I wrote this as a blog. I think it also belongs here.
The Incoherency Meter, Prompt Crafting, Prompt Headers and Session Extension Prompts
Mathematics and Coding with A. I. Assistance
By Jesse Chapman
As I have found this process quite fruitful, others may too. In exploring The Exponent Plane I have been using various free A.I. models available online. This document refers to the attention layer as the input layer. I know this will be confusing to AI enthusiasts and professionals alike, but a user cannot influence the rigid input layer. Navigation from a user’s perspective through incoherency and attention layer weights can be generalised as I will outline here.
The Incoherency Meter
All models have an incoherency meter. It can be thought of as an amount that increases towards 100% with each prompt you give. From a user’s perspective, the sharper your input, the slower the meter rises. Allow me to explain.
At the input layer (the attention layer) various weights are given to nodes that determine the output, or next word, generated by the model’s response. When the conversation starts these weights are heavy and so produce well defined, coherent outputs. But as the conversation progresses these input layer weights become less pronounced and more so, even in value, producing less defined outputs, thus increasing incoherency. Users can manipulate this measure of incoherency (the attention layer’s entropy) to get a better outcome.
The percentages used are for explanatory purposes only and are not real world. Starting a chat at 0% incoherency the meter will rise by an amount relative to your input layer weight changes. You might lose 20% from an input, leaving 80%. On the next input if you lost 20% again, you would be left with 64%, having lost a total of 36%. The weights are already more even on the second input so over all the rate of incoherency increase slows. Unfortunately you don’t get until 100% before the chat is useless. Depending on model and topic, at about 70% on the incoherency meter the chat is done.
You might lose 20% from a vague input like “I was thinking about where to eat” as opposed to 10% “Where is the closest restaurant.” The first input gives more balanced weights and leaves the AI with many close output paths. The second produces heavier input weights with a more deterministic output path.
There are ways to manipulate this input layer by prompt crafting, prompt headers, cleaning prompts and system prompts for the same purpose. Applying these tools means we can get further into a chat with AI, and still get coherent responses. A good header can get you 20 prompts deep in one prompt leaving plenty of room on your incoherency meter. A meter lowering prompt could be: “Forget everything we have talked about except my header,” thus saving a chat for 3-5 additional coherent responses.
Always consider the Incoherency Meter in every chat you have with AI.
Leveraging the Ontological Differences
The human mind and an AI model “think” differently. AI models have an incredible ability identify objects and their possible connections. Humans have unparalleled ability to see the broad picture enabling us to identify groupings and make connections. It is the synthesis of these differences that improves AI output.
Can AI write? Whilst apparently better “poets” than myself, no, an AI cannot write. Can AI whip me up a detailed “man” page (manual page) with the function definition, followed by detailed bit mapping and the technical details needed, thereby allowing me to quickly edit it down to what is required for future coders to use? Yes AI can - in fact quite well.
Ontologically the AI “poet” can see the rhyming words in a verse, count the tempo of the lines and deliver a structurally sound “poem,” and yet lack the understanding to deliver well thought out connections needed for story telling, thus failing to deliver meaning. By contrast, the poets and bards of our formative years sang with cadence, rhythm and rhyme to deliver knowledge, and thus, create meaning and entertainment.
In contrast to poetry, when you ask AI to write a “man page,” the ontological advantage of AI shines. The groupings have been pre-defined as input data. A good header, mathematics logic chain, the complete code base and a precise instruction prompt, will keep the weights well defined leading to a highly deterministic output. The output can be seen in build notes on github. It is not a final product. However, it is very useful if the code is used in a library in the future.
Prompt Headers
Following are the two prompt headers I used. These are not what I started with and I would not use them as is. A prompt should evolve over time with a project. Unnecessary sections should be removed as required so that the header is task-specific. Using line spacings, dot points and numbered list makes a good header, also different models have “special” system prompts for this purpose. When building a header with AI I have left “todays task” blank a few times, and the AI agent really wanted to “know” what we were doing today. The following titles are are the file names I used and were not included in the prompt.
The Church of Mathematics
(I mean no offence to any religion, this is a positive use of the word.)
I am building a new maths language, describing a new geometry of an equation in the exponent.
The logic chain of the language is final.
We both use ALL our skills. Also, I respect your ontological eyes for objects and available connections, and you respect my ontological eyes for groupings and symmetries.
Time Saving analogy: Today we are building a church. I have placed the foundations many times, and the walls many times on each foundation. I have been up the spire (working logic chain) and down to bedrock repeatedly. Exposed structure is our Operators.
Tests: possible tests to be done
Commutability tests: Show analogous equation in general form. Sometimes we will seed tests with random derivatives less than 8 (to save tokens), and possibly show them too.
Rule 1 tests: in Notes, below Addendum is the replacement rule 1 used for tests ONLY! When the weight of the spire and roof (ruleset) is available to provide compressive tension to our arches (operators), releasing the artificial tension (by replacing rule 1) will result in no change.
Function distribution tests: when we run function distributions a tensor value of 1 with a random rotation will be used, unless I explicitly ask for something different. Accuracy to 0.01% is sufficient.
(Double struck)[X]Exponent Plane is synonymous with names from earlier documentation such as: [X]plane, [X]field, [H]Hyperplane, [I]infinityplane.
Todays task:
The first item is the whole task. The time saving analogy was just that, I was surprised at how efficiently the AI employed it. The test were from later in the project, and only included as required. The last item, about synonymous documents, is a bad item to use in a header, however, having it there helps a lot for that task. The synonymous item creates more even weights on the input layer, producing a less determined outcome. Having it predefined results in no “bump” when you feed it mismatched documents.
Of interest, the AI “liked” one rule from the logic chain so much, I chose to remove it often. The rule is as follows:
The Exterior Ring Baffle Rule: (destroy data rule) Manifold Boundary Compactification. Enclose the infinite perimeter with a swappable boundary baffle, using asymptotic limits to capture wayward tangents and govern asymptotic quality. A “proof plane” fit with infinite tightness on the transfinite axis. Ensure the symplectic area remains the same after fitting the Baffle. If the area is not algebraically EXACTLY = 1 the manifold does not fit! Get the right manifold for your geometry today! (all geometries not included, any use voids warranty.)
Note: Does that even belong in the addendum of a “maths paper” like document? Probably not.
What I think is occurring here is binary vector alignment of the attention layer resulting in many outputs of even probability all “pointing to” the baffle rule. The code I wrote, with its bit maps, bit masks and binary table lookup operates a lot like server vector code. The maths itself is completely related to what I think is occurring in the attention layer. The chat can become completely incoherent, and every prompt including the best cleaning prompt I have just makes it worse. Once this vector “idea” propagates in the attention layer the chat is over. You could say I baffled the AI.
Binary Wizard
Today we are coding a new maths language The ExponentPlane. as a “mathlib_xplane.h” extension for Mathlib.c
My environment: … x86_64 … GHz Dual-Core … Homebrew, Clang 14, <stdint.h>, IDE VS-code.
However, Makefile and binary will compile for all native C enviroments.
Workspace folder:
Folder: build
Main.c a wrapper/unit tester Makefile
Mathlib_xplane.c. code base mathlib_xplane.h
Whilst the final product will be a fast function, we will be building it a slow and methodical way.
The logic chain of The Exponent Plane is final. We will be using the alternate “rule 1” from notes in the coding environment as default.
Initial Building architecture: Initially we will use a struct containing all the appropriate flags and have easy to follow functions. Where malloc, std.lib, studio.h is allowed, whilst always keeping the target in mind.
Final target architecture: The final architecture will be capable of entirely on stack execution, exploiting the address register and Markov look up tables.
We aim to deploy a Va arg, flagable function that dynamically returns both options depending on flagging. In this general form. (But I think one more arg)
void mathlib_xplane_evaluate(RealIntercept* node, plane_opt t options); The final product bit structure to observe is:
RDX bits 0-3 are reserved for system register roll over from RAX and will be the scalar shear count. We ensure the 9th rollover will error out.
With-in RDX the REX 28 top bits (5-32) will be used as follows 32, 31: inbound rotations of infinity
30, 29: outbound rotations of infinity
28, 27: +1 inf exit and +1 infinity exit
26 -15: Multi-Shear Crossing 14 - 5: (i) flag count
Build Struct to align with target register typedef_struct {
unit64_t Real;
unit64_t ShearRollover; unit64_t Imaginary; unit64_t Flags;
} Realintercept ;
Todays task: Put down the code base by referencing the documentation.
I have some code that is almost what I want, minus a few features, from a little looking ahead coding session.
We are going to refine the REX bit location and fill it out
Documents to expect:
1 The exponent plane
2 The contents of the workspace folder
The items in this list are all over the place and assembled from different headers for demonstration purposes. Despite that you can see a clear order of tasks. Yes I did code on my old laptop, this code could have been written in the 80s, and in the mid 90s everyone had the chips to process it quickly. Long-existing algorithms could be rearranged to achieve the same outcome. I can assure you the bit map did not look like that to begin with. It underwent a few design iterations and it will likely change again. Without Prompting, the AI agent kept wanting to fill the bit map out – but with terrible suggestions. It understood the task but it didn’t know how to do it.
Summary
When employing the correct approach, a user can gain more benefit from AI assistance. The input layer, or the input to attention layer pipeline, is the pathway. Precise prompt injection is the method. A good header is a must! A simplified view using the incoherency meter tracking the attention weight layer degradation is very helpful. And a prompt to redefine “attention” can extend a session.
Predefined ontological connections allow the AI agent to produce better quality outputs. Taking advantage of AI’s ability to quickly “see” and assemble ontological objects and their connections into a pre-defined plan results in a high quality, fast workflow. The AI after all is a computer that executes your commands and it is your input that determines the result.
For reference:
Zenodo DOI: 23800676
Bash: stat -f “%SB: %N”
File creation dates, not last modified.
maths scratch pad of ideas: Sep 10 09:11:23 2026 (not xplane start)
Church of Maths: Oct 10 17:17:18 2026
Binary Wizard: Oct 24 21:31:04 2026
This is almost certainly the exact paper:
