Paper IX · Euler Congruences
Explore the 23 visuals
Read the complete guide to congruences and sharp dyadic geometry, followed by eight worked arithmetic and lifting examples. All 20 original figures have descriptions and selectable labels; recorded data are available in accessible tables.
This guide explains the indexing, valuation gains, exact periods, and remaining boundaries through exact arithmetic and theorem-derived figures. The main paper supplies the proofs; the Lean companion supplies the formal source and evidence map.
Congruences and dyadic geometry
- The object: rows, degrees, and normalization · PDF section 1
- What the original conjecture asked · PDF section 2
- Removing the starting-row threshold · PDF section 3
- Each step contributes its own precision · PDF section 4
- The final rank-four congruence · PDF section 5
- Why address two has a different slope · PDF section 6
- When mixed steps attain the baseline · PDF section 7
- First differences determine an exact metric · PDF section 8
- Least periods, rather than sufficient periods · PDF section 9
- A full period visits every allowed residue · PDF section 10
- Seeing the scaled isometry · PDF section 11
- How far the simple metric law extends · PDF section 12
- From factorial gain to analytic radius · PDF section 13
- Which evidence does which job · PDF section 14
- Reading the conclusion at the right strength · PDF section 15
Worked arithmetic and lifting
- Compute the reciprocal before estimating it · PDF section 16
- The normalization loss is a real boundary · PDF section 17
- Odd primes require the sign twist · PDF section 18
- Newton coordinates turn precision into a finite computation · PDF section 19
- A compatible root is built one digit at a time · PDF section 20
- A singular residual can stop a lift · PDF section 21
- Separate corrections need not be one shared correction · PDF section 22
- A derivative weight can cost one digit of root precision · PDF section 23