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Cached Magnitude — Simple Instructions
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1. What it is
Cached Magnitude is a way to represent extremely large numbers without writing out all of their digits.
Instead of storing the entire number, we store its location within a hierarchy of increasingly large numerical scales.
Think of it like a map.
> The larger the number, the larger the map scale we use.
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2. The magnitude hierarchy
The scales are:
Magnitude → Stack → Pillar → Tower → Block → Vault → Chunk → Field → Environment → Logos → ...
Each level is vastly larger than the one before it.
The hierarchy can continue indefinitely:
Logos → Undecacache → Duodecacache → Tridecacache → etc.
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3. Magnitude
Magnitude is the basic scale.
Mₙ = 10ⁿ
So:
M1 = 10
M2 = 100
M3 = 1,000
M9 = 1,000,000,000
M10 = 10,000,000,000
And importantly:
M10 is also S1.
The scales can therefore overlap in value while providing a more useful way to organize the enormous-number space.
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4. Higher caches are recursive hyperoperations.
A Stack is exponentially built from Magnitude.
A Pillar is exponentially built from Stack.
A Tower is exponentially built from Pillar.
And so on.
The important idea isn't that the lower levels become invalid.
They don't.
Instead, a higher cache gives us a way to describe increasingly enormous ranges while the lower caches continue to describe the detail inside those ranges.
Each tier uses the same metric progression: reaching the 10th level of the current tier establishes the first level of the next tier. The lower tier does not disappear; it continues to provide progressively finer levels between higher-tier boundaries.
M10 = S1
M100 = S2
M1000 = S3
S10 = P1
P10 = T1
T10 = B1
Thus, a higher cache is not a replacement for the lower cache. It is a larger recursive magnitude context containing the lower cache's progression.
Further unfolding:
S1 = M10 = 10^10
S2 = M100 = (10^10)^10 = 10^100
S3 = M1000 = (10^100)^10 = 10^1000
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5. Think of it like a map
Imagine finding a house.
You don't describe it by listing every millimeter between you and the house.
You say:
> Country → State → City → Street → House
Each step gives you a more precise location.
Cached Magnitude does something similar:
> Huge scale → smaller scale → smaller scale → raw number
The higher cache tells you where you are in the enormous numerical landscape.
The lower cache tells you where you are within that larger region.
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6. The notation
The notation alternates between:
raw number → magnitude cache/level → next raw number → next magnitude cache/level
The pipes identify hierarchical positions; they are not multiplication symbols. A cache indicator provides the magnitude context used to interpret the adjacent numerical value. In its simplest form, N|Mx evaluates as N + Mx.
For example:
|6|S1|8|M11|2|
means:
6 is the raw number at the S1 scale.
S1 identifies that magnitude cache and its level.
8 is the next raw number.
M11 identifies the next, finer magnitude scale.
2 is the remaining raw value.
The pipes are separators; the cache indicators provide the magnitude context used to interpret the adjacent numerical value.
A cache indicator contributes the numerical value of its indicated cache to the expression. In its simplest form, N|Mx means N + Mx.
10|M1 = 10 + 10 = 20
246|M2 = 246 + 100 = 346
900|M1 = 900 + 10 = 910
The order of reading is essentially
(|#|L#)(|#|E#|)(|#|F)(|#|C#)(|#|V)(|#|B)(|#|T#)(|#|P)(|#|S#)(|#|M#)#.(reverse order for decimals)
If a resulting value exceeds the representable range of its current cache context, the representation graduates to the next appropriate cache level while preserving the lower-order remainder.
900|M1 = 910 = 9|M2|10
Which can be written fully and most formally as
9|M2|1|M1|0.0
Less formally as
9|M2|1|M1
while the shorter:
9|M2|10
All denote the same exact 910.
(LEFCVBTPSM#.#MSPTBVCFEL)
Any absence of an integer in the |pipeline| is assumed to be zeros and any undefined cache tier is assumed to be its minimum value.
Absent pipeline integers are implicitly zero. An undefined cache tier assumes its minimum level. If a leading integer is omitted where the syntax permits it, the leading value is presumed to be 1. The rightmost unmarked integer represents the remaining raw value.
SM.MS becomes 10,000,000,010.01000000001
THIS IS JUST DESCRIBING RAW EXISTENCE SHATTERING NUMBERS as if like basic 1-1,000,000.
The only "Math" in presenting them alone, is knowing what magnitude caching order you're looking at.
The math performs, otherwise, like ordinary carrying, but with extra steps.
Cached Magnitude favors the shortest representation that preserves the required numerical information. A fully expanded pipeline may be used when formal structure or individual cache levels need to be made explicit.
There is more than one way to write sums:
“The same value can have multiple valid representations, ranging from compact notation to fully explicit formal pipeline notation.”
346 has 4 ways to be written, including plain numbers. The other 3:
246|M2
3|M2|46
|3|M2|4|M|6.0|
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7. Why the system is useful
Cached Magnitude is not intended to replace ordinary numbers or conventional scientific notation when those forms remain simple and useful.
Ordinary decimal notation is preferred for ordinary values. Scientific notation is preferred when a large value can still be conveniently expressed with a single exponent.
Cached Magnitude becomes useful when scientific notation itself becomes recursively nested or unwieldy—when the exponent or magnitude descriptor requires another magnitude description.
In simple terms:
Scientific notation compresses a large number. Cached Magnitude compresses the nesting of large-number descriptions.
Cached Magnitude can represent enormous numerical magnitudes without requiring the complete expanded decimal integer to be written or stored.
The computer stores something more like:
> “This number is here, at this enormous scale, with this much detail inside it.”
The underlying number can therefore be vastly larger than the computer could ever physically expand into ordinary digits.
10^10 → scientific notation is sufficient.
10^100 → scientific notation is still sufficient.
10^(10^10) → the exponent is itself a large magnitude expression; this is where Cached Magnitude begins to become useful.
10^((10^10)^(10^10)) makes cached magnitude significantly more useful.
Eventually, with enough exponent nests, CM notation becomes almost necessary.
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8. Reverse-tier extrapolation
When a number is represented at a very high cache level, we can move downward through the hierarchy to resolve additional detail when that detail is represented.
For example:
Pillar → Stack → Magnitude → raw number
Each lower level gives additional resolution.
This means the system can work with enormous values without expanding them completely.
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9. Hardware Theories
Cached Magnitude can be implemented entirely in software, but it could also become a hardware-supported numeric format.
A processor could have specialized operations for:
comparing Cached Magnitudes
adding them
subtracting them
multiplying them
normalizing them
moving between cache levels
converting them into human-readable numbers
The hardware doesn't need to become capable of physically storing the enormous number.
It only needs to become capable of manipulating its compact representation.
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10. The simplest possible explanation
If you only have ten seconds to explain the whole thing:
> “Cached Magnitude is a hierarchical number system. Instead of storing every digit of an enormous number, we store its position on progressively larger numerical scales, using smaller scales to describe the detail inside the larger ones.”
And the analogy:
> “It's basically GPS for huge numbers.”
That, I think, is the sentence I'd build everything else around.
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Research and speculation
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Arithmetic Potential Speculation:
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1+9=10 = 1|M|0.0 = M1 = M
10|M1 = 20 = 2|M1 = 2|M
900|M1 = 910 = 9|M2|1|M1|0.0 = 9|M2|M1| = 9|M2|10
Conceptual projection for how it has to work to be useable:
- Locate: any juggernaut number gets pinned to a coordinate (tier + position) instead of sitting there as an unmanageable blob of digits — that's the GPS/addressing move.
- Decompile: once located, you can break the number into partitions — the piece that lives in this tier, the piece that lives in the tier below, etc. — rather than treating it as one monolithic value.
- Partitioned arithmetic: you operate on the manageable partitions separately (tier-level shortcuts, coarse comparisons, tier-preserving operations) instead of trying to do the operation on the full expanded number.
- Reconstitute: fold the partitions back together — respecting the carry/rollover rules between tiers — into a single coordinate (or, in principle, back into a raw expanded number if you ever needed it).
🥱🧐
Additional information will be added as notation reaches completion.
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Magnitude → Stack → Pillar → Tower → Block → Vault → Chunk → Field → Environment → Logos
1-9; 11-99; 111-999; context subject to cache indication.
M1-10
Magnitudes. Number of zeros.
S1-10
M100=S2; S10 = P1 = M10,000,000,000
So now we're using Cached Magnitude index notations. P1 = M(1[e]10)
T1=P10=M(1[e]100)
B1=T10=P(1[e100])=M(1[e]10,000,000)
So now we have some basic syntax principals to address.
Any absence of an integer in the |pipeline| is assumed to be zeros and any undefined cache tier is assumed to be its minimum value.
Absent pipeline integers are implicitly zero. An undefined cache tier assumes its minimum level. If a leading integer is omitted where the syntax permits it, the leading value is presumed to be 1. The rightmost unmarked integer represents the remaining raw value.
So SM.MS would read 10,000,000,010.01000000001
If fully unfolded.
Therefore S2M is effectively 10,000,000,000^10 and I don't think I can copypasta that many zeros in any meaningful capacity.
And every successive Tier Graduation is progressively higher.
For example 9+1= M (or M1, or |1|M1|0.0|
But it takes 99+1 to reach M2.
Similarly, there are M11-99 between S1 and S2 and M111-999 between S2 and S3
Therefore it is necessary to use sub-cache "scientific notation" indicators after a period of tier progressions to even fathom where we are anymore.
So now we're using Cached Magnitude index notations. P1 = M(1[e]10)
T1=P10=M(1[e]100 <I THINK!>)
B1=T10=P([1e100])=THE REASON I'M TRYING TO BUILD THIS SYSTEM 🤣😭🤣🤓
Decimals anywhere but the center pipeline are irrelevant and have no meaningful impact.
🧐
WHAT'S IN YOUR WALLET?
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Continuing recursive operations, I presume M-L is as good as 0-9, we've begun Anew;
I think? Maybe? We're "magnitudal squaring"?
Logos → Logic Gate → Primordia → CAO
L10^L10=LG1
LG1^LG1 I THINK would continue the pattern....
Wherein LG10^LG10 SHOULD represent the next container. Primordia...? If I'm actually continuing the recursive hyperoperation pattern correctly and constraining to metric graduation.... I think?
Or Logoi...?
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