Introduction
Mathematics almost always cares about the result of a calculation — the answer at the end of the line. But what if counting the steps taken to reach that answer reveals something the result alone cannot? What if two calculations that produce identical outputs are actually fundamentally different, and we have simply been ignoring the evidence?
The Theory of Effort is a key concept of 4D maths. It appears in the 4D calculator as the length of a number line, tracking the result of specific settings as the value dissolves towards its limit. Using the effort value we can qualify calculations in a completely revolutionary way, with deep implications for mathematics, science, and computer technology.
The Theory of Effort forms a part of our new concept of 4D maths. Using this powerful technology, we are able to track the number of numerical iterations and quantify calculations that would otherwise appear to produce the same results. This is quite different from standard mathematical conventions, which tend to reiterate values to infinity without quantifying the process at each step. We can also quantify calculations that repeatedly exhibit the same answer, such as 1 × 1 = 1 (effort 1) or 1 × 1 × 1 = 1 (effort 2), using the distinction to explore certain calculations in greater depth.
Key takeaways
- The effort value counts the number of multiplicative or iterative steps in a calculation, not just its final result — allowing two equations with identical outputs to be mathematically distinguished.
- Irrational constants such as e have a natural effort boundary: up to effort 7 the value can be expressed as a rational fraction, but beyond that point only a decimal approximation is possible, a limit that standard mathematics does not recognise.
- Binary computing states follow the same odd/even effort pattern as negative-number multiplication, suggesting that effort-aware systems could form the basis of advanced digital locks in cybersecurity.
Counting Calculations
The Theory of Effort is found in the construction of the 4D calculator. Whenever a value is entered into the time, space, and dimension fields, the 4D calculator reiterates an equation. As it does, the results dissolve towards a specific value. Each iteration is counted as a single effort value. By noting the effort value, we can identify a specific value for any calculator setting. This is important because, more often than not, all space values will dissolve towards the same number — only by tracking the effort values are we able to differentiate one number from another.
Quantifying an Equation
The Theory of Effort is a very simple idea, yet also very powerful when it comes to qualifying a mathematical equation. Take a simple example involving negative multiplication:
The two equations above produce two kinds of result, one positive and one negative. In the first case -1 × -1 = 1, as two negatives make a positive. As the first number is multiplied by a second one, we can say it goes through a single effort value. In the second example the result is -1. As the original number goes through two calculations it has an effort value of 2. We could extend the calculation following the same pattern:
There is a very simple pattern here. If the effort value is odd then the result will be positive, and if the effort value is even then the result will be negative. This holds regardless of how many times we multiply by -1. Notice that we do not count the first number in the effort value — we only count the number of times a multiplicative action is performed.
For further exploration of how number behaviour changes across dimensional boundaries, see our work on dimensionless science.
Where the Fraction Ends
Next we can consider a slightly more advanced use of the Theory of Effort. Let us consider the equation that generates the number e. The number e is found by adding together the reciprocal values of all factorial numbers plus 1.
We can see from the image above that there is a limit to the number e in terms of its representation as a fraction comprised of two numbers. After effort 7 all subsequent values can only be expressed as a decimal fraction. This notion is strangely absent from traditional mathematical thinking. The importance of being able to express a number in terms of a rational fraction is normally not recognised. However, as the universe operates in a quantised manner, being able to qualify numerical constants to a certain degree does play a significant role. You might notice that effort 0 to 3 are musical ratios. The atomic structure is limited to 7 S-orbitals — all of which are important clues in recognising the nature of reality and the periodic table.
The fraction 685/252 is expressed as decimal 2.71825396825396825396, which exhibits a recursive series (highlighted in bold) that excludes the numbers 1, 4, and 7. As a side note, 825 ÷ 369 provides a very close approximation to √5.
As numbers like e are so important in mathematics, it is surprising that no one has taken the time to evaluate them in terms of effort values, or made any attempt to qualify them over time. It is worth noting that recursive functions play an important role in a vast amount of mathematical solutions, including the field of fractal geometry. The concept of infinity as a bounded, qualified phenomenon is explored further in Aleph 0.5 and Solving Infinity.
Effort Values and Computer Science
The Theory of Effort also plays a role in the realm of computer science. Computers operate on the basis of binary code, which can only be expressed by the numbers 0 and 1. This is because computer hardware is made of small transistor switches that can either allow an electrical signal to pass or accept a lower voltage that prevents electrical flow. In simple terms, this is a switch that can be turned off or on. As a computer operates over time, a binary switch changes position, which processes information and allows programs to be executed.
Take the example of a single switch. Starting in the off position, the switch can be changed into the on position — that is an effort value of 1. When the switch changes back to 0 this is effort 2. When switched again into the on position, this is effort 3. Any ON position will be an odd effort number, whereas any OFF position will be an even effort value.
Notice that this is very similar to the previous example of -1 × -1, where all odd effort values equal a positive and all even effort values equal a negative number. This example is the simplest possible using a single binary switch. However, when we increase the number of switches to 2 we find that the complexity of each state increases exponentially.
With two switches — call them left and right, both starting in the off position — we have two options: we can switch either the left or the right first. At effort 1 the switches will therefore always be opposite to each other. At effort 2, if we change the same switch again both return to their original position; if we change the other switch, both will be on. It takes effort 2 to turn on both switches simultaneously. Whilst there are only two possible end-states (both on or both off), the pathway by which each state is reached can differ. Either the left or the right was changed first. This means there are four possible pathways at effort 2, even though the number of end-states is just two.
The point is that only by tracking changes over time can we know which pathway is the correct one to arrive at a particular state. If we change the position of a switch a third time (effort 3), one switch will be off and the other on — there is no other option. However, the number of possible pathways doubles again to eight. Only by tracking the effort values can we be sure which switch was changed and in which order.
Odd effort values always result in the switches being in opposite positions; even effort values always result in the switches sharing the same orientation, either both on or both off. Only by tracking each step can we be certain of the pathway.
In this simple example we have examined only two binary switches. As the number of switches increases, the number of variations and pathways grows exponentially. The Theory of Effort is a methodology for tracking changes over time, with a variety of uses in computer science, particularly in the field of cybersecurity. By developing computer systems that can distinguish the pathway by which bits of code alter states, we can envision advanced digital locks that can only be correctly decoded once the effort value of the final state is known.
Conclusion
Why is the Theory of Effort important?
More often than not, mathematics is concerned with the result of an equation. The Theory of Effort allows us to quantify each step that a calculation performs. In terms of the 4D calculator, we can identify numbers that are devolved from a particular setting as the value dissolves towards its limit. The effort value gives every calculation a temporal signature — a quality that conventional mathematics has, until now, discarded entirely.
What are the consequences of the Theory of Effort?
Because we are able to track calculations through time we have a novel way to examine many mathematical functions. This includes any reiterative equation, fractal geometry, and binary systems. The approach connects naturally to broader questions about the quantised structure of reality — questions explored in depth across dimensionless science, Aleph 0.5, reciprocal number space, and solving infinity.
FAQ
This idea seems relatively simple. Has no one thought of it before?
As far as we are aware, no. The standard convention in mathematics is to reiterate values towards infinity without quantifying each step. If you find a theory that covers similar ground, we would be very interested to hear about it.
What do you mean by tracking changes over time?
In 4D maths we differentiate between numbers and calculations. Numbers exist regardless of whether they are counted. Calculation is a process of number. For example, a computer hard drive begins its life formatted at zero, even before it is energised with electricity. Only when electricity flows into it can the state be changed. The physical drive has a specific state (number) that changes over time (calculation).
What is 4D maths?
4D maths is a framework that we developed at In2Infinity, which extends conventional mathematics by treating time as a fourth dimension of number. Rather than evaluating equations only at their final result, 4D maths tracks how values evolve step by step through recursive processes, revealing structural properties that are invisible to standard analysis.
How does an effort value differ from a simple iteration count?
An iteration count merely records how many times a loop has run. An effort value is a precise qualifier: it identifies which specific state a number occupies along a dissolving number line, distinguishing calculations that produce identical results but arrive there by different pathways. Two computations can share the same result yet carry different effort values, making them mathematically distinguishable for the first time.
