🟩TOKEN ORDINALS

ORDINAL THEORY APPLIED TO META-PROTOCOL TOKEN

ORDINAL THEORY

ORDINAL THEORY ASSIGNS A UNIQUE, SEQUENTIAL NUMBER TO EACH SATOSHI, THE SMALLEST ATOMIC UNIT OF BITCOIN. THESE NUMBERS REFLECT THE CHRONOLOGICAL ORDER OF SATOSHI CREATION, STEMMING FROM MINING REWARDS AND TRANSACTIONS AS DICTATED BY BITCOIN'S PROTOCOL.

MATHEMATICAL FOUNDATION

THE FUNDAMENTAL FORMULA USED IN ORDINAL THEORY TO NUMBER SATOSHIS IS BASED ON THEIR SEQUENTIAL APPEARANCE IN EACH MINED BLOCK AND TRANSACTION.

$Sn=n(n+1)2Sn​=2n(n+1)​$ WHERE $nn$ REPRESENTS THE TOTAL NUMBER OF BLOCKS MINED UP TO A CERTAIN POINT, AND ​ CALCULATES THE CUMULATIVE SUM OF ALL SATOSHIS GENERATED TO THAT BLOCK.

ADAPTATION TO META-PROTOCOL TOKENS

THE TOKEN ORDINALS MODEL APPLIES THE PRINCIPLES OF ORDINAL THEORY TO META-PROTOCOL/META-LAYER TOKENS. HERE WE EXPLORE THIS USING NON-ARBITRARY TOKENS (NATS) AND UNIQUE NON-ARBITRARY TOKENS (UNAT) DERIVED FROM THE DIGITAL MATTER THEORY (DMT) AND FRAMEWORK. THIS CAN BE APPLIED BROADLY FOR ANY NAT OR ARBITRARY TOKEN (AT) AS WELL BUT SINCE GROK42 TOKEN SUPPLY IS BASED ON THE HEIGHT NUMBER OF EACH BITCOIN BLOCK, RATHER THAN THE NUMBER OF BITCOIN SATOSHIS GENERATED PER BLOCK, THIS DOCUMENT FOCUSES ON THIS TOKEN CONFIGURATION.

EACH BLOCK $ii$ RESULTS IN THE CREATION OF $ii$ TOKENS, ALIGNING WITH THE BLOCK'S HEIGHT NUMBER. THE SERIAL NUMBERING OF THESE TOKENS FOLLOWS THE ORDINAL METHOD, PROVIDING A CLEAR AND TRACEABLE SEQUENCE OF TOKEN ISSUANCE.

FORMULA FOR GROK42 TOKEN SERIAL NUMBERING

THE SERIAL NUMBERS FOR TOKENS IN EACH BLOCK ARE DETERMINED USING THE SAME CUMULATIVE SUM FORMULA:

$Ti=i(i+1)2Ti​=2i(i+1)​$ WHERE $ii$ IS THE BLOCK HEIGHT.

THIS FORMULA GIVES THE TOTAL NUMBER OF TOKENS GENERATED UP TO AND INCLUDING BLOCK $ii$, ALLOWING FOR SEQUENTIAL NUMBERING OF TOKENS FROM ONE BLOCK TO THE NEXT.

FOR BLOCK $ii$, TOKENS ARE NUMBERED FROM $Ti−1+1Ti−1​+1$ to $TiTi​$, where $Ti−1Ti−1​$ IS THE TOTAL FROM THE PREVIOUS BLOCK.

EXAMPLE WITH BLOCK 42

TO ILLUSTRATE THE APPLICATION OF OUR TOKEN CREATION MODEL TO A SPECIFIC BLOCK, CONSIDER BLOCK 42. ACCORDING TO OUR MODEL, BLOCK 42 WOULD GENERATE EXACTLY 42 TOKENS. USING OUR ADAPTED ORDINAL THEORY FORMULA, WE CALCULATE THE TOTAL NUMBER OF TOKENS GENERATED UP TO AND INCLUDING BLOCK 41 (THE PREVIOUS BLOCK) AS FOLLOWS:

$T41=41×(42)2=861T41​=241×(42)​=861$

THIS SUM REPRESENTS THE TOTAL NUMBER OF TOKENS FROM ALL PREVIOUS BLOCKS. THUS, FOR BLOCK 42, THE TOKENS GENERATED WOULD BE SEQUENTIALLY NUMBERED STARTING FROM 862 (SINCE THE LAST TOKEN FROM BLOCK 41 WAS 861). THE LAST TOKEN FROM BLOCK 42 WOULD THEN BE NUMBERED: $861+42=903861+42=903$

THEREFORE, TOKENS GENERATED FROM BLOCK 42 WOULD BE ASSIGNED SERIAL NUMBERS FROM 862 TO 903.

THIS EXAMPLE DEMONSTRATES HOW THE PRINCIPLES OF ORDINAL THEORY CAN BE SEAMLESSLY INTEGRATED INTO OUR TOKEN DISTRIBUTION MODEL, ENSURING A SYSTEMATIC AND TRACEABLE METHOD OF TOKEN ISSUANCE BASED ON BLOCK HEIGHTS.

CONCLUSION

BY APPLYING THE ORDINAL THEORY TO TOKEN CREATION BASED ON BLOCK HEIGHTS, WE UTILIZE A PROVEN MATHEMATICAL PRINCIPLE TO INNOVATE WITHIN THE BLOCKCHAIN SPACE. THIS METHOD ENSURES A TRANSPARENT, SEQUENTIAL DISTRIBUTION OF TOKENS, PARALLELING THE UNIQUENESS AND TRACEABILITY OF SATOSHIS IN THE NATIVE BITCOIN MODEL BUT FOCUSING ON TOKEN ISSUANCE AS A FUNCTION OF BLOCK HEIGHTS.

IN THE PRIMER INTRO PAGE, THE USE CASE IS OUTLINED IN REGARDS TO THE GROK42:SPLIT PROTOCOL OPERATION TO ASSIGN EXISTING ART INSCRIPTIONS TO EACH INDIVIDUAL TOKEN BY SERIAL NUMBER. LIKEWISE, A GROK42:SPLIT CAN BE INCLUDED AS PART OF NEWLY CREATED INSCRIPTIONS AS WELL. THERE ARE OTHER USE CASES THAT CAN BE EXPLORED THAT GO BEYOND THE SCOPE OF THIS DOCUMENT.