Simple schedules need too many singleton exceptions

Even with a separate best key in every section, the five schedule families from Session 44 require at least 46 of the 83 singleton tokens to be exempt from the A/I rule. Allowing A, I and O still requires 36 exemptions. Removing the two documented red singleton tokens leaves the minimum at 46 among the remaining 81.

This is a conditional exclusion of those schedules, not evidence that 46 tokens really are markers. There is no new verified plaintext.

What changed in this test

Session 44 required every training singleton to decrypt to A or I. Here we ask a different, retrospective question: how many exceptions would be necessary even after choosing the best parameters using all singleton locations?

The search family is unchanged: constant shifts, linear rune counters, linear word counters, and repeating two- or three-shift keys, in both arithmetic directions. Each section has its own key. Rune order, counters and apostrophe handling are unchanged. Every coefficient or shift ranges from 0 to 28.

An exemption removes the A/I constraint at a location. It does not delete the rune, join adjacent words or change the counter. Treating markers as instructions that alter the cipher is therefore outside this test.

We minimize exceptions exactly. Linear and constant settings are enumerated. For periodic keys, each phase's shift is optimized independently, then the phase minima are added. All tied optimum keys are retained. A section may select a different family from every other section, making this a generous lower bound within the declared search.

Minimum exceptions

Thus at most 37 of 83 singleton locations, about 45%, can satisfy A/I under any combination of the tested section-specific schedules. These counts are not small enough for a handful of abbreviations or red initials to repair the mismatch.

O is an explicit additional allowance, not a newly identified plaintext letter. The A/I/O sensitivity is a count-only experiment; its tied keys were not ranked on longer text.

The red-token sensitivity excludes the heading M on LP2 page 6, offset 75, and the initial A on page 54, offset 0, only from the constraints. The lower bounds in their sections remain unchanged. Counter positions retain both tokens.

Do the best fits help with longer words?

We ranked all 146 tied parameter settings that achieve their section's minimum A/I exception count. Overlapping families and arithmetic symmetries mean this is a count of settings, not necessarily distinct decryption maps.

The ranking uses only words of at least four runes. For those words, we average within-word conditional trigram log probabilities from a small rune-language model trained on the five known decoded LP1 groups, with additive smoothing of 0.1. Singletons and two- or three-rune words contribute nothing to that score.

Higher scores are better under this model. For comparison, leaving each known LP1 group out of model training and scoring its longer words gives scores from -2.964 to -2.280. Every real-text winner scores below that small benchmark range. Their inspected openings remain incoherent; no candidate has been verified as plaintext. Full winning outputs and their exceptional locations are retained in the results file.

This is exploratory ranking, not an independent validation result or a calibrated language test. All singleton locations are used in fitting, and all longer words are used to rank ties. A candidate allowed more than the minimum number of singleton exceptions could score differently. Other languages and writing conventions are not assessed by this English-derived model.

Controls

We constructed ten encrypted controls from the decoded LP1 groups, one for each family and arithmetic direction. Before encryption, we changed two known singleton plaintext tokens to F and TH. These deliberately violate A/I while preserving the longer words.

In all ten controls, the exact minimum is two exceptions. Ranking the tied settings by longer-word score selects a key that recovers the entire modified plaintext exactly, including the two planted exceptional tokens. This demonstrates that the implementation can recover these schedules when a few exceptional singletons really are present.

The language model was trained on that same known corpus, so the control's ranking advantage is intentionally favorable. It validates implementation and recovery on these constructed examples; it does not establish general cryptanalytic power.

Does the amount of fit exceed chance?

We shuffled singleton rune values among their positions within each section, preserving each section's inventories and counters. Each of 999 seeded trials repeats the full minimum-exception search and sums the section minima.

• Observed minimum: 46 exceptions.
• Shuffled median: 46.
• Shuffled central 95% interval: 43–49.
• Lower-tail comparison, with the standard plus-one correction: 0.513.

The observed fit is ordinary under this limited permutation comparison. It does not identify a statistical lead. The comparison tests the arrangement of these singleton values against fixed positions; it is not a realistic generative model of encrypted English.

The two arithmetic directions have the same attainable A/I membership counts after reflecting the key and swapping A with I. Consequently, the permutation loop need only evaluate one direction. Both directions are retained for longer-word ranking because their complete outputs differ.

Implication for the hunt

“A few singletons are abbreviations or markers” does not rescue this bounded family. A viable explanation would need substantially different state evolution, different boundaries or plaintext conventions, or many more exceptions. Nothing here demonstrates that the visible separators are wrong or that an ancient language is present.

This closes the small-exception loophole for the tested schedules. Expanding them without another constraint risks fitting arbitrary noise. A next model should make a checkable prediction involving both the rare repeated neighbors and longer passages, rather than fitting singleton locations alone.

Evidence and reproduction

• Complete minimum counts, all tied keys, ranked outputs, controls and shuffled outcomes
• Reproduction script
• Schedule definitions and shared helpers
• Session 44: strict singleton constraints
• Session 43: source-image boundary audit

Run python singleton_exceptions.py with NumPy installed and the companion notebook files in the same directory. The output records script and transcription hashes. All page numbers and rune offsets are zero-based.