, which has all the properties of a classical angular momentum. But because of its half-integer spin it cannot be explained in a classic way by rotary motion.

Above, both proton and electron were described as the smallest possible and persistent spinning gyroscopes in the quantum world with a four-dimensional diameter 2λ_w, but a different three-dimensional structure that determines the inertial mass. The same must apply to their antiparticles and the neutron and its antiparticle, as inevitably results from the equality of spins in all three fermions. It is known that the fermion spin can assume the sub-values ħ/2 and –ħ/2 with respect to any direction and that those spins can take on an odd multiple of ħ/2, namely ħ/2, 3ħ/2, 5ħ/2, etc., where ħ = h/2π. Evidently the unit ħ = h/2π forms the basic rule of the spin to a certain extent. It is known that this angular momentum cannot be traced back to a rotating object, which itself consists of revolving, non-rotating material particles, but must be an intrinsic property of the particles to which we now turn.

With the rotation diameter 2λ_w of the fermions and h = λ w 2 in the simplest case the four-dimensional angular momentum (spin) takes the value s = ℏ / 2 = h / ( 2 × 2 π ) = λ w 2 / ( 2 × 2 π ) at the zero point of the three-dimensional embedding of λ w 2 / ( 2 × 2 π ) . The angular momentum in the spatially four-dimensional R⁴ would therefore be 2 π s = λ w 2 / 2 = 2 λ w 2 / 4 , so that λ w 2 / 4 rotates on both sides of the zero point in four-dimensional. However, since according to the above length λ_w is retained on both sides of the zero point, both surfaces λ w 2 / 4 must be triangular surfaces with a common axis λ w = 2 λ x ≡ 2 λ 1 . Thus, the four-dimensional angular momentum 2 π s can only be perceived as point s = λ w 2 / 4 π = ℏ / 2 in three-dimensional space-time R³.

From the fact that the fermions proton and electron as well as their antibodies are the smallest four-dimensional permanently stable rotational bodies, it follows immediately, that two full rotations of 360 degrees each (2 × 360˚) are required when the rotational axis is rotated to reach the initial value again.

This doubling of the rotation of an axis of rotation in the four-dimensional, which appears strange in three dimensions, is a compelling mathematical consequence. Mathematicians proved, that symmetric rotation in R⁴ space has two invariant axis-planes which are completely orthogonal, intersect at a point, are left invariant by the rotation and return after rotation by twice the angle around the axis (2 × 360˚). These mathematical-physical facts on the quantum level are therefore also compelling evidence that the three-dimensional space of our world is embedded in a four-dimensional space (see Chapter 5.)

The four-dimensional structure of the fermions also explains the fact that the amount of spin is always the same, while its axis direction can change in a strange way. This alternating spin direction between zero (0) and one (1) results because, as described above, the four-dimensional surfaces λ w 2 / ( 2 × 2 π ) on both sides of the zero point of the three-dimensional embedding λ₁ × λ₂ × λ₃ rotate rigidly in R⁴, whereas the R^{3 −} embedding with 2 × λ₁ as the axes of rotation in R³ freely movable is. In this way, the axes of rotation can be freely pivoted around the zero point, so that in the simplest cases, λ₁ parallel to λ_w, λ₁ perpendicular to λ_w and λ₁ inclined by 45˚ to λ_w, the spin Figures 1 − cos0˚ = 0, 1 − cos90˚ = 1 and 1 − cos45˚ = 0.29 … on the sphere 4 π λ 1 2 (Gaussian sphere). This also shows that the spin behaviors of the fermions and the associated probability calculations in the Hilbert space have a causal physical cause.

In ref. [15] has been demonstrated, on quantum scale of the space of the three-sphere S³ locally each dimension or manifold to be associated with a fundamental length, namely λ 1 = 1 / 2 × h in R¹, λ 2 = 2 × λ 1 in R², λ 3 = 3 × λ 1 in R³ and λ w ≡ λ 4 = 2 × λ 1 in R⁴ so that must be valid:

2 λ 1 = 2 λ 2 = 2 λ 3 3 = λ w = h (18)

These fundamental lengths really seem to constitute the very limit spatially of nature on Euclidian microscopic scale. Also has been proposed the following variations of the latter fundamental lengths to owe the so-called anomalous conductance features in two-dimensional quantum point contact experiments [17 , 18]:

e 2 / λ 1 2 = 4 e 2 / h = 2 . 0 0 × 2 e 2 / h

e 2 / λ 1 λ 2 = 2 × 2 e 2 / ( h 2 ) = 1 . 4 1 × 2 e 2 / h ,

e 2 / λ 1 λ 3 = 2 × 2 e 2 / ( h 3 ) = 1 . 1 5 × 2 e 2 / h ,

e 2 / λ 1 λ 4 = 2 e 2 / h = 1 . 0 0 × 2 e 2 / h ,

e 2 / λ 2 2 = 2 × 2 e 2 / ( 2 h ) = 1 . 0 0 × 2 e 2 / h ,

e 2 / λ 2 λ 3 = 2 × 2 e 2 / ( h 6 ) = 0 . 8 2 × 2 e 2 / h ,

e 2 / λ 2 λ 4 = 2 e 2 / ( h 2 ) = 0 . 7 0 × 2 e 2 / h ,

e 2 / λ 3 2 = 2 × 2 e 2 / ( 3 h ) = 0 . 6 7 × 2 e 2 / h ,

e 2 / λ 3 λ 4 = 2 e 2 / ( h 3 ) = 0 . 5 8 × 2 e 2 / h ,

e 2 / λ 4 2 = e 2 / ( h ) = 0 . 5 0 × 2 e 2 / h .

Thus, the normal integer conductance plateaus

G = n e 2 λ 2 2 = n 2 e 2 h (19)

are integer multiples of the right-hand members of the above listed respective ratios. The “anomalous” conductance plateaus e.g. G ≈ 0.5 × 2e²/h, G ≈ 0.7 × 2e²/h, G ≈ 0.85 × 2e²/h and G ≈ 1.42 × 2e²/h, respectively, then result from the above variations of the quanta of dimension-dependent lengths in dependence on the respective experimental condition. Clearly all “anomalous” conductance features show up. Even the observed variation of the actual position of the 0.7 feature [19] between 0.6 and 0.8 is neatly explained by the narrow neighborhood of the 0.58, 0.67, 0.7 and 0.82 structures, respectively, to the former. Furthermore, from this novel microphysical notion results the “additional” factor of two to e²/h quite unconstrained as a consequence of quantum-geometry without any recourse to a special physical model.

In accord with this physical notion the so-called half-plateaus, G = (n + 1/2) × 2e²/h come into existence if λ₃ (instead of λ₄) is unity so that λ_iii = 1 and it must be valid λ i = h / 3 , implying that the above Coulomb formula e²/λ² for the first conductance step has to be written in the form

e 2 λ i 2 = 3 e 2 h = 1.5 × 2 e 2 h (20)

If all further conductance steps switch to the above normal mode, then we receive from the latter Equation (n + 1/2) × 2e²/h and, therewith, the conductance half-plateaus.

The quantum-geometrical picture also offers a convincing explanation of the expression for calculating the magnetic flux quantum, i.e. the quantum of magnetic flux passing through a superconductor, given by

ϕ 0 = n × λ 2 2 e = n × h 2 e (21)

Furthermore, the puzzling properties of the integer quantum Hall effect (von Klitzing 1980 [20])

R H = R H ( n = 1 ) n = λ 4 2 e 2 × n = h e 2 × n (22)

are fully explained as well as the fractional quantum Hall-effect (Strömer, Hill 1984 [21]), where n = m/(2m + 1) and n = 1 – m/(2m + 1) = (m + 1)/(2m + 1) and we, owing to the particular physical situation, obviously observe a fractionalization of λ 4 2 such that only the sum of the fractions’ reciprocals results

again in ( λ 4 2 ) − 1 , namely:

R H ( m ) − 1 + R H ( m + 1 ) − 1 = e 2 m λ 4 2 ( 2 m + 1 ) + e 2 ( m + 1 ) λ 4 2 ( 2 m + 1 ) = e 2 λ 4 2 = e 2 h (23)

I.e. the sum of these fractionally quantized Hall resistances remains constant. In this quantum-geometrical notion the fractional quantum Hall-effect observed in the new material graphen in steps of n = 4(m + 1/2) with the lowest plateau h/(2e²) even at room temperature, i.e. ≈20˚C (Novoselov 2004 [22]), is just a special case of the integer quantum Hall effect. It owes its existence to the fact that at the particular experimental conditions the first step of the Hall resistance is not λ 4 2 but rather λ 2 2 = 1 / 2 h , where m = 0 , 1 , 2 , ⋯ , m . Therefore it is clear that the above resistance formula really is valid with n = 4(m + 1/2).

1) Already in 1878 the mathematician and initiator of set theory, Georg Cantor, showed the mapping to be unique from a line segment into a square, unique from a square into a line segment, i.e. bijective and discrete [23]. With other words: Cantor proved that unique, one-to-one mappings can be constructed between spaces (manifolds) of different dimensions.

2) Whereas his colleague Giuseppe Peano in 1890 especially proved, the mapping firstly to be unique from a line segment into a square, secondly ambiguous from a square into a line segment, i.e. a point in the square generally corresponds to various points in the line segment, and thirdly continuous [24].

3) The final result of this mathematical research has been the widely known theorem of the invariance of dimension, which first has been proven by Luitzen Egbertus Jan Brouwer in 1911 [25]. According to Brouwer’s proof can a piece of a k-dimensional manifold bijective and absolute continuously mapped only again into a piece of a k-dimensional manifold, but never into a piece of an m-dimensional one if m ≠ k. It should be remarked that there have been given many other proofs of the theorem since.

Bohr remarked (see Chapter 4.3.) that the wave behavior in quantum physics only occurs in very special physical situations or constellations (experiments), but that mostly the classical notion of physics is observable. Possibly, this Janus-faced side of quantic nature might be the very reason of Bohr’s vague concept of complementary.

The “normal” undisturbed trajectory of a stable particle or photon, travelling through geometrically three-dimensional space-time, starts with spontaneous emission from four-dimensionality and ends with reception at a location in space-time. I.e. the latter will be recorded by three coordinates of space and one time coordinate. Thus, this topological mapping V 4 ∈ R 4 → V 3 ∈ R 3 is a point to point mapping but where the way of the mappings carrier, i.e. its trajectory, between the manifolds according to Cantor is discrete: Normally, in nature time and direction in space of the spontaneous emission are non-predictable physically.

Hence, Cantor’s mapping law seems to be responsible for physical mappings from R⁴ into R³ in those cases where the undisturbed trajectories of all particles as carriers of the mappings in space-time, whether with or without rest mass, follow Cantor’s rules from their spontaneous emission till reception. Accordingly the propagation of microscopic material particles in space must be thought of as motion of geometrically smallest possible objects with three-dimensional basis in R³ and the proposed four-dimensional height being associated with the former’s rest mass. Photons however are emitted and received by four-dimensional atoms at their respective also four-dimensional Bohr radii (see Chapter 4.3.) and also moving through three-dimensional space in accord with Cantor’s mapping rules. This topological-physical notion of particle and photonic trajectories as bijective but discrete mappings V 4 ∈ R 4 → V 3 ∈ R 3 and, therewith, as classical (including special relativity) inertial translational motion and photonic propagation, respectively, implies its causality. Hence, this kind of mapping is by no way wavelike, neither in the form of Schrödinger’s wave-packet nor of the de Broglie wave for single particles or of photons according to Einstein. In so far current paradigms that the propagating particle, whether light or material, is either a pure wave-packet (e.g. Schrödinger) or some crossing of wave and particle (e.g. the early de Broglie) are likewise not true physical notions. This also implies that the entanglement of quantum systems in the case of mappings according to Cantor does not exist (nor to Peano; see below). The latter non-existence of entanglement also must be valid in the case of the widely discussed correlation of physical properties of two or more particles (photons) separated by any given distance. In all those cases also a Peano ambiguity cannot occur (see below). Therefore, if a physical property a₁ of a quantum system S₁ is measured (recorded in R³), then the value a₂ of the correlated physical property of the separated system S₂ automatically follows. I.e. from the non-existence of entanglement in the sense of current theory automatically follows non-existence of instantaneous non-local collapse of the putative wave-function. Einstein’s reproved spooky superluminal action at a distance is not a realistic physical notion in the case of physical Cantor mapping but neither in Peano mappings, as will be shown in the following chapters. Also will be demonstrated, the very recently published “disproof of theories based on classical trajectories” not to discriminate the preceding, i.e. the physical existence of classical trajectories of material particles and photons in the case of Cantor mappings.

Richard Feynman: “It will take just this one experiment, which has been designed to contain all of the mystery of quantum mechanics, to put you up against the paradoxes and mysteries and peculiarities of nature one hundred per cent. Any other situation in quantum mechanics, it turns out, can always be explained by saying, You remember the case of the experiment with the two holes? It’s the same thing.”

Note, that in all quantum-mechanical and optical experiments, where the wavelike distribution of particles and photons becomes manifest, neither the physical built-up of the respective particle(s) is measured nor its above discussed undisturbed trajectory in the course of Cantor mapping, but rather e.g. position (coordinates), momentum, time or spin, i.e. exclusively physical properties, recorded after the Cantor trajectory’s interruption in nature or by experiment. This will say, that the wave pattern only occurs if the classical trajectory’s Cantor mapping is interrupted through e.g. scattering, the passage of a measuring device, a crystalline structure or the edge of some material, or, if an electron orbiting about the atomic nucleus is considered and such those physical phenomena are brought into contact with the latter’s four-dimensional topology. Of course, these are only paraphrases of the most popular physical phenomena in the quantum domain, namely e.g. scattering, diffraction of particles and photons by a double slit, of photons and particles by a crystal and of particles and photons passing the edge of some material about in a distance analogous to the width of the slits in the double-slit experiments and the generation of the atomic spectra.

We maintain that in all those cases, and others more (e.g. Chapter 4.4.), we directly observe the four-dimensional structure of material particles and, in association with this, Peano’s ambiguity of topological mappings through physical carriers (material particles and photons) from R⁴ into the spatially three-dimensional space-time R³. From the preceding and the history of quantum physics it is obvious, that it will suffice to show, the de Broglie wave as well as Heisenberg’s uncertainty principle directly to follow from the latter suppositions. This means, that especially the de Broglie wave directly expresses the magnitude of ambiguity of a mapping from R⁴ into R³ in dependence on the momentum of a single particle as carrier of the mapping.

Thus, in association with the preceding discussion referring to the spatially four-dimensional built-up of fermionic matter as well as the of the Universe, Peano’s proof seems to be of special physical interest although with a scale difference of

P 2 λ w ≡ P 2 h = c A 4 ( 1 − sin A ) cos 5 A = 4.88 × 10 40 (24)

We consider a mapping from spatial dimension k (=4) into dimension k − 1 (=3), abstract from dimensions k − 2 and, then, in any case Peano’s finding can directly be applied such that, according to the preceding, continuity of the mapping is postulated with the consequence, that a point in volume V^k (V⁴) corresponds to various points in V^k−1 (V³) and the bijection of the mapping, i.e. one-to-one correspondence, gets lost, implying its ambiguity. In the case of material particles we proceed for reasons of simplicity from a single proton as the carrier of the topological mapping. By rewriting the de Broglie formula λ_B = h∙p⁻¹ of the moving proton, with rest mass m₀, we receive in the relativistic borderline case, where the Lorentz factor γ ≈ 1, in association with Equation (2):

λ B × m 0 v = h ≡ λ w 2 (25)

Factor 2(1 − sinΑ) in Equation (2) is neglected. We rewrite this expression into

λ B × λ w v c = λ w 2 (26)

Obviously is the latter formula the physical transformation of Peano’s above introduced topological mapping from (four-dimensional) plane λ w 2 as the product of the (four-dimensional) line segment λ w v / c = τ w = m 0 and the (three-dimensional) ambiguous line segment λ_B = λ_wc/v, such that really applies:

λ B = λ w 2 c λ w v = λ w 2 τ w v = h p (27)

If any particle with mass m < m₀ is considered, then λ_w in the left-hand side of Equation (26) and in the denominator in the middle of Equation (27) becomes n × λ_wv/c, where n = m/m₀, because only for the proton λ₁λ₂λ₃ = 1 is valid. From this is obvious that not the mass of the mapping’s carrier be the decisive factor but the momentum p = λ_wnv/c = mv/c. Now, if a proton with dimensionless momentum p = λ_wv/c according to Equation (26) approaches another material object with four-dimensional height λ_w then, at three-dimensional distance λ_wc/v, one obtains

λ w c v × λ w v c = λ B × λ w v c = λ w 2 = h (28)

Thus, from this (four-) two-dimensional plane λ w v / c × λ w c / v = λ w 2 a Peano mapping into (three) one-dimensional line segment λ_wc/v with ambiguity λ_B in accord with Equations (26) and (27) is received. Note that Equation (28) can also be rewritten as:

T w × T w c 2 = λ w 2 ,     λ B ( T ) = λ w 2 τ w c 2 = h E p (28a)

This means, that the already experimentally known quantum geometric effect, dubbed tunneling effect, follows also directly as ambiguity effect in accordance with the above.

It is proposed, this topological effect to be the very cause of all known quantum-physical phenomena, especially of the above mentioned interference of material particles (and photons; see below) by a double slit and diffraction by traversing a crystal or passing the edge of some material about in a distance analogous to the width of the slits in the double-slit experiments, furthermore, of the generation of the atomic spectra. In the following especially the physical phenomena referring to the double-slit experiments and the origin of the Bohr-radius’ wave-structure shortly shall be discussed whereby, as widely known, in the former case three physical situations are to distinguish: firstly the double-slits shall be open simultaneously, secondly the slits shall be open successively and third the double-slits shall be open simultaneously with detectors behind the slits.

It is obvious that according to Peano’s proof every quantum-mechanical event in R⁴ occurs at a unique four-dimensional point (coordinates), which in R³ only ambiguously can be recorded with likewise ambiguous coordinates if the unique Cantor trajectory is interrupted through the above discussed physical situations where the four-dimensional structure of matter plays the decisive role. Thus, topologically two open slits for approaching and the slits passing particles according to Equation (28) in association with Equation (27) on passing will deliver two parallel ambiguous mappings λ_B implying the generation of an interference pattern of two ambiguity amplitudes according to wave physics. One should notice that, whilst approaching one slit according to Equation (28), the particle’s ambiguity with respect to the other slit must be

λ ′ B = λ w 2 c 2 v 2 + d 2 = λ w c v 1 + d 2 v 2 λ w 2 c 2 (29)

where d denotes distance between the two open slits, such that for the ambiguity ratio of the open slits referring to one single incident particle applies

λ ′ B λ B = 1 + d 2 v 2 λ w 2 c 2 (30)

This implies that for all particles, directly approaching one slit, according to Equation (28) the probability exists of value

P ′ B = ( 1 + d 2 v 2 λ w 2 c 2 ) − 1 / 2 (31)

to eventually pass the other slit such that, referenced to one slit, the total probability is the sum

P B = ( 1 − P ′ B ) + P ′ B = [ 1 − ( 1 + d 2 v 2 λ w 2 c 2 ) − 1 / 2 ] + ( 1 + d 2 v 2 λ w 2 c 2 ) − 1 / 2 = 1 (32)

This implies occurrence of the well-known two-slit interference pattern of the two ambiguity waves at the receiver behind the slits even if only one particle (photon) is involved (see below). Whereas according to Equations (28) and (27) with only one slit open, only one ambiguous mapping λ_B and, therewith, the normal single-slit diffraction pattern necessarily must occur. Obviously ratios (31) and (32) can be used to test the proposed theory by varying distance d.

Note, that a recent double-slit-diffraction experiment, which recorded single electron detection events diffracting through a double-slit, with the result that a diffraction pattern was built up from individual events, exactly complies with the preceding theoretical expectations [26] (see below).

In the third case, with detection of the incident particle, in what form ever, after passing the slits, are the three-dimensional coordinates of both particles uniquely recorded in R³ (space-time) and, therewith, ambiguity λ_B in all cases terminated such that we now have bijective mapping V 3 ∈ R 3 → V ′ 3 ∈ R 3 , implying classicality, such, that diffraction and, therewith, an interference pattern cannot occur at reception. Note the similarity of the latter topological termination of the ambiguity wave λ_B with Bohr’s otherwise physically unfounded assumption, that observation (measuring) causes the collapse of the wave. I.e., according to Kopenhagen follows from the intended action by itself (per se) non-observation of the wave, with the conjectured spooky (Einstein) consequences (see Chapter 4.4.). It is clear, that in the latter case all forms of the above discussed interference pattern will revive, if after recording the three-dimensional coordinates both particles again would undergo a physical situation as described in Equation (28) and the following.

Besides, if we require Heisenberg’s uncertainty relation for an approaching proton,

Δ x × Δ p ≥ h (33)

to take the form

Δ x × Δ p = h (34)

then, according to the above, this can be rewritten into

Δ x × Δ p = λ B × m 0 v = λ w c v × λ w v c = λ w 2 = h (35)

I.e. Δx = λ_B and Δp = m₀v, because on this elementary level protonic mass must be m₀ = λ_w/c. Clearly relation (35) in association with (33) contradicts Equations (25) to (27) if p = m₀v increases and λ_B equivalently decreases, with h = λ w 2 remaining constant. This also is true if we switch to the relativistic version, with non-negligible Lorentz factor γ > 1:

Δ x × Δ p = λ B × m 0 v = h γ − 1 (36)

Thus, it follows the uncertainty relation (23) to be the relativistic version of the de Broglie Equation (15) such that applies

Δ x × Δ p = λ B × m 0 v × γ ≥ h (37)

Heisenberg was of the opinion that it is not possible to measure energy and time as precisely as desired and therefore “physics should only formally describe the context of the perceptions”, furthermore, “because all experiments are based on the laws of quantum mechanics and thus the Equation (33) are subject, then quantum mechanics definitively establishes the invalidity of the causal law.”

According to the above, Heisenberg's “measurements” are of no importance at all. They can only ever be accurate to a certain extent. Whereas the topological effects described above are conditioned by natural laws and therefore completely independent of any measurements. Therefore, in any case the physical terms “time” and “impulse” are not arbitrarily precise in nature, but always “sharp”. In particular, this applies to the time quantum τ_w and thus for any period of time anyway, but the impulse and speed are also always “sharp” in terms of direction and speed. In all cases, wave phenomena are only triggered, when quantum objects come so close that their four-dimensionality triggers the effects described above. These are completely causal, but not directly traceable due to the Peano’s mapping ambiguity between R³ and R⁴.

In the case of a single photon we proceed from Einstein’s equation of the photon energy E p h o t = m p h o t c 2 = h ν = h c / λ p h o t = λ w 2 c / λ p h o t by rewriting the latter formula to

λ p h o t × m p h o t c = λ p h o t × Δ τ w c = λ w 2 (38)

since according to Equation (2), equivalently must be valid m_phot = Δλ_w/c = Δτ_w. Hence, in the case of the photon as physical carrier of Peano’s topological mapping, we obviously have the ambiguous mapping from two-dimensional plane λ w 2 into one-dimensional line segment Δλ_w = λ_wτ_w/Δτ_w with Peano ambiguity λ_phot.

If analogously to the foregoing a photon with momentum Δτ_wc according to Equation (38) approaches a material object on quantum scale, then at distance λ_wτ_w/Δτ_w we have Δ τ w c × λ w τ w / Δ τ w = λ w 2 = h and, thus, from this two-dimensional plane, also a Peano mapping into ambiguous one-dimensional λ_wτ_w/Δτ_w = λ_phot, in accord with Equation (38), being the basic topological cause of all known wavelike phenomena of light. From the preceding one derives the propagating photon’s “frequency” and energy to be of value:

ν = Δ τ w τ w 2 = Δ τ w c 2 h ,         E p h o t = Δ τ w c 2 (39)

Notice the analogous physical built-up of the proton’s rest energy E = λ_wc = τ_wc² and of the photon’s energy E_phot = Δλ_wc = Δτ_wc², respectively.

Of course, the latter reasoning could have been abridged through referring to de Broglie’s above mentioned introduction of the wave-concept for material particles as analogue to the wave behavior of single photons.

Finally it will be shown that the increase of “waves” in steps of n = 1 , 2 , ⋯ , n of the by Bohr assumed electron’s circular orbit in the case of the hydronic atom, such that λ_B = 2πr_n/n, solely is due to the previously discussed quantization in steps of elementary length λ_w. We rewrite Equation (25) in accord with Equation (23) to

λ B = h m e v e = λ w 2 k 2 τ w ( 1 − sin A ) v e = λ w k c 2 ( 1 − sin A ) v e (40)

where 2τ_w(1 − sinΑ)/k = m_e. Through insertion of v_e = h/(2πm_er₁) into the second left-hand member of the latter eq. we receive λ_B = 2πr₁ and hence

λ B = λ w k c 2 ( 1 − sin A ) v e = 2 π r 1 (41)

where r₁ denotes first Bohr-radius. The latter mid-ratio can be extended in steps of n/v_e only, so that speedv_e decreases accordingly, since λ_w must be conserved, such that for any radius applies:

λ B ( n ) = n λ w k c 2 ( 1 − sin A ) v e = 2 π r n (42)

I.e. the circumference of the Peano-ambiguities of the four-dimensional Bohr-radii is an integer multiple of the de Broglie wave due to quantization in steps of value λ_w. As before, it is obvious that not a real wave is associated with the orbiting electron, nor is the Peano-de Broglie ambiguity rotating about the nucleus. Rather, the electron’s whole four-dimensional orbit ambiguously is hidden for the R³-observer.

Eventually it should be emphasized that by inversion of the preceding topological argumentation, from the quantum-mechanical experimental findings conclusively follows the four-dimensionality of the micro cosmos to be the simplest and most plausible interpretation.

According to the previous it is also obvious that due to the fact that λ_w intersects local three-dimensional XYZ-plane (λ₁λ₂λ₃ on elementary scale) of the three-sphere at zero-dimensional point (singularity), rest time and rest mass (rest energy) must be scalars which always take well determined values in the rest frame, implying that they cannot be subject of quantum mechanical rules, as already remarked by Dirac and Feynman as “sharpness”.

Thus, the conclusion is unavoidable: The ambiguity of physical-topological Peano-mappings V 4 ∈ R 4 → V 3 ∈ R 3 is not associated with a real wave or wave-packet propagating in spacetime, with real group velocity, phase speed, momentum and energy of the wave. Rather it is a “stationary” topological phenomenon, bound solely to the previously discussed and many more natural or experimental physical situations, where a mapping through a material or photonic carrier from the four-dimensional structure of matter into spatially three-dimensional space-time becomes manifest. This implies that the Schrödinger equation and related wave-functions are specialized mathematical models of the Peano ambiguity of accumulations of material and photonic carriers or of physical properties of the latter in four-dimensionality, as e.g. the spin. I.e. wave-functions are mathematical devices or computational tools with no real physical significance. This obviously implies non-existence of entanglement in the sense of current theory and of the associated instantaneous non-local collapse of the respective wave-function (Einstein’s “spooky action at a distance”), even in cases where this apparently seems to be proven. Rather, it can be assumed that two entangled but known quantum states are more or less “frozen” and that they keep this state when separated, so that even after a considerable distance, when one state is measured, that of the other immediately follows. In other words, each of these two quantum states, which are now separated, is not uncertain, but rather certain, though unknown, so that when one of the two states is disclosed (observation, measurement) that of the other results immediately. This applies to any distance, no matter how long. A “spooky action” (Einstein) with a high speed incredibly faster than light is therefore not required.

Finally is to stress: The fact that already a single quantum carrier linearly triggers topological ambiguity in the form of the de Broglie-wave for material particles and Einstein’s equation for photons, respectively, is obviously the very cause of the algebraic linearity of the wave-functions as summations of many single linear ambiguity events.

E.g., consider the above mentioned “Experimental proof of nonlocal wave-function collapse for a single particle using homodyne measurements” [27].

The authors claim for the first time to have demonstrated the instantaneous non-local collapse of the wave-function of a single photon by splitting the latter photon by means of a beam splitter of adjustable reflectivity R (reflection and deflection) between two laboratories and experimentally testing if the choice of measurement in one lab really causes a change in the local quantum state in the other lab. This for they have been using homodyne measurements with some different measurement settings. Notice that the authors maintain to have been splitting a single photon (into two portions of the single photon), the entanglement of the photon’s splits and the non-local collapse of the wave-function.

According to the preceding not a photon has been split but Peano’s ambiguity wave, according to Equations (15) to (17), when the incident photon approached the beam splitter according to Equation (28). Hence, it must be valid

λ p h o t = λ w τ w Δ τ w = λ ′ p h o t + λ ″ p h o t = λ w τ w Δ τ w R + λ w τ w Δ τ w ( 1 − R ) (43)

where R denotes reflectivity as rational number ≤ 1. According to the preceding, must the probabilities to observe the photon in one or the other lab ambiguously add up as to

P = P ′ + P ″ = R + ( 1 − R ) = 1 (44)

Now in the case of homodyne detection or non-detection of the incident photon in one of the both labs i.e. unique record of its presence or non-presence implying immediate switch to bijective mapping V 3 ∈ R 3 → V ′ 3 ∈ R 3 , associated with termination of ambiguity λ p h o t = λ ′ p h o t + λ ″ p h o t , the former ambiguous probability to observe the single photon promptly takes the unique form

P = P ′ + P ″ = 1 ( 0 ) + 0 ( 1 ) = 1 (45)

It is clear that the photon (particle) analogously to the double slit experiment always is traveling only one definite way, but which, owing to Peano mapping V 4 ∈ R 4 → V 3 ∈ R 3 , ambiguously is hidden for the observer in R³.

As already mentioned, has recently apparently been proven non-classicality of a cesium atom’s motion on a discrete lattice, i.e. a so-called quantum walk [28]. The latter atom moves in one of the two standing optical waves of laser light with λ_phot = 866 nm that have opposite electric-field polarizations. In this connection should be noticed that the photons of laser light, due to the enforced stimulated emission and, hence, non-random (non-discrete) parallel propagation of all photons into one definite direction are no more carriers of discrete and one-to-one Cantor mapping as introduced above, but of ambiguous Peano mapping instead. The experiment consists of measuring correlation between the atom’s positions at different times in a complicated manner. The further details are not of interest for our purpose to show that the experiment does not disprove classicality of trajectories, i.e., unique and bijective Cantor mapping, since it really is a further subset of the previously discussed ambiguous Peano mapping. Owing to the atom’s enclosure in an ambiguous photonic lattice relative to a resting photonic lattice, the movement of the atom from its rest position in either direction in jumps of one or more wavelengths to another lattice will in the simplest case be ambiguous

λ B = λ w 2 m C s v = λ ′ B + λ ″ B = λ w 2 2 m C s v + λ w 2 2 m C s v (46)

whereby m_Cs denotes mass of Cesium atom. Thus, e.g., for one jump we have Peano ambiguity of the probability in the order of

P = P ′ + P ″ = 1 2 + 1 2 = 1 (47)

After measuring, i.e. detection or non-detection and associated prompt switch to bijective mapping V 3 ∈ R 3 → V ′ 3 ∈ R 3 , this takes the unique form

P = P ′ + P ″ = 1 ( 0 ) + 0 ( 1 ) = 1 (48)

Therefore, the experiment’s set-up is unsuitable to prove or disprove classicality of trajectories but rather shows that the Peano ambiguity is valid in this very special physical situation too.

The above reasoning, that the photonic enclosure of the cesium atom enforces Peano mapping of its propagation in units of the laser’s wavelength, also is directly verifiable. Consider that according to the

preceding Equation (28) and (38) it must be valid λ B × λ w v c = λ w 2 and λ p h o t × Δ τ w c = λ w 2 . If one demands λ_B = λ_phot, then this implies λ w v c = Δ τ w c = τ w v . Therefore, if instead of λ w v c the real cesium atom’s

dimensionless momentum p C s = n 2 λ w ( 1 − sin A ) v / c is introduced, where n = m_Cs/m₀ = 131.9454, then automatically follows 2 λ B = n λ w ( 1 − sin A ) v / c = λ p h o t = Δ τ w c , or, λ_B = 1/2λ_phot, so that can be written:

λ B = λ p h o t 2 = λ w 2 m C s v = λ w c 2 n ( 1 − sin A ) v (49)

It is clear that owing to the ambiguity of the atom’s location within the ambiguous photonic lattice, the velocity of the “jumps” in steps λ_B = 1/2λ_phot in the denominator of the most right-hand side of Equation (49) can be anything between –1/2 ≤ v ≤ 1 of the wavelengths in cm∙s⁻¹. For this exists the evolution ln ( 1 + v ) = v − v 2 / 2 + v 2 / 3 − ⋯ such that especially is valid ln 2 = 1 − 1 / 2 + 1 / 3 − ⋯ and e^v = e^ln2 = 2. Thus, the atom’s ambiguous velocity (in the standing wave: quantum-walk) must be v = ln2 cm∙s⁻¹ = 0.69315 cm∙s⁻¹. With the proper values of v, c, λ_W in cm, n and angle Α inserted into the latter formula we receive

λ B = λ p h o t 2 = 0.814013 × 10 − 13 × 2.99792 × 10 10 2 × 131.9454 × ( 1 − sin 43.788 ˚ ) × 0.69315 = 433.15   nm (50)

I.e. nearly exactly the reported 1/2λ_phot = 433 nm of the laser light.

Above has been introduced the proposal the Universe to be an Einsteinian near spherical static rotation hyper ellipsoid spinning in a geometrically four-dimensional space with Euclidian E⁴ metric [15]. Furthermore, that every observer at any point within the three-sphere all ingoing light from any distance will perceive as tangential Minkowskian pseudo Euclidian projection such that the four-dimensional Euclidian distance spatially is transformed into time associated with apparent recession velocity, apparent time dilation and associated redshift (the latter explains the deviations of distance calculations on the grounds of supernova data from the standard expansion model – in the latter explained through “dark energy” as apparent ones). This implies that any receiver at any point of the three-sphere will perceive the emitter’s time to be apparently dilated and, thus, the ingoing light redshifted. The CMB has been explained as Planckian radiation resulting from cyclic red-and blue-shift of all light circling the Universe, and from the former’s energy density its correct temperature 2.72 K has been calculated, where K denotes Kelvin.

Now, how does red-shift of light in curved space, other than in expanding plane Minkowskian space through the optical Doppler-effect, happen physically? This for, consider a geodesic ingoing at a resting receiver from a far-away emitter in distance ΔΡπ/2 and, therewith, the ingoing light considerably red-shifted, as we know since the pioneer work of Georges Lemaitre (1927) and Edwin Hubble (1929). According to the law of optical deflection must the light ingoing from curved space for any resting observer be reflected or mapped into a plane pseudo-Euclidian Minkowski space. Since self-motions in the great distances within the hypersphere can be neglected, and the transmitter and receiver can therefore be viewed as fixed relative to one another, it is clear that the measured redshifts cannot be explained by the optical Doppler effect of the prevailing theory (expansion of flat space). Rather it must be due to another hitherto unknown physical effect, to which we now will turn. With Equation (5) the quantum-geometrical origin of time has been derived as the mapping of the four-dimensional radius λ_w of the proton (electron) in the form of the dimensionless tie-point − λ x 2 + λ y 2 + λ z 2 − λ w 2 = 0 in R³ as the quantic continuum τ_w, or sequence of points dτ_w, with highest possible velocity ω P ≡ c = const . , such that is valid d λ w = ω P d τ w and so the improper integral Equation (5). The fact that λ_w, and therefore τ_w, is always perpendicular to its embedding in R³ means that in the curved space of the Einsteinian hypersphere the time quanta τ_w of the receiver’s rest system and the respective time quanta τ ′ w of a very distant emitter must be inclined one to another in direct relation to the curvature by angle Α of the geodesic ingoing at the receiver. However, because according to the above the respective geodesic ingoing at the receiver is mapped as a pseudo-Euclidean Minkowski space, will the zero points of the R³-embedding of τ ′ w and τ_w come to lie on a Euclidean line. The time quantum τ ′ w of the distant emitter is thus rotated by the angle Α in relation to the time quantum τ_w of the receiver in the 3D-sphere so that the time quantum τ ′ w appears stretched analogously to the above special-relativistic Equation (4) and it follows c / ν ′ = c τ ′ w = λ o b s . Therefore, the geodetic curvature around the angle A must become visible in the receiver’s pseudo-Euclidean Minkowski space through extension of the received light’s wavelength by the factor:

λ o b s = λ 0 ( 1 + sin A cos A ) = λ 0 ( 1 + tan A ) = λ 0 ( 1 + z ) ,     z = tan A (51)

Therefore follows directly:

λ o b s λ 0 = 1 + z = 1 + v a p p c 1 − v a p p 2 c 2 ,       z = tan A = 1 + v a p p c 1 − v a p p 2 c 2 − 1 (52)

This implies, v a p p c , the dimensionless velocity derived from the latter redshift, be only an apparent one:

v a p p c = ( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 = ( 1 + tan A ) 2 − 1 ( 1 + tan A ) 2 + 1 (53)

In this way the fixed four-dimensional distance of the emitter’s position within the three-sphere projected into the pseudo-Euclidian Minkowski projection by the ingoing light, at the receiver is transformed into the simulacrum of the emitter’s apparent dimensionless recession velocity.

Consequently every receiver within the sphere can the spatial four-dimensional coordinate of the emitter also perceive as pseudo-Minkowskian in the form of dilation of the emitter’s time by the Lorentz-factor:

( 1 − v a p p 2 c 2 ) − 1 2 = { 1 − [ ( 1 + tan A ) 2 − 1 ( 1 + tan A ) 2 + 1 ] 2 } − 1 2 (54)

Since in the hypersphere all geodesics can be considered to have the same curvature and all λ_w, and thus τ_w, are perpendicular to the zero point of their respective three-dimensional embedding, any geodesic of any length can also be expressed with the same angular dimension as that used in the above equations, namely by the Angle “Α”.

The universe is assumed here as a rotating Einsteinian hypersphere, but it is clear that the distribution of centrifugal forces within the sphere rather suggests an elliptical shape. Recent observations of different redshifts at roughly the same cosmic distances e.g. would do this for redshifts in a hyper-ellipsoid. From Equation (53) follows

( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 = v a p p c = 2 r A P π (55)

where r_Α denotes apparent distance in tangential Minkowski projection on the grounds of the dimensionless apparent recession velocity v_app/c. This delivers the constant value v_app/r_A = 2c/(πΡ) = const. The latter relation enables one to calculate radius Ρ and, therewith, the circumference of the three-sphere S³ by considering that, to mention it again, v_app denotes apparent recession velocity per distance r of 1/4σ = 1/2πΡ, which is identified as the Hubble constant H 0 ≡ v a p p / r A = 2 c / ( π P ) = constant , wherefrom applies

P = 2 c π H 0 (56)

Independent measurements of the cosmic parameter H₀ have all led to different values, the highest value being H₀ ≈ ≥74 km∙s⁻¹∙Mpc⁻¹ and the lowest ≈ 67 km∙s⁻¹∙Mpc⁻¹, where km denotes kilometer, s second and Mpc Mega parsec. The values ≈ ≥74 km∙s⁻¹∙Mpc⁻¹ were all obtained with space telescopes, whereas H₀ ≈ 67 km∙s⁻¹∙Mpc⁻¹ from the Planck satellite on the basis of repeated measurements of the “local” CMB radiation and by earth-bound telescopes was determined. Experts consider H₀ ≈ 67 km∙s⁻¹∙Mpc⁻¹ as the correct value, while highest H₀ ≈ ≥74 km∙s⁻¹∙Mpc⁻¹ is controversial. Based on the above Equation (15) a relativistic solution of the discrepancy has been proposed, with the result that H₀ ≈ 67 km∙s⁻¹∙Mpc⁻¹ from the Planck satellite is correct, whereas the measurements of the space telescopes are relativistically distorted [29]. But because the most observations and therefore also most numbers in the literature stem from space telescopes, the value H₀ = 75 km∙s⁻¹∙Mpc⁻¹ is chosen as the basis for the following calculations.

With H₀ = v_app/r _A = 75 km∙s⁻¹∙Mpc⁻¹ one calculates Ρ = 2.9 × 10³ Mpc = 10.8 × 10⁹ lightyears (ly).

Obviously, due to the four-dimensional deflection of the emitter from the three-dimensional tangential pseudo-Euclidean Minkowski space of the receiver, the mapping by light signals according to Peano must also be associated with an ambiguity of the latter mapping analogous to the Broglie formula on the quantum scale. Thereby, the cosmic redshift and the associated dimensionless apparent velocity v_app/c according to Equation (42) in dependence on the magnitude of deflection from the receiver’s tangential Minkowski space projection (and, therewith on distance) play the crucial role. From the preceding it is obvious that the Peano ambiguity of the cosmic redshift be referred to the latter’s background rest frame (CMB).

Since in the closed S³-hypersphere the redshift is observer-centric, exists for every observer a light horizon at distance 1/2πΡ, where (1 + z) → ∞, with geodesics Δσ = 1/4σ = 1/2πΡ from the emitter to the receiver, both resting. According to ref. [15]: “… it is clear that the emitter’s position, thus, must be deflected by value Ρ from the receiver’s line of sight in tangentially projected Minkowski space, which is orthogonal to Ρ”. I.e., the bending of light-path through the warped space of the S³-sphere causes the receiver to observe pseudo Euclidian projections (Minkowski space) of all ingoing geodesics Δσ = 1/2πΡ in the direction of the tangent vector to the latter such that must be valid:

λ P × π P 2 = 2 π P 2 λ w × π P 2 = π 2 P 3 λ w (57)

I.e., we have a topological Peano mapping λ_Ρ from π²Ρ³, the half of the hypersphere 2π²Ρ³, with four-dimensional height λ_w of the emitter via the four-dimensional geodesic 1/2πΡ into the flat tangential Minkowski projection 2πΡ² with height λ_w of the receiver, whereby the latter projection must be ambiguous since of the higher dimensionality of π²Ρ³λ_w. Note that the farthest light is in-going at the receiver from distance 1/2πΡ such that (1 + z) → ∞ and he, therewith, observes a two-dimensional semi-sphere on the plane of the sky. Hence, really applies:

λ P = 2 π 2 P 3 λ w π P = 2 π P 2 λ w (58)

Other than in quantum physics, the cosmic Peano ambiguity is not bound to the carrier of the mapping associated with a special physical situation on quantum scale, but only on the latter mapping λ_Ρ in the form of fluctuations of the cosmic redshift. One calculates λ_Ρ = 36.0886 × 10⁶ ly being according to Equation (58) equivalent to apparent recession velocity v_app = 819.96 km∙s⁻¹.

Clearly ambiguity waves with redshift amplitudes equivalent to the latter apparent recession velocity signify the Peano mapping of geodesics ingoing at the receiver from light horizons at distance 1/2πΡ redshifted to (1 + z) → ∞. For distances smaller than 819.96 km∙s⁻¹ from the receiver in accord with Equation (58) are lower (decreasing) harmonics of the former ambiguity wave to expect in the order of

λ ( v a p p c ) < λ P = ( z P 2 i ln 2 + 1 ) 2 − 1 ( z P 2 i ln 2 + 1 ) 2 + 1 (59)

Reversely, for larger distances higher (increasing) harmonics should arise in the form of

λ ( v a p p c ) > λ P = ( z P 2 i ln 2 + 1 ) 2 − 1 ( z P 2 i ln 2 + 1 ) 2 + 1 (60)

where i = 1 , 2 , ⋯ , n and z_Ρ denotes redshift associated with apparent recession velocity 819.96 km∙s⁻¹ according to Equation (58):

z P = 819.96 c + 1 1 − 819.96 c − 1 = 2.73884 × 10 − 3 (61)

In the following calculations on the grounds of Equation (59) are compared with a survey of astronomical observations of redshift periodicities (e.g. Tifft, 1996 [30]):

λ ( v a p p / c ) < λ P

# i: 3 4 5 6 7 8 9 10

λ_Ρ(i) km∙s⁻¹: 148.01 74.08 37.04 18.52 9.26 4.63 2.32 1.16

Obs. λ_Ρ(i): ≈146.38 ≈73.19 ≈36.60 ≈18.30 ≈9.15 ≈4.57 ≈2.28 -

For #s i = 1 and 2 correspondingly is calculated 590.53 km∙s⁻¹ and 296.48 km∙s⁻¹, respectively.

Quite obviously are most of the observationally derived redshift periodicities of Tifft and other researchers in accord with the preceding calculations according to Equation (59) (60).

According to Equation (60) basically also periodicities in the redshift of far-away galaxies and quasars should be observable, which indeed have been found by some researchers. The following synopsis compares predictions according to Equation (60) with some astrophysical observations [31 - 33]:

λ ( v a p p / c ) > λ P

# i: 3 4 5 6 7 8 9

λ ( v a p p / c ) : 0.0151 0.0299 0.0589 0.1142 0.2142 0.3766 0.5909

λ_Ρ(i) z: 0.0152 0.0304 0.0607 0.1215 0.2431 0.4836 0.9720

Obs. z: - ≈0.03 ≈0.06 ≈0.11 ≈0.258 ≈0.44 ≈1.0/1.1

The also in the literature mentioned redshift periodicity at z ≈ 0.312 e.g. can be explained by z ≈ 0.312 ≈ 5 × 0.0607 ≈10 × 0.0304 ≈ 21 × 0.0152 etc.. But especially the observed accumulations of redshifts around z ≈ 0.63, 1.2 and 1.8 could be due to the accumulation of redshift periodicities at those values resulting from Equation (60). E.g. is calculated from the above λ_Ρ(i)z:

z ≈ 0.63 ≈ 42 × 0.0152 ≈ 21 × 0.0304 ,

z ≈ 1.2 ≈ 5 × 0.2431 ≈ 10 × 0.1215 ≈ 20 × 0.0607 ≈ 40 × 0.0304 ,

z ≈ 1.8 ≈ 1.9423 ≈ 2 × 0.9720 ≈ 15 × 0.1215 ≈ 30 × 0.0607 ,

where z = 1.9423 is referred to # i = 10 of Equation (60). Obviously, other reported periodicities can be explained analogously. For the succeeding #s i = 11 and 12 is calculated z = 3.8894 and 7.7821, respectively.

Note that especially the observed periodicities λ ( v a p p / c ) > λ P are not undisputed [34]. Anyhow, in the range λ ( v a p p / c ) > λ P the observational situation is much less definite than in the case λ ( v a p p / c ) < λ P .

Eventually, it should be emphasized that the harmonics in the form of observer-centric redshift periodicities, according to the preceding, are not to understand as real accumulations of the respective observed cosmic objects, as e.g. galaxies or quasars, i.e., quantization of their distances. Rather, the former resonances represent Peano ambiguity of the redshift, implying ambiguity of the distances. This will say, that the distances derived from those redshifts are ambiguous too, i.e., they are defective. Furthermore, deliver the above equations a convincing reason for the anisotropic, undulating redshifts measured in the CMB (instead of the gravitational waves introduced in the early universe as part of the big bang model).