In the year 2005-2007 the HiRes Collaboration reported unequivocal evidence for a break of the cosmic-ray spectrum close and above to 5 × 1019 eV . This suppression was confirmed by the Auger Collaboration at a significant lower energy  in the range (2 - 3) 1019 eV. Presently this fundamental feature of the spectrum can be further investigated by the Telescope Array (hereafter TA) and Auger Collaborations which operate the two largest instruments gathering data above 1019 eV.
The TA experiment located in Utah, North America, has a collecting area of about 700 Km2 and is deployed in the historical site of the Fly’s Eye experiment which first recorded the florescence light of air showers in order to measure the energy of primary cosmic nuclei. The technique improves the energy resolution achievable otherwise.
The Auger experiment located in Argentina, South America, has a collecting area of about 3000 Km2. The difference in the collecting areas of the two instruments is reflected in the error bars of the measurements. In fact both instruments use the same hybrid technique which jointly exploits the florescence light yield and the muon density at ground of giant air cascades. Although the data samples collected by the Auger apparatus for measuring the energy spectrum and the chemical composition have an intrinsic superior statistical precision, this study necessitates the outcomes of both experiments.
The calculation of the cosmic-ray spectrum in the range 1019 - 2.4 × 1021 eV presented in this paper is based on two empirical and one theoretical inputs designated by A, B and C. The limited size of this paper impedes a critical examination of the empirical inputs A and B. Consequently they are concisely summarized by two statements: (A) the chemical composition of the cosmic radiation evolves from light to heavy in the range 5 × 1018 - 1020 eV    ; (B) there is a suppression in the energy spectrum above 2.6 × 1019 eV   with respect to a power-law extrapolation from the vast, adjacent, lower energy band, for instance the energy decade 2.6 × 1018 - 2.6 × 1019 eV .
Let us anticipate that the energy scales of the instruments are of critical importance for the validation of the energy spectrum computed in this paper above 2.6 × 1019 eV because imperfections in the calibration of the energy scales affect also the observed fluxes. The Auger apparatus in the hybrid mode of operation at 1019 eV has an energy resolution of about 15 per cent and 10 per cent at 1020 eV. The TA instrument has a comparable energy resolution. As a matter of fact, if the energies reported in the published measurements are regarded as real and not fleeting, there was (2007) and there is (2016) an evident mismatch in the energy scales of the HiRes, Auger and TA instruments. Probably the inter-expe- riment calibration is not resolvable by a rigid shift in the range 1019 - 1020 eV but involves non linear adjustments of the energy scales.
2. The Power Law of the Energy Spectrum of the Cosmic Radiation with a Single Index
A brief description of the third input C follows. Two decades of research via numerical simulation of the properties of Galactic cosmic rays have led to the explanation of the knee and ankle of the energy spectrum of the cosmic radiation . The knee and the ankle are effects caused by the particle transport in the Galaxy (propagation effects) and not by the acceleration mechanism. This explanation is achievable only by introducing a notable assumption called Principle of Constant Indices  which states:
The physical process accelerating cosmic rays in the Galaxy releases particles with an energy spectrum featured by a power law and a constant index of 2.67 ± 0.05 in the energy range 109 - 5 × 1019 eV.
Cosmic-ray propagation in the Galaxy has a negligible effect (see Chapter 17 of ref. 10) on the index inherent the acceleration process. Why this assumption is called principle is justified elsewhere .
The cosmic-ray spectrum shown in Figure 1 in the limited range 109 - 2.6 × 1019 eV is comprised between two rails, the blue lines, which differ by a factor 38 marked in Figure 1. The theoretical framework which explains the knee and ankle features also provides the characteristic gap expressed by the factor 38 .
Measurements of the energy spectra of 12 individual cosmic nuclei in the preknee region 1011 - 1015 eV indicate that the spectral indices are compatible with a common value of 2.67 × 0.05 . Above the ankle energy the all-particle spectrum measured by Haverah Park, Yakutsk, Fly’s Eye, HiRes, AGASA, Auger and TA experiments is also compatible with the common index of 2.67 × 0.05 observed in the preknee energy region (see for example Figure 2 of ref. ).
Presently (2016) the physical process in the Galaxy accelerating cosmic rays in the range 109 - 2.6 × 1019 eV is unknown, nevertheless it has some identified fea-
tures  : (1) it is distributed in space; (2) the accelerated particles obey a power law compatible with a single index of 2.67 ± 0.05 ; (3) it operates in the range 109 - 2.0 × 1020 eV. For conciseness in this paper the ensemble of these features is designated by Galactic Accelerator. Thus the Principle of Constant Indices  has been rephrased here by the properties of the Galactic Accelerator.
Other parameters of the Galactic Accelerator, conceivable a priori, could be the maximum energy of operation Emax and the efficiency of the acceleration cycle versus energy denoted here F. The efficiency may be a function of some variables such as the energy, the time interval elapsed between the birth and the extinction of the cosmic ray, the nuclei abundances at the injection stage of the acceleration cycle, and others. The acceleration cycle denotes a sequence of subprocesses that convert nuclei of the quiescent matter at the injection stage, up to the highest observed energies of 3 × 1020 eV   and eventually beyond this empirical limit.
The Galactic Accelerator is expected to attain Emax above 2.6 × 1019 eV since below this energy the spectrum conforms perfectly to a power law with a parameter of 2.67 (see Figure 36 ref. 10). The tentative energy Emax reached by Uranium nuclei is likely 2.4 × 1021 eV which is regarded in this paper as the extreme limit to the Galactic Accelerator.
3. The Failure of Particle Injection to the Galactic Accelerator
Data useful for the following inference are the flux and the chemical composition in the range 1019 - 1020 eV condensed in the statements (A) and (B) in the Introduction. A scheme of the logical paths of the inference is summarized in Figure 2. Simplicity is an ingredient of the reasoning and intervenes in more
Figure 2. Logical scheme with four biforcation levels based on experimental data (energy spectrum and chemical composition) leading to the conclusion that quiescent nuclei are suppressed at the injection of the galactic accelerator (see also Figure 3) above the energy threshold ELI (Z) proportional to the atomic number Z of the nucleus.
than one link of the logical chain. The intrinsic acceleration process is also assumed not to alter the chemical composition at the injection, which is a common assumption recurrent in the literature.
A priori any incipient deviation from a power-law spectrum with a single parameter could be a depression or an enhancement. The data clearly manifest a depression above the energy 2.6 × 1019 eV marked by a vertical green segment in Figure 1. This particular energy is designated by ELI(H) and shortly by ELI where H is for Hydrogen or protons and LI for Lack of particle Injection to the Galactic Accelerator. These wordings are justified below.
According to the Auger Collaboration the flux depression is halved with respect to a power-law extrapolation with a single parameter at the
energy of (4.01 ± 0.21) × 1019 eV  which is quite consistent with ELI = 2.6 × 1019 eV being this value the lowest energy where the deviation manifests itself (see Figure 36 ref. 10).
Since the hypothetical extragalactic component would yield an enhancement relative to the extrapolation, the observed spectrum rules out the extragalactic component (level 1 of Figure 2). It follows that the Galactic Accelerator is still operating at this energy but a subprocess of the entire acceleration cycle initiates to fail, or equivalently, that the Galactic Accelerator is loosing its global efficiency exhibited and demonstrated below 2.6 × 1019 eV via the constant index of 2.67 ± 0.05 . Since the break in the spectrum at the energy ELI does exist   either the intrinsic acceleration process is becoming inefficient or nuclear abundances are changing with the energy. The constraint of the simplicity dictates that the chemical composition and the acceleration efficiency do not vary simultaneously in the same energy range.
Notice the following important circumstance about the chemical composition of the cosmic radiation around and above to 1019 eV. The fractions of cosmic nuclei above 1019 eV cannot change by nuclear interactions as they do at lower energies. In fact the matter density in the interstellar medium is about 1 g/cm3. A nucleus crossing the Milky Way Galaxy at very high energy accumulates an average grammage between 6 to 8 millig/cm2. Since interaction cross sections are known, it follows that the nuclear abundances cannot change by significant amounts. Due to this circumstance the fractions of nuclei at 1019 eV cannot be very different from those measured in the preknee energy region. Detailed calculations of the chemical composition in the range 1012 - 1019 eV confirm this crude estimate (see Figure 7 of The new horizon disclosed by the measurement of the chemical composition of the cosmic radiation above the ankle energy, Proceedings of Science, ICRC 2015, paper 466, by A. Codino).
The measurements of the Auger and TA experiments indicate that the nuclei fractions change with the energy in the range 1019 - 1020 eV (level 3 in Figure 2). Moreover they change in a very selective manner: the fraction of heavy ions augments with energy, or equivalently, the fraction of light ions decreases with energy (level 4 in Figure 2). The trend of the chemical composition with energy is certain but the absolute fractions of individual nuclei composing the cosmic radiation are still not known. The simple conclusion to be drawn is that: (1) the acceleration mechanism performs with the same efficiency above ELI = 2.6 × 1019 eV; and (2) the relative abundances of cosmic nuclei at the sources, before acceleration, change with the energy.
Major alternatives excluded by data as structured in the scheme of Figure 2 in the energy interval ELI − Emax are: (first alternative) (1) the efficiency of the acceleration process changes with the energy and (2) the relative abundances of the cosmic nuclei are energy independent; (second alternative) (1) the efficiency of the acceleration process does not change with the energy and (2) the relative abundances of cosmic nuclei are energy independent.
According to the conclusion given in Figure 2 the cosmic-ray flux is expected to abruptly fall as the energy increases and then to stabilize in plateaux (staircase pattern) as shown in Figure 3. The gap between adjacent plateaux are directly proportional to the abundances of the nuclear species in the interval, 1 ≤ Z ≤ 92. Thus the observed flux above ELI can deviate from the lower rail in Figure 1 by discrete amounts; amounts related to the fractions of nuclei composing the cosmic radiation. After the flux fall of a given nucleus, the spectrum regains the same slope of 2.67. Virtually each nucleus gives an intensity step. As the energy increases the lighter nuclear species disappear and the total flux grows thinner and thinner (see Figure 3).
4. The Predicted Energy Spectrum in the Range 1019 - 2.4 × 1021 eV
The determination of the energy spectrum above ELI ELI(H) = 2.6 × 1019 eV requires two additional parameters: (I) the abundances of cosmic-ray nuclei around 1019 eV at the injection; (II) the rule by which cosmic-ray nuclei are filtered at the injection.
Figure 3. Illustration of the mechanism giving rise to a staircase pattern in the energy spectrum. The mechanism is designated in this paper by lack of particle injection to the Galactic Accelerator and the lower energy where it manifests itself is ELI = 2.6 × 1019 eV. As far as the energy increases above ELI quiescent atoms at the injection are somehow filtered, initiating with the lightest nuclei, i.e. protons and terminating with the heaviest one, Uranium at the energy ELI (U) = 92 × ELI(H) = 2.4 × 1021 eV.
The fractions of nuclei of the cosmic radiation normalized to Iron adopted in this calculation are: H 0.40, He 0.22, CNO 0.17, Ne-S 0.10, F-Ca, 0.012, Sc-Ni .089 and Zc-Ge 7.4 × 10−4 based on a variety of measurements   . They are expressed in total energy per particle and not in energy per nucleon. This unit is appropriate for very high energies. Nuclei are bunched in small groups with adjacent atomic numbers to simplify the calculations.
The relative amounts of nuclei at the injection do not suffice to calculate the energy spectrum because the particular energy of a nucleus Z for which the injection process is hampered, is not assigned. The suppression mechanism delineated in Figure 3 only implies the disappearance of lighter nuclei before the heavier ones, as the energy increases. The simplest rule reflecting the fact that the chemical composition is becoming heavier above 1019 eV is to admit that nuclei of atomic number Z are depleted above the threshold energies ELI(Z), that is, ELI(Z) = Z × ELI(H). The implication is that above the energy ELI(Z) the element Z is not available as cosmic-ray source matter for any reason whatsoever. According to this linear relationship between E and Z, Helium is expected to be depleted above the energy, Z × ELI(H) = 5.2 × 1019 eV, Nitrogen above 1.75 × 1020 eV, Silicon above 3.5 × 1020 eV, Fe nuclei above 6.7 × 1020 eV eV and so on. If the rule were a linear relationship between E and A, e.g. ELI(Z) = A × ELI(H) the incipient energies for the flux fall would change accordingly (A is the atomic weight of the nucleus).
The predicted spectrum in the range 1019 - 2.3 × 1020 eV is given in Figure 4 (green squares) along with the positions of the threshold energies ELI(Z) of the
Figure 4. Computed spectrum up to the energy of 2.3 × 1020 eV where the Oxygen depletion is expected to take place. The horizontal blue line is the extrapolated spectrum based on the Auger data  in the energy interval 1018 - 1019 eV.
abundant elements (H, He, C, N and O). The observed spectrum is multiplied by E2.67 and the blue line represents the resulting flux and its extrapolation to high energies. The blue line is normalized to the Auger data at the intensity of 798 par/m2 s srGeV1.67  which corresponds to 1.159 × 10−25 par/m2 s sr GeV in the flux unit reported in Figure 1. Note that this normalization to the Auger data, and not to those of the TA experiment, is arbitrary.
Of course the flux in Figure 4 continues to decrease with the power law, which is the characteristic of the Galactic Accelerator, but there is an additional decrement caused by particle injection failure generating a first break above 2.6 × 1019 eV (proton depletion).
In Figure 5 the computed spectrum is extended up to 2.4 × 1021 eV where Uranium injection failure is predicted to occur. Above the Fe threshold, ELI(26) = 26 × ELI(H) = 6.7 × 1020 eV Iron disappears, giving rise to an almost vertical flux fall, due to the paucity of trans-iron nuclei. The trans-Iron break is not out of reach of the planned JIM-EUSO experiment .
5. Comparison between Computed and Observed Spectra
The spectra measured by TA and Auger Collaborations are compared with the predicted spectrum (green squares) in Figure 6. In order to better focus on the minute aspects of the spectrum, a linear scale of energy is used. Below 8 × 1019 eV the computed spectrum lies between the TA and Auger data and above this
Figure 5. Computed spectrum (green squares) up to the energy of 2.4 × 1021 eV where Uranium depletion is expected to commence. Fluxes above Rubidium nucleus are out of scale, due to their extreme paucity.
Figure 6. The predicted spectrum (green squares) is compared with the recent measurements of Auger (blue dots) and TA experiments (red squares) . Notice that the last blue dots are upper limits. The horizontal blue line is the extrapolated spectrum based on the Auger data  in the energy interval 1018 - 1019 eV.
energy is compatible with the TA data while those by Auger are deficient. Notice that the average gap between TA and Auger fluxes above 8 × 1019 eV attains almost an order of magnitude signaling severe problems in the measurement procedures.
Figure 7 and Figure 8 show the comparison with the spectra measured by previous experiments having the largest exposures. An example of the spectrum measured by the Fly’s Eye Collaboration, reported in the year 2000  is shown in Figure 7 (black dots). In the half energy decade, between 5 × 1019 eV and 1020 eV there is a hint for the proton depletion. Above 1020 eV some events are observed and the related flux is slightly above the prediction (green squares).
The spectrum measured by the AGASA Collaboration  is shown in Figure 8. With some imagination a small valley is visible in the interval 5 × 1019 eV 1020 eV, compared to the extrapolated AGASA spectrum rooted in the range (1 - 5) × 1019 eV and represented by the black horizontal segment. Data points above 1020 eV in Figure 8 exceed the computed spectrum (green squares).
The four quoted experiments together do not disprove the calculations described in this paper. If the comparison is limited to the data collected by the hybrid techniques of the TA and Auger instruments, the predicted flux is too high against the Auger data but not against TA data (Figure 6). Should the comparison include the fluxes reported by AGASA and Fly’s Eye experiments, the data are evenly scattered, above and below, around the predicted spectrum and no firm conclusion emerges, neither for rejection nor for validation. The re-
Figure 7. The energy spectrum of Fly’s Eye experiment (black dots)  is compared with calculation (green squares) in the range 1019 - 2 × 1020 eV.
Figure 8. The energy spectrum measured by the AGASA experiment (black dots)  is compared with the calculations (green squares). The blue line is normalized to the Auger data  while the black horizontal segment is an average value of the AGASA flux in the interval (1 - 5) × 1019 eV. The highest AGASA data point around 2.3 × 1020 eV is out of scale and does not appear in the figure.
jection of the predicted spectrum only on the basis of the Auger flux above 8 × 1019 eV is premature until the large gap between TA and Auger spectra will not be clarified (Figure 6).
A second fact in favor of the computed spectrum reported in Figure 4 is that the Auger spectrum above 5 × 1019 eV deviates from the extrapolation (blue line Figure 4) not by a power law with a single parameter but by steps, as the Auger data in Figure 9 demonstrate.
An alternative explanation of the observed cosmic-ray spectrum above 1019 eV recurrent in the literature is that a hypothetical extragalactic component of the cosmic radiation would suffer a depletion by the impact with the ubiquitous photons of 6.7 × 10−4 eV and density of 411 particles/cm3. The depletion would commence above 6.0 × 1019 eV according to the original calculations made sixty years ago and is usually described by a power law with a single parameter, much softer than 2.67. This hypothetical phenomenon is known as GZK effect. Let us mention that HiRes, Auger and TA Collaborations have explicitly claimed evidence for the GZK effect   .
Figure 9. Energy spectrum of the cosmic radiation in a linear scale of energy up to 2.3 × 1020 eV measured by the Auger Collaboration  . The horizontal blue line represents the extrapolation of the spectrum with a universal index γu = 2.67. The normalization is slightly different from that of the previous figures. The green horizontal line is a visual guide which marks an arbitrary suppression factor of 12.2 relative to the upper blue line. The data suggest the staircase pattern of the spectrum compatible with the calculation reported in this paper.
The heavy chemical composition of the cosmic rays above 1019 eV makes this alternative explanation quite unlikely as explained elsewhere (The absence of the GZK depression in the energy spectrum of the cosmic radiation, by A. Codino, ICRC 2013, Rio de Janeiro, Brasil).
 Codino, A. (2017) The Principle of Constant Indices and the Knee and Ankle Features of the Energy Spectrum of the Cosmic Radiation, Submitted for Pub- lication and Available Soon in the Astrophysical World Network.
 Codino, A. (2015) Progress and Prejudice in Cosmic Ray Physics until 2006. Società Editrice Esculapio, Bologna. http://www.editrice-esculapio.com/codino-progress-and-prejudice-in-cosmic-ray-physics-until-2006/