JATZEK PERIODICAL #3
π-Seeded BASE Repeat-Pattern Results

Research report | Archived one-million-character streams | November 11, 2026

Philip John Jatzek (project supervisor) and STORM (programming and analysis collaborator)

(c) November 11, 2026, Philip John Jatzek. STORM acknowledged as AI programming and analysis collaborator. All rights reserved.

Executive summary

The 36-BASE exploratory reanalysis is complete. Each BASE was evaluated using its archived 1,000,000-character stream from the November 8, 2026 loop-search package; all archived SHA-256 stream digests matched. This report measures repeated digit windows and repeated CLIMB/FALL event-label sequences, retaining the earlier coverage and MATCH scores for comparison.

The primary coverage BASE 28.406505541 produced 176 additional occurrences of exact 12-character blocks, compared with 614 for the separate MATCH-winning BASE 19.264. Three of the 36 archived streams contained a duplicated 16-character window; none contained a duplicated 32-character window at the tested length. These finite patterns do not establish an infinite loop or recurrence of an underlying numerical state.

1. Scope and definitions

These are not ordinary digits of π. Beginning with π minus 3, the numerical recurrence repeatedly multiplies the fractional state by an exact rational BASE, records the integer part, and carries the fractional remainder forward. The emitted integers are written in decimal notation and concatenated; a digit window may cross emitted-integer boundaries. Coordinates in this report are one-based and inclusive.

For exact digit-window lengths 4, 8, 12, 16 and 32, a repeated unit is an identical substring beginning at two or more positions, including overlapping positions. Additional occurrences sums counts beyond the first occurrence of each repeated unit. Repeated CLIMB/FALL event motifs are ordered sequences of event labels—not repeated digit substrings.

This reanalysis used all 36 archived streams rather than generating new independent numerical streams. Historical coverage is the union of positions in which at least one of fifteen four-digit CLIMB/FALL pairs has equal, nonzero cumulative counts. MATCH is the summed active-pair count at 500-character checkpoints.

2. Comparison of the principal BASE values

3. Notable repeated blocks

For 28.406505541, 018221913191 appears at positions 183,635 and 775,889. Its most frequent eight-character unit, 12121121, occurs ten times; 1111 occurs 2,124 times.

For 19.264, 116111714121 appears at 182,720, 526,439 and 897,459. Its most frequent eight-character unit, 12111111, occurs 30 times; 1111 occurs 5,028 times.

Precisely three of the 36 selected streams had any duplicated 16-character substring:

Across 36 selected BASE streams, the median count of additional 12-character occurrences was 180.5 (range 149–614); the maximum was attained by BASE 19.264. No exact 32-character window was duplicated at the tested length.

4. CLIMB/FALL event-pattern repeats

The scanner detects fifteen four-distinct-digit ascending combinations from digits 0 through 5, together with each reverse FALL pattern. An M event indicates a transition into a nonzero balanced state for one pattern pair. Multiple event slots can refer to the same printed-character coordinate.

For the center BASE, the most frequent length-two motif is `C:0124`, `C:0124` (18 occurrences). Its leading length-three motif `C:0124`, `C:1245`, `F:0123` occurs four times. One five-event sequence appears twice, while no six-event motif repeats. For BASE 19.264, there are 666 additional length-three motif occurrences and 37 additional length-four motif occurrences. These counts differ partly because the event streams differ in size and composition.

For 28.406505541 the four historical GAP intervals remain 1–3,240; 3,777–6,937; 9,943–11,753; and 13,244–14,100. All positions 14,101–1,000,000 are covered under the cumulative balanced-pair rule; this does not mean a repeated digit loop persists throughout that interval.

5. Verification and limitations

• Every newly rescored stream SHA-256 matched the corresponding archived SHA-256 in results.json.

• The center results for five digit-window lengths and five event-motif lengths, along with coverage, MATCH, and GAP endpoints, exactly matched the prior independent November 8 relationship analysis.

• BASE 19.264 coverage, MATCH, and repetition metrics agreed with a separately executed full repeat_relationships.py pass.

• This is a deterministic reanalysis of archived streams, not independent regeneration of the π-seeded recurrence.

• The 36 BASE values were selected in advance. Related BASE values, shared prefixes, skewed printed-digit frequencies, overlaps, and multiple comparisons can affect counts. No p-value, predictive claim, mathematical periodicity or statistical significance is established.

6. Next experiment for Periodical #3 continuation

Freeze a primary statistic (recommended: additional occurrences of exact 12-character windows) and a specified BASE comparison before testing a longer continuation. Preserve CLIMB/FALL cumulative counts across the one-million-character boundary. Compare future-data results with independently generated, prespecified controls and distinguish: (a) finite duplicated substrings, (b) a repeated numerical recurrence state, and (c) eventual mathematical periodicity.

The earlier November 8 consecutive-loop scan used unit periods 1–256 with at least three complete consecutive copies; its longest multivalue printed-character span was 14 characters. The current nonconsecutive-window study answers a separate question and should not be confused with that earlier consecutive-loop result.

Appendix A. All 36 BASE values: comparable repeat summary

Coverage uses the historic balanced-pair rule. 12-char and event-3 columns show additional occurrences; 16-char indicates whether any exact repeated 16-character digit unit was present. The underlying CSV/JSON archive retains all five window lengths, hashes, GAP coordinates and event summaries.

Appendix B. Reproducibility record

Source archive: STORM_Loop_Search_2026-11-08.zip, maintained separately. Reanalysis outputs: Pi_Repeat_36_BASE_Summary.csv; Pi_Repeat_36_BASE_Results.json; and repeat_batch.py. The executable can be rerun with the archived source extracted into repeat_test/ beside the script.

Research status: 36-BASE archived-stream reanalysis completed November 11, 2026. New-stream replication, controls, and held-out continuation remain unperformed.