Arbitrary page-reset substitutions with period 42 cannot produce a rune-frequency profile like our pooled solved-page sample from these two sections. This test does not assume one-rune words mean A or I. No new plaintext.
Session 52 left one limited possibility: a page-reset period of 42 with arbitrary substitutions could satisfy the singleton constraint within sections 6 and 8, which contain the repeated phrase. Its additive specialization already failed. This session tests the more flexible version against every rune in those sections.
The family and an exact bound
There are 4,840 runes: 1,524 in section 6 and 3,316 in section 8. For each rune, the state is its zero-based page rune offset modulo 42. Every state may use any bijective substitution of all 29 runes. One table is shared across the two sections in the primary model.
For any chosen output rune, a state can send only one ciphertext rune to that output. Its maximum contribution is therefore the largest ciphertext count in that state's row. Sum these row maxima to obtain the exact largest possible count of any output rune, over all permitted substitution tables.
The bound is attainable: map each state's most frequent input rune to the same output rune, then complete each substitution bijectively. It does not depend on a search heuristic, guessed keyword, singleton interpretation or language model.
More generally, sort each row's 29 counts and add them by rank. The resulting histogram bounds every cumulative top-k frequency and gives the minimum possible unigram entropy. It is an extremal histogram, not a recovered plaintext.
Results
We also check period 21, which survived Session 52's narrower A/I/O sensitivity, and allow separate substitution tables for the two sections as a more permissive comparison.
| Period | Substitution tables | Maximum count of any output rune | Maximum frequency |
|---|---|---|---|
| 21 | Shared across sections 6 and 8 | 289 / 4,840 | 5.97% |
| 21 | Separate for each section | 338 / 4,840 | 6.98% |
| 42 | Shared across sections 6 and 8 | 354 / 4,840 | 7.31% |
| 42 | Separate for each section | 430 / 4,840 | 8.88% |
Separate tables relax the bound, but then equal positional phases alone no longer guarantee a shared substitution at the repeated phrase. That comparison deliberately gives the family more freedom.
What the solved text looks like
These are empirical comparisons in the same 29-rune representation, not universal thresholds for English.
| Solved group | Runes | Most frequent rune's share |
|---|---|---|
| Warning | 184 | 14.67% |
| Wisdom | 157 | 8.92% |
| Koan | 778 | 12.60% |
| Divinity | 755 | 13.64% |
| Instruction | 89 | 11.24% |
| Pooled sample | 1,963 | 12.63% |
The shared period-42 ceiling falls below every listed group. Its gap from the pooled sample is substantial. The separate-table ceiling is only slightly below the smallest individual group, so that tiny difference should not be treated as strong statistical evidence. No probability or universal language rejection threshold is assigned here.
The exact conclusion is about what the cipher family can output. The inference that it is a poor explanation depends on expecting the unknown plaintext to resemble the solved material. Unusual wording, another language, an intermediate encoded message or an extra layer could have a flatter distribution.
Controls
Two constructed controls place repeated copies of the solved koan and divinity rune streams into the same 4,840 positional slots. Each is encrypted with 42 randomly generated phase-specific permutations. The known inverse recovers every rune exactly, and each computed bound permits the true plaintext histogram:
| Control | True plaintext peak | Calculated upper bound |
|---|---|---|
| Koan repeated to fill the slots | 12.60% | 13.72% |
| Divinity repeated to fill the slots | 13.78% | 14.15% |
These controls validate the bound, not unknown-key recovery. Repeated source text is deliberate and does not provide independent language samples. A separate three-symbol, two-state fixture exhausts all 36 substitution tables and agrees with the analytic peak and entropy extrema.
What this changes
The residual position-only hypothesis now has two separate limitations: its additive version conflicts with the singleton premise, while even arbitrary substitutions impose a low maximum rune frequency. We should deprioritize this specific period-42 explanation rather than search more keys within it.
This does not rule out state depending on preceding text, skipped key positions, different periods, reordering, inserted symbols, or an additional transformation. Nor does a repeated ciphertext phrase prove repeated plaintext or repeated complete state.
The next useful direction should change an explicit structural assumption and predict behavior outside the known phrase. Merely adding more substitutions to this fixed phase schedule cannot overcome the bound.
Reproduce
- Exact bound and controls
- Phase counts, full bounds, benchmark histograms and controls
- Page-position inventory, corrected parser, rune utilities, and known solved groups
Run python -B phase_frequency_bound.py with NumPy and the companion files. The source hash is recorded and unchanged. The test uses all runes in the two sections; no singleton is excluded and no plaintext is fitted.