Full state alphabets expose a letter frequency gap

The three frozen section rules that retained full-section capacity in Session 70 fail a stronger common letter screen, if every state must encode the same declared alphabet of at least 20 plaintext types. A broader, deliberately favorable scan allows new selectors but finds no bank with up to eight tables and 25 through 29 declared plaintext types passing the saved reference profiles.

Some 20–24-type settings remain. The result does not rule out restricted alphabets that change by state, larger banks, other repeat rules or English with a different frequency profile. No real decoding table, key or new plaintext is recovered.

The working source was restored from the saved notebook before this test. Its master hash and Session 70 results are unchanged.

The new premise

Session 70 reserved a group of ciphertext codes for a designated plaintext letter R:

g(C) = ((a*C+b) mod 29) mod k
state = g(previous ciphertext rune)
D[state,C] = R exactly when g(C) = state

Each section could have independent codebooks. Homophonic choices within R's group could suppress exact neighboring repeats even if R were a common letter, as the known-text controls demonstrated.

This session adds an encoder-domain requirement: one declared d-type plaintext alphabet, including R, must be available in every state. Each decoding row maps onto that same alphabet. It may map several ciphertext symbols to one plaintext symbol, but it cannot leave a declared letter unencodable in a state.

This is a new assumption, not something established by the source images. A method that permits different plaintext alphabets in different states, or imposes restrictions on message sequences, need not satisfy it. The counterexample below shows why that distinction matters.

We examine d = 20 through 29. The five saved solved groups individually contain 20 to 25 observed rune types; their pooled scored text contains 26. Observed types in a sample do not establish the full declared alphabet of its unknown encoder.

An exact row capacity

A state group with q codes reserves all q for R. Its remaining 29 minus q codes must cover the other d minus 1 plaintext types. Therefore:

d <= 29 - q + 1

For the three frozen rules from Session 70:

The five-table bank has four rows capped at 24 types and one at 25. A six-table bank has five rows capped at 25 and one at 26. A 26-type alphabet available everywhere would require at least eight tables in this affine grouping family. None of the three frozen rules meets that condition.

These are codebook capacities, not claims that the actual plaintext uses exactly that many symbols.

Bound all possible letter-frequency profiles without fitting D

For a chosen selector, count current ciphertext symbols separately within each state. R's total is fixed by same-group transitions. Each row's other code counts can be assigned to the d minus 1 non-R outputs, subject to each output receiving at least one code.

If a row has L non-R codes, it has L minus (d minus 1) extra codes beyond one per output. Its most concentrated permissible histogram is obtained by assigning the largest extra-plus-one code counts to one output, then assigning the remaining counts individually to the other outputs. Align these ranked outputs across rows and add the fixed R count.

This construction attains the maximum cumulative count of the largest r output bins for every r simultaneously in the relaxed family. To see why, any chosen r non-R outputs must leave at least one code for each of the other outputs. Their total can therefore collect at most the largest extra-plus-r code counts in each row. The construction reaches that total; aligning row ranks reaches the summed bound. Adding the same fixed R count preserves the resulting sorted partial-sum bounds.

Consequently, no other permitted decoding table can exceed these cumulative counts. The construction also minimizes single-rune histogram entropy in this relaxed family. This is an empirical histogram bound, not a language entropy-rate estimate.

The bound ignores singleton table assignments and actual plaintext letter identities, giving the model more freedom than a complete decoder would have. Anonymous tables attaining the bound are retained and verified. They are extremum witnesses, not proposed English keys.

What is compared

The existing eight substantial LP2 sections are used, with the first rune of each omitted. Pages and words do not reset the state. There are 12,939 scored transitions, as in Session 70. The nine-rune number label remains outside this scope.

The five solved groups are also scored with their first rune omitted. A pooled reference combines their 1,958 scored runes. We compare the first ten sorted cumulative proportions of each reference with the maximum permitted proportions for a candidate. This focuses the screen on common-letter concentration. Full bounds are retained, but zero-frequency tails are not used as the primary screen.

A reference containing more than d observed types is excluded from that d comparison because the declared alphabet cannot represent it. Otherwise, a deficient partial sum among the first ten fails that empirical profile. Integer cross-products are used in the implementation to avoid rounding decisions at the threshold.

These are reference-dependent necessary screens. They are not universal limits on English, a probability that a cipher is correct, or a statistical significance test. The models, text and references have all been examined previously.

The frozen survivors fail the full-section profile screen

The selectors from sections 1, 5 and 7 remain frozen at their Session 70 values. We do not replace them with new settings when they fail. For every encodable d from 20 upward, each fails every compatible full-section reference in at least one of the first ten partial sums.

A 24-type alphabet gives a concrete example:

These are maxima over the permitted D rows, not scores from a guessed key. Different references fail at different rank counts; the complete comparisons are saved.

At d = 25, the five-table rule is unencodable. The two six-table rules can offer the alphabet, but their maximum combined top-two shares are only 20.29% and 20.65%, versus 22.52% in the koan reference. Their non-R peak ceilings are 6.76% and 6.99%. Having enough codes for individual letters therefore does not guarantee enough freedom to reproduce several common letters together.

At d = 20, the reference failures are much narrower. The three top-nine ceilings are 71.03% to 71.47%, versus 71.59% in the instruction sample. That reference contains only 88 scored runes. These small differences are particularly sample-dependent and do not justify broad claims about English.

Continuation-only profiles are also retained. Some of those shorter samples pass particular reference screens even though the complete-section profile fails. The result above is a full-section constraint on the frozen rules, not an assertion that every continuation diagnostic fails.

Could another small-bank selector escape?

The secondary check deliberately grants additional flexibility: all 812 affine settings for each k = 1 through 8 may be selected using the complete section. It retains Session 70's more permissive A/I/O reference screens and both possible categories for R. It then applies the new alphabet and first-ten-bin bounds.

This is 6,496 parameter settings per section, with duplicates retained. It is an optimistic capacity audit, not independent validation or permission to call a refitted setting a recovered key.

No setting with a declared alphabet of 25 through 29 types passes any compatible saved reference in any section. Sections 4 and 8 have no survivor at any tested alphabet size.

The 20–22-type survivors compare only with the short, 20-type instruction reference. Section 0 also admits 24-type settings matching the divinity profile: 20 parameter settings at seven tables and 260 at eight. These counts are not numbers of distinct ciphers or evidence of readable plaintext.

The available references change with d, so passing any reference is not a nested test as d increases. For a fixed compatible reference, requiring more output types tightens the code-allocation constraint. No actual D bank has been fitted for any secondary survivor.

Controls and the boundary of the conclusion

Three small-alphabet fixtures exhaust 4,052 complete decoding tables that cover their declared alphabets in every state. Every maximum sorted partial sum agrees with the formula, and exhaustive minimum histogram entropy agrees as well. A separate infeasible fixture verifies the row-capacity rejection.

Fifteen known-text controls reuse Session 70's retained independent codebooks at six, eight and 29 states, one for each of the five solved groups. They re-encode the known messages with a fixed new random seed, reconstruct state from completed ciphertext in a separate decoding pass and recover all plaintext exactly. The declared alphabets range from 20 to 25 types. Every true histogram lies within every partial-sum bound. Scalar and vectorized implementations agree, and each retained anonymous extremum table attains its claimed histogram.

The new formula also reproduces Session 33's bijective limit when k = d = 29: 12,939 transitions, 86 repeat-channel positions, maximum output share 5.5337%, and minimum histogram entropy 4.789442 bits. Its complete sorted histogram matches the saved earlier result. That is a regression check of the generalized formula, not a new decipherment.

A separate scope counterexample has four ciphertext symbols, two states and five global plaintext types. Each state can produce only three of those types. A restricted message decodes and re-encodes exactly under the same group-repeat rule, but the full-alphabet-per-state premise is false. Applying this session's capacity rejection to that example would therefore be incorrect.

Restricted state alphabets remain an opening. The counterexample supplies no Cicada key and no reason to prefer that mechanism in the real text. Similarly, fewer than 20 declared types, banks larger than eight, different repeat channels, changing keys and other boundary semantics remain outside the numerical conclusion.

Reproduce and inspect

Run with Python and NumPy from the shared evidence directory:

python3 -B state_alphabet_bound.py

• Alphabet capacity, exact concentration bound, candidate scan and controls
• All bounds, anonymous witnesses, reference comparisons and secondary settings
• Session 70 model and original screens
• Unchanged Session 70 results and retained control codebooks
• Earlier bijective bound used for regression
• Unsolved source and singleton definition
• Rune loader
• Solved reference groups
• Unchanged master transcription

Master SHA-256: b0b2e13c3cd35512ee5955a2d9a56c22d7410e283d35dd9c90abbaa7b99279eb. The script checks it and records its own hash and the unchanged Session 70 result hash. Prior reports and evidence are preserved.

Decision

Deprioritize the small, full-alphabet group-repeat family under these reference assumptions. The few remaining capacity settings need a concrete decoding rule and evidence for their smaller or restricted alphabets before another plaintext search is justified. Passing a short reference profile does not identify a language, literary source or ancient civilization.

No verified new plaintext is found. The gain is a sharper, tested boundary on what the surviving model can encode.