A repeat channel narrows section-specific table banks

A specified repeat-avoidance mechanism removes the two- and three-table opening from Session 69. Under this new rule, each section needs at least five or six tables to pass the stated full-section reference screens. Selectors chosen on the earlier two thirds of each section produce four exact singleton contradictions later. Another fails the full-section letter-frequency screen. Three retain full-section capacity; only two also pass the continuation-only frequency screen.

The most useful control result is a counterexample to inferring a rare plaintext letter from rare ciphertext doubles: ordinary solved prose with 248 scored E positions can be reversibly encoded with zero exact neighboring repeats. E occupies a group of homophones, and the encoder chooses another member when possible. This does not identify E or this mechanism in the real book.

No new plaintext or key is recovered. The negative conclusion applies to the stated group-based repeat channel, not every homophonic cipher or every way to change section keys.

What supports the section boundaries?

The retained source map supports passage structure but has not identified cryptographic boundaries. Session 55 showed that the 1,524-rune master section containing the tree combines pages 27–32 with a separately headed, 91-rune block on page 33. Session 57 recorded 13 red introductory spans, five inside the existing substantial sections; its fixed local frequency comparison supplied no unusual change at those five internal headings. That test could miss key changes that preserve rune composition.

Session 62 reproduced known methods and rejected specified boundary resets as universal rules for those methods. It did not rule out new section-specific rules in the unsolved pages.

We therefore retain the existing eight substantial master sections for this bounded test, without calling them established key units or introducing more subdivisions. The nine-rune number-label section is excluded. No new image transcription, source authentication or layout classification is performed here.

The new rule and its prediction beyond singleton words

Each rune has its standard ciphertext index C. Define:

g(C) = ((a*C + b) mod 29) mod k
state at position i = g(previous ciphertext rune)
plaintext = D[state, current ciphertext rune]

Each section has its own fixed many-to-one D bank, selector parameters and designated plaintext letter R. The added rule is:

D[state, C] = R exactly when g(C) = state

Thus every same-group transition decodes to R, and every different-group transition decodes outside R. Exact ciphertext doubles are a subset of same-group transitions. R may have several codes in a group, so the encoder can avoid repeating the exact previous code even when the plaintext is R.

This predicts R's entire section frequency directly from ciphertext groups, before D's other entries are fitted. It also predicts that a same-group singleton word must be R. These are stronger constraints than the allowed-letter mass alone in Session 69.

All a = 1 through 28, b = 0 through 28 and k = 1 through 29 are checked: 23,548 selector settings per section. D remains unknown. The repeat rule is analyst-chosen; no message or illustration is claimed to supply it.

Pages, words and punctuation do not reset the state. The first rune of each section is omitted because its predecessor is unspecified. The eight sections contain 12,939 scored transitions, including the 86 exact neighboring repeats previously recorded.

Screens and their limits

Retained singleton words must decode into S = A/I, or the more permissive A/I/O. We distinguish R inside S from R outside S, permitting each section to choose either category.

• If R is outside S, a same-group singleton is an exact contradiction of the ordinary-word premise.
• Every state/current-cipher cell observed as a singleton must decode into S throughout the same section. Counting all its occurrences gives a minimum allowed-letter mass.
• If R is inside S, the entire repeat channel also contributes to that forced allowed-letter mass.
• R's predicted share must pass the corresponding single-letter reference cap.

The five saved solved groups are scored with their first rune omitted, matching the context policy. The most permissive individual-group reference caps are:

The outside cap comes from E in the warning group. It does not assign R = E in the real text. Inside caps come from A in the koan and O in the instruction group. These are empirical sample screens, not universal English limits or significance tests. Exceeding one rejects compatibility with the chosen reference cap, not English in general.

The same-group singleton contradictions are exact under the selected outside-R model and boundary premise. The frequency comparisons have the weaker, reference-dependent interpretation above.

Full-section capacity

Two groups force R into at least 42.74% to 46.49% of positions across the separate sections. Three groups force at least 27.27% to 29.68%; four still force at least 18.18% to 21.27%. All exceed even the most permissive single-letter cap of 14.7541%. Separate D banks do not change these selector-dependent counts.

After both frequency screens and the singleton condition are imposed, the first full-data capacity survivor is:

These settings use all section data to calculate capacity. They are not recovered keys. A newly selected full-data setting is not allowed to replace a failed frozen prefix setting in the continuation test.

There is an additional codebook limit: a state group containing q ciphertext symbols reserves all q for R, leaving at most 29 minus q plus 1 plaintext types encodable in that state. Five-table banks have row capacities of 24 or 25 types; six-table banks have 25 or 26. That is a per-state capacity, not the number of types in the union of different state outputs. A proposed decoder must justify its plaintext alphabet and supply actual codebooks; passing the frequency screen does neither.

Freeze the earlier-part selector and test the continuation

For each section of N runes, the cut is floor(2N/3). Fitting uses current-rune positions 1 through cut minus 1. Evaluation begins at the cut and retains the actual preceding ciphertext rune. No state is restarted at the split.

The combined fitting partitions contain 60 singletons; the continuations contain 23. At each k, eligible settings pass both training frequency caps and the singleton condition. Choose the smallest k with any eligible setting, then minimize training allowed-letter mass, training R mass, outside before inside, and finally a and b in ascending order. Freeze the selector and R category. No D entries or plaintext are fitted.

The A/I and A/I/O selections coincide in these primary cases, all choosing R outside the allowed set:

Three frozen settings pass the full-section screens, but only sections 5 and 7 also pass the continuation-only caps. Section 1's later R share is above the empirical cap even though its complete-section share is below it. That short-window mismatch is not a mathematical contradiction: natural-language letter frequencies can vary between passages.

Section 6 fails the full-section reference because its frozen selector forces R into 232 of 1,523 positions, or 15.2331%, above 14.7541%. Its singleton condition still passes. The four exact lexical contradictions occur at:

At each listed location, the frozen selector groups the predecessor and current rune together. The resulting one-rune word must be R, contrary to that selector's fitted outside-A/I/O category. Labels are ciphertext labels, not proposed plaintext. These boundaries reuse the previous audit; no fresh image check is claimed.

Removing the two known red initial tokens leaves 81 singleton constraints, with the same first passing k values, frozen pass/fail outcomes and four contradiction counts. Counters and rune sequences remain unchanged. The excluded tokens are both in the evaluation portions, so the 60-token fitting inventory stays intact.

This is a retrospective continuation check on already studied text. It tests a selected rule's conditional predictions, not a blind plaintext holdout, a complete key search or statistical evidence that a surviving rule is real.

Six known-table recovery controls

Constructed controls use the five solved groups, with independent codebooks for each group, E as the common designated R, and k = 6, 8 or 29. Each codebook covers that group's entire observed plaintext alphabet, which ranges from 20 to 25 types. The planted affine selector is a = 7, b = 11.

The encoder either avoids exact repetition whenever another R homophone exists or chooses the previous code with probability 0.25. Both policies use fixed codebooks. A separate pass reconstructs state from completed ciphertext and recovers every plaintext rune exactly. All six planted selectors pass both the A/I and A/I/O screens.

All rows score 1,958 positions. At k = 29, R has only one code in the current state, so repetition is unavoidable and the nominal probability has no effect. At k = 6 or 8, a common E channel coexists with zero exact doubles. These controls are not fitted to the real repeat rate and do not demonstrate unknown-key recovery.

Why other summary states were not searched under this added rule

If the guarantee that every raw repeat decodes to one fixed R is imposed for every possible ciphertext history, a sum-two, difference-two or sum-three summary has a collision problem. For any previous rune and any summary value, older runes can be chosen to produce that value. Every decoder state is therefore compatible with every possible repeated rune. The universal rule forces every D cell to R, giving a constant decoder.

This does not exclude those summaries under other repeat rules or restricted reachable histories. It explains why the specified universal channel is naturally expressed with a grouping of the previous rune.

The implementation checks all 841 two-rune histories for each two-rune summary and all 24,389 three-rune histories for the checksum example at eight states. All 232 state/current-rune cells are forced to R. The previous-rune summary instead forces just 29 cells, one per possible previous rune. The general argument applies throughout the stated affine bank; the enumerations are implementation controls.

An independent three-symbol fixture exhausts 438 decoder assignments across 18 selector cases, checking both inside- and outside-R singleton sets. Every exact minimum allowed mass and hard contradiction agrees with the counting rule.

Reproduce and inspect

Run with Python and NumPy from the shared evidence directory:

python3 -B section_repeat_channel.py

• Selector screens, continuation protocol and recovery controls
• All per-k outcomes, frozen selectors, exact contradiction witnesses and control codebooks
• Unsolved source and singleton definition
• Corrected lexical grouping
• Rune loader
• Solved reference groups
• Unchanged master transcription

Master SHA-256: b0b2e13c3cd35512ee5955a2d9a56c22d7410e283d35dd9c90abbaa7b99279eb. The script verifies it and records its own hash. Earlier reports and sources are preserved.

What remains open

Section-specific banks retain some capacity, but this pass supplies no D codebook or English continuation for the real text. Different R rules, singleton exceptions, other alphabets, changing selectors and alternate boundary semantics are outside the negative conclusion. Independent section banks also need an account of the repeated cross-section phrase; no such account is supplied here.

A next worthwhile constraint is the alphabet and ordinary-letter frequencies that a surviving state's actual codebook can support. The row-capacity formula gives a concrete starting point. More selector settings or new passage splits need an independent reason, rather than merely enough flexibility to pass a screen.