No tested position-only schedule both aligns the repeated phrase and satisfies the one-rune-word constraint across the corpus. A narrow section-local exception survives for arbitrary substitutions, but fails as a simple additive key. No new plaintext.
This follows the structural mismatch in Session 51. We ask whether a reused positional state could account for the phrase on pages 27 and 55. This is a conditional explanation, not an assumption that equal ciphertext must encode equal plaintext.
What “same state” means in this test
At each position, encryption is an arbitrary bijection of the 29 rune values, selected only by a declared positional counter. It may be nonlinear. It has no additional dependence on preceding plaintext, preceding ciphertext, random choices, skipped symbols or an unobserved internal state. This is a deliberately bounded position-only family.
Counters reset at a word, explicit text-line separator, page, paragraph, marked section, or transcribed full stop; the global counter never resets. We test each counter directly and modulo every period from 1 through 64: 455 specifications, including equivalent cases. A candidate must put all six corresponding phrase runes in identical states, not merely align their first rune.
The period range and global/page/section alignment idea were already examined in Session 6. The added test here is an exact necessary condition for arbitrary per-state substitutions, rather than a language-scored search for an additive key.
Measured positions
All offsets and page numbers are zero-based. Full stops refer to . in the retained transcription; this encoding does not preserve every visual distinction between red and black marks.
| Counter at phrase start | Page 27 | Page 55 | Difference | Aligning periods up to 64 |
|---|---|---|---|---|
| Global rune position | 6,555 | 12,950 | 6,395 | 1, 5 |
| Section rune position | 28 | 3,310 | 3,282 | 1, 2, 3, 6 |
| Page rune position | 28 | 70 | 42 | 1, 2, 3, 6, 7, 14, 21, 42 |
| Paragraph rune position | 28 | 302 | 274 | 1, 2 |
| Since transcribed stop or paragraph/section break | 9 | 293 | 284 | 1, 2, 4 |
| Explicit text-line rune position | 8 | 2 | -6 | 1, 2, 3, 6 |
| Within-word rune position | 0 | 0 | 0 | Every tested period; direct counter too |
Both phrases consist of two three-rune words, so the within-word states are 0,1,2,0,1,2. The other counters increment across the entire phrase. In total, 88 specifications align all six states, including 65 highly redundant within-word variants. The number 42 is an offset difference, not evidence of a literary reference or intended key length.
An exact necessary condition
If every one-rune plaintext word is A or I, then any one shared state can emit at most two distinct ciphertext runes at those word locations. A bijection cannot map two plaintext values to three ciphertext values.
This requires no guess about which symbol means A and no fitted cipher key. For each state, count the ciphertext types. More than two is a contradiction under the premise. Allowing O as a third possible word raises the limit to three.
We use the corrected 83-token inventory from Sessions 42 and 43. The result remains conditional on these being plaintext words with preserved boundaries and rune lengths. Initials, structural symbols, nulls, another language or different boundary semantics may invalidate the premise.
| Scope or sensitivity check | Phrase-aligned schedules passing the singleton condition |
|---|---|
| One shared mapping per state throughout the corpus; A/I | 0 of 88 |
| Same, allowing A/I/O | 0 of 88 |
| A/I after excluding the two previously identified red singleton tokens | 0 of 88 |
| Sections 6 and 8 share mappings; other sections get separate mappings but use the same schedule; A/I | 0 of 88 |
| Only sections 6 and 8 are constrained; A/I | 1 of 88: page-reset period 42 |
| Only sections 6 and 8 are constrained; A/I/O | 2 of 88: page-reset periods 21 and 42 |
Passing this count test is necessary, not sufficient, for a real decipherment. It neither supplies the substitutions nor establishes plausible longer words.
The surviving exception is limited
The two phrase-bearing sections have 27 singleton tokens occupying 20 of the 42 page-reset phases. Seven phases contain two singleton occurrences. An arbitrary substitution at every phase can accommodate their A/I assignments. Twenty-two phases have no singleton constraint at all.
A simple additive key does not pass, however. For P = sign*C - K mod 29, A and I have values 24 and 10. Two different ciphertext values at the same phase must therefore differ by 14 or 15 modulo 29. Six of the seven paired singleton phases violate this requirement.
For example, phase 32 contains:
| Location | Cipher rune label | Rune value |
|---|---|---|
| Page 28, rune 32 | ING | 21 |
| Page 45, rune 32 | O | 3 |
Their difference is 18 modulo 29, so they cannot both decrypt to A/I under the same additive phase key, in either arithmetic direction. Allowing O still leaves five incompatible phases in each direction for the period-42 model.
The broader arbitrary-substitution model also cannot be extended to all sections under the tested common schedule. In section 4, page-reset phase 21 contains three distinct singleton runes: D at page 16/rune 21, I at page 17/rune 147, and TH at page 19/rune 189. That section alone needs more than two singleton plaintext values at the phase. These are transcription-based witnesses; this session adds no new image inspection.
Interpretation
A shared clock value does not establish a shared complete cipher state. The simplest position-only explanations face explicit contradictions under the ordinary-word premise. The one section-local arbitrary-substitution exception is too flexible to count as a supported lead, and its additive specialization fails.
These results do not exclude feedback-dependent state, variable key consumption, different schedules in different sections, non-bijective encoding, or equal ciphertext arising from unequal plaintext. They also do not retroactively prove the independent-random-key model false. We have narrowed two specified explanations; we have not identified the actual mechanism.
The useful next distinction is between a genuinely repeated state and different states that happen to emit the same six runes. Any proposed state-reuse rule needs evidence outside this phrase and must account for the singleton contradictions or explicitly change the plaintext/boundary premise.
Controls and reproduction
A constructed control replaces the 83 singleton values with encryptions of A/I under 42 randomly generated phase-specific permutations. The inverse recovers all 83 inputs exactly, and the capacity check reports zero required exceptions. A small counterexample with a third ciphertext type requires an exception. These validate the necessary-condition calculation; they are not unknown-key recovery tests.
- Audit implementation
- All 455 specifications, sensitivities and contradiction witnesses
- Corrected parser, rune utilities, and word utilities
Run python -B reset_state_audit.py with NumPy and companion files. The source hash is recorded and unchanged. The results include the minimum number of tokens that must violate the allowed-singleton premise for each aligned model; these are model requirements, not proposed transcription corrections.