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Supernovae as Empirical Evidence for a Curved, Static and Spatially Closed Cosmos

Supernova photographed by the Hubble Space Telescope in 2006. Credit: NASA, ESA, P. Challis and R. KirshnerSpirit of Truth, Enlighten and Guide our Research
O Mary Conceived without Sin, Pray for us who have Recourse to Thee

An alternative to the Big Bang paradigm was presented at the International YouResAstro 2012 Workshop for the physical part and at the X International Ontology Congress for the metaphysical part. The physical part is reviewed here, with some further clarifications and updates. Tighter correspondence of results occurs between the alternative paradigm values and the empirical data. A topology of four and mathematically interlinked frames of reference is proposed. The analysis uses the measurements of the distance of the Supernovae, in particular the discrepancy between the measurements obtained using the shift of the light towards longer wavelengths (i.e. redshift) and those using the degree of brightness attenuation. Such discrepancy is read in the Big Bang paradigm as evidence of acceleration of expansion. The alternative paradigm reads the discrepancy as a measure of curvature in a static curved cosmos, implying physics with tilting time axes, without need of dark energy. Such physics would be applied to the observed universe plus the act of observation, implying the need of both modern standard physics on the observed universe and an alternative physics perspectives. The updates below imply a shift of the main frame that would provide direct empirical evidence of raw data.


Previous articles proposed how to read empirical evidence, which is used in the standard Big Bang paradigm, in order to support an alternative paradigm (Benazzo, 2010, 2011). The latter article may generate skepticism as it argues there is evidence for superluminality, while Einstein’s fixed maximum speed of light c has been recently confirmed by scientists. That article is far from claiming that Einstein’s relativity in modern standard physics needs corrections. It rather provides some evidence that, in addition to such physics - where Einstein’s relativity with the fixed maximum speed of light c holds - there would be an additional physics representation where the same empirical findings would provide evidence of superluminality. If these two representations were applied to the same physical domain, the two physic views would contradict each other such that only one of the two could be possible. Those articles analyse evidence implying that the two physical readings would apply to two different physical domains, such that they may be concurrent. The traditional physics would be applied to the observed universe, while the alternative paradigm would apply to the observed universe plus the scientist’s observation act. It is impossible for the observation act to be concurrently empirically studied and instrument of that same empirical study. In the defined cosmos domain therefore, the collection of empirical evidence would be unavailable. This domain would therefore be transcendental, normally unreachable by empirical evidence. The 2011 article proposes four frames of reference that would allow relating mathematically the modern standard physics applied to the observable universe, with the physics applicable to the transcendental (i.e. unobservable) cosmos. The previous articles read the empirical findings both in a flat observed universe domain and in a curved cosmos domain. The measurements of the two domains provide interlinked findings that show many corresponding measurements. The flatness of the observer’s light cone would virtually flatten the curvature of the cosmos, such that the observed universe would be a virtually deformed frame of the actual unobservable cosmos frame.

Without this interlink and without passing through modern standard physics, the application of empirical evidence to a transcendental domain as the defined cosmos would be unfeasible. The key feature of the alternative paradigm is to consider the time and space axes to tilt, while traditional and modern standard physics keeps the time axis vertical and the space axis horizontal at the large scale. The comparison between the two paradigms is in Figure 1. “U” stands for flat expanding universe; scaled down to be compared with “C”; “C” stands for curved static cosmos. The time and space axes are preceded by “U” or “C” to mark to which of the two paradigms they refer. The origin in each of them is marked by “0”. The “C” paradigm has multiple tilting time axes Ct around a curved space Cs.

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F1/ Scaled down Big Bang Universe (U) vs alternative paradigm Cosmos (C), on space (s) and time (t)In the cosmos domain, the time axis tilting provides for varying speed of light in a curved static cosmos. With two physical domains, the stationariness of the cosmos, in agreement with Einstein’s (1917) vision of a static universe, would apply to the cosmos (universe+observation act), without a fixed maximum speed of light (see Ellis, 2007, Magueijo, 2003 and Moffat, 2002 on varying speed of light). The relativity he was studying provided evidence of a fixed maximum speed of light in the observed universe domain. Its math used in the Big Bang paradigm provided evidence of an expanding universe. The addition of the cosmos domain would allow reconciling these two aspects of Einstein’s work (Benazzo, 2010, 2011a, 2011b). Einstein’s stationariness would fit the cosmos domain, while a fixed maximum speed of light would fit the standard universe domain. The analysis below on the alternative paradigm would consider some empirical evidence for this.

In these pages, the paradigm is examined for finding more precise readings of its structure and for further clarifying the implications.


The topology proposed uses the standard representation of the light cone (see Figure 2 and Figure 5). More experienced readers may wish to skip or glance quickly though this section about light cones. An accessible explanation of how the light cone works is depicted by Davies (1982) and one proposed simple reading of it is in a previous article (Benazzo, 2011b) in two paragraphs at the beginning of the section “Graphical calculations on the alternative topology”.

F2/ Standard light cone: red arrow = star in space receding at the speed of light cIn few sentences, the reader may consider a fixed distance from Earth, and consider that distance as the radius of a celestial sphere. Then such sphere may be considered as the three dimensional view accessible on the internet, like the full sphere 360 degrees view of a room of a historical palace (which may be navigated through by dragging the mouse). The mouse would correspond to the observer moving the sight direction on different parts of the sky on such sphere. The sphere may then be considered as a photo negative, and such spherical negative would need to be considered to move towards the observer at the speed of light c, shrinking in the process to arrive surrounding the Earth’s atmosphere and then to arrive further shrinking until the observer’s telescope and eye (the negative would need to be imagined somehow as able to be traversed by the stars without tearing apart, like for example a very thin thread net dragged against a flowing wind). While the spherical negative shrinks, the impression of the traversed stars leaves the light mark on the negative, which acquires therefore more and more tiny dots of light. Time is represented here by the flow of shrinking of the spherical negative. If the vertical space dimension is taken away, the Earth and the spherical sphere (the negative) become circles on a plane, and the observer becomes a small dot at the centre. The shrinking of the negative becomes thus a shrinking of a circle. If time is represented vertically to flow upwards (on the vertical axis freed by photographing 3D space on a 2D plane), the shrinking circle negative draws a cone upwards and the top vertex represents the observer receiving the shrank circle negative. Such cone represents the past arriving to the observer, and is called past light cone. When light is emitted from the Earth surface (e.g. from cities, or reflected from the water surfaces) and from a strong light house near the observer, such light is emitted in space as an expanding sphere, which expands in time, in the future. Such dynamic may be represented by an inverted light cone, called future light cone. This may be fitted on the top of the past light cone, such that the two together show light arriving from the past and then flowing away in the future.


Main empirical characteristics are considered in order to provide an alternative topology. These are based on the redshift and on the stars luminosity.

More experienced readers may wish to skip or glance quickly through this paragraph about redshift and luminosity. The redshift is an effect of light that is parallel to the Doppler Effect for sound. This effect generates a difference of sound for moving objects, like a car, approaching the observer, overtaking her/him, and then receding away. When approaching, the pitch of sound is higher than when receding away. The wavelength of sound shortens when the sound source approaches and extends when the source recedes. The Doppler Effect occurs also for light emitted by a source moving with respect to the observer, in the same way. For faraway stars, there is a wavelength effect parallel to the Doppler Effect. The stars leave a signature of different light intensity at different frequencies, like the irregular signature of the indentation of a door key. For faraway stars, this signature moves to longer wavelengths. In the visible light, longer wavelengths are towards the red colour, therefore the effect is called redshift. This effect is larger the larger the distance of the star from the observer. The Big Bang paradigm reads the redshift as indication of recession velocity of the stars, likewise the Doppler Effect. In the Big Bang paradigm, this recession velocity would add up by accumulation of successive expanding distances and would be accelerating. Another possible measure of the distance of a star is its brightness. This depends on the distance and the star luminosity emitted on its surface. The luminosity depends on other variables, i.e. the temperature and the radius. Due to the aggregation of these variables, the contribution of distance to brightness is normally unobtainable. For some stars under certain conditions, it is possible to determine their surface luminosity and therefore to use their brightness as a means to measure distance. They are called standard candles. Supernovae, those of type Ia in particular, are such candles. This possibility applies also to gamma ray bursts (GRB). Most of them are believed to be a narrow beam of intense radiation from a particular collapsing form of supernova (S. E. Woosley and J. S. Bloom, 2006).

For supernovae type Ia, comparison is possible between the distance computed by using redshift and the distance computed by using brightness. Saul Permutter, Brian P. Shmidt and Adam G. Riess who were studying supernovae as standard candles have first empirically discovered in 1998 a discrepancy between the two (S. Perlmutter et al., 1998, A. G. Riess et al., 1998). A readeable account of the cosmology implied is done by A. G. Riess and M. S. Turner (2004). The measurements by redshift indicated a shorter distance than the measurements by brightness. Since then the measurements have progressed. The Supernovae Legacy Survey (Conley et al., 2011) is one of the surveys that are currently providing recorded measurements data of redshift and brightness for a large number of Supernovae. In the Big Bang paradigm expanding universe, this discrepancy meant an acceleration of the expansion. This acceleration requires the addition of dark energy, an even more elusive type of energy compared to the dark matter. C. H. Lineweaver and T. M. Davis (2005) decode the very technical language and physics involved in the Big Bang paradigm and provide an excellent account, quite readable by the general public, of its main characteristics.

The alternative paradigm assumes that the redshift may be due to a different effect than spatial recession (Figure 2). It would originate from the tilting of the time axis (Figure 1, Cosmos frame C), seen thorough the observer’s flat light cone (Figure 1,Obsever’s  frame and Figure 3), such that the farther the star, the more of the speed of time would be seen. The stars would instead be static in a curved space time (Benazzo, 2003, 2010, 2011). Einstein (1917) was initially for a static and spatially closed universe.

The hypothesis made by the alternative paradigm is that the discrepancy between the two measures would provide evidence of the measure of curvature of space time in a static cosmos, while reading the expansion as a virtual expansion in modern standard physics where time is considered to flow vertically at the large scale. Such modern standard physics would need to be the one applied to raw empirical data, as the signals arriving to the observer are received though a flat light cone. With a flat light cone peering through a curved cosmos, there would need to be relations that transform a curved geometry into a virtual flat geometry. In flat space-time geometry, light would travel a certain space horizontally in a given time vertically, such that a right triangle defines the speed of light (Figure 1). At longer and longer distances flown by light, the triangles considered are always self similar and this would indicate a constant speed of light c. The alternative topology uses such triangle and is pictured in Figure 4, as follows.


In the topology where the variables are situated, there are four frames. With reference to Figure 4, frame O is the frame of the lightcone of the Observer. Frame C is the frame of a Curved Cosmos. Frame E is an Expanded frame due to the curvature of frame C. It represents how the faraway local observer would see the physics of the tilted space time locally in that faraway location. In addition, it would allow the Observer in frame O to receive information from the hidden Cosmos frame C, through transformations of curved arcs into flat sides of right triangles. Frame P is a Projection of frame E into a geometry with space and time axes respectively parallel to those locally felt by the Observer in frame O, for which standard physics determines a fixed maximum speed of light c. To make an analogy, the Earth may be represented with a faithful view on a sphere, while transformations need to be done in order to represent it on a flat poster without tridimensionality, through a projection which holds fixed certain proportions (even if it may need to compromise on others).

F4/ Alternative topology with its frames: cosmos (C), observer (O), expanded (E), projected (P)The comparison with the Big Bang paradigm may be done as pictured in Figure 1 above. The red line represents the Big Bang paradigm,  with scaled down acceleration, starting from the bottom with an extremely fast inflation (almost horizontal expansion in a lot of horizontal space in very little vertical time), then the expansion passes to a rate more comparable to the current rate and then the curve shows a successive acceleration of the expansion. It is apparent that the alternative paradigm follows a comparable curve, explaining it in a different way. The observer would be forced to gaze at frame C looking through the flat light cone of the frame O. Given that C is curved and O is flat, then the observer would receive empirical data though transformations that may be represented by frame E and frame P.

On each triangle of the figure, there is one blue side which represents space, indicated with “s”, one violet side which represents time, indicated with “t”, one yellow side representing the light vector, indicated with “l”. They represent flat geometry. Frame C is curved in a circle representing actual space in the Cosmos domain. Time flows there from the centre of the green space circle outwards as radii of ever fractally expanding circles. A bigger green circle would represent the same curved space in a future time with respect to the smaller circle. The circles look bigger and bigger in the future, however they would be exactly the same, without space expansion. As the frame O provides the geometry of how the Observer’s light cone intersects the Cosmos frame C, a choice needs to be done on where this light cone intersects the circled space. This intersection would generate a chord and its arc (see Figure 4); and two radius touching their two endpoints define an angle at the centre.

A hypothesis is done that that intersection marks a chord on the light cone corresponding to the Hubble length where the stars result as receding at the speed of light c when the Big Bang paradigm is used to read empirical data. Additionally in the light cone, its surface representing the speed of light c is assumed tilted away from its central time axis by 60 degrees. Figure 3 describes this choice. In the Big Bang paradigm represented by Figure 2 (and Figure 5 below), the red line with red stars is pictured superposed to the light vector at the same 60 degrees, representing thus a star moving at the same speed of light c. In the other Figure 3, the entire light cone is tilted until its time axis superposes the observer’s future light ray towards the astronomical body. What would move at the speed of light would then be the time of such faraway star. The rest would be stationary and this implies that the star would be stationary in space. The future light vector from the star towards the observer would be parallel to the time axis of the observer. This would be taken to mean that the signal from the faraway star would normally never reach the time axis of the observer, that is, it would never reach the observer. It would become invisible. In special relativity, when a body is receding at the speed of light, it becomes invisible, implying an infinite redshift (Lineweaver and Davis, 2005).

F3/ Alternative: tilted time axis = stationary star with time receding at the speed of light cSuch representation describes therefore a special relativity type dynamic. Before passing to describe a general relativity type dynamic, such special relativity, applied to the alternative paradigm, requires that the observer’s point of view be inverted without changing the laws of relativity. As such, inverting the perspective, the future light vector from the light cone to the right is parallel to the time flow axis of the light cone to the left, as in the reverse case. In addition, with the 60 degrees angle chosen mentioned above (see figure 2 and 3), comparison may be done on the length of the light vector before it reaches the observer’s light vector. In the modern standard physics representation of Figure 2, the future light vector from the star towards the observer reaches the vertex of the light cone where the observer is situated. In the tilted alternative representation of Figure 3, it is vertical and it reaches the observer’s light vector vertically on top. The 60 degrees angle defined above, imply that these two vectors have the same lengths, meaning that in their respective frames of reference, they flow for the same amount of time. As the triangle formed between the two light cones of Figure 2 and Figure 3 is equilateral, the point of view may be reversed providing an equivalent time of flight. This implies also that modern standard physics and the alternative paradigm may be read and related on the same topology. Figure 5 represents an equivalent however rotated view of the modern standard representation of Figure 2, to compare it with Figure 3 of the alternative paradigm.

F5/ Standard light cone (rotated): red arrow = star in space receding at the speed of light cThis double reversibility would provide a relativity equivalence principle for both aspects. This would imply full interchangeability of time with space displacement at the speed of light c. Time would flow at the speed of light. With the hypothesis that time tilting provides redshift, then the light cone of Figure 3, tilted at 60 degrees, would provide the same redshift of an object receding at the speed of light (Figure 2 and Figure 5). Changing from such 60 degrees to other angles between the light cone surface and the central light cone time axis, the mentioned triangle would be different than equilateral and the light vector considered to flow at the speed of light c would flow for a different amount of time in the two frames of Figure 2 and Figure 3, losing interchangeability of point of observation and of interpretation between modern standard representation and the alternative representation.

Such angles relation determines a fixed angle between the light vector with the space axis (30 degrees) and with the time axis (60 degrees) (Benazzo, 2003, 2010, 2011), as apparent in Figure 4. This allows expressing space, time and light vector with the same unit of measure (Benazzo, 2010, 2011). In the calculations below, the space units are used. This is important in the topology to allow triangulation relations among different frames that are tilted differently.

Starting from these special relativity considerations, light from stars would need to become invisible at the Hubble length, however empirical evidence shows that stars are seen also beyond the Hubble length. There is the assumption that there would be topological transformations that allow seeing the light even at greater than those 60 degrees angles. Relativity allows exchanging space with time, depending on which of the moving object is taken as fixed point of observation. There is the space-time triangle of the vertical light vector at the Hubble length in the topology (Figure 3 and Figure 4 Frame E) that protrudes towards and reaches the time axis of the observer. The signal is considered to undergo general relativity type transformations for which the signal is received by the observer when first the space-time triangle of the light vector touches the time axis of the observer. This consideration is pictured by frame E triangles. This is assumed to represent in the alternative paradigm the dynamics of general relativity described by Lineweaver and Davis (2005). In this case light is received from the Hubble Length and its redshift is below infinity.

For marking the starting and ending points of the segments, the starting is indicated with “i”, standing for interception of the Curved Cosmos light signal, by the Observer’s flat light cone. Then the arrival, defined by “a”, occurs when the light signal on the light cone surface reaches the Observer’s time axis. The axes and the vectors are marked in the same way, and the difference between them consists in the vectors being added of the mark “_ia”, meaning they have an interception for start point and an arrival point, delimiting their length. In the topology, if there is only “i” or “a”, then this indicates that the caption indicates a point, either the starting intersection, or the arrival at the receipt time axis.

In frame C, time flows from the centre outwards throughout the 360 degrees. There are two green arches, the external one represents the present, the internal represents space at a given starting point in the cosmological past. The two are identical physically, however they are fractally different. Previous analysis (Benazzo, 2011) has brought to consider that brightness would discount fractality, such that the two green arches measure exactly the same, both the starting one at interception and the arrival one (larger). On the other side redshift would be unable to discount fractality and therefore would show the initial smaller space one, i.e. as smaller. The larger one is therefore considered to reflect brightness dimming, and the smaller to reflect redshit. Therefore, for each of the three flat frames, O, E and P, there are two triangles, the smaller one is called “r”, to provide the measure of the redshift. The bigger one is called “b”, to indicate the brightness dimming due to distance.

In the curved frame of the Cosmos, C, there are some measures without difference between redshift and brightness dimming. These are indicated without differentiating between “r” and “b”. In frame C, the two different arches differentiate rather between start and arrival cosmological time, such that the radius difference between them provides the cosmological time passed between interception and arrival. There would as such be two different times: The universal time would be the classic one as the vertical axis of the space-time triangle of the light vector (Figure 1, the one axis Ct that is parallel to Ut). The cosmological time would be the radial difference between the intercepted and the arrival arches. This is described more in detail in Benazzo (2011).

Calculations can be performed on such topology as done in the previous article. The formulae of the previous article are reconsidered with this updated notation, and they are characterised by a rationale supporting their definition.


A reader wishing to skip mathematical formulae may wish to glance through this section for the description of the rationale behind the formulae and then leap to the next section on the empirical implications of the formulae.

Previously proposed formulae (Benazzo, 2011) are reviewed with some additional considerations. In order to avoid recursive back and forth with the previous article, formulae are here proposed in full. In addition, given the intricate topology implying different frames of the same variables, the following paragraphs propose a redefinition of the variables in order to simplify their reading, proposing categorization by frame. Most of the formulae are identical, even if with renamed variables. Reference to the previous formulae is done by adding a number of the previous formula in the format e.g. {cpf9} standing for “compare with the previous formula number 9”. Variables of the formulae below are pictured in Figure 4.

When a parenthesis with a number is put after the variable, it indicates at which angle, called sigma, the variable is considered, i.e. which is the time axes tilting angle at the centre of the topology that is used in that formula. When this is absent, the formula is considered generally as applying to different angles. Angles considered in this article are from 0 to 90 degrees tilting from the observer’s time axis.

The Hubble length defines how far from the observer is the region where the stars recede at the speed of light when the Big Bang paradigm is considered. The arc space Csb_ia is considered as pertaining to the transcendental curved frame C, therefore invisible. The observer would rather receive instead the Hubble length through the flat geometry as Esb_ia. Therefore the following calculations are performed.

Concerning reference to the previous article (Benazzo, 2011), calculations from {cpf1} to {cpf6} correspond, in both calculations and description of the rationale. Calculations {cpf7} to {cpf9} are the same as before, however they are described by an updated rationale. Formula {cpf10} is performed differently, to obtain the same results. Calculations in the subsequent formulae use the same basic structure, however they are updated by the addition of deformation rectification factors of the Expansion of frame E and the Projection of frame P.

[1] {cpf1} Definition: Esb_ia{60} = 13.70 billion light-years = Hubble Length 
Hawking and Mlodinow (2010) and NASA Science Team (2011) use 13.70 billion light-years

With Esb_ia as Expanded space relative to brightness in frame E

In this case with sigma=60degrees, the cosmological time Ctb_a of the curved space Csb_ia where the observer is situated is the same time Ctb_a as that of the curved space where the observed star was situated at emission of the signal Csb_i. In the curved space domain of frame C therefore the speed of light at such angle tilting would be absolute (in standard physics it would be said infinite).

[2] Definition: from the section above, the vector of the light cone of the future (called null vector) is defined above as tilted outwards at 60 degrees from the observer’s central time axis (Figure 2 and 3). This occurs therefore also for the past light cone: the past light vector forms with the observer’s central vertical time axis in Figure 4, a 60 degrees angle.

[3] {cpf2} Formula: Ctb_a{60} = Esb_ia{60}/SIN(RADIANS(sigma))*SIN(RADIANS(180-90-sigma)) = 7.909698688 billion light-years

With sigma fixed = 60 degrees

Ctb_a(60) defines any other Ctb_a{sigma} as it is the radius of the Curved Cosmos space circumference Csb.

[4] {cpf3} Variables: Ctb_a = radius of the curved space relative to the observer = time “t” in frame “C” for the “b” brightness space “Csb” at arrival time “a”. This is considered the cosmological time of the present of the observer.

sigma = angle at the centre, between faraway tilted time axis Ct{sigma} considered and the local observer’s vertical time axis Ct(0)

The described topology implies four frames, with differently tilted time-space reference frames. To relate the frames, space vectors need to be related and swapped with time vectors and vice versa by means of a measure conversion factor. This poses a challenge that has been subject of research. Carlo Rovelli (2008) analyses the formulation of classical mechanics, of general relativistic systems, and quantum mechanics avoiding singling out time as special independent variable, while treating it as the other physical variables. Lee Smolin (2000), with whom he has collaborated, is also among researchers in this field. The alternative topology, while encountering such challenge, would also provide a criterion for a path to a solution. Fixing the angle at the centre for the Hubble length at 60 degrees, between its time axis and the observer’s time axis, it provides possibilities of triangulation determining a unit conversion factor between time and space.

[5] {cpf4} Equivalence relative to measures, determining the topology unit conversion factor:

Ctb_a measures also the time of the triangle Ol{90}_0_Csb{0}. The full triangle is obtained extending Olb downwards until it reaches Ol{90}. The horizontal Os{90} extends from Ol{90} to the origin 0 and is considered as the expansion form the Big Bang in presence of a simplifying assumption of constant rate of expansion, i.e. without neither an initial fast inflation, nor acceleration. This simplifying assumption is done assuming that inflation and acceleration would be due to the curvature of the cosmos frame C. The corresponding time Ctb_a on Ct{0} axis is considered as the time from the Big Bang in such a Big Bang paradigm.

In perspective O, Ctb_a = time from the Big Bang, without acceleration = 13.70 billion years

13.7 billion years of frame O = Ctb_a = 7.909698688 billion light-years of frame E

1 billion year = 0,577350269 billion light-years

This allows expressing all the measurements, time, light vectors and space in terms of one of their units. The following formulae use space, in terms of billion light-years (Benazzo, 2010).

[6] {cpf5} Formula Ctb_i{sigma} = radius of the internal (start cosmological time) circumference on the Ct{sigma} time axis. It is determined as follows. From where the external curved space Csb intercepts the Observer’s time axis Ct{0}, the light cone vector Olb is drawn downwards until its intercept with the time axis Ct{sigma}, defining Ctb_i{sigma}.

Ctb_i{sigma} = Ctb_a/SIN(RADIANS(180-60-sigma))*SIN(RADIANS(60)) 
In frame C, time-brightness intersection at start time equals time-redshift at arrival time, i.e. Ctb_i{sigma} = Ctr_a{sigma}

[7] {cpf6} Ctr_i{sigma} = radius where the internal (start cosmological time) light cone intercepted the curved space relative to redshift, Csr. It is determined as follows. From where the internal curved space Csr intercepts the Observer’s time axis Ct{0}, the light cone vector Olr is drawn downwards until its intercept with the time axis Ct{sigma}, defining Ctr_i{sigma}. 
Ctr_i{sigma} = Ctb_i{sigma}/SIN(RADIANS(180-60-sigma))*SIN(RADIANS(60))

[8] {cpf7} Formula of Ptr_ia{sigma}: rather than virtual time felt, here it is called Projected time in frame P. The arrival point of the time span is determined as follows: the Observer’s present (brightness) light cone Olb intersects the time axis Ct{sigma}. This point is called Ctb_i{sigma}. Then a circumference is chosen, with centre in the origin 0, that passes through Ctb_i{sigma}. The tangent in that point is projected towards the Observer’s time axis Ct(0), and the so determined Esr_ia{sigma}, represents virtually expanded space in frame E. It determines the arrival point Ptr_a{sigma}. In short: the arrival time in frame P is calculated as the first intercept on the observer’s time axis Ct(0) by the redshift “r” space-time triangle of frame E, in Ptr_a{sigma}. The start time in frame P is determined by the redshift “r” triangle of the Observer’s frame O. It is established by considering the above determined internal circumference and its intersection of the Observer’s time axis Ct(0) in Olr_a{sigma}. A light vector Olr{sigma} is drawn from this intersection downwards and determines its intersection with the time axis Ct{sigma} in the point Ctr_i{sigma}. The horizontal line from there onto the Observer’s time axis Ct(0) determines the start time Ptr_i{sigma}.

Ptr_ia{sigma} = (Ctb_i{sigma}/SIN(RADIANS(180-90-sigma)) * SIN(RADIANS(90))) - (Ctr_i/SIN(RADIANS(90)) * SIN(RADIANS(180-90-sigma)))

[9] {cpf8} Variables:

An updated description of the variables is here proposed: The Expanded triangles of frame E represent the faraway local dynamic. The points Ptr_a{sigma} and Ptb_a{sigma} represent when first the space time triangles of frame E reach the time axis of the observer Ot(0), and determine general relativity perception of the faraway tilted space time. The time relative to the expanded frame E is represented on the top left with the segments Etr_ia{sigma} and Etb_ia{sigma}. Given the needs of the below formulae to determine the redshift with respect to frame E, frame P is used as an intermediate step, as frame P has the light vector Pl{sigma} parallel to the Observer light vector Ol{sigma} in frame O.

[10] {cpf9} Relative wavelength projected Plr_ia{sigma} in frame P, to be then compared in next formulae with the parallel relative wavelength Olr_ia{sigma} in the Observer’s frame O, so that two redshift referred triangles are compared.

Plr_ia{sigma} = Ptr_ia{sigma}/SIN(RADIANS(30))*SIN(RADIANS(90))

[11] Formulae of the relative wavelength Olb_ia{sigma} felt locally in the Observer’s frame O, at the Cosmological time Ctb_a. This latter is defined as the Observer’s present, where the observer is present when receiving the signal. Therefore, for the denominator of the redshift, the brightness reference “b” is considered rather than the frame “r” of the redshift. The redshift triangles are instead used for the terms of the numerator, as they are considered those that determine redshift:

In parallel to the above, the formula to obtain Olb_ia{sigma}, passes through the calculation of the projected time on the Observer time axis, Otb_ia{sigma}:

Otb_ia{sigma} = Ctb_a{sigma}-(Ctb_i{sigma}/SIN(RADIANS(90))*SIN(RADIANS(180-90-sigma)))


[12] Olb_ia{sigma} = Otb_ia{sigma}/SIN(RADIANS(30))*SIN(RADIANS(90))

{cpf10} An alternative formula with equivalent result is as in the previous article.

In addition to the brightness measurements, the corresponding redshift measurements for the same Observer frame O have the following formula:

[13] Otr_ia{sigma} = Ctb_i{sigma}-(Ctr_i{sigma}/SIN(RADIANS(90))*SIN(RADIANS(180-90-sigma)))

Olr_ia{sigma} = Otr_ia{sigma}/SIN(RADIANS(30))*SIN(RADIANS(90))


Formulae for the redshift in frame E are as follows. In the previous article this calculation was absent and the redshift was calculated in frame P.

The usual standard formula for redshift is z = (relative received wavelength of emission signal – relative local wavelength of the observer) / relative local wavelength of the observer

In the previous article, the two parallel segments used for the nominator were Plr_ia{sigma} and Olb_ia{sigma}, comparing a redshift space time triangle with a brightness space time triangle. This was done to have the same Olb_ia{sigma} at the numerator and denominator, as in the usual formula.

It is considered important that the numerator compares redshift with redshift measures, in order to compare ‘kiwis’ to ‘kiwis’ As before, it is also important to calculate the redshift comparing two segments that are parallel in the topology. To allow measuring redshift relative to the tilted expanded frame E, it is assumed that the parallelism requirement applies only to the terms of the numerator.

To calculate the nominator with respect to frame E, Elr_ia{sigma} needs therefore to be calculated, as follows.

[14] Esr_ia{sigma} = Ctb_i{sigma}/SIN(RADIANS(180-90-sigma))*SIN(RADIANS(sigma))

[15] Elr_ia{sigma} = Esr_ia{sigma}/SIN(RADIANS(60))*SIN(RADIANS(90))

The other term of the numerator needs to be a transformation of Olr_ia{sigma} to become parallel to Elr_ia{sigma}. The transformed Olr_ia{sigma} could then be named TOlr_ia{sigma} and is calculated through a proportionality factor as:

[16] TOlr_ia{sigma} = Olr_ia{sigma} / Plr_ia{sigma} * Elr_ia{sigma}

then the redshift in frame E is named Ez and is calculated as:

[17] Ez{sigma} = (Elr_ia{sigma} – (Olr_ia{sigma}*Elr_ia{sigma}/Plr_ia{sigma})) / Olb_ia{sigma}

{cpf11} The previous article formula of redshift calculated in frame P would need to be updated, using on the numerator Olr_ia{sigma} rather than Olb_ia{sigma}, as follows:

[18] Pz{sigma} = (Plr_ia{sigma}-Olr_ia{sigma})/Olb_ia{sigma}

In addition to the calculation of the redshift, the previous article calculated a value that was analysed as the discrepancy of supernovae, between the measurement done using the brightness measure, and that done using the redshift.

The analysis led to consider the ratio of discrepancy between the brightness curved space arc Csb_ia{sigma} and the redshift curved space arc Csr_ia{sigma}, as corresponding to the Supernovae discrepancy. The rationale is that the brightness arc space intercepted in Olb_i{sigma}, would measure Csr_ia{sigma} that then moves and arrives fractally to arc Csb_ia{sigma} when the Observer receives the signal in Csb(0). The same would occur for the redshift signal; the arc would be intercepted in Olb_i{sigma}, measuring Csr_ia{sigma}, and then move fractally to Csb_ia{sigma}. The difference between the two would be that for brightness, the fractality is discounted such that Csb_ia{sigma}=Csr_ia{sigma}, therefore in the topology, the arc Csb_ia{sigma} is used. For redshift, no such fractal discounting would work, such that Csr_ia{sigma} would remain fixed, therefore Csr{sigma} is the chosen arc to represent the redshift.

Calculations are as follows. Formula {cpf12} and {cpf13} remain the same:

Formula: space arc relative to brightness:

[19] {cpf12} Csb_ia{sigma} = Ctb_a{sigma}*RADIANS(sigma)

Formula: space arc relative to redshift:

[20] {cpf13} Csr_ia{sigma} = Ctb_i{sigma}*RADIANS(sigma)

Formula for Supernovae Type Ia - and for gamma ray bursts - discrepancy between measurement by brightness and that by redshift: This formula starts from the discrepancy in the Curved Cosmological frame C, which is as such considered transcendental to empirical raw data investigation:

[21] {cpf14} Delta_Cs{sigma} = (Csb_ia{sigma}-Csr_ia{sigma})/Csb_ia{sigma}

This gives the same result of the corresponding formula for flat space in the Observer’s frame O.

[22] Delta_Os{sigma} = (Osb_ia{sigma}-Osr_ia{sigma})/Osb_ia{sigma}

There are noticeable turning points in this curve. There is a coasting point where the difference becomes maximum before it decreases and then there is an inversion point where the difference is zero and then gets increasingly negative. The coasting maximum point with the Delta_Os or Delta_Cs formula corresponds quite tightly with the graphs by Wright (2011) on measuring the curvature of the Hubble diagram using the data by Conley et al. (2011) and by Kowalski et al. (2008). With increasing redshift instead, these two identical curves Delta_Cs and Delta_Os have a negative increase in values at a much smaller rate than the Conley et al. and Kowalski et al. data. The hypothesis is made that in traditional physics, the Expanded frame is considered as if it was the actual representation of the universe, without virtual deformations involved. In a way, the traditional physics representation would start from the Expanded frame E and a scale down factor between frame E and frame C would be needed in a process to find back frame C measurements. As the reverse of this is done in the formulae used, a scale up is performed.

From here the calculations are additional with respect to the previous article.

The Delta_Os is expanded to frame E, by a proportionality factor (variable scaling) determined by the relation between the two:

[23] Delta_Es{sigma} = (Delta_Os{sigma})*Esb_ia{sigma}/Osb_ia{sigma}

F6/ E. L. Wright plot 2011; supernovae discrepancy  plus DEs superposedThe result is compared with empirical data. In the standard paradigm, such data may be plotted in a curve that compares the condition of a universe of zero discrepancy between brightness and redshift measurements with the condition of discrepancy. E. Wright provides such coloured readable diagram (E. Wright 2006 and 2011 using updated data). The 2006 one is used in both Figures 8 and 9. The 2011 one is used in both Figures 6 and 7. The curve may be different with different parameters of dark matter and dark energy (S. Perlmutter and Brian P. Schmidt, 2003).

The curve of this alternative paradigm is superposed on the diagram with the curves plotted by Wright (2011) in Figure 6, up to z=2 (from the Conley et al. (2011) and Kowalski et al. (2008) analysis of the Supernovae Legacy Survey plus ESSENCE survey for Kowalski et al.); and on the right (Wright, 2006), in Figure 8, up to z=7. The light blue Des curve of the alternative paradigm, superposed to Wright’s diagram, has the same shape of the dashed magenta Closed (Non-Flat) Dark Energy Model curve. It is shifted a bit larger.

F8/ E. L. Wright plot 2006; supernovae discrepancy  plus DEs superposedClosed space characterises the alternative paradigm, and this correspondence highlights this. Flatness is rather a standard feature of the Big Bang paradigm and in the alternative paradigm it would correspond to the flat virtual expanded frame E. A question is then posed if it is possible to establish a rationale to transform it to the Flat Dark Energy Model. Considering the empirical flat frame E, as the calculations start from the defined hidden curved cosmos frame C, there is a rationale to check if there is another scaling factor to consider.

The coasting point of Delta_Es in Frame E occurs at 35.264 degrees, and it measures a discrepancy of 0.184, considered as difference between the distance determined from brightness discounting fractality and the distance computed from the redshift lacking such discount factor.

The corresponding coasting point of Delta_Os in Frame O occurs at 30 degrees, and it measures 0.134 of such discrepancy.

F7/ E. L. Wright plot 2011; supernovae discrepancy   plus DEs_scaled superposed The factor used is therefore the ratio of the Os coasting value in relation to the Es coasting value:

[24] Constant scaling for frame E = Delta_Os_Max/Delta_Es_Max = 0.134/0.184 = 0.730

Each value of Delta_Es is then multiplied by this factor:

[25] Delta_Es_scaled{sigma} = Delta_Es{sigma}*Delta_Os_Max/Delta_Es_Max = Delta_Es{sigma}*0.730

The resulting intense green curve of the Delta_DEs_scaled of the alternative paradigm is superposed to Wright’s diagram (Figures 7 and 9). It fits nicely with the magenta curve of the Flat Dark Energy Model plotted by Wright (2011) based on Conley et al. (2011) and by Kowalski et al. (2008) up to z=2 and the one by Wright (2006) up to z=7. This is a tighter matching that the previous one.

F9/ E. L. Wright plot 2006; supernovae discrepancy  plus DEs_scaled superposedThe alternative paradigm, providing for a virtual flat perspective of a curved cosmos, would justify this tighter match with a higher suitability, of the modern standard physics to deal with flat perspectives rather than different perspectives. The alternative topology, rather than requiring a choice between curvature and flatness, would provide for the concurrency of a curved cosmos with a flat universe. The supernova discrepancy calculated with the alternative topology shows two different curves, matching respectively the Closed Dark Energy Model and the Flat Dark Energy Model. The difference between the two would be the constant scaling of formula [24] defined here above. 



Frame E space of the previous article, was previously related to redshift calculations from frame P. This mixing of frames generated impossibility to precisely support considerations. Frame E now uses its proper Ez redshift calculation defined above (formula [17]). After the reviewed formulae here above, the following topology calculations from frame E (Table 1) are compared with empirical data from the Big Bang paradigm.

Frame E: values for different angles at the centre (sigma)

E_FEATURE 1) Appearance of recession at the speed of light c, that is the Hubble length, occurs at a redshift (z) of Ez=1.33 (60 degrees angle from the observer) with little discrepancy from empirics showing z=around 1.39 (Wright, E., 2012).

E_FEATURE 2) The supernova discrepancy between the measurement by brightness and by redshift is zero at redshift of Ez=1.33 (60 degrees angle from the observer) without particular discrepancy from empirics (z=around 1.33). See the binning performed by Wright (2011) on the combined data file provided by Conley et al. (2011) on the Supernova Legacy Survey, and which provides e.g. a discrepancy near zero, i.e. 0.0961 at a redshift of z=1.32375.

E_FEATURE 3) The discrepancy concerning supernovae, between measurements by brightness and measurements by redshift, has a coasting point, between its increase and decrease that is at around 1/3 of the distance from the observer in relation to the total distance from the Big Bang, considered as 13.7 billion light-years. Around 1/3 is then at about 4.57 billion light-years from the observer, i.e. 9.133 billion light-years from Big Bang (Riess, A. G. and M. S. Turner, 2004). The supernovae discrepancies coasting point agrees as it occurs at (1/3)*1.065 of the distance from the observer in relation to the total distance of 13.7 billion light-years, in agreement with empirical findings.

E_FEATURE 4) This coasting point occurs at Ez= 0.535 (35.264 degrees angle from the observer), in agreement with the empirical measurement (from around z=0.35 to around z=0.58 as from Wright (2011) on the combined data file provided by Conley et al. (2011) on the Supernova Legacy Survey, also depending on the model chosen (see Wright 2011)).

In comparison, the Projected frame P, which relates to the frame C of the previous article, looks as follows:

The redshift Pz is calculated by formula [18]. Delta_Ps{sigma} uses the same formula structure as [23]

[26] Delta_Ps{sigma} = (Delta_Os{sigma})*Psb_ia{sigma}/Osb_ia{sigma}

where Psb_ia{sigma} is obtained from Ptb_ia{sigma}. Ptb_ia{sigma} is obtained using the same formula structure [8], applied to brightness triangles, as:

[27] Ptb_ia{sigma} = (Ctb_a{sigma}/SIN(RADIANS(180-90-sigma))*SIN(RADIANS(90))) -(Ctb_i{sigma}/SIN(RADIANS(90))*SIN(RADIANS(180-90-sigma)))


[28] Psb_ia{sigma} = Ptb_ia{sigma}/SIN(RADIANS(30))*SIN(RADIANS(60))

In addition, in order to calculate the other measurement, the Delta_Ps_scaled{sigma}, a fixed scaling with structure as in [24] is calculated:

[29] Constant scaling for frame P = Delta_Os_Max/Delta_Ps_Max = 0.134/0.234 = 0.572

Where Delta_Ps_Max occurs at Pz = 0.774, with Pz calculated with formula [18]


[30] Delta_Ps_scaled{sigma} = Delta_Ps{sigma}* Delta_Os_Max/Delta_Ps_Max = Delta_Ps{sigma}*0.572

Calculated noticeable points (Table 2) as follows are compared with empirical data from the Big Bang paradigm.

Frame P: values for different angles at the centre (sigma)P_FEATURE 1) Appearance of recession at the speed of light c occurs at a redshift of Pz=1.397 with the updated formula used in this paper (52.403 degrees angle from the observer), in close agreement with empirical evidence showing z=about 1.39 (Wright, E., 2012).

P_FEATURE 2) Zero supernovae discrepancy has resulted as a feature that needs to occur where the distance from the observer is the Hubble length, taken here at 13.7 billion light-years. It should then occurs at a redshift of Pz=1.397 (52.403 degrees angle from the observer), while it occurs at redshift Pz=2.00 (as with the formula from the previous 2011 article) without agreement with empirical evidence (see the binning performed by Wright (2011); e.g. a discrepancy near zero, i.e. 0.0961 at a redshift of z=1.32375).

P_FEATURE 3) The supernovae discrepancies coasting point occurs at 53.1% of the distance from the observer in relation to the total distance of 13.7 billion light-years, without agreement with empirical data (should be around 33%). It is at (1/3)*1.59 of the distance.

P_FEATURE 4) Such coasting point occurs at redshift Pz=0.774 with the formula updated in this article, without agreement with empirical evidence (from around z=0.35 to around z=0.58).

The plotting of the curves for Delta_Ps and Delta_Ps_scaled, given these discrepancies, is here overlooked. Such calculations lead to consider Frame P as a projection of a curved frame onto a flat perspective that maintains correspondence with a part of all the proportions and loses though correspondence with some others.

In addition to such turning points, the above formulae for Delta_Ez and Delta_Ez_scaled in relation the additional formula for Ez, allow the plotting of such curves, with a considerable degree of matching with the Closed Dark Energy Model and the Flat Dark Energy Model respectively.

As analysed above, this constitutes an important additional indication that the actual empirical frame would be the Expanded frame E. Nevertheless, the Hubble Length occurs in the projected frame P as a value more tightly connected with empirical data. Could it be that the Hubble Length is calculated more on a flat projection, i.e. Frame P, in relation to the Expanded frame E? Frame P would be in any case a useful representation for a ration that allows tilting the Olr_ia{sigma} and transform it to its corresponding TOlr_ia{sigma} (formula [16]) geometrically parallel to the Elr_ia{sigma}, for measuring the redshift Ez in frame E.

The discussed correspondence of Frame E with empirical data provides important evidence for the alternative paradigm and its topology. These findings are taken as empirical evidence of mathematical interlink between the frames, thus empirically interlinking the empirical observed universe of virtual frame E with an unobservable (i.e. transcendental) Cosmos of actual frame C. This implies that empirical evidence may be applied to such transcendental cosmos passing through traditional physics based on the observable universe. The empirical frame E, as virtual flattened vision of frame C, would be necessary to the unobservable frame C calculations.

Further investigation may be done to verify the paradigm and formulae, to check if the data calculated in the topology agree with the empirical data, and to verify the implied level of sigma significance.

Given the analysis on the large scale structure in terms of geometry, some implications may be analysed.



The values of the formulae above for space are still difficult to match, apart from the close matches at the turning points. As the alternative paradigm considers virtual distortive effects, it could be possible that space measurements versus redshift would need some formulae of adjustment for reverting such distorsion. A hypothesis is such direction is investigated by A. Feoli, L. Mancini, V. Rillo, M. Grasso (2012). They apply corrections in order to subtract virtual effects within the modern standard physics. The alternative paradigm approach is the search for adding virtual effects starting from frame C, by means of frame E and/or P, to represent the Big Bang paradigm, for allowing better matching to empirical spaces vs redshift measurements. Further analysis could clarify.


The supernovae discrepancies, first empirically discovered by Saul Permutter (1998), Brian P. Shmidt and Adam G. Riess (1998), needed an update of the theoretical frame. Theoretical physics based on the Big Bang paradigm has led to read the data as evidence for acceleration of an expansion of the universe. Such acceleration needs a repulsive energy, which has been called dark energy. This has been called alternatively as cosmological constant in the standard model (recalling Einstein’s cosmological constant) or as a scalar field with energy density varying in time and space.

The defined curvature, implying variable speed of light (VSL), may be as substitute to the cosmological constant, which can become zero (Moffat, 2002). The adjustments analysed by A. Feoli, L. Mancini, V. Rillo, M. Grasso (2012) provide also for situations of absence of cosmological constant, directly in the modern standard paradigm.

The cosmological constant in the alternative paradigm would be a feature of its virtual flat universe frame, and therefore it would need to be called “universal constant” rather than “cosmological constant”. The Curved Cosmos frame C explains the supernovae discrepancy with the curvature and its fractality feature. From Figure 1 above in addition, it can be seen that inflation is a feature of a flat universe, and it is as well accounted for by the curvature of frame C, as a feature of the inversion of the supernovae discrepancy between 60 and 90 degrees of time axis tilting. VSL theories would allow an alternative to inflation (Moffat, 2002). The alternative paradigm as such proposes a substitute for both acceleration and inflation.

As frame C implies the addition of the act of observation, further research could be performed on the role of such act in the balancing of the outlined transcendental cosmological forces.


The alternative topology would allow selecting a preferred system of units (J. Magueijo, 2003) for VSL theories, for determining what varies. In Benazzo (2011) there is a precise proposed relation between the varying speed of light and redshift.


Such radiation is considered in the Big Bang paradigm as evidence of the remnants of the big bang. Recently, V. G. Gurzadyan and R. Penrose (2011) provide empirical evidence of concentric structures in the radiation. This is incompatible with the standard interpretation of CMB radiation. In modern standard physics, they elegantly analyse this as evidence of continuation of the universe from previous aeon eras. The alternative paradigm here proposed, with a curved static cosmos and closed space, reads the CMB radiation as spherical twilight from stars beyond the horizon in a four dimensional curved space-time cosmos. This is taken to be analogue to the horizon arc twilight at sunset on the three dimensional spherical Earth.


Gravitational waves are a prediction of general relativity and are implied by the presence of a maximum speed of light c. When the speed of propagation is instantaneous, then gravitational wave would be impossible. The topology of the alternative paradigm involves varying speeds of light (Benazzo, 2011, see Ellis, 2007, Magueijo, 2003 and Moffat, 2002 on varying speed of light theories), including instantaneous propagation, therefore the requirement of gravitational waves would be absent in frame C of the alternative paradigm when instantaneous propagation occurs. There are frames of reference with propagation at varying speeds, and here the gravitational waves could be a valid representation. Further research could clarify this.

The topology represents curvature of space time, while, expanding fractally, it concurrently constitutes an accelerated frame of reference. Albert Einstein (1907) under “Principle of Relativity and Gravitation” (Part III) presented an argument recalled by P. M. Brown (2002) as follows: “A complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system”, known as Equivalence Principle. This step, to consider the reference system as accelerated, allowed Einstein to pass from special to general relativity. In the alternative topology, the fractal expansion shows that time and space further back in the past get fractally smaller and smaller. Time and space in each successive time get larger, even if just fractally. The progression accelerates constantly, in the sense that there is a fractal enlargement of the space and time unit. In other words this can be seen as longer time and longer space, giving an unchanged relation, which implies unchanged speed of light c. Such enlargement of space and deceleration of time could be related to gravity as acceleration in the general frame of reference.


Calculations have been analysed from 0 to 90 degrees from the observer. Analysis at higher than 90 degrees could be performed.

In the alternative paradigm, Permutter’s (1998), Riess’ and Shmidt’s (1998) discovery would continue to constitute excellent evidence of acceleration of the observed universe. Such type of acceleration would be virtual in the alternative topology. That discovery on supernovae would additionally be a fundamental empirical finding allowing the measurement of degrees of curvature of the Cosmos. Without such milestone discovery, there would be little empirical base on which to develop and support the alternative paradigm presented. In such alternative paradigm, acceleration is rather a fractal feature of a curved static cosmos, for which time and space accelerate constantly in the general frame of reference (frame C above). Further calculations, refinements, consistence and significance controls could be performed.


An alternative fractal topology with multiple frames provides for an unobservable cosmos defined as the aggregate of observed universe plus the act of its observation. The cosmos is characterised by a continuously bending time axis at increasing distances. Its unit of measure is defined by a 60 degrees bending at the Hubble Length. The observer would be obliged to gaze at the curved cosmological frame through the flat observer’s light cone. Because of this difference, the former would be unobservable and the observer would collect empirical data on a third virtually expanded frame. The supernovae discrepancy is interpreted as a feature of the curved cosmos fractality. The brightness would discount fractality, compensating it, while redshift would overlook this, considering space in the past as smaller. Curves of this discrepancy, generated in the topology, are compared with empirical data from the supernovae discrepancy curves, those of modern standard physics. A high level of matching is found, providing evidence for the reinterpretation of the Big Bang paradigm. This would be an excellent description of a virtual however empirically observable universe, mathematically interlinked with an actual curved cosmos. This interlink thus provides empirical evidence of such hidden cosmos frame.


The author is grateful to his wife and family and to editor in chief Guido Zeccola for unbiased mindset. In general, content wise, the author is grateful for all the occasions of inspirations received from the cosmos.
This article is intended to remain in the public domain and no other or successive decision whatsoever may change this.

Piero Benazzo




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