R12 Query-Distributional Context No-Go

Status: tight average-case context law; reject as a novel context-scaling mechanism.

Model

Let X_1,...,X_n be independent uniform source bits. A one-pass encoder commits to b source-dependent bits before an independent late query Q is drawn from known probabilities mu_{n,i}. Source tokens, KV cache, retrieval, external memory, and source-dependent oracle access are unavailable after commitment. The answer is X_Q and loss is average bit error.

For coordinate errors e_i, every randomized encoder obeys

b >= I(X;S) >= sum_i (1 - h_2(e_i)).

The exact task rate-distortion problem is therefore

R_n(D) = min sum_i (1-h_2(d_i))
         subject to sum_i mu_i d_i <= D,  0 <= d_i <= 1/2.

If mu_(1) >= ... >= mu_(n), a useful finite converse is

D_n*(b) >= max_{m>b} mu_(m) m h_2^{-1}(1-b/m).

Storing the k most likely source bits gives

D_n*(k+O(log n)) <= (1/2) sum_{i>k} mu_(i).

Concentration characterization

Define

K_n(delta) = min{|A| : mu_n(A) >= 1-delta}.

For independent source bits, sublinear retained memory and vanishing average error exist exactly when uniformly computable sets of size o(n) carry 1-o(1) query mass. For necessity, let G={i:e_i<=sqrt(D)}. Then

mu(G) >= 1-sqrt(D)
|G| <= b / (1-h_2(sqrt(D))).

Thus b=o(n) and D->0 force query concentration on o(n) coordinates. The error falls because discarded context is almost never queried; conditional error on discarded positions and worst-case error remain 1/2.

Tight scaling control

For recency rank r=n-i+1, let mu_{n,i} be proportional to r^-alpha with alpha>1, and keep the newest k=n^beta bits for 0<beta<1. A circular buffer achieves

D_n = O(n^{-beta(alpha-1)}),

while the converse gives D_n*(b)=Theta(b^{1-alpha}). Online updates and queries cost O(1) word operations. The matching implementation is a sliding window cache or one-way streaming sketch.

Collapse audit

• the committed state is a weighted INDEX/random-access sketch;
• the objective is functional source coding with query side information;
• PSR/AIS and predictive rate-distortion cover structured predictive variants;
• reservoirs or weighted caches implement the same average-case retention;
• hierarchical summaries help only when the source/query function has a compact sufficient statistic, which is source structure rather than a new mechanism.

Exact all-query memory remains n bits. Reopen only for a learnability or computation separation on structured sources against a resource-matched comparator. No CPU falsifier or Shohin fit is authorized.