The Same Equation, Different Worlds
Meditations on Oneness · Local folio mark 11111111
One inscription, three folds
Meditation 8 preserves one proof copied into three internally coherent folds of the Manyfold. A margin claim says that always has only the two solutions and .
“The inscription survived. Which inference did not?” — Vivian Meridian
The disputed proof
The final arrow uses the zero-product rule:
Fold I · Ordinary numbers
Use separate paper for the proof and check.
-
The zero-product rule is true for integers and for complex numbers. Use the Archive argument to find all integer solutions and all complex-number solutions of .
Check that every solution you list works, and explain why there can be no others.
Fold II · Arithmetic on fifteen positions
The 15-position dial
On this dial, calculations are made modulo 15. Here is the same position as 14.
Meditation 5 records
at exactly .
Audit the proof at
Record the audit on this page.
-
Use to audit the Archive proof.
Check Your calculation or conclusion Remainder of modulo 15 modulo 15 Product of the two factors Which arrow fails? Show that the factor product is 0 modulo 15 although neither factor is 0. Identify the precise arrow in the Archive proof that fails.
Fold III · Matrix arithmetic
The matrix world
For matrices, use
The matrix versions of 1 and 0 are
For every real number , define
An infinite family
Use separate paper for the matrix products and proof.
-
Prove all of the following.
- for every real .
- and .
- Different values of give different matrices, so has infinitely many matrix solutions.
- Both and are nonzero, but .
Finish with one sentence: name the step that stayed valid in all three folds, and the inference that did not.