I’m about discontinuous innovation. I’m asked at times to define discontinuities. Well, Kuhn’s crisis science is one answer. Mathematical holes is another. Anything on the other side of a constraint that nobody knows how to cross, yet another. Or, a logic that drives across an inconsistent space. Or, the line approaching the horizon of a hyperbolic space that never arrives because it goes to a limit and can’t pass that point of convergence. Or, the simpler case of anything not continuous.

As marketers, our answers are simpler. If we can bring it to an existing market from within an existing category, it is continuous. We’ll make some money, but we won’t change the world. We put our innovation into the late mainstreet, the device or the cloud market and starting there we leave a lot of money behind. Don’t worry, everyone else left that money behind as well.

But, if we have to take the long road of complete adoption in a nonexistent category and face the nascent bowling alley, the B2B non-IT early adopter, and the so-called non-existent Chasm. We don’t leave any money behind, and we have to create careers and a value chain, those being outcomes from generating a category which in turn generates larger financial returns than what our own intra-firm managerial accounting tells us, then the innovation is discontinuous.

I posted this after hearing an interview about the *Mittag*–*Leffler’s theorem.* * *The Mittag-Lefler Theorem is about holes or more to the point how to make holes with a single function. Or, simpler, how to describe them. The holes exist. The function doesn’t. The point of the function is to process the holes to some end. This function can be described in a manner that it deals with all the holes, not just one. Green’s theorem deals with one hole you encounter while doing the integration.

The holes of the Mittag-Lefler Theorem show up in complex analysis. Kicking it back to marketing, we seek one hole, just one, but this theorem tells that there would be many holes. That’s the point of the bowling alley in Moore’s technology adoption lifecycle. We put the technology into a product built for one early adopter in their vertical market. This being one lane of our bowling alley. This being one hole. Then, when we have the capacity, we put the technology in another product for another early adopter in another vertical market. This being another lane of our bowling alley. This being another hole. We have to do this a total of six times across seven years. Discontinuous innovation is not fast. There is an eventual point, success in the tornado we face as we enter the horizontal maybe ten years later. But there is a point.

Those six lanes would fill six holes with six different value propositions in six different vertical markets, but the same underlying technology. Each of those client engagements would be in a different place at different heights in the industrial classification tree. That puts the holes at different heights from the complex plane.

But, back to the math. One kind of function that requires the theorem are Meromorphic functions like the one below. The cool thing is that you can write a function that describes all those holes. This is a relatively simple function. The holes could be all over the plane, and still, a single function would handle them. I can imagine using a Fourier sum to get this done. A Fourier analysis would give us a collection of trig functions describing a wave that hits these holes. That sum would be a sum of different waves where each wave hits some of the holes.

This example is simple. It only requires one frequency. The cosines go to zero at each hole. We take the reciprocal of that, aka we divide by zero, and a hole results. These would be more involved because the holes are of different sizes. I don’t know how to do that yet. But, this is a start. This is a good mystery if your math isn’t there yet.

I drew the red line to say where the wave would be oscillating. The thick black line is the wave, a cosine wave. The reciprocal of the cosine gives us the hole. Those holes are where we will be making our money for a few years, seven years plus development time and value proposition development and execution, plus two years or more in that vertical. This if you’re in the last of the six lanes of your bowling alley.

The function f(x) is not a complex function on the complex plane, but if the red line has an angle of zero degrees, the function is complex. The origin is at (0,0).

Fourier analysis can break down a much more complicated signal into a wider set of waves, and sum all of those waves into a single function. I’ll add a few more complications to the figure.

So this was my first try. It’s wrong. This is complex analysis, so the waves are on an axis through (0,0) and would have a different complex variable multiplying each wave. Regardless of where you put the holes a sum of complex trig functions can get us there. The figure shows the component waves that a Fourier analysis would deliver.

Here, I have put a hole out there in complex trig. I should have drawn the black ellipse centered at the origin. This polar complex view is far simpler than the waves shown in the previous figure. There might be more waves to add up here, but it is clearer. I’m not sure the trig functions are correct, but this is my best and last attempt for now.

I raised it for some clarity, but that puts the height in the equations. I deliberately drew the height of the new hole some depth w below the whole, so for this wave the height adds v and subtracts w. The reason I put the height in the equation will take us back to the marketing. Back to a vertical issue relative to where we enter the vertical market associated with the hole.

Verticals are organized by the industrial classification tree. Every vertical is a subtree of the classification tree. Don’t enter the vertical at the top of the subtree, nor at the bottom of the subtree. Try to leave yourself some room to generalize towards management at the root of the subtree or to specialize towards the detailed work in the leaves of the subtree. The most difficult work would be to implement and sell to siblings. There will be enough to do for the early adopter client and their company.

The height of the hole, w, would match with the vertical height of the client’s business in the industrial classification tree.

We will look at the vertical in the technology adoption lifecycle (TALC). The vertical is just one of several normals that are summed into the TALC. They have not been drawn to scale. Keep in mind that the device/laggard and cloud/phobic markets are small and short in terms of time.

The hole is shown in the top layer of the figure showing individual normals that get summed into the TALC shown in the bottom layer of the figure. The hole is on the far right. The normal for the vertical would replace the Meromorphic function we used in the previous figures. The hole is associated with a single lane in the bowling alley.

There would be six lanes for a given discontinuous innovation. They would be entered into successively until the company could afford and is staffed to do more projects at once. One early adopter engagement, particularly the early ones for a given technology, would take two or more years. That these engagements are stretched out over time, satisfies the requirements of the Mittag-Lefler theorem that insists on the holes being clearly separated.

Now, we’ll fill the bowling alley.

I’ve used a fragment of the industrial classification tree to find a B2B early adopter in the middle of their vertical. Then, I measured the depth of the early adopter business in that tree to the total depth in the classification tree. Then, I put the hole for their position on the normal of the aggregate TALC. All six verticals were measured in the same way and placed on the aggregate TALC. Then, I used the polar form to build the hole accessing functions. There are six verticals, one for each early adopter. We need six different verticals, rather than six engagements in one vertical. I then set up a rough schedule for getting those six applications of the underlying technology done.

Once all six verticals have been built, we ensure that the early adopter’s value expectations are met. Then, we help them write their business case. We will use that business case when we market and sell to the early adopter’s network through the first three degrees of separation.

Once we have built successful applications in those six verticals, we can sell the underlying technology more directly into the IT horizontal. It takes quite a while. It is not the flash in the pan miracles we see in the consumer phase. Time is money, earned money.

Enjoy.