APPLIED ALGEBRA: if (X-Y)=7, then...

Therefore, thinking algebraically, if Model 2 predicted a value of X=12 but in this case, the result is (X-Y)= 7 (where "Y" represents "deleted or missing frames") we can only conclude that at least 5 frames are missing, for the result of our equation is (Y=5).

If X=12 and (X-Y)= 7, then Y=5.


At Z-155, the right flasher light appears instantly and brightly illuminated and the left side instantly darkened without the transition frames, which should have shown momentarily equal luminosity in both flashers. This phenomenon of equal luminosity is easily seen at Z-135, Z-147, Z-166, and Z-177 despite the glare of sunlight reflecting from the chrome fixtures.


Studying frame Z-155 closely, one can note that Ms. Willis, the little girl seen running before, has come to complete stop. In standstill, her feet, firmly planted and wide apart, are plainly visible behind Mr.Robert Craft, a tall black cloaked man who was also strangely teleported onto the scene from nowhere at Z-132 instantly into Frame Z-133. At least 5 frames are missing at this point, Z-15/4 & 1/55 (note that in this notation [Z-15/4 & Z-1/55] the diagonals [/] represent the splice marks.

After the splices, two more lost frames (Z-156 & 157) are indicated as the Z-Frame count jumps suddenly to Z-158 where the right side flasher unit is to be seen still "On". It remains "On" for another 9 frames, totalling a 12 frame sequence of continuous illumination (Z-155 to Z-166 inclusive) before extinguishing completely at Z-167. Were it not for some missing frames and the presence of the splices at both Z-15/4 & Z-1/55, this sequence might have served to confirm Mr. Zapruder's estimate and would have passed for a normal unit flash sequence.


The following notation shows the sequential 12 frame, right flasher "on" count" (as opposed to the actual uncut 10-frame span seen from Z-136 through Z-146) of continuous right side illumination between Z-1/55 and Z-166. Beginning with the splice at Z-15/4, we observe the following anomalies:

1. Z-15/4 (*splice*)

2. Z-1/55 (*slice)

3. (Z-156 & Z-157 *missing*)

4. (Z-158 *extremely blurred* but Ms. Willis is seen standstill)

5. (Z-159 *temporal anomaly*-False leg movement painted in!)

6. (Z-160)-(Z-161)-(Z-162)-(Z-163)-(Z-164)-(Z-165)-(Z-166)

This 10-frame span with several frames obviously deleted is much too long to be natural considering the abrupt jump at the spliced frame, Z-1/55, and the absence of transition frames.

The truncated 7-frame span between Z-148 and Z-15/4 is also suspect, being too short for either mode. At least one series, possibly more than one, of "left-side-on" frames has been removed at each juncture.

Transitioning to left side flasher "on" at Z-166, the left flasher shines steadily for the next 10 frames (Z-167 through Z-176) transitioning to right side "On" again at Z-177 and remaining "On" through Z-182 when the right side flasher disappears behind the Stemmons Freeway sign.

As we look at the data in the table above, we can create a very accurate timeline of micro-events based on the flash rate observed and the rate predicted. When we visualize the timeline based on the number flashes observed (cited above) versus the number of flashes predicted, we find the Z-Film deficient. We can see clearly where temporal anomalies exist and these anomalies should be reflected in the odd movements of persons and objects in the Z-Film.

Whereas, within the 2.295 second interval of time after Z-136 through Z-177, we should see 5.6 "On/ Off" unit flashes (ufl) alternating between the two lights (2.295 sec X 2.439 ufl/sec = 5.597 or 5.6 flashes). We observe only 3 transitions at Z-147, Z-166, Z-177 and NONE between Z-147 and Z-166.

5.6 (predicted) - 3 (observed)= 2.6 flashes (missing)

We find that 2.6 missing flashes, comprised of a minimum 10 frames each (2.6 X 10) reveals a MINIMUM deficit of 26 frames. Twenty-six (26) frames would constitute a significant loss within the short time span being studied, 2.295 seconds, signifying a loss of more than full second of time and action (26/18 = 1.444 sec). This would mean in addition to the 2.295 seconds, we rightfully ought to be seeing 1.444 seconds more of information instead. If this is so then we are seeing time contracted by a factor of 68/42 = 1.619. This means that the remaining 42 frames (properly synched) would make the President's limousine appear to be travelling 1.619 times its actual speed. Thus, the reduction ratio was 1.619 to 1.

The figure of 68 frames is close to our theoretical calculations of 67.2 frames (5.6 cycles of approximately 12 frames each in 2.295 seconds with the expected number of t*frames using Mr. Zapruder's 24 fps values). Therefore, many t*frames are not visible due to the cutting, deletion of segments and multiple splicing.


Later, I discussed the "1.619" discovery with my associate, Mr. Marc Wolf, he exclaimed "Oh God! That's the Golden Mean!". "But...isn't the Golden Section 1.6180339...?", I rejoined. It was a perfect example of "not seeing the forest for the trees".

This jolted me into several realizations simultaneously for, indeed, something about this number had been intriguing and vexing me for many days since I had first derived its value. So much did it distract me that almost every night over the previous week my sleeping mind would reel, my body tossed and turned with images from the segment in my mind's eye, replete with dream voice saying "Wake up! It's right there, you're not seeing it! Look again! It's right there!"

I did wake up many times but "what It was" was difficult to grasp consciously because "It" was not just one thing. It was many things from time dilation to time contraction and temporo-spatial distortions to time/motion anomalies and contradictions. All of which can be summed up in two words, "cognitive dissonance".

Induced cognitive dissonance such as that which we have discovered in the Z-Film occurs when the mind and/or the senses (in this case vision) are presented with conflicting information, temporo-spatial distortions or with physically and/or intellectually contradictory information, thus, "dissonance". However, the dissonance is partially disguised and most times overlooked due to a series of distractions built in by the editing of the Z-Film. This leads most viewers to "forget" the discrepancies in the film by distracting the eye of the viewer into overlooking the splices in the film, along with inexplicable jumps in temporal continuity and anomalies in human movements in the background of the film. The viewer is almost always left somewhat intrigued and mystified by something intangible and "not quite right" about the events just witnessed in the Z-Film.


On December 14, 1996, during a discussion with Dr. Raymond Singer, a specialist in neurotoxicology (in whose light the Z-Film can also be studied, see below), I pointed out the discovery of the use of the "phi" ratio in contracting the Z-Film while showing him the various splices in the Z-Film. As I was doing so, I conducted precise measurements of the Z-Frames in question (Z-15/4 & Z-1/55, Z-20/8 & Z-2/12, Z-324 & Z-325).

To my further astonishment, in comparing the areas occupied by the inserted frame segments at Z-15/4, Z-1/55, and Z-20/8, I found that they had been ALL inserted precisely into the film using a formula employing the same 1.619 ratio as had been demonstrated earlier as the "frame reduction ratio" (68 predicted frames/42 observed frames = 1.619 = 68/42).


Both an 8 mm. and a digitized version of the Z-Film obtained on cd-rom from Medio Multimedia's "The JFK Assassination: a visual investigation" were analyzed. A Teac 4X Cd-Rom Drive in a MARS Cydonia 486DX2-66 PC computer equipped with 32 megs of Ram, a Number Nine 9FX Motion 771 2 meg VRAM Hawkeye SVGA video card and high resolution 14" SVGA monitor were used to view the film. The Z-Film was analyzed frame-by-frame, with each frame enlarged to a standard window height of 5.25 inches for all measurements.


When this researcher measured the ratio of the total height of frame Z-15/4 to the height of the segment spliced to it, we found the insert height to be 2 inches from the top edge of the black splice mark to the bottom of the frame. The ratio of the insert height to the total frame height (2:5.25) resulted in a rather obscure and mystifying figure of ..3809523. With expectations of more immediate and spectacular results the figure of .3809523 seemed at first a bit disappointing, so we took careful measurements again and, of course, with the same results.

However, this time, after verifying our second set of results, we realized that if we subtracted our result (.3809523) from UNITY (1), (1 minus .3809523), the result was .6190477. This was

certainly a more encouraging result, indicating that our intuition was reliable and proving to be correct. This writer then recalled that the "Golden Section" represents a relationship between two (2) objects, 2 measures, or 2 aspects of a single object and that subtracting our result (.3809523) from DUALITY (2), (2 minus .3809523) the figure that emerges is PRECISELY 1.6190477!

When we measured the segment inserted along the top of Z-1/55, we arrived at identical results. When we measured the relationship between both inserts to the total height of the 2 frame sequence, we arrived at 4" (Insert Ht.): 10.5" (2-frame height) or .3809523.

We then conducted measurements of frame Z-208, once again we found THE IDENTICAL RESULT, .389523! In other words, the ratio of the size of the inserted segments IN ALL THREE SPLICED FRAMES to the height of each frame produces a value of .389523, or the complement of "phi", the complement of the "Golden Section". Phi has been known since antiquity as an essential element in a human being's perception of something visually as "beautiful" or "natural". One might describe the phi proportion as nature's measure for organic movement, growth and expansion, exemplified in nautilus shells, plant growth, leaf distribution, whirlpools and Texas tornadoes.

The following diagrams demonstrate visually the use of a variable approximation of the Golden Section (1.619) to alter the Z-Film BOTH temporally through the 1.619 to 1 "frame reduction ratio" and geometrically (spatially) at THREE specific frames with each one spliced to out-of-synch frame segments. When the film was analyzed and measured according to area and height, the results show that a.) the splice marks; b.) inserts of out-of-synch frames; and,c.) the use of black marker to vary the thickness and thinness of the splice line of inserted segments were ALL used to join the film artificially according to a "phi" proportion, or "Golden Section".

Diagram 1

The Use of Phi as a Frame Reduction Ratio

in the Zapruder Film (Z-133 to Z-182)

(* = 1 frame in series; / = splices)

Predicted Frames = 68

***************************************************************** **

Observed Frames = 42 {68:42}= 1.619


The above diagram represents the relative lengths of the 2 versions of the Z-Film, the original at 24 fps (68 frames) and the currently accepted altered version at 18.3 fps 42 frames.

In addition, as noted by researcher Jones Harris, one of the reasons for the removal of segments at Z-154 & Z-208, may well have been the removal of information which would have appeared in the sprocket area of the film. When measured with a metric ruler, we found that each sprocket hole was exactly 1 mm. wide and that the sprocket area comprises 1.5 mm. of the 8 mm. width of the film strip. The average 8 mm. projector further masks approximately .25 mm at the frame edge, thus 21.875 percent of the information on each frame is not visible to the viewer (1.75 mm/8 mm = 0.21875). This is consistent with the accounts of Mr. Zapruder & the Newmans that something transpired then.

Diagram 2

The Use of Phi as Insert Height Ratio

in Frames Z-15/4 (left) & Z-1/55 (right) of Z-Film

(Frame Height on SVGA Monitor = 5.25 ")

Frame Z-15/4 Frame Z-1/55

[--------------------------------------------------------------] I 2 Inches = Insert{I Inserted\ GOLDEN I

I Height{I Fragment/ SECTION = 2-phi I

I I>>>>>>>>>Splice<<<<<<<<<<<<<<<<I

I Z-15/4 I Z-1/55 I


I Frame Height = 5.25 Inches I Insert Ht:Fr.Ht= ratio I

I Insert Height = 2.00 Inches I 2 in.:5.25 in. =.38095= 2-phi I

I I (2-1.619 =.38095 I

I>>>>>>>>Splice<<<<<<<<<<<<<<<<I I

I Inserted \ GOLDEN I}Insert = 2 inches I

I Fragment/ SECTION = 2-phi I}Height I



When this researcher measured the proportions of the frame segments inserted between Z-208 & Z-212, they were found to bear similar proportions as those demonstrated above for frames Z-15/4 & Z-1/55. When we study the segment of the Z-Film showing the mortal head wound(s) to President Kennedy in Z-313 ff., we found that the unusual framing of the subjects (JFK, Governor Connally and the limousine) which many viewers took to be an accidental loss of the subject by Mr. Zapruder was not accidental.

Though Mr. Zapruder filmed his subjects in center frame throughout the opening seconds, he suddenly began to lose the subjects and the limo around Z-270. The limousine drops completely out of frame and Zapruder almost "loses" the subjects completely by frame Z-291 at which they descend to the lowest point .11904 of the frame height or "latitude" from the bottom, a figure obtained cross-checking the findings of mechanical engineer Noel Twynam, done in his unique study in this field and presented to the Z-Panel.

Coincidently, this segment contains the moment (Z-295) where the vapor trail of a bullet which struck both men was discovered by this writer in 1992 (see above). When shown in close-ups, the area showing the strike to Governor Connally is usually masked by tight zoom-in shots focusing on JFK or artificially as in Robert Groden's versions. One segment of Groden's "Secret Service version" used in "The Men Who Killed Kennedy" series shows the upper torsoes of the two men above what appears at first glance to be a close-up of the area between two frames with a dark horizontal border between them.

However, upon slowing the film speed digitally in video mode and stopping Robert Groden's version of events, this writer discovered that the segment below what appears to be the horizontal frame border IS NOT THE TOP OF THE NEXT FRAME (which should be GREEN showing the grassy area at the top of the "real" next frame). It actually shows the lower torsos of JFK and Governor Connally. The existence of the area below showing the lower portions of the two men's suits and ties demonstrates conclusively that Mr. Zapruder did not "lose" his subjects at all. Using a dark horizontal strip to mask the area in question an illusion has been created. The illusion in this case is that we are seeing the area between 2 sequential frames. In fact, we are seeing a "cover-up" right before our eyes but few have eyes quick enough to perceive this.

The area masked by the horizontal strip in Groden's so called "Grodenscope" version would show the lower portion of a visible vapor trail left on the film by a bullet which struck both men, at Z-295, In unravelling this illusion we can begin to discern some shadowy faces behind the cover-up. Unmasked frames and close-ups of JFK and Governor Connally alone show the trail of the bullet with fragments striking Connally's wrist and ricocheting into his thigh (Z-295).


Between Z-291 and Z-312 the occupants of the limousine begin to rise again into the frame but remain "oscillating" up and down between the lower .15 and .11 of the frame area until suddenly rising to the .38095 at Z-323-324 (just before Mrs. Kennedy's bolt from the back seat) indicating a significant deletion at that juncture also. Interestingly, the figures of Mrs. Kennedy and Clint Hill, also demonstrate kinesic anomalies while oscillating in wave-like motion along the border of the ..38095 latitude. The motion Mrs. Kennedy made to retrieve a piece of the President's skull has been deleted as well as her pushing off the dead President's body before leaping out of the car. Also removed was the President's final gesture, an extension of his right arm as she was holding the left. "It was beautiful" Mrs. Kennedy later said.


This discovery forced me to consider its magnitude and it's full implications, i.e., that someone had consciously applied (in reverse), "phi", a "sacred measure" known as the Golden Section, one of the cornerstones of science, which gives rise to the Fibonacci Series, perverting and twisting its purpose to the cause of "Antiscience". Since the phi-section has always been known a hallmark of beauty and proportion, one can understand that its use here in reversal of its purpose and principle nonetheless served "to beautify" a grisly and heinous event. It, thereby, makes the edited film's aspect and falsified perspective of events more palatable and acceptable psychologically to the general viewer.

Thus, we must acknowledge that we are faced with a cunning adversary or group of adversaries of extremely high but perverse intellect. Whoever edited the Z-Film knew that the film could be altered successfully if the changes were kept subliminal, i.e. were kept below the threshold of human visual perception. The key was to keep the changes within "proportion" to what is "natural". This was achieved with the use of "phi-sectioning".

It is self-evident that the presence of this proportion (1:1.619) temporally and spatially in the editing of the Z-Film in at least FOUR DISTINCT FRAMES (Z-1/54 & Z-1/55 and Z-2/08-Z-2/12) and across a 49-frame span could never have occurred by chance. 1.619 is less than 1/1000th off the infinitely repeating variable measure of the Golden Section, known as "phi", or 1.6180339..., a sacred measure of proportion known since the time of the building of the Great Pyramid and expounded by Pythagoras. Furthermore, as "phi" is calculated to ever longer decimal places (1.6180339...), though it never reaches this value, it is infinitely expanding toward 1.619 as it expands toward its next dimension.


If we think of the alternating flasher cycle as a stroboscopic metronome we can conduct a very exact study of frames Z-133 through Z-182. If we compare what is there with what should be there in terms of the flasher cycle, our examination becomes like comparing two sheets of music notation which should mirror each other identically. Using the flashers to count "beats" like a metronome we can easily find not just missing beats but missing notes and missing time as well.

In Diagram 3, we have an idealized visualization of the flasher cycle which should appear in a 2.295 second period of time represented by frames Z-136 through Z-182:

Diagram 3

An Idealized 10 Frame per Flash Sequence

Z-136 Z-156 Z-176

LEFT ********** ********** *******

RIGHT ********** ********* Z-182

Z-146 Z-166


The following (Diagram 4) is a schematic demonstrating the observed flash frame sequence of left and right flash units as well as points of splicing (/), insertions (S1 & S2), deletions (->) and visual anomalies (#!). "*t" designates transition frames. Follow the asterisks from left to right.

Diagram 4

Actual Flasher Frame Distribution Z-136 to Z-182

(A Schematic of Time Contraction)

Z-136 Missing-> Z-156&7 /Z-159 (#!)Anomaly

*********** Z-1/55*-/-*/********** ******Z-182 Z-147*t (#!) /S2 Z-166*t Z-177*t

*******/S1_\Inserts **********

Z-15/4 (#!)Anomaly Z-167


Herein, we have a partial answer to the question posed earlier, "How is it possible for a vehicle travelling at no more that 11.2 mph appears to be travelling at 30+ mph?". The figures above when coupled with the average projection speed for 8 mm. film of 18 fps unveils the optical illusion.

If we calculate using Mr. Zapruder's 24 frames per second figure, we find that out of every 24 frames 12 frames should have remained lighted continuously for each flasher cycle. Thus, the remaining 12 frames when projected at 18 frames per second would make President Kennedy's limousine appear to be moving 1.5 times faster than actual speed or 11.2 mph x 1.5 = 16.8 miles per hour.

If our calculations employ the FBI/Warren Commission figure of 18.3 frames per second, we find that 9.15 frames would have to show light for the duration each unit flash. For any second of film wherein a complete cycle of flasher frames were to be removed (18.3 - 9.15 = 9.15), the remaining 9.15 frames shown at standard 18 fps projector speed would create the appearance of the car to be travelling 1.9672 times faster than actual speed or 22.03 mph.

However, neither figure alone, 16.8 mph above nor 22.03 mph in the second calculation, can account for the illusion of 30+ mph speed alone. There are two other factors: wind and pulse or cadence.


A great contribution to the illusion of speed in the Z-Film was rendered by the southwest wind which blew through Dealey Plaza on the morning of November 22, 1963 at a minimum 10 mph, gusting at times to 16 and 20 miles per hour. The wind caught the left side Presidential Flag at a propitious angle several times unfurling it to its fullest while the American Flag on the right side drooped at times across the right fender and in a forward direction. This is indicative of a rapid deceleration. When we couple the apparent speeds of the vehicle under the two camera speeds, 16.8 mph and 22.03 mph, with the effect of a minimum 10 mph headwind, we can compute a minimum relative wind speed approaching 26.8 and 32.03 miles per hour which contributes significantly to the illusion of velocity in the Zapruder Film.

Though significantly influenced by wind effects, the Z-Film's illusion of velocity is brought to completion by something more unusual which we term "visual pulse", cadence. A modulated visual pulse has been induced in the Zapruder Film by the deletion of several series of right side flasher frames and the transposition of key frames.

The removal of these flasher frames result in pulsing motion in the apparent and illusory acceleration of the Presidential limousine as it approaches and passes the Stemmons Freeway sign. This rhythmic visual pulse of the "stroboscopic acceleration" described above gives the viewer an odd impression of a lurching, or galloping, rhythm relative to foreground and background information.

An acoustic analogy to this "modulated pulse" is the auditory

effect of hearing a few bars of "The Lone Ranger Theme" in Giaccomo Rossini's "William Tell Overture". Expressed visually in Morse Code:

... ... ... ..

Though the listener may be perfectly still, the rhythm of the music can instantly communicate to the listener a sense of movement at galloping speed returning to "those days of yesteryear." One can see visually that it is the pause (space) between the sets of dots that gives the theme its cadence and rhythm. For contrast compare the above Morse Code with other possibilities such as:

.... .... ... or ...........

In a manner similar to the first cadence, the deletion of the right flasher frames creates a rhythmic visual pulsation in the Z-Film.


Is there any concrete and measurable evidence to support this in the Zapruder Film? If so, where would one most likely find it? For the car does appear to move quite smoothly? Too smoothly, in fact! Because the car is a rigid and roughly rectangular form apparently rolling along on other symmetrical forms, the wheels, it seems unchanged and appears to move in a rhythmically pulsed acceleration. At first glance, little seems strange except 5 blink-like splices in the opening 11 seconds of the film at Z-132-133, Z-154-155, Z-207 -Z-212. The right side of President

Kennedy's face and shoulder have been blacked out artificially at Z-168 with black ink or "Magic Marker".


Between the frames numbered Z-207-212, we found artificially

inserted and spliced in a manner almost exactly like that described above at Z-154-Z-155, pieces of 2 frames which by normal concepts of space-time should have expressed themselves between Z-189-190.


As stated earlier, we found that the temporal anomalies in the film revealed the proportion called "Phi" or Golden Section, a ratio of 1:1.61818..., had been used to alter the viewers sense of the passage of time by masking it, using an organic formula. However, this insight led to a more remarkable discovery: The "phi" section had been used TO CUT AND SPLICE IN THE INSERTS DISCOVERED BETWEEN BOTH Z-154 and Z-155 and Z-208 and 212! That is to say that these frames had been geometrically sectioned and inserted according to the formula for the Golden Section:


-- = -- = 1.6180339...


This was a stunning revelation. Although we had noted that the technique used to cut the Z-Film at both Z-15/4 and Z-1/55 was similar to the one used to cut it at Z-208-212 (utilizing the insertion of an out-of-synch frame), what we had described above turned out to be a spatial approximation of the Golden Section as exact as the frame reduction ratio (68/42 = 1.619).


At Z-189-190, frames appear to be missing at a point where the cameraman stopped panning left to right momentarily after panning too far right and returns abruptly left as if the car had stopped behind the Stemmons Freeway sign. The removal of these frames

(dubbed "Z-189.5/A&B") from their natural position after Z-189 and before Z-190 to Z-212 creates a remarkably constructed optical illusion masking the car's arrest and slow acceleration thereafter.

Frames have been deleted to remove evidence that the car slowed and/or stopped (see Senator Yarborough's testimony above). Its deceleration would be easily revealed by a change in the shape of the tires as sinking of the front of the limousine on its spring suspension would visibly alter the spatial relationship of the wheel wells to the wheels as can be seen in the Nix Film.

Witnesses, Mr. & Mrs. William Newman, saw the President "jumped up" in the car "as if going along with a gag or something" and Zapruder claimed to have seen this, though lawyers claimed the sign was in the way. The Newmans have also "disappeared" from the Z-Film.


Because the car's shape and wheels remain uniform in motion, it is difficult to discern the alterations in the Z-Film if one looks only at the limousine and its occupants. This researcher therefore chose to analyze human body movements in the film to test the theory of stroboscopic acceleration. For while wheels appear uniform in movement, human body movements are reciprocal, opposite and complementary, alternating right and left. Thus, a false step on film is easily discernible.

Using this premise as a springboard, this researcher hypothesized, that the effect caused by the deletion of the right side flasher frames could be thought of as making the film more "left sided" due to the absence of the right side flasher frames as the viewer is presented with all or more left sided information and no or little right side information.

As this was self-evident, wheel movement would appear unchanged despite the "stroboscopic acceleration" but human body movements captured on the film should likewise manifest the same degree of "left-sidedness", thus rendering as well some markedly visible strangeness to human movements and non-movements, as well. Thus, as was predicted by the logic of our hypothesis, the ultimate proof of tampering was found in the strange behavior and erratic movements of the bystanders on either side of Elm Street. The most famous of these is Ms. Rosemary Willis, the little girl seen running alongside of JFK's limousine for several frames before she stops in the background precisely at Z-189-190. This point in time is discussed above as the point from which a frame (Z-189.5/A&B) was deleted and transposed to Z-208-Z-212.