C505 — The529 family is first a duration grammar

Question

What does two-stage E expansion guarantee without any pivot choice?

Inputs

{
  "E": "25/23",
  "small_seed": 1058,
  "macro_seed": 10580
}

Sources

[
  "C480 checkpoint",
  "File52c §§3.8–3.9,6",
  "C481"
]

Opened utc

2026-09-27T22:45:10.631228+00:00

Results

{
  "small": [
    1058,
    1150,
    1250
  ],
  "macro": [
    10580,
    11500,
    12500
  ],
  "general_integer_width_rule": "k*529 -> k*575 -> k*625",
  "proof": "D_k,a(y)-D_k,a(x)=k*(y-x); second pivot cancels from the width too"
}

Finding

Every supplied529-multiple width follows this ladder under two E expansions. Tenfold scaling commutes with it. Width agreement therefore connects C480 and the rounded macro family while leaving their source roles and pivots distinct.

Reassessment

Derive the additional congruence condition needed for integral endpoints.

Checks

{
  "small": true,
  "large": true,
  "scale": true
}

Closed utc

2026-09-27T22:45:10.631494+00:00

Record provenance

Exact record hash: aaca498f1807d3744a9f95aa62a07a6b8de9dd53b097979c73df677fb1b93bc7

Source journal: Research_Cycles/C0482_C0531/journal.json