Assessment 178: final diversity forces paid future work

9 October 2026. Research assessment. Argument and readable checker.

Outcome

A final bound of $K$ count classes forces a future completion cost. The exact weighted cost is an ordered contiguous-partition minimum; $d$ current classes force quadratic fresh-pair work and, for fixed $K$, $\Omega(d^2\log(d+2))$ future encoded symbols. This can fund a repair worker only if its progress survives updates. Absolute-counter expansion, global alias restart and a quota based only on $d$ fail explicit tests.

The exact counter-completion formula groups sorted current counts into at most K contiguous blocks and raises each block to its maximum. Positive row prices permit ordinary-name costs to be included. The formula is an elementary ordered partition argument, not a claimed new optimization method. It supplies a lower bound for the radial language; the abstract minimum need not itself be a radial completion.

Distinct current counts force triangular movement costs. Balancing block sizes yields the sharp pair bound. Repeating distinct binary row names along enough long moves strengthens the future input bound without name comparability or target-width gates. Exact completion debt decreases by at most a paid update’s price. The weaker function of d alone lacks that property, as the eight-to-seven-class example demonstrates.

Dependent construction attempt

A worker with bounded local steps and cost within the forced future window would meet accepted-input deadlines if its pending interpretation stays valid through intervening updates. This identifies a plausible source of repair time. The construction is still missing its name-driven launch, compact class state and update-stable work invariant. It is not legitimate to supply the optimal partition, sorted counts, numerical class count or ready handles as advice.

Two continued attacks fail precisely. Absolute unary expansion can cost exponentially more than the actual remaining input in a one-profile radial completion. A whole-name-dictionary rebuild can cost arbitrarily more than the last twelve-symbol merge in a two-profile radial completion. These witnesses preserve the intended final promises, so their objections cannot be dismissed as irrelevant malformed prefixes. Shared gaps, prepared differences and readable pending actions remain escapes.

Draft audit and evidence

The writer audited the actual proof; no independent reviewer is claimed. The optimal-destination argument uses positive prices and only nondecreasing counts. Same-count rows may be reunited at the smallest feasible destination without increasing cost. Introducing later rows cannot lower the work required for the old ones. The encoded bound counts only real future syntax and name bits. Lower bounds on accepted continuations are not themselves an input recognizer or a stopping test. The conditional worker includes preparation, routing, observations and allocation; overlapping jobs and restarts cannot double-spend the same input.

Finite checks passed 1,536 weighted optimal-cost comparisons against independent Dijkstra searches over legal named unit moves, 4,608 one-sided debt-drain checks, 2,004 ordered states including common-offset invariance, 112 name-price assignments, 46 explicit radial offset witnesses and nine last-merge witnesses. The predecessor shared-predicate checker remains the separate control. Python arithmetic and the dynamic program are semantic oracles, not native instructions.

No new external theorem was imported. Prior input-funded compilation, rectangle deficits, fixed observation banks, bounded-live class repair and causal/replay controls retain their exact hypotheses. The new application proposes mandatory future coalescence as a deadline certificate, while leaving the physical worker unresolved. Entry129’s established results and the reversal direction of any later PEG transfer are unchanged.

Seven-surface disposition

Custody: original proof and checker in GitLab, no primary papers or scratch. Claims: supporting written cost theorem and conditional scheduling implication, no CLAIMS.yaml admission. Evidence: general proofs and finite semantic checks. Failures: expansion, restart and local-quota tests with reopen gates. Frontier: update-stable gap-sensitive shared predicates or final-only K=2 certification. Reader synthesis: checkpoint, program and agenda updated together. Site: generated HTML/KaTeX note and exact-byte syntax-highlighted checker; source/deployment parity belongs in integration state. No new paper or homepage significance promotion.

NEXT: Construct a restart-free, gap-sensitive shared class-predicate worker whose pending work is bounded by a retained completion-debt certificate. Obtain the launch condition and class information from ordinary names, incorporate entry/leave and shared gap changes, and preserve immediate readout and final selection. Avoid absolute-count expansion, all-alias restarts and treating the d-only lower bound as a local potential. A final-only K=2 certificate remains the alternative.

ACCOUNTING: Scheduled 82->83 once; next 84 editorial, then dedicated literature review 85. Last dedicated 175 retained. Mathematical 116, window 0 and last reassessment 116 unchanged. Supporting written theorem and conditional schedule, no root admission or major-result count.

Editorial 84 comes next. Review rendered recent research and code pages, retain this exact mathematical target, and resume it after the editorial work. Literature 85 must retain its substantive multi-paper primary-proof duties.