Assessment175: exact observations and moving count classes

9 October 2026. Dedicated three-paper primary-proof review. The written synthesis and constructions are maintained in Exact counter observations and persistent count classes. Reader 51’s conditional result remains intact.

Result and classification

Exact negative observations are compatible with real-time computation. A fixed finite family of linear observations closed under input actions reduces to augmented counter copies and bounded translations, giving a native monitor through the existing tape compiler. This is a written supporting consequence of imported classical results, not a new general persistence theorem.

For the radial-row frontier, an ordered count histogram avoids moment arithmetic entirely. Each literal causes only a unit move between neighboring counts; unit stacks implement all necessary gap and multiplicity changes. The mutable construction checks final count diversity and supplies late selectors even with arbitrary intermediate diversity. Its persistent transfer is conditional on coherent current bucket access from arbitrarily many ordinary-name aliases. This narrows the obstacle from generic variance evaluation to a precise shared-metadata operation.

Primary-proof review

All three linked editions were read through their proofs and concluding sections. Primary PDFs and extracted text stay outside the repository. The augmented-counter paper explicitly invokes its 1985 companion’s counter-code invariants; no fresh full review of that separate companion is claimed.

Seiferas and Vitanyi, Counting Is Easy

JACM 35(1988),985-1000; author arXiv edition,15 pages. SHA256 a2f4f3b38de60786411ab6b997cf2d139a3ea0bc48bc72dea4ede048cbf846a0.

The command set is increment, decrement and sign inquiry on a fixed number of counters. Section 1 absorbs constant response delay using bounded low-order remainders. Sections 3-6 schedule signed-digit propagation by permuting radix positions; primes and directional marks eliminate recursive control. The opportunity argument bounds how many lower-level visits occur before a higher-level repair. Significance marks expose zero locally while carry work remains unfinished. Sections 7-8 connect the local rewrite rules to that invariant and prove radix 4 suffices by bounding carries between opportunities. Section 9 optimizes storage and distinguishes oblivious from value-dependent schedules.

Application: finite labels and worst-case delay fit the Lab’s tape compiler. The theorem does not include adding a row’s entire cardinality to a moment register, or managing an input-growing bank addressed by arbitrary names. Its positive lesson is that pending representation work need not delay exact zero readout.

Vitanyi, An Optimal Simulation of Counter Machines: The ACM Case

SIAM J. Comput. 14(1985),34-40; author edition,15 PDF pages. SHA256 d0d48020534304386dd8f61d7a88473ec12663851c8009b34aa5b5ebbe9e200a.

Section 2 first defines an ideal simultaneous coordinate-selection update on all digit positions. Delayed switches then apply selections before later carries reach those positions. Invariant E records ordered switch queues; the normalized matrix used in Lemma 1 denotes the same integers as the ideal machine, while its low cells already agree with the implemented machine. Lemma 2 replaces every queue by its product in the finite semigroup of k-coordinate selections. Storage is therefore finite per cell despite arbitrarily many pending copies. Operations include bounded translations and exact zero bits; the track count k is fixed.

Application: this supplies the counter-action interface in the new observation-closure argument. A finite semigroup compresses delayed work because updates preserve the representation’s interpretation. It does not authorize an input-growing set of tracks, arbitrary snapshot imports, or arithmetic multiplication. Initialization in our consequence is fixed, not an input-supplied large integer.

Mehlhorn, Sundar and Uhrig, Maintaining Dynamic Sequences under Equality Tests in Polylogarithmic Time

Algorithmica 17(1997),183-198; author edition,16 pages. SHA256 fbe1b9b563fa4fbe82f3a20bd5cf179f4d5604cab90a529c648ee499cdca4eaf.

Section 2 builds locally determined block decompositions: deterministic color reduction gives bounded blocks with an O(log* m) dependency neighborhood. Section 3 interns symbols, powers and pairs into exact signatures; Lemma 5 proves signature equality equivalent to sequence equality. Section 4 repairs only the affected neighborhoods across O(log n) hierarchical levels, with persistent tree copying. Lemmas 6 and 11 control propagation at concatenation seams; Lemmas 7-10 analyze randomized costs, and 12-13 give deterministic time O(log n(log m log* m+log n)) and incremental space O(log n(log n+log* m)). Equal is constant time after signatures exist.

Application: n is sequence length and m is operation count; growing word arithmetic and signature comparison are RAM operations. Exactness is not the problem. Signature production, dictionary access and word comparison are not fixed-radius finite-label primitives. The proof supplies neither a constant-time native interner nor a final-short-name budget for its hierarchical repair.

Comparison and dependent attack

The histogram proof carries the arithmetic replacement through to validation and late reuse. Exact pair lookup supplies the only increment trigger; gaps change by one even when an empty bucket is deleted, because its destination is the adjacent integer. Final radiality remains separately checked. At the boundary, at most K count buckets receive finite labels in bounded work, so no row scan is needed. Mutation broadcasts those labels through all aliases. This broadcast is exactly what has not been supplied by an immutable representation.

A shallow-copy counterexample retains two row references to count 1. Incrementing one row creates count 2 and changes count 1’s metadata. The other row’s old reference still has no next bucket and the old multiplicity. Its next increment can duplicate the count 2 bucket. The failure concerns naive current-view repair, not all native machines. Reader 49’s bounded-live metric repair and reader 50’s final-one-profile rectangle are controls that already avoid this failure under their own hypotheses.

Evidence and continuity

The new checker passed 9,432 histogram sequences and 77,790 prefix checks, including duplicates, count gaps, disappearing groups, modulus 1 and long unequal counts. It also passed 10,125 observation-lift transitions and a stale-alias witness. Python integers appear only in semantic oracles and the algebraic finite audit. The histogram algorithm uses mutable links and unit stacks; it is explicitly not a native implementation. The written argument establishes the general constant local bound.

Seven surfaces: original source/checker only; supporting claims mapped without changing CLAIMS.yaml; proof plus finite evidence retained; failed direct interning/import and shallow-copy routes recorded; native moving-class frontier preserved; reader synthesis in the existing program and this linked argument; Site deployment remains deferred because the unrestricted native construction is still open and the current public K=1 result remains accurate. No source-parity or deployment claim is made for this addition.

NEXT: Construct or refute a specific persistent current-count-bucket service for unit-incremented ordinary-name rows. Test shared finite correction actions against stale aliases, disappearing buckets and short repeated names. Preserve final-only K=2, literal radiality and late pins; do not return to generic moment arithmetic without a concrete operation-level advantage.

ACCOUNTING: Scheduled 79->80 once; dedicated review completed with three primary proof comparisons. Last dedicated 175, next 81 research, next dedicated 85. Mathematical 116 and window 0 unchanged; last reassessment 116 retained. No new root admission or major-result count.

Validation receipt

The new finite checker, unchanged residue-selector control, Python compilation, JSON/cadence checks and whitespace checks passed. No shared verifier or root claim changed, so the narrow research checks were used; the previously recorded full-suite failures were not rerun or reclassified. No public deployment was performed.