The Society for the Diffusion of Knowledge P.O. Box 964, Kaunakakai, HI 96748

    The Random Field is virtually a sea of waves passing in every direction. The larger the waves, the more readily they may travel without decay. This is the decay attributed to auto-convolution. Despite their greater ability to propagate through the field, their production and thus frequency in occurrence is less for any given finite region of the field. Though smaller waves occur more frequently, as by-product of larger waves undergoing auto-convolution, they suffer band extinction: waves smaller than 3 s-units being unable to traverse the average distance between surfaces.
    Remember. a wave in free space can travel approximately three times its wave diameter before convoluting back onto itself and has over 0.9 probability of undergoing interchange with at least one surface in a set of ten surfaces.
    Given two surfaces which might be intersecting (thus describing a line of intersection), being that the surfaces are very close in and around the line of intersection; potentially closer than the average distance between surfaces throughout the field, a greater portion of smaller waves can propagate in and around lines of intersection.

    In the case that IDDIs are not working across a line of intersection between surfaces, their directions will not be opposed, and thus cancellation cannot be expected.
    Since IDDI, as motion, occurs in discrete, indeterminate and instantaneous steps, at any given instant, an additive process of these impulses is impossible, even in the rare occurrence (presumably) that two or more such impulses occur concurrently, in which case, the surfaces merely move that discrete distance without regard to combined, so to speak, IDDIs. In a sense, each IDDI masks the other. If these indeterminate impulses are dissimilar, then I might imagine that those of greater magnitude supersede those of lesser magnitude. Admittedly, this is all conceivably difficult stuff.

    Without the possibility of the presence of field configurations, all of this would be of little value. Given the presence of two field configurations, one may then at least anticipate some sort of interactive traction between such configurations, or in the least an inkling of these prospects.
    For example, imagine two surfaces passing through and belonging to two different radial configurations; glued to them by lateral restorative mechanism. Without exception and in every case of the same, these surfaces will somewhere intersect in space, and if their intersection is sufficiently shallow, the effect of IDDIs produced by field waves traveling through the field in-and-around the places of intersection, should be felt at each radial core.
    Besides the direct connection of radials, intermediary surfaces also play a role in the matter. For each intermediary surface connecting two connecting radials, an additional set of IDDIs which are generated in the zone of the acute angle between intermediary surface and one of the radials is generated. As in the case of every intersecting radial, the magnitudes of IDDIs generated by secondary surfaces also vary, the average being the same as the average magnitude for IDDIs caused by connecting radials. Connecting radials are those radials passing through and associated with the radial configurations.
    In this process, each radial is treated as being perfectly straight, which they are not: perfect straightness being a unique condition. In the long run, because of the great number of radials involved, some bending one way, others another, and all wavy with all sorts of textures and variations, the average of which would be closely represented by straight radials, an essential simplification enabling expedient utilization of these trigonometric relationships, without harm to the outcome.
    In the accompanying illustration, radial configurations M and N are shown with their corresponding radials. The radials are shown generally straight, though they most likely will be curved this way and that. The waviness of radials is an instantaneous snapshot of a dynamic system of moving radials (remember, radials are field surfaces); the size of this waviness, its texture so-to-speak, directly representing the size of the waves causing the motion in the first place. What we have is a vast and dense sea of waves, moving from one surface to the next, in this case, from one radial to the next, causing them to bump into each other. And, as we have learned, each time they bump, technically, converge into one another, there is a fifty percent chance of interchange and IDDI. In proximity to the line of intersection between radials, not only are these chances enhanced, particularly for smaller waves, the magnitude of IDDI is affected.

    Enlarging our view of two such radials (Insert A), we can see the actual bumping of the two radials at what is designated as point O, which is associated with point X. If interchange occurs at point O, its affects (IDDI) will be transmitted along each radial to the respective centers M and N where it will be felt.

    In the case of perfectly orthogonal intersection of radials m and n, four zones are shown (A, B, C and D), none of which are closer than 0.6 s-units to X (representing the innermost limit of possible interchange) nor further out than an arc distance of 3 s-units (representing the outermost limit of possible interchange). These zones represent and are a rough approximation of the activity zones where interchange takes place surrounding one point of intersection. f is the angle of intersection between radial m and n.

H = (L + 1.2) x tan (f/2).

    H tells us the average distance, approximating the arc distance between radials, that waves must travel between radials m and n, in order to impinge, thus setting up the prospects for interchange. In the same way as the length (L) of a given zone can be directly related to the frequency of occurrence (f) of interchange, the average distance between radials (H) within a given zone can be related to the probability (P_X) that interchange will at all occur, where:

P_X = ( 1 - H/3), where H cannot be greater than 3.

    What this means is that if the radials are very close, virtually, touching in this zone, the probability of interchange is virtually unity, whereas if the radials are separated by the extreme limit of wave travel, 3 s-units for average waves of unit wave diameter, then the probability is virtually zero.
    The last important relationship is the magnitude of displacement caused by interchange for a given zone. Since the distance between radials represents the total distance both radials must travel; each therefor traveling one-half this distance. If interchange occurs at the innermost zone boundary, which is the shortest distance between radials, then the maximum allowable displacement would be one-half this distance for each surface. The same is true at the outermost boundary, which would produce the greatest magnitude of displacement. Rather than calculating each one of these and all possibilities in between, the average is taken, where the average displacement (d) is determine as: d = H/2; H being the length of the mid-line for a given zone.
    The final displacement (I) generated within any one of the four regions, is the result of many discrete displacements occurring within a region. Though these may arise anywhere along the length (L) of a region, for simplification they are said to occur at the mid-line (H), where, I = P_X x f x d.
    This displacement does not represent an additive compounding of motion in terms of magnitude, but is instead treated as a compounding of motion in terms of time, though time itself has not been demonstrated to exist. But consider this.
    What we are trying to achieve is the development of some sort of interaction between radial configurations. Since a configuration is held together by each rotation of its counter rotating waves, each wave sort of gathering in wandering radials and approaching ambient field waves, the actual affect of any displacement to the group as a whole is not immediately transferred to the group as a whole, but is dependent upon the time rate of the rotating waves.
    Excusing what seems to be an obvious play on words, for the first time the appearance of time has appeared upon the scene; not as some subjective concoction, but in the stark reality that rotating waves with finite speeds do not get around their respective radials instantaneously. In other words, the instantaneous glue that holds radial configurations together is really slow glue; or in physical terms, sticky.
    What we have then is this central core whose radials are constantly being nudged this way and that, with the net effect being averaged over a finite duration directly dependent upon the period of rotating waves.
    In this light, I have taken the liberty to incorporate this behavior in the simulation, where within a certain period of time, designated a t-unit, a great number of displacements caused by separate and independent interchanges within a region can occur, and though each instantaneous in their duration, would eventually begin to overlap each other; essentially becoming saturated. Since there effects are not additive, their overall effect within a given period of time would reach a maximum, where even more displacements produce no greater effect.

    If the variable term f in the expression for I is altered so that,

I = P_X x (K₂^-1/f) x d, where K₂ > 1,

the expression becomes asymptotic, as f approaches infinity. In this example constant K₂ is set equal to two, the factor (K₂^-1/f) can never exceed 1.0 for any and all positive values of f. In order to offset this limit, a new constant K₃ is introduced, where K₃ may be assigned any positive value. Our final expression for displacement now becomes this:

              P_{X x d x K3}
      I = -------------------.
                 K₂^-1/f

(The above is still quite preliminary and is not incorporated in actual computer simulations as its validity remains uncertain.)

    This resultant displacement (I) is magnitude and direction at a specific location (the mid-line H). In other words, the displacement (I) is the final displacement affected at either radial at the mid-line H, generated by all discrete displacements of the local zone around H. Since there is no hypothesized theoretical mechanism which might attenuate this displacement, it is felt everywhere at once, with its original magnitude (I) over the entire radial, even an infinite distance from its location of origination at H. This displacement affects the two surfaces simultaneously everywhere along them, and is simultaneously felt at both radial cores (M and N).
    Though instantaneously felt at each core, its manifestation as a net displacement of the core itself requires the elapsing of a finite duration which produces a modest displacement in conjunction with the virtually multitude of radials which also comprise the core.
    At this point, what we have is a triangle M-N-X. At one of its apexes, designated as X we have discrete displacements being generated in and around X causing displacement of its sides m and n. The sides m and n are radials which can translate this motion to the respective radial centers M and N. The base of the triangle is the distance (D) between radial centers.
    The entire triangle is free floating, viz., none of its elements, M, N or X be locked in position, such as in the case of mechanical fulcrums. As it turns out, long-term average net displacements acting on M and N, which can be considered a couple separated by the distance D, are symmetrical to both the X-axis as well as a mid-point between M and N lying on the X-axis, such that this couple will neither rotate nor move upward or downward, or in any direction. The only anticipated change will be a net displacement of M and N towards or away from each other directed along the X-axis.
    In order to derive some tangible results from IDDI generated at and around X, configuration N is considered to remain at rest, thus any displacement acting against it and generated at X is considered to cause X to move, rather than N. In other words, N is a fulcrum point. Subsequently, any displacements caused by X and acting on N are added to M's motion; M therefor being the recipient of the combination of both motions.
    These motions are vector motions, N wanting to move in this example in the direction I_N (shown downward at slightly left). Since N is considered to be fixed, X moves instead with the same magnitude in the opposite direction. Here we see X moving from its position at X to X'. This motion though, reflected at M, is restricted to only orthogonal motion D, where D is the orthogonal component of the vector I_N at X relative to the radial m passing through M and X. Normally, relative to X, M will move in the vector direction IM at M. Since both D and IM act orthogonal at M, they may be added or subtracted as though scalar values.
    The distance D is a variable constant which can be assigned any positive value greater than zero. In some cases then, this value may be set so small as to cause the positions of M and N to fall inside the active regions A, B, C or D for iterations of certain Xs which are very near M or N. In the case this happens and either or both M and N fall inside the mid-line H of any one of the four regions, in which case IDDI, because of the cancellation of opposing zones, becomes zero.

    This represent the most important aspect of this work in that theoretical premises have been quantified and simulated with excellent results by computer. Over a fifteen year period, five distinctly different programs have been performed, the first of which, three exploratory simulations, were done by an Olivetti P-602 wire-memory computer. These first three simulations, which more-or-less exposed the light at the end of the tunnel, will not be explained here, since they contain some error in fundamental logic. The fourth simulation performed independently in 1973 on an IBM 360 at UCLA will also not be explained, since it was not that comprehensive, despite yielding superb results. (Actually the results were based upon invalid premises.) Those interested in flow charting and documentation of these first four computer programs should contact the Society. The last simulation, comprising reasonably valid logic, extreme range and comprehension, as well as good results (based upon incorrect premises), will be explained here. Amazingly, it was performed on a Texas Instruments Programmable TI-59 hand-held calculator capable of only 900 program steps and having only thirty-two registers for data storage. I believe its range was 10^+/-99, with thirty-two bit precision. Unquestionably, Charles Babbage would be thrilled beyond delight by its indirect addressing capability!
    The actual program used required only 666 program steps and took about three and a half hours to run.
    The idea is to take two radial configurations, designated as M and N, separated by a certain distance and calculate the net displacement acting upon them caused by IDDI between their respective radials. In order to do this, radial configurations are restricted to the following specific orientation. They are in line and their central axis parallel. The central axis of each configuration is an imaginary straight line passing through or nearly through the center of the central core and perpendicular to the orbital plane of the counter rotating waves. In other words, the orbital plane of both radial configurations must be coplanar to a common plane, designated here as the K-plane. Also designated is an imaginary axis passing through each configuration's center and coincident to the K-plane. This axis is purely imaginary and is designated the X-axis. At the mid-point along the X-axis between centers M and N providing the origin of a rectangular frame of reference utilizing the X-axis as basis, are two other mutually orthogonal axes: the Y-axis (vertical) and the Z-axis (front and back).

    These results are wholly anticipated since the number of samples increases with the square of the distance D, all IDDIs within the M-N circle produce a divergent moment at M (relative to N), and because each new sample within this circle is complimented by the presence of another point of intersection between radials and secondary field surfaces. In other words, as the divergent circle grows, for a given field density (comparable to sampling density) the divergent impulses will increase fourfold.
    If the distance between M and N was to be made infinite, thus making the divergent circle infinite, one can expect a fourfold increase of the total IDDI acting at M or N for every doubling the circle's diameter, to become infinite as well, exactly offsetting all convergent IDDIs generated throughout the universe. If on the other hand, the diameter of the circle is brought to zero, M and N thus coinciding, the resultant convergent IDDIs derived from the entire field would be as well infinite. One might expect then, that as field configurations are brought ever so close together, so close that they are virtually coinciding, the sum of all convergent IDDIs, being infinite, would make it impossible to separate these configurations with a finite magnitude of IDDIs. This especially true if one considers zero to be the ground state, but it is not true if infinity is the ground state. Clearly, in dealing with infinitesimals such as zero and infinity, both options are acceptable for those who reside in the finite realm between.
    Thus, one can assume infinity to be the ground state, and as the circle becomes larger and the divergent IDDIs become greater in their sum total, the actual differential is infinity minus the finite total of impulses generated inside the divergent circle. At an infinite distance away, both the divergent and convergent impulse would become offset, resulting in a net zero IDDI.
    Comparing two conditions, the first when M and N are separated by say, 3,200 units and when M and N are separated by 1,600 units, the net total of all IDDIs both divergent as well as convergent would be infinity minus 364,812 at 3,300 units and infinity minus 90,827 at 1,600 units.
    In algebraic terms, the difference between the two, would be (& - 364,812) - (& - 90,827), which simplifies to -273,985; infinities canceling out. This is important, for it tells us that a finite differential between finite variations based upon an infinite ground state is acceptable. Don't worry about the minus sign, since sign convention has not been established.
    On the next page is a plot of these results. This curve represents the increase in the net total IDDI acting on radial configuration M caused by interchanges occurring within the divergent circle of diameter M-N, where the distance (D) between radial centers is equivalent to this diameter. Though values where D is less than 50 s-units were computed, their results are deemed unreliable at this time, being that some of the more subtle expressions were not utilized, such as the cancellation of IDDIs between the mid-line H between zones of activity.

    As early as 1987, refinements were made in order to accommodate close-in interaction between two radial configurations;  distances analogous to nuclear range.  The results showed exceptionally "powerful" repulsive and attractive net impulses at close range, and relatively weak attractive net impulses at all greater ranges whose ratios closely, though not precisely, adhered to inverse square ratios.
    Given an unusual particle fine structure of radiating surfaces, simple geometric relationships can explain both attractive and repulsive forces demonstrating powerful close range influence and greatly weaker long-range forces of attraction.
 
 

     Using a home PC, a very simple (less than 400 steps) program can be executed, demonstrating a single force continuum linking at least, nuclear and gravitational forces to a common mechanism.  Granted that a more powerful and faster computer, such as the Cray, could do the job quicker, the small and inexpensive PC at my disposal, a TI Programmable 59, churning out 666 iterative steps, simulating the force between two singular masses - presumably two neutrons - can get the job done in about 3.5 hours.
    Results already show a weak attractive force obeying the inverse square law with great precision (+/- 0.000000005) at distances ranging from several millimeters to over 10¹³ light-years.  Presumably, since both objects represent masses, this weak attractive force is the force of gravitation.
    At even shorter distances, as close as 10^-13 meters, the gravitational constant slowly increases by only about 3%.  But at even shorter distances, as close as 10^-15 meters, which is now nuclear range, the constant begins to take erratic swings, increasing threefold, and then declining sharply, becoming negative (thus representing a repulsive force).
    As this distance between these two simulated masses is reduced, the constant takes more than a dozen swings between being positive and negative, reaching a maximum negative peak twenty-two thousand times greater than its long range magnitude.  This negative peak occurs at a distance of about 6.5 x 10^-16 meters.   A maximum positive peak, sixty times greater than the long-range constant, occurs around 10^-15 meters.  (Refer to Fig. 1)

    The premise of this singular force continuum lies in a rather new study of surface mechanics.  In this study, the fine structure of simple masses, such as a neutron, consists of a large number of flexible surfaces aligned in a radial order, providing a magnetic moment along a central axis, and a nuclear-gravitational field radially symmetrical to this axis.  Under special conditions, in which the surfaces are bent or spiraled, Coulomb forces may occur.
    Since these surfaces are flexible, they neither remain stationary nor flat (both conditions of rest and being flat are unique), but instead freely undulate in all directions away from the central axis.  This theoretical behavior provides a special interaction to occur between surfaces, creating a slight impulse, which is transmitted along the surfaces to their respective masses from which they radiate.
    The underlying premise for this impulse is somewhat too complex to be discussed here. Essentially, in theory, everywhere two or more surfaces intersect, numerous discrete impulses are generated, causing the surfaces to converge.  In the case of surfaces which are orthogonally intersecting, these impulses tend to balance (Fig. 3-a), resulting in no displacement.  In the case of surfaces intersecting at a very acute angle, these impulses tend to drive the surfaces together (Fig. 3-b).  This is because of the direct result of theory in that impulses can be generated in a region no closer than about 0.6 s-units to the line of intersection between surfaces, nor at a distance where the arc distance between surfaces exceeds three s-units.  These limits are borne out as practical limitations in wave interaction and propagation in the study of surface mechanics.

(An s-unit is an arbitrary unit of distance used in conjunction with another arbitrary unit, a t-unit (time), such that motion may be equated:  s (speed) = s-unit/t-unit.  Though no exact relationship has been found, an s-unit is somewhere equivalent to 2-3 x 10^-14 cm.)

    The above illustration (Fig. 4) shows the more complex relationships not detailed in Figure 3.  As a matter of simplifying convention, the surfaces, denoted as S₁ and S₂, are shown to be perfectly straight, rather than wavy.   Also, the axis of intersection (X) between both surfaces is orthogonal to the paper, which means that both surfaces at this location must also be orthogonal to the paper.  The relationship between the arc distance 3 (the outer boundary of each region) and the inner boundary at a radius of 0.6 from x, is exaggerated.
    In the case of orthogonally intersecting surfaces (f = 90^o) or nearly orthogonally intersecting surfaces (44^o < f  < 136^o), four regions: A, B, C and D, are formed.  On the other hand, if angle phi is acute (actually less than 44^o), or nearly acute, only two regions are formed:  A and B.  Since by convention angle phi is always the inferior angle formed by the intersection of two surfaces, regions A and B are not by limitation the only regions formed when phi is acute.  If phi were to face the bottom of the page, regions A and B would not be formed, rather regions C and D.  In any case, impulses generated, associated with this region, tend to drive the surfaces together.  For example, impulses generated in region A (Fig. 4-b) causes surface S₁ to move downward, and surface S₂ to move upward.  Within region A the relative motion of surfaces is therefor convergent, but since motion is directed every the same on a given surface, the same motions produces a relative divergent motion of surfaces on the other side of X;  the axis of intersection X having no affect on this motion.  This is typical of all motion produced by this region.  In the case of orthogonally intersecting surfaces, the motions produced by each region generally offset each other, resulting in a zero net motion.

    In every case, this motion is always directed laterally to its given surfaces, and is unaffected by distance, its magnitude being everywhere the same on that surface.  This of course provides for a dynamic field around mass particles, forcing the particle elements, the surfaces from which they are derived, to move in all sorts of directions.  Because such a structure can be shown to be cohesive, the surfaces, though having diverse sorts of motion, cannot tear away from it, resulting in a collective motion of the particle.
    This implies of course that though a surface may want to move, it cannot, resulting in impulse pressure, or what is commonly called, force.
    Another way of putting it, the force at-a-distance exerted between singular masses or ponderable bodies of mass, is the diverse magnitude and direction of impulses generated around the axis of intersection of many such sets or pairs of surfaces emanating from each mass.
    The probability of any discrete impulses occurring is directly dependent on the arc distance* between surfaces where these impulses might occur, and if such a distance were to exceed three s-units, these impulses are thought not to occur.
    *This is a simplification of the real theoretical behavior, which is even further simplified by allowing this distance to be the base of a triangle formed by A-B-X, in Figure 6.
    This marks the outer boundary of any region, whose distance (O-X) from X may be determined as,

    The innermost boundary of any region is as mentioned, a distance of 0.6 s-units from the axis (X).
    An important variable which determines the rate of impulse generation, is the length of a given region (l), where,

    The probability that any impulse will occur, directly relates to the distance between surfaces.  For simplification, rather than determining impulses occurring throughout a region, an average impulse representing the average of all impulses is seen to occur at the mid-line (b) between the inner and outer boundaries.  Since the probability of its occurrence directly relates to the length of this mid-line, the value for b must be determined, where,

and where the probability of impulse (P_x) may be expressed as,

    As mentioned before, associated with each region, there is the rate or frequency of impulses (F_x), where the value of F_x is directly dependent on l:

    Finally, there is a magnitude of impulse (d_s) based on the distance between surfaces.  Again an average magnitude is taken, occurring at the mid-line (b), where,

    Only one half the value for b is allowed, because the distance one surface travels is reduced by the distance its opposing surface travels.  Since both will travel with the same but oppositely directed speeds, both travel 1/2 b.
    The total impulse (I) generated within any one of the four regions, is the result of many discrete impulses occurring within a region.  Though these may arise anywhere along the the length (l) of a region, for simplification, they are seen to occur at the mid-line b, so that,

    This impulse does not represent an additive compounding of motion in terms of magnitude, but instead a compounding of motion in terms of time.  In essence then, concurrent motions do not amplify each other.  Thus for very large regions experiencing many concurrent motions, a maximum saturation is achieved, where even more impulses produce no greater effect.
    If the variable term F_x in the expression for I is altered so that:

I = P_x w d_s w (k₂-1/F_x) where k₂ > 1, the expression becomes asymptotic, as F_x approaches infinity.  If in this example constant K₂ is set equal to two, the factor (K₂-1/F_x) can never exceed 1.0 for any and all positive values of F_x.  In order to offset this limit, a new constant K₃ is introduced, where K₃ may be assigned any positive value.  Our final expression for impulse now becomes:

    This resultant impulse (I) is magnitude and direction in time, at a specific location (the mid-line b).  In other words, the impulse (I) is the total impulse affected at either surface at the mid-line b, generated by all discrete impulses of the local region around b.  Since there is no hypothesized theoretical mechanism which might attenuate this impulse, it is felt everywhere at once, with its original magnitude (I) over the entire surface, even at an infinite distance from its location of origination at b.  This however is only true if one considers a singular impulse on one surface.  In the case of two or more surfaces, such as a field of surfaces, a general attenuation of this impulse will occur, as the result of a neutralization of displacement (or motion) between surfaces.
    For example, if two simultaneous resultant impulses (A and B) occur at separate and distinctly different locations on a surface, two different results can occur.  If both impulses are directed to the same side of the surface, the affect of the nearest impulse will override the farthest impulse, forcing the surface to conform to both.  Since surfaces cannot be discontinuous, this conformation to dissimilar impulses must be smooth and continuous, resulting in intermediate impulses along the surface of varying magnitude in the same direction.

    If on the other hand impulse B is directed to the opposite side of the surface than impulse A, the surface's conformation to dissimilar impulses will yield intermediate vectors directed to both sides, with a null vector somewhere between.

    In neither case does an impulse greater than the original impulses occur, since this is not an additive process, and in both cases, many intermediate impulses of lesser magnitude occur, suggesting that in a field of many surfaces, the magnitude of any given impulse undergoes a field interactive attenuation with distance, such that at the exact source of this impulse, its magnitude is maximum, and at an infinite distance away, its magnitude declines to nothing.   This relationship may be simply expressed as,

where I is the magnitude of the impulse at its origination, and where I_s is the resultant magnitude at any distance s.  Here we see that if s approaches infinity, I_s approaches zero, and if s approaches zero, I_s approaches I.
    There is of course no proof that this expression is valid nor accurate.  It is a matter of applied mathematics seeking an expression faithful to the construct without the presence of numerical data.  Much more complicated expressions could be devised, yielding similar results, which would not necessarily be more valid or accurate.  This is merely my first choice because of both expediency and simplicity.
    This impulse affects the two surface simultaneously, and it affects both particles, P₁ and P₂ associated with these surfaces (Refer to Figure 5.), and is customarily measured as a force acting on one particle only.  This is also true for ponderable masses as collections of particles.  In both cases, the overall impulse measured at one body in opposition to the other body at rest, results as the attenuation of impulse along both of the two surfaces, one of which radiates from one body, and the other surface from the other body.
    In this case, a triangle is formed with one mass, designated as M at one apex, the second mass N at the second apex of the triangle, and the line of intersection (X) between radials d_m and d_n at the upper apex.  (Refer to Figure 9.)  The distance d_m and d_n are the distances associated with the attenuation of impulse I, and may be substituted into the previous expression for s.

    More specifically, an impulse generated in any region A, B, C or D, around x, will be attenuated at M and N respectively, such that,

and

    If N were held fixed in space, the attenuated impulse at M would therefor be,

providing the customary measurement of force of one object relative to another.  Here we have taken a product association of I_m and I_n, rather than the sum, since I_m (n=fixed) cannot exceed either.
    Applying this rule, for very great distances in D, both d_m and d_n will be large (or at least one of them will be large), resulting in I_m (n=fixed) being small.  Conversely, if D is very small, one or both of d_m or d_n, will be smaller, resulting in I_m (n=fixed) being in general greater in value.
    By dropping I from the previous expression, a useful coefficient, K_c, is obtained, where,

This coefficient (K_c) may be called the coefficient of attenuation, and for all positive values of D, ranges between unity and zero.
    As we shall find, it is this relationship which produces iterative results obeying the inverse square law, where in general,

    Another thing to mention, not only is the original impulse subject to attenuation, it can undergo re-direction.  Since there is no rigid and fixed frame of measure, impulse direction is always orthogonal to its respective surface, and surfaces can (and usually are) curved, the impulse direction will vary in space.

    Presumably, this movement must be considered to be oppositely direct (-I_n) than I_n at N.  This motion though, is reflected at M as D, where D is the orthogonal component of motion I_n relative to S₁ at X'.  Normally, mass M will move with motion  I_m, but since N is fixed, it requires the additional component D (unattenuated at M), causing mass M to move I_m + D.  This is of course a unique example where S₁ and S₂ are straight rather than curved;  curvature being a more common condition.
    Before attempting iterative simulation of force, one more behavioral condition must be included.
    If masses M and N draw so near, as to fall with the mid lines of b of any of the four regions, a neutralizing of impulse results. Under this condition, the attenuation of impulse by the presence of the field is no longer a major factor, since M and N are so close, few, if any, field surfaces intercede them.  In this case, the coefficient of attenuation is abruptly altered to a new expression:

where a is the straight line distance from mid-point b to the line of intersection x.  (Refer to Figure 5.)

    Under the very special condition where M and N are concurrent, or coincident, (D = 0), R_m would equal zero, resulting in K_c being equivalent to zero.
    Quite unlike conventional physics, subscribing wholly in the inverse square law, which would make the force of gravity immensely great in close quarters, approaching infinity as two mass objects begin to merge to coincidence, this hypothesis denies such bizarre illimitability of force;  reducing it to zero at zero closure.
    In order to simulate the the impulse exerted against mass M, a variety of trigonometric relationships must be established.  Given a triangle with masses M and N at its lower apexes, its base would represent the distance (D) between these two masses.  (Refer to Figure 12.)  Its upper apex (x) would represent the line of intersection between two radial surfaces emanating from masses M and N, whereas its two sides d_m and d_n would represent these two radial surfaces, as well as the distance between x and M and N, respectively.
    Associated with sides d_m and d_n are angle a and angle b.  Essentially, during the iterative process, sides d_m and d_n are stepped through various position of angles a and b from >0^o to 180^o for angle b, and from 0^o to some value less than angle b, for angle a.

    For each one of the iterative positions, a value for I_x  is determined and summed to an overall value SI_x.
    Initially, angle a is set equal to zero, meaning that side d_m is lying flat along the x-axis.  Side d_n is set at a slight incline (angle b>0^o) so as to intersect with side d_m, thus forming a line of intersection at x.  It is not necessary that the line of intersection is orthogonal to the paper, but merely passes through the plane of the paper at x.  The impulse at M is then calculated, and its moment along the x-axis is then calculated, such that,