{
  "step": "C372",
  "primitive_integer_family": {
    "scale": "m positive integer",
    "inner_input": "69m",
    "inner_output": "70m",
    "pivot_separation": "230m",
    "outer_input": "299m",
    "outer_output": "300m",
    "whole_side_if_source_first_segment_equals_pivot_separation": "529m"
  },
  "proof": [
    "Assume integer input and output gaps, positive expansion, and equal gains under the two specified ratios.",
    "For J, x maps to x+x/299. Integer x and integer output imply m=x/299 is a positive integer; hence x=299m and output300m.",
    "For P, z maps to z+z/69. Equal gain m gives z=69m and output70m.",
    "The pivot separation is x-z=230m. Conversely these formulas satisfy both conversions for every positive integer m.",
    "When a separate source first segment equals that pivot separation, the whole side is (230+299)m=529m.",
    "The actual 460-year source separation forces m=2. This is a conditional algebraic explanation of the existing1058=2K side, not a source command to pair the ratios."
  ],
  "integer_input_only_family": {
    "input_outer": "13t",
    "input_inner": "3t",
    "pivot_separation": "10t",
    "common_gain": "t/23",
    "integer_outputs_iff": "t divisible by23",
    "diagnostic_not_source": {
      "D": 10,
      "outer_input": 13,
      "common_outer_output": "300/23"
    }
  },
  "source_scale": 2,
  "source_applications": [
    {
      "field": "cumulative",
      "i": 2,
      "gain": 2,
      "source_pivot_separation": 460,
      "first_source_segment": 460,
      "input_gaps": [
        138,
        598
      ],
      "output_gaps": [
        140,
        600
      ],
      "whole_side": 1058,
      "primitive_scale": 2,
      "repaired_partition": [
        458,
        600
      ]
    },
    {
      "field": "cumulative",
      "i": 3,
      "gain": 2,
      "source_pivot_separation": 460,
      "first_source_segment": 460,
      "input_gaps": [
        138,
        598
      ],
      "output_gaps": [
        140,
        600
      ],
      "whole_side": 1058,
      "primitive_scale": 2,
      "repaired_partition": [
        458,
        600
      ]
    },
    {
      "field": "cumulative",
      "i": 4,
      "gain": 2,
      "source_pivot_separation": 460,
      "first_source_segment": 460,
      "input_gaps": [
        138,
        598
      ],
      "output_gaps": [
        140,
        600
      ],
      "whole_side": 1058,
      "primitive_scale": 2,
      "repaired_partition": [
        458,
        600
      ]
    },
    {
      "field": "cumulative",
      "i": 5,
      "gain": 2,
      "source_pivot_separation": 460,
      "first_source_segment": 460,
      "input_gaps": [
        138,
        598
      ],
      "output_gaps": [
        140,
        600
      ],
      "whole_side": 1058,
      "primitive_scale": 2,
      "repaired_partition": [
        458,
        600
      ]
    },
    {
      "field": "regular",
      "i": 3,
      "gain": 2,
      "source_pivot_separation": 460,
      "first_source_segment": 460,
      "input_gaps": [
        138,
        598
      ],
      "output_gaps": [
        140,
        600
      ],
      "whole_side": 1058,
      "primitive_scale": 2,
      "repaired_partition": [
        458,
        600
      ]
    },
    {
      "field": "regular",
      "i": 5,
      "gain": 2,
      "source_pivot_separation": 460,
      "first_source_segment": 460,
      "input_gaps": [
        138,
        598
      ],
      "output_gaps": [
        140,
        600
      ],
      "whole_side": 1058,
      "primitive_scale": 2,
      "repaired_partition": [
        458,
        600
      ]
    }
  ],
  "source_counts": {
    "complete_cumulative_rows": 4,
    "primary_regular_rows": 2,
    "companion_controls_rejected": 2
  },
  "identity": "2*299-69=529; at sourced scale2:460+598=1058, repaired458+600=1058, and138→140.",
  "interpretation": "A primitive arithmetic consequence of imposing equal P/J gains. The inherited ratios and source geometry supply it; no new historical witness, textual pairing instruction, date anchor or operator is established. Primitive230 is a scalar normal form, not an adopted chronology endpoint or a reopened primer."
}