**Topic:** A non-statistical, axiomatic, & mathematically-derived mechanistic model of copulatory thrusting and its evolutionary origin.

## Preface

This article is beyond the scope of my work on genopolemology, although if I ever discover some relationship between that and copulatory thrusting I might very well reference this (but I find it unlikely). For this reason, this will be more of a chill article, despite all the calculus. Nevertheless, there is an important point I would like to get across. I want this article to serve as an example of a point I have been making for a long time, pertaining to the decrepit state of academia and its abuse of statistics as a substitute for logical deduction. The second reason for me writing this is that sex sells and I figured “hey, why not + it’s interesting and informative”.

Let me start by saying that I don’t, in fact, randomly think about d*cks all day (lol). I discovered the patterns described below because I have a brain that sees patterns everywhere and makes all kinds of connections, and my wife and I are trying for a baby which is why sex has been very much on my mind. Why do human males thrust and why would this behavior emerge?

More importantly, why is the scientific literature on this subject matter so painfully dismal? One would think that after more than a century of research and a cumulative multi-million dollar budget, the experts in the field would have developed a definite explanation by now. So, I decided to solve this problem. And it took me about 1 hour to develop the model in my head, 30 minutes to write up the math, and a few hours to write this article. Before I proceed, let’s take a look at what the experts have contributed thus far.

**Note:** I exclude from my criticism the tiny minority of intellectuals who do contribute greatly to our understanding of the world. In fact, my criticism is to be understood as the degree to which the institution of academia is held together by this tiny minority.

## The Literature on Copulatory Thrusting

Here, I do my best to steelman the current consensus:

**Evolutionary Advantage:**One of the primary reasons behaviors evolve is because they confer some advantage to the organism. In the case of copulatory thrusting, it’s believed that this behavior might increase the likelihood of successful fertilization. By ensuring deeper deposition of sperm, it might increase the chances of sperm reaching the egg.**Sperm Competition:**In species where females mate with multiple males in a short time frame, there’s competition among sperm from different males to fertilize the egg. Thrusting might help in displacing sperm from previous mates, thereby increasing the chances of a particular male’s sperm fertilizing the egg.**Stimulatory Role:**Copulatory thrusting might also play a role in stimulating the female and ensuring her continued participation in the act. In some species, female orgasm, which can be facilitated by thrusting, might aid in sperm uptake or retention.**Locking Mechanisms:**In some animals, thrusting helps in ensuring that the male’s reproductive organ is securely locked with the female’s, ensuring successful sperm transfer. This is especially true in species where the act of copulation is brief.**Evolution of Morphology:**The evolution of reproductive organs in many species might have co-evolved with the behavior of thrusting. For instance, the shape and size of the penis in many animals might have evolved to be more effective in combination with thrusting.**Emergence of the Behavior:**Like many behaviors, copulatory thrusting likely emerged gradually. It’s possible that ancestral species had a rudimentary form of this behavior, and over time, as it conferred reproductive advantages, it became more pronounced and refined.**Neurological and Hormonal Factors:**The act of copulation and associated behaviors are also regulated by various neurological and hormonal mechanisms. The evolution of these mechanisms in tandem with physical behaviors could have played a role in the emergence and refinement of thrusting.

And here are the problems with these explanations:

- Thrusting does not ensure a deeper deposition of sperm. In fact, the post-ejaculatory continuation of thrusting (which is not uncommon) scoops out the deposit. Even in the case of post-ejaculatory cessation of thrusting, there is no guarantee that the deposition of sperm occurs at any particular depth of penetration. In other words, it can occur on the way out. So the error of point 1 is that thrusting as a means of achieving depth of deposition is far too unreliable to consistently increase the likelihood of successful fertilization; this cannot have a significant effect size on the evolutionary trajectory of thrusting.
- As explained in point 1, thrusting (in humans) does scoop out recent prior deposits (i.e. sloppy seconds, thirds, …). But this is only true in the case of species who have a glans shaped like that of a human. There are species who do not have this shape and, as such, this does not explain why even those animals thrust. Moreover, the explanation provided is a reversal of causality. The human glans evolved to have the shape it has because when human males thrust, the shape which is best at scooping out deposits is most likely to be inherited by the next generation. The shape of the glans cannot be the cause of thrusting if thrusting came first. Total non-sequitur and a lazy one at that.
- Why would a female evolve to enjoy thrusting if there were no prior thrusting taking place? Again, this is a reversal of causality. The evolution of thrusting must have preceded the evolution of female orgasm.
- Locking is irrelevant to the emergence of thrusting. For starters, animals who do not lock thrust regardless. Furthermore, this is yet another reversal of causality. One cannot evolve to lock without first evolving to thrust.
- There is no error in point 5 but it also isn’t really saying much. So my 1st thought is “duh”. We get it, thrusting and penile shape can influence each other but the question is WHY. Point 5 isn’t really answering it.
- Again, “duh”. We are well aware that early forms of thrusting lead to later forms of thrusting. We want to know why we started doing this.
- Again, “duh”. We get it. There is an interplay between hormones / neurotransmitters and thrusting. But WHY.

I’m not impressed.

## Mathematical Models of Phalli

Let’s start with a simplistic model of a phallus and gradually add parameters to eventually represent real-world phallic dynamics.

### Phallic Model V1

There is a phallus-shaped mass *A*. There are negatively charged filaments attached to it all over. There is a positively charged point *B*. What happens as *B* approaches *A*?

Opposite charges attract. As *B* approaches *A*, the negatively charged filaments on *A* will be attracted to *B*. The strength of this attraction will increase as *B* gets closer to *A*. As *B* gets closer, the filaments on *A* that are closest to *B* will experience a stronger attraction than those farther away. This could cause the filaments to orient themselves towards *B*, especially if they are flexible. At some point, as *B* gets very close to *A*, other forces (like repulsion due to electron clouds of atoms, or mechanical forces if the filaments have some rigidity) might come into play, preventing *B* from merging with *A*. This would be an equilibrium position where the attractive electrostatic forces are balanced by other repulsive forces.

### Phallic Model V2

Add parameter to V1: the filaments in mass *A* are increasingly more frequent at the tip where point B approaches. They are less frequent at the shaft or base. The interaction between point *B* and mass *A* will now be influenced by the varying density of the negatively charged filaments on *A*.

As before, the negatively charged filaments on *A* will be attracted to the positively charged point *B*. However, since the filaments are more frequent at the tip of *A*, the attraction will be stronger at the tip compared to the shaft or base. The filaments at the tip, being more numerous and closer to *B*, will have a stronger tendency to orient themselves towards *B* compared to the filaments on the shaft or base.

### Phallic Model V3

Add parameter to V2: filament attraction at the tip compresses mass *A* at the tip, creating a high pressure zone.

As point *B* approaches the tip of *A*, the increased filament density results in stronger attraction. This attraction causes the filaments to pull inward, compressing the material of *A* at the tip and creating a high-pressure zone. The filaments at the tip will still have a stronger tendency to orient themselves towards *B*, and this orientation might be further emphasized due to the compression and increased pressure. The compression at the tip and the resulting high-pressure zone might lead to mechanical behaviors in *A*. For instance, if *A* is elastic, it might try to expand back to its original shape once *B* moves away, leading to oscillatory behaviors.

### Phallic Model V4

Add parameter to V3: point *B* is a hole and inside it is shaped like a vaginal canal where its wall is positively charged.

As mass *A* approaches the hole (point *B*), the negatively charged filaments, especially those at the tip of *A*, will be attracted to the positively charged walls of the vaginal canal inside *B*. This attraction will still cause the filaments to pull inward, compressing the material of *A* at the tip and maintaining the high-pressure zone. The filaments at the tip of *A* will orient themselves towards the positively charged walls of the vaginal canal in *B*. This orientation might be more pronounced due to the shape and charge distribution inside *B*.

The compression at the tip of *A* and the resulting high-pressure zone might still lead to mechanical behaviors in *A*. Additionally, the interaction between *A* and the walls of the vaginal canal in *B* might lead to other mechanical behaviors, such as friction or resistance to movement. The depth to which *A* can penetrate into *B* will depend on the balance of forces: the attractive electrostatic forces pulling *A* in and the repulsive forces (from the high-pressure zone and the mechanical properties of *B*) pushing A out.

As *A* penetrates deeper into *B*, the tip of *A* will experience increased compression due to the attraction of the negatively charged filaments to the positively charged walls of *B*. This is especially true if the filaments are denser at the tip, as they will be pulled more strongly towards the walls of *B*. The walls of *B* will exert a resistance against the penetration of *A*. This resistance can arise from:

- The elasticity of the walls of
*B*. - Friction between
*A*and the walls of*B*. - Any other mechanical properties of
*B*that resist deformation or displacement.

The increased compression at the tip of *A* will lead to an increase in the internal pressure of *A*. This pressure will be highest at the tip and will decrease towards the base. The pressure gradient might cause the material of *A* to be pushed backward (towards the base), especially if *A* is elastic. As *A* penetrates into *B*, it will displace some volume inside *B*. If *B* is filled with a fluid or is not completely hollow, this displacement can lead to an increase in pressure inside *B*, which can further resist the penetration of *A*.

### Phallic Model V5

Add parameter to V4: the filaments on mass *A* are now negatively charged points along mass *A*.

Unlike the filaments, the charged points won’t have a physical orientation towards *B*. However, the region with a higher density of charged points (e.g., the tip) will still be more attracted to *B*.

### Phallic Model V6

Add parameter to V5: upon 1st penetration, the air inside *B* exits such that it creates suction on mass *A*.

As mass *A* penetrates into *B* for the first time, the air inside *B* exits, creating a suction effect. This suction will act in conjunction with the electrostatic attraction between the negatively charged points on *A* and the positively charged walls of *B*. The combined effect will enhance the compression at the tip of *A*, intensifying the high-pressure zone there.

The rate and manner in which the air exits *B* during the first penetration can influence the strength and duration of the suction effect. For instance, if *B* has a small opening through which the air exits, the suction might be stronger and last longer compared to a scenario where *B* has a large opening.

### Phallic Model V7

Add parameter to V6: point B is moist on the inside.

The lubricating effect of the moisture will introduce a counteracting force that reduces the resistance to penetration. The compression at the tip of *A*, the resulting high-pressure zone, the suction effect, and the lubrication will lead to complex mechanical behaviors in *A*. The lubrication might allow *A* to move more freely within *B*, potentially leading to oscillatory or sliding behaviors.

### Phallic Model V8

Add parameter to V7: mass *A* is rigid but can withstand some degree of bending and compression.

The tip of *A* will still experience compression due to the electrostatic attraction and the suction effect. The rigidity of *A* will determine how much it can compress before resisting further compression. Once the forces causing bending or compression are removed or reduced, *A* will try to return to its original shape due to its inherent rigidity. The rigidity and flexibility of *A* mean that it will experience stress when subjected to forces (like bending or compression). If the stress exceeds a certain threshold, *A* might undergo permanent deformation or even breakage.

### Phallic Model V9

Add parameter to V8: there are no more charges; the points on mass *A* are now neurons and they respond to stimulation by encouraging mass *A* to expand at the point of stimulation, and the inside of point *B* is stimulatory and warm.

The neurons on *A* might form a feedback loop where continuous stimulation leads to continuous expansion until a certain threshold is reached. This can lead to oscillatory behaviors or rhythmic movements of *A* inside *B*. When the neurons on mass *A* come into contact with the stimulatory and warm interior of *B*, they promote expansion at the point of contact. This localized expansion will increase the internal pressure in the stimulated regions of *A*, especially if the expansion is rapid or significant.

### Phallic Model V10

Add parameter to V9: there is a malleable tube inside mass *A* and this tube ends with a hole at the tip.

The internal pressure of the tube will be influenced by the flow of its contents, the external pressure exerted by *A*, and any suction or pressure changes inside *B*. The moisture inside *B* can introduce hydrodynamic effects, especially if *A* moves rapidly or if there’s a significant amount of moisture. Additionally, any fluid inside the malleable tube can introduce its own hydrodynamic effects, especially if it’s expelled or sucked through the hole at the tip.

### Phallic Model V11

Add parameter to V10: the tube connects at the base to a sack of liquid but the initial pressure at the tip is too high for the liquid to exit.

The sack containing the liquid will have its own internal pressure. This pressure will be influenced by the volume of the liquid, the elasticity of the sack, and any external forces applied to the sack (e.g., compression of *A*). The initial high pressure at the tip of the tube prevents the liquid from exiting. However, as *A* interacts with *B* and undergoes various mechanical and neural stimulations, this pressure might change. If the pressure in the sack becomes significantly higher than the pressure at the tip of the tube (due to external forces or other factors), the liquid might start to flow towards the tip, even if it doesn’t exit immediately.

Let’s apply Model V11 to a scenario: after the initial penetration, mass *A* thrusts in and out.

As *A* thrusts deeper into *B*, it will encounter varying degrees of resistance from the walls of *B*. The neurons on *A* will be further stimulated by the warm, moist, and stimulatory interior of *B*, promoting localized expansion. This can increase the pressure inside the malleable tube and potentially push the liquid towards the tip. As *A* is pulled out, the suction effect and the lubrication from the moisture inside *B* will influence the ease of this motion. The neurons on *A* might experience reduced stimulation as they lose contact with the interior of *B*, leading to reduced expansion. The liquid inside the tube might flow back towards the sack due to decreased pressure at the tip.

### Phallic Model V12

Add parameter to V11: as soon as the pressure inside the sack equals the pressure at the tip, strong muscular spasms compress the sack with each spasm.

Each muscular spasm compresses the sack, leading to a sudden increase in its internal pressure. This will force the liquid inside the sack to be pushed out of the tip of the tube at a high velocity, especially if the pressure from the spasm exceeds the resistance at the tip; e.g. compressing the tip of a water hose.

Now, we have a model that fits the physical conditions of actual genitalia. Let’s represent V12 mathematically.

**Note:** if you’re not interested in the math, you can skip to the next section.

#### Definitions

t: time

P_{sac}(t): sac pressure at time t

P_{tip}(t): tip pressure at time t

P_{spasm}(t): pressure from spasm at time t

P_{thrust}(t): pressure from penetrative thrust at time t

P_{acc}(t): accumulated pressure at time t from thrusting

V_s(t): volume of sac at time t

k: elasticity constant of sac (pressure over volume)

Q(t): intrasaccular flow rate at time t

R: resistance at the tip => pressure drop per unit flow rate (Pa \cdot s / m^3)

\beta: proportionality constant relating pressure difference to flow rate (\frac{s/m^3}{Pa})

\omega: periodic frequency of thrusting (radian)

## Intrasaccular Pressure Dynamics

**EQ1:**\frac{dP_{sac}(t)}{dt} = \frac{ dP_{acc}(t) }{dt} + \frac{ dP_{spasm}(t) }{dt} – \frac{d( k(V_s(t) – V_{s0}) )}{dt}

## Tipular Pressure Dynamics

Q(t) = \beta(P_{sac}(t) – P_{tip}(t))

\frac{dQ(t)}{dt} = \beta( \frac{dP_{sac}(t)}{dt} – \frac{ dP_{tip}(t) }{dt})

\Delta P(t) = RQ(t)

\frac{d \Delta P(t)}{dt} = R\frac{dQ(t)}{dt}

\frac{ d P_{tip}(t) }{dt} = R \beta( \frac{ dP_{sac}(t)}{dt} – \frac{ dP_{tip}(t) }{dt} )

\frac{ d P_{tip}(t) }{dt} = R \beta( \frac{ dP_{sac}(t)}{dt}) – R \beta(\frac{ dP_{tip}(t) }{dt} )

\frac{ d P_{tip}(t) }{dt} + R \beta(\frac{ dP_{tip}(t) }{dt} ) = R \beta( \frac{ dP_{sac}(t)}{dt})

\frac{ d P_{tip}(t) }{dt}(1 + R \beta) = R \beta( \frac{ dP_{sac}(t)}{dt})

**EQ2:**\frac{ d P_{tip}(t) }{dt} = \frac{R \beta}{1 + R \beta} \frac{ dP_{sac}(t)}{dt}

## Spasmic Pressure Dynamics

**EQ3:** P_{\text{spasm}}(t) =
\begin{cases}
P_{\text{spasm\_max}} & \text{if } P_{sac}(t) = P_{tip}(t) \\
0 & \text{else}
\end{cases}

## Thrusting Pressure Dynamics

P_{thrust}(t) = P_{thrust\_max} \cdot sin(\omega t)

\Delta P_{thrust}(t) = \gamma P_{thrust\_max} \cdot sin(\omega t)

Where 0 \le \gamma \le 1 is the proportionality constant.

P_{acc}(t) = \int_{0}^{t} \Delta P(t) dt

P_{acc}(t) = \int_{0}^{t} \gamma P_{thust\_max} \cdot sin(\omega t) dt

**EQ4:**P_{acc}(t) = -\frac{ \gamma P_{thrust\_max} }{\omega} \left[ cos(\omega t) – 1 \right]

## General Copulation Equation

From EQ1: \frac{ dP_{sac}(t) }{dt} + \frac{ dk(V_s(t)-V_s0) }{dt} – \frac{ dP_{spasm}(t) }{dt} = \frac{ dP_{acc}(t) }{dt}

P_{sac}(t) + k(V_s(t)-V_s0) – P_{spasm}(t) = P_{acc}(t)

From EQ4: cos(\omega t) = 1 + \frac{ \omega P_{acc}(t) }{\gamma P_{thrust\_max} }

\omega t = arccos(1 + \frac{\omega P_{acc(t)} }{\gamma P_{thrust\_max}} )

t = arccos(1 + \frac{\omega P_{acc(t)} }{\gamma P_{thrust\_max}} ) \cdot \frac{1}{\omega}

**EQ5:**t = arccos(1 + \frac{\omega P_{sac}(t) + k(V_s(t)-V_s0) – P_{spasm}(t) }{\gamma P_{thrust\_max}} ) \cdot \frac{1}{\omega}

## Maximizing Copulatory Efficiency

(i.e. minimizing time-to-ejaculation)

**Deductions from EQ5:**

- \therefore lim_{\omega \to \infty} t = 0
- \therefore lim_{P_{thust\_max} \to \infty} t = 0
- \therefore lim_{P_{spasm}(t) \to \omega P_{sac}(t) + k(V_s(t) – V_{s0}) } t = 0
- \therefore lim_{k \to 0} t = 0
- \therefore lim_{V_{s0} \to \frac{ \omega P_{sac}(t) – P_{spasm}(t) }{k} – V_s(t) } t = 0
- \therefore lim_{P_{sac}(t) \to \frac{ P_{spasm}(t) – k(V_s(t) – V_{s0})}{\omega} t = 0}

## Movement Optima

Every movement, such as the flapping of wings or the slithering of a snake, has an optimum permutation. In turn, those organisms which move in a more similar manner to the optimum permutation increase their chances of passing on their genes. This means that, over time, the specie in question will evolve to mimic more and more closely the theoretical optimum, partly due to the selection for certain neurological factors but also due the selection of certain anatomical traits (e.g. position vs length of wings vs muscle fiber composition…).

Copulatory thrusting is subject to the exact same kind of selective pressure and its optimal constraints are described in the prior section “Maximizing Copulatory Efficiency”. From the deductions made, we can infer the selection of certain traits:

- The greater the thrust frequency, the lower the time-to-ejaculation. One can infer selection for a minimal level of aerobic capacity to accommodate higher frequencies of thrusting and the thrusting itself to its cumulative effect on intrasaccular pressure (EQ4).
- The greater the intrasaccular pressure increase from thrusting, the lower the time-to-ejaculation. One can infer selection for a minimal level of anaerobic capability to provide for more forceful thrusts.
- The closer the increase in intrasaccular pressure from pubococcygeal spasms to the constraint outlined in point c., the lower the time-to-ejaculation. One can infer selection for an optimal level of muscular strength and size of the pubococcygeus, i.e. one of the so-called kegel muscles in the pelvic floor.
- As intrasaccular elasticity approaches 0, the time-to-ejaculation approaches 0. This suggests stabilizing selection in favor of lower elasticity, but only to a point where it remains sufficiently malleable for shock absorption – an excess of intrasaccular inelasticity may increase the risk of injury from shocks.
- The closer the initial sperm volume to the amount outlined in point e., the lower the time-to-ejaculation. This suggests selection in favor of an optimal amount of fluid production by the seminal vesicle and Cowper’s gland.
- The closer the intrasaccular pressure to the constraint outlined in point f., the lower the time-to-ejaculation. This confirms the previous inferences.

Given that the female orgasm increases the likelihood of fertilization, the selective pressures above in non-humans are balanced by the selection for not ejaculating too early before the female orgasm; in humans, however, this might not be the case due to the propensity to resort to non-penetrative means of stimulating the female and achieving orgasm in the case where the male ejaculates first.

## Conclusion

We now have a conclusive true-by-definition model of copulatory thrusting which attributes its evolutionary origin to phallic pressure dynamics, rather than anything I steelmanned earlier. In fact, by virtue of how the General Copulation Equation was derived, any interpretation of statistical data which does not fit with it can accurately be described as false. Newton’s mechanics improved and was expanded upon not thanks to statistics, but due to even more of the same axiomatic methods later used by Einstein and the like.

Statistics can confirm trends and relationships. But axiomatic theories confirm statistical interpretation and not the other way around. I hope the experts in this field take heed of what I’ve done here and recognize how embarrassing this is. This ought to be the foundation of your field and nothing I’ve done here is really all that complicated, so… what exactly have you people been doing this whole time?