R12 Dynamic Frontier Compression No-Go

Status: tight context-scaling law, rejected as a new reasoning primitive. The surviving construction is frontier dynamic programming plus ordinary recurrence.

1. Address-aware frontier theorem

At a processing cut t, let B_t be the possible active dependency frontiers. Factor v has q_v possible states and the already closed portion has s_t distinguishable summaries. If every future answer depends on the past only through this frontier, then the residual-state count obeys

N_t <= s_t * sum_(B in B_t) product_(v in B) q_v,
b_t = ceil(log_2 N_t).

The bound is exact when future queries distinguish every support, assignment, and closed summary.

Consequences:

• fixed public width w: b_t = w log_2 q + log_2 s_t;
• unknown named frontier among n factors: add log_2 binomial(n,w);
• deterministic k-local updates cost O(k) once the structure is known;
• general constraint/probabilistic messages may require Theta(q^w) storage and work.

Minimizing the maximum frontier over processing orders is the established vertex-separation/pathwidth boundary.

2. Nonlinear order-sensitive control

The discrete Heisenberg action

(x,y,z) star (a,b,c) = (x+a, y+b, z+c+x*b)

is nonlinear and order-sensitive. For A=(1,0,0) and B=(0,1,0),

A star B = (1,1,1),
B star A = (1,1,0).

Length-T histories have polynomially many residuals, so exact state grows only logarithmically in T; a free-group control has exponentially many residuals and needs linear-in-T bits. This is the classical polynomial-growth group boundary, not a new context law.

3. Discoverability boundary

Passive recovery of a dependency hypergraph needs bounded interaction order, an observable faithfulness margin, positive probability for every required contrast, and a unique minimal factorization. Under symmetric label noise, the sample scale has the ordinary inverse-square dependence on the faithfulness margin and 1-2 eta.

Without those assumptions, one unseen event-query edge defeats safe deletion. For every finite trace radius, a finite residual quotient can match a free action on the entire observed ball while having radically different long-run growth. Finite ordinary traces therefore cannot certify future residual innovation.

4. Collapse audit

• Pathwidth/treewidth supplies the same frontier law.
• Factor graphs and dynamic Bayesian networks send the same separator messages.
• Tensor-network contraction cost is governed by the same cut width.
• OBDD width counts the same residual subfunctions at a cut.
• Segment trees accelerate associative composition without reducing summary information.
• Recurrent memory directly stores frontier assignments or group coordinates.
• Runtime robustness reduces to ordinary error-correcting or fault-tolerant computation.

5. Decision

Dynamic-frontier summaries beat raw-history storage and arbitrary transition tables, but not the strongest structure-aware comparator. A CPU experiment would reproduce known dynamic programming or algebraic accumulation. No R12 implementation is authorized. A future candidate must expose behaviorally observable structure that uniquely identifies itself from ordinary traces, survives runtime noise, and beats a matched structure-aware recurrent model.