The rational bound is valid independently of LP optimality or floating-point
precision. For a set Q of useful ordinary slots with weight w, let lambda lie
between0 andw. If S is the chosen alphabet, then
  w * indicator(S intersects Q) <= (w-lambda) + lambda*|S intersects Q|.
Summing gives a constant plus the loads of the chosen slots. At mostK chosen
slots cannot exceed the sum of theK largest nonnegative loads. The review
recomputes this with exact integer numerators and a common positive denominator.

For gap d, let c_d(p) be the independently audited minimum exceptional touches
needed to produce its next output at endpoint slot p. The free exceptional
endpoint is p=0. For any genuine alphabet S, the required touches are at least
min(c_d(0), min_{p in S}c_d(p)). Counting layers below the exceptional cost
expresses this minimum as a constant minus coverage of threshold endpoint sets.

When the exceptional endpoint has no finite cost, the implementation chooses
b=max(finite endpoint costs)+1. For a feasible S, some ordinary endpoint is
finite and the layer representation is still exact. For an infeasible S, the
finite b clips an infinite requirement downward. This is another sound
relaxation. Therefore an unconditional identity for EVERY S would be too strong,
but the lower-bound result remains valid. 2,909 of the2,916 random test alphabets
hit this harmless clipped case; all clipped values were <= direct values.

Sum the interval touch lower bounds over all observed consecutive returns.
At any one time, a particular card belongs to at most one strict-interior
consecutive-return interval. An off-top transposition touches exactly two
cards, so it contributes at most two counted touches. It cannot contaminate
the return interval at either endpoint: an emitted card is at top0 while the
exception swaps two nonzero positions. Hence total exception events E satisfy
  2E >= sum of all required tracked-card touches.
Rounding the rational touch lower bound upward after division by2 is valid.
Events in different messages are physically distinct, and no shared initial
deck or plaintext alignment is assumed.

All729 saved certificates matched rebuilt threshold sets and exact arithmetic.
Their bounds range87..127;521 exceed the separate universal88 bound. Combining
the two yields per-pair bounds at least88, up to127. These are necessary
frequencies, not feasible cipher constructions or complete model exclusions.
