Do the rare doubles follow a repeating counter?

Date: 20 September 2026.

Result: no convincing shared periodic concentration within the tested range. The strongest single peak is a page-based period of 30, but its full-scan adjusted comparison is 0.2781. That does not justify selecting a 30-rune key or treating the number as a clue. No new plaintext or cipher was identified.

Session 29 quantified the shortage of repeated neighbors. This session asks a different question: given the 86 repeats that do occur, are their positions concentrated at a particular phase of a repeating counter?

Fixed scope

Before executing the scan, the script fixes:

• Periods 3 through 32, inclusive. The earlier parity check already covered period two.
• Two counters: reset at each marked section, or reset at each original page.
• Every possible phase within each period, producing 1,050 comparisons in total.
• A one-sided test for an excess of events at a phase, with a maximum-statistic correction across the complete family.
• 9,999 simulations, seed 3301030. No extension of the range or extra counting conventions after inspecting the results.

The data are the same eight substantial sections and their 12,939 eligible edges. All 86 repeat events, their section offsets and their original page offsets were checked against the event inventory from Session 27. No fresh full-image transcription was performed.

Position is the zero-based index of the left rune of each pair. A page counter retains the actual page offset when a section begins partway through a page. Non-rune marks do not advance either counter. Cross-section edges are excluded; cross-page edges inside a section remain eligible. As before, rune-stream adjacency can span non-rune material rather than physically neighboring glyphs.

A phase of one at period 30 therefore means left-rune offsets 1, 31, 61 and so on within the page. It is not a search restricted to the first 30 runes.

Conditional comparison

Within each section, uniformly choose the observed number of repeat-event positions without replacement. This preserves all eight section counts: 4, 6, 9, 10, 11, 13, 12 and 21.

Each simulated event configuration is reused across all periods, phases and both counter rules. This retains the dependence between overlapping tests, including related periods. For every phase, the expected event count and variance account for its actual number of eligible edges in each section using the conditional hypergeometric formulas.

The score is the observed excess divided by its conditional standard deviation. The largest score across all 1,050 comparisons is the full-scan statistic. Sparse phase counts are calibrated by simulation, not by treating those scores as normally distributed.

This comparison relocates event labels, not the actual runes. It assumes that event locations are exchangeable within each section. It does not preserve rune-specific double counts, local event spacing or unknown encryption dependencies.

Strongest observed peaks

Rows are ranked by standardized excess, not by the single-test tail probability; those rankings can differ because the discrete distributions differ.

The winning score is 4.1162, below the simulated 95th percentile of the full-scan maximum, 4.8502. Of 9,999 simulated complete scans, 2,780 produce a peak at least as large as the observed winner. The plus-one comparison is therefore 0.2781, roughly 28%.

The raw 0.0006 for the winning row answers the question of a single specified phase. It is misleading if reported as evidence after selecting that phase from 1,050 possibilities. The corrected result supplies no persuasive reason to choose period 30, 31 or another tested period. It is not a probability that a particular cipher is true or false.

Constructed controls

Two artificial event sets keep every section's repeat count but place all 86 events into one chosen phase:

Both deliberately strong patterns are detected. These are event-label controls, not simulated encrypted books. They show that the implementation can detect conspicuous periodic concentration, not that it reliably detects weak or changing cycles.

Interpretation and limits

The broad repeat shortage remains intact. This pass adds no convincing shared timing pattern to explain its exceptions. It does not rule out periodic ciphers: an actual periodic key need not make doubles cluster at one phase, and independent phase offsets between sections could obscure a pooled pattern.

Untested here are periods outside 3–32, independent section-specific phase choices, word or line counters, non-rune counter advances, selective skipped symbols, combinations of phases and changing periods. These are limitations, not a recommendation to expand the search arbitrarily.

The correction covers this declared family only, not every hypothesis considered across the entire investigation. All events were already present in earlier sessions, so this is retrospective analysis rather than confirmation on untouched data. Non-detection is especially limited by the small total of 86 events and the exchangeability assumption.

Decision: retain the measured anti-repeat tendency as a structural constraint. Do not promote the 30-rune peak into a key, literary reference or numerical interpretation. No new key search was justified by this pass.

Reproduce and inspect

• Analysis code
• All 1,050 comparisons, top candidates, event inventory and controls

Run with Python and NumPy beside lp_core.py, the retained master transcription and double_context_results.json. The code makes no network calls and records the transcription hash. It verifies the 86-event inventory and original page offsets before the scan.