by means of relative selection

Using the Weyl and Ricci tensors to prove Darwin’s theories of competition and evolution

and to refute creationism and intelligent design

The

We use our *Brassica rapa *experiment to prove that we can measure the exact mass and energy a population expends upon Darwinian fitness and competition.

The American Heritage Dictionary defines ‘carrying capacity’ as:

“the maximum number of individuals that a given environment can support without detrimental effects” (Houghton Mifflin Harcourt Publishing Company, 2011).

The carrying capacity is the maximum population numbers a species can indefinitely maintain as its equilibrium size and set of behaviours in that given environment. If it is below its carrying capacity the species will tend to increase in numbers; and if it is above its carrying capacity it will tend to decrease in numbers. We therefore use our experiment to measure both:

- the processes some species uses to establish itself at its carrying capacity; and
- the equilibrium age distribution values in numbers, masses, and energies that define that carrying capacity.

We selected *Brassica rapa* as our demonstration population because it is an extremely well-known fast-growth plant whose genome has been thoroughly studied. It is very commonly used in scientific experimentation, and a small industry has grown to provide it. Our experiment and our results can therefore be very easily replicated—and validated—in any competent laboratory.

Since we wanted it to be as self-evident and indisputable as possible that we successfully isolated all number-dependent effects—and only number-dependent effects—we grew our *Brassica rapa* under tightly controlled and uniform conditions throughout. We maintained constant pot sizes, constant light, and provided a constant water supply. Those growing conditions are also easily replicated.

We began at a seed density below *Brassica rapa*'s estimated carrying capacity. We then let the plants follow their intrinsic and natural cycle and rhythm … which we wanted to measure. We let them develop naturally to flower, fruit, and seed. We let the seed density at the end of each generation determine the seed density to plant at for the beginning of the next. We recorded all masses, energies, and numbers. We also recorded the generation time, *T*. As is standard in such situations, *B. rapa* suitably oscillated, over successive generations, in search of its equilibrium and its carrying capacity.

We normalized our results so that the mean number per each distinct generation was always expressed per 1,000 plants. We called this normalized population, centred on 1,000, a “unit population”.

We then used the standard techniques of population biology to calculate the equilibrium age distribution population. We calculated equilibrium mass and energy values for both (A) the individual *Brassica rapa* plants (*m̅* and *p̅*) in each unit population; and (B) the whole population (*M* and *P*). We also expressed all those values in our unit and normalized terms.

We called the normalized equilibrium age distribution population of generation length *T* that we calculated a “unit Liouville population”. We then applied tensorial analysis to (A) and (B) and to the unit Liouville population using the following equations:

- ∇ •
*M*=*M*⁄*n*; - ∇ •
*P*=*P*⁄*n*; - ∇ ×
*M*= (∫*M dT*/∫*dm̅ dn*) = ∂*m̅*⁄∂*t*- ∂*n*⁄∂*t*; and - ∇ ×
*P*= (∫*P dT*/∫*dp̅ dn*) = ∂*p̅*⁄∂*t*+ ∂*W*⁄∂*t*- ∂*n*⁄∂*t*.

Our data, the four equations, and (A) and (B) tell us, through the partial derivative ∂*n*⁄∂*t*, which is the aberrancy, that *Brassica rapa* expends exactly 0.19 grams of mass and 1.225 joules of energy, over a 36 day period, upon Darwinian fitness and competition … and therefore that it does not follow any creationist and/or intelligent design template.

- CONTENTS

- Before we begin
- The Experiment

- Introduction
- 1: Setting the Scene
- 2: The Requirements
- 3: Reproducing the Cycle
- 4: The Sun and the Cosmos
- 5: The Earth
- 6: The Ecology
- 7: The Biological Entity
- 8: The Genome
- 9: The Energy
- 10: Existence and Equivalence
- 11: Physiology and Metabolism
- 12: Death and Dissipation
- 13: Reproduction
- 14: The Anomalies
- 15: Measuring the Anomalies
- 16: The Basis
- 17: The Biological Potential
- 18: How to Measure Variations
- 19: The Refutation
- 20: The Proof
- 21: Conclusion
- Biography

An algebraic and geometric topology based proof.

A vector calculus based proof.

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