Computational definitions, startup and transport sensitivity

This edition extends the graph and constructor review. All source results remain claims at pinned revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb; no trusted proof run or production implementation is inferred.

A finite algorithm may activate after the useful horizon

The uniform k-server construction canonicalizes a finite rational metric and simulates its finite table constructor from scratch for floor(log2(t+1)) elementary transitions at request t. If the constructor needs T_P transitions, activation is exactly t*=2^T_P-1. Earlier requests always use the first label. The constructor is guaranteed finite in the source argument; its running time was not measured here.

The displayed movement constant is B=(t*-1)D+(2a+2)kD, with D the metric diameter and a the constructed coefficient. The distinct-start zero-additive clause belongs to the separate policy-existence theorem. It does not eliminate startup loss from this uniform implementation. Request processing polynomial in input encoding and log request number does not guarantee polynomial activation delay or useful short-horizon loss.

The exact movement reference evaluates three ordinary named policies and a conventional offline DP. A three-point line starts at (0,1) and alternates requests 1 and 2. At 30 requests, fixed-first-label cost is 30 and the attained offline optimum is 2. At only two requests the optimum is 1; the test preserves this horizon distinction. These are finite controls, not an implementation of the source policy. They motivate a replay/startup assurance product and defer a literal backend until useful constructor and latency measurements exist.

Uniform introduction, schedule and constants, formal scope, 137 checks, saved examples.

An exact transport sensitivity benchmark

The cube example uses u=max(-x1,x1,a) and v=max(-x1,x1,a+(a/2)x2), for 0<a<1/2. Both pushforward targets have outer masses (1-a)/2 and central mass a. Only the central atom moves, by b=a/2. Its target distances are exactly W2^2=ab^2 and W1=ab; the maps reassign source mass b/2, giving squared L2 map distance b/2+ab^2.

The new reference clips affine halfplanes and integrates the nine cell intersections with rational polygon areas. It compares these independent geometric results with the displayed formulas. For a=4^-j, the square of the constant required for a half-exponent bound is 2^(j-1)(1+a^2), which grows without bound. For the one-third exponent, the sixth power of the ratio is (1+a^2)^3/16. The source dimensions above two factor out; the implemented geometry is two-dimensional.

The manuscript also gives a usable explicit domain formula, improving the earlier dossier's unresolved-constant statement: C_*^2=12dR^2(1+sqrt(162))^2+28L^2 P_K/volume(K), where P_K sums coordinate projection volumes. For K=Y=[-1,1]^2, using R^2=L^2=2, P_K/volume(K)=1 and sqrt(162)<=13 gives C_*^2<=9464; the rational choice C=98 is sufficient if the source theorem is accepted. This can be very loose and proves no pointwise, discrete-source, entropic or learned-map guarantee. The reference is a benchmark control rather than a general transport solver.

Sharpness construction, domain constant, 131 checks.

Decision derandomization and compiler readiness

LogspaceEquality compares binary language classes using finite-control machines, endmarked input, local binary work tapes and fresh fair bits. Work space counts all traversed writable cells, including blanks. The randomized classes require a polynomial worst-case clock on every coin tape and the specified acceptance gap. Language equality does not preserve a random output law, generate cryptographic entropy or derandomize arbitrary ML.

The paper's compiler assumes supplied positive time/space parameters a,b and specializes one fixed deterministic separator library. It does not decide those semantic resource promises. The selected Comparator target proves the class-equality claim; the compiler and its numerical resource bounds are not separately selected.

The explicit bounds use P=1000(d_code+2)^4(a+b+2), rho=100(P+2), S=1+(t+1)(2rho+1)(g+2), K=4S, c=2S+1, and H=2^((u+100)^2)*(q_D+1). Four auxiliary work tapes are named. Conservatively taking only d_code,a,b>=1, t>=4, g>=2, u>=1 and q_D>=1 yields declared K>=5,184,032,084, c>=2,592,016,043 and H>=2^10202 (at least 3,072 decimal digits). These are lower envelopes for deliberately loose declared upper-bound parameters, not lower bounds on actual runtime/space or proof of impracticality for redesigned simulations. No valid program is asserted to attain minimum description length, and no compiler/library transition list was executed.

Parameter arithmetic, compiler source, formal scope.

Semantic evidence and earning boundary

41 source-level definition observations compare KServer, Brenier and LogspaceEquality challenges and selected solution definitions. No intended-model mismatch was detected there. A lexical control also finds the displayed logspace definition/structure prefix identical after comment/whitespace removal. This does not check Lean elaboration, imported dependencies, the proof or executable software.

The thirteen-contract registry now includes those model distinctions, with 12 linter checks. Seven configurations/24 declared entries remain queued. The commercial lead stays one shared evidence/model-assurance workflow, with movement replay and transport calibration as narrower modules. Pricing, demand and recurring delivery remain unvalidated.