C506 — Integral endpoints require pivot compatibility

Question

What extra condition distinguishes an integral width ladder from integral dates?

Inputs

{
  "map": "D_E,a(x)=(25x−2a)/23",
  "integer_source_and_pivot": true
}

Sources

[
  "Strategy §5C",
  "C505; exact algebra"
]

Opened utc

2026-09-27T22:45:39.171525+00:00

Results

{
  "one_stage_iff": "x congruent a mod23",
  "two_stage_condition": "first outputs integral; second pivot congruent to every first output mod23",
  "pair_condition": "if original width belongs to529Z, integral first outputs share one residue mod23",
  "same_pivot_two_stage_iff": "x−a divisible by529",
  "proof": "25x−2a ≡2(x−a) mod23 and2 is invertible mod23; for a repeated pivot D_E,a²(x)=a+(625/529)(x−a), gcd(625,529)=1"
}

Finding

Widths lose the pivot term; endpoints retain it. A529-multiple width guarantees a shared second-stage residue after an integral first stage, but it does not supply a licensed pivot of that residue.

Reassessment

Apply this criterion to the already supplied C480 pivots and the fixed12026 trial, without choosing new pivots.

Checks

{
  "coprime": true,
  "common_residue_width": true
}

Closed utc

2026-09-27T22:45:39.171772+00:00

Record provenance

Exact record hash: 260e6718bf7692c169bc4c7e10531a4f8684d9ad927a762ce3e5610c50872518

Source journal: Research_Cycles/C0482_C0531/journal.json