# C498 — Decimal gains explain the path increments

## Question

Which algebraic properties of decimal reversal generate the observed increments?

## Inputs

```json
{
  "three_digit_core": "10*(100a+10b+c)",
  "two_digit_core": "100*(10a+b)",
  "source_segments": [
    1650,
    1050,
    9170,
    3430,
    1400
  ]
}
```

## Sources

```json
[
  "File52c §1.2",
  "C481",
  "C495–C497"
]
```

## Opened utc

2026-09-27T22:43:04.561288+00:00

## Results

```json
{
  "gains": {
    "1650": 3960,
    "1050": 3960,
    "9170": -1980,
    "3430": 0,
    "1400": 2700
  },
  "three_digit_gain": "990*(c-a)",
  "two_digit_gain": "900*(b-a)",
  "general_invariant": "digit sum mod9; with10^k placeholders, gain divisible by9*10^k",
  "not_23_preserving_example": {
    "input": 230,
    "inverse": 320,
    "inverse_mod23": 21
  }
}
```

## Finding

The primary990 increments and the2700 tail increment are decimal-place effects. Reversal does not by itself preserve the23-lattice; its conjunction with the Key families still requires particular source inputs.

## Reassessment

Determine the exact compatible scaling and translation rules for the rounded path module.

## Checks

```json
{
  "gains": true,
  "negative23": true
}
```

## Closed utc

2026-09-27T22:43:04.561498+00:00

## Record provenance

Exact record hash: `80b38f1933d3c37d67d4e2333ccff17c6e0df5268b132a6a2c803e23a6f197f4`

Source journal: `Research_Cycles/C0482_C0531/journal.json`