Assessment161: live dependencies and coordinate recycling

8 October2026. Ordinary research, scheduled67. Written extension109/window5, last reassessment104 unchanged. Next68 is research; dedicated review70 remains. No root admission or literature-priority claim.

Result and strongest exact statement

Reader41 owns the complete proof. A fixed finite permutation can preserve all surviving row masks exactly when old/new joint observation profiles have equal multiplicities. That criterion alone could require reading many rows. Its linear instance has a charged compact implementation: the surviving coefficient space has two injective coordinate embeddings; their isomorphism extends to GL(r,2). The current reader absorbs this invertible transport and all old masks remain unchanged. Reference-counted coefficient contents supply the constraints without scanning old row keys or payloads.

This yields native recognition for arbitrary intermediate-key lengths and one common target width, with a checked live registered rank bound. A ready candidate registers during its next own target spelling while its key capsule is open. The exact prefix learner of40 guarantees that spelling before the final row certificate. The earlier final-rank-only same-width class is preserved, not silently generalized: the new condition is order-sensitive and stricter in rank. The finite small-target-width branch uses literal per-row target masks and final rank counting; it does not bound the intermediate-key universe.

Sustained attack and dependent closure

A direct occupied-key rebuild would still cost one operation per old key. The latest dense row can leave only a width-dependent suffix, independent of that population. Repairing a row when its name is looked up also fails as stated: its coefficient strings become available only at the leaf, after the spelling that was supposed to fund their scan has ended. No general impossibility follows.

A monotone historical basis avoids reinterpreting old masks, but can retain arbitrarily many obsolete directions even when only one is live. Discarding and reusing an axis without a transport loses surviving answers. The successful mechanism instead preserves every surviving coefficient equation and extends that partial coordinate isomorphism to a finite invertible map.

The first schedule considered a global registration queue. Its result could arrive after the affected row key had closed, leaving an uncharged remote payload repair. The completed construction waits for the candidate’s next own record. Its key is already open when its target spelling begins; all coefficient work fits that target word; the normal capsule close commits the payload and reader together. This carries the algebra through its native routing obligation. There is no supplied handle or deferred key replay.

Proof audit

The writing agent audited the actual draft. No separate review is claimed.

• The invariant is on all of F2^r, including syndromes outside the current physical image. It is initialized by finite local-table substitution and preserved by invertible character transport. Agreement merely on the current image would not justify future expansion.
• With raw row coordinates c_old and c_new, Q satisfies c_old Q=c_new. The reader update is C_new=C_old Q; the new stored row mask uses C_new inverse. These directions matter. The finite experiment checks nonidentity transports.
• Old/new ranks may differ. Complete independent bases of the surviving embeddings to bases of F2^r; this proves an invertible extension exists. Padded physical coefficient rows can be zero; they are not claimed independent.
• The coefficient index counts each registered row’s original basis references, not just one global basis. Zero counts permit deletion. At most2^r-1 distinct nonzero vectors remain after a legal transaction; at most that plus2r are temporarily considered. Equal vectors are identified by bit scans.
• All surviving rows’ spectral dependencies lie in the span of their retained basis references. Constraints on the finite indexed set therefore suffice for every old row, including nonlinear truth tables and repeated shared directions.
• Unary multiplicities allow unboundedly many rows. Their updates are constant pointer operations; coefficient matching, dependence tests, basis coordinates and construction take O_r(n) bit/local work, charged to actual target symbols.
• No other row registers within one target spelling. Ready-at-start selection avoids charging a late learner completion to already consumed bits. Row name capsules close at the actual current record, with no ancestor walk or remote callback. A short key supplies no assumed width budget.
• A new tier marks the row uncertified but keeps its old registered dependencies until replacement. They count toward the gate even if provisional or stale. This makes the input restriction explicit and avoids a free withdrawal.
• The last legal tier retains enough own target bits for a complete registration word and punctuation. Final certification changes no global coordinates. At the right boundary every accepted row is certified and has its final registered mask. There is no global compilation or EOF job.
• For bounded target widths, per-row literal masks are fixed-size even with unbounded key names. A fixed array of unary Fourier-support counts checks final rank. This replaces the old finite-whole-right-relation shortcut, which would be invalid once intermediate keys have arbitrary lengths.
• Old versions retain old reader roots and masks. This is structural persistence, not a grant of free arbitrary version selection. Actual names continue to pay current recognition routes. The PEG conclusion retains reversal.

Finite experiment

tools/research/check_live_rank_transport.py uses literal target point sets and direct character sums as semantic oracles. It passed:

• 65,536 old/new pairs of two-observation profiles on four coordinate atoms, checking profile multiplicities against all actual permutations.
• 1,920 registration proposals at capacities1-3 and widths up to6:1,345 admitted, 575 rejected for rank, including60 nonidentity transports.
• 672,176 row membership comparisons,38,492 OR-product queries and128 retained versions. Every surviving stored mask is asserted byte-for-byte unchanged.
• 4 forced nonidentity transports preserving a nonconstant survivor while another half-row becomes full;88 recycling updates in8 growing historical-rank examples;3 final-rank-zero histories that exceed the live bound;2 failed one-bit permutations after forgetting a still-needed independent direction.
• 24 symbolic preparation/registration horizon inequalities at unbounded-branch cutoff widths. Keys in the semantic experiment have unrelated lengths.

The script does not implement native pointer timing. The written charge proves locality and the per-symbol bound; symbolic clock inequalities test its stated margin. The experiment tests successful transports and explicit failures, not only examples constructed in one permanently fixed basis.

Scholarly judgment and relation to controls

Classify this as an extension using standard linear algebra and the Lab’s existing persistent scheduling constructions, with high confidence in that classification. Extending a subspace isomorphism to an automorphism and composing changes of basis are known methods. The additional content is their immutable-mask invariant, reference-counted surviving-dependency contract and fully charged same-record registration. No literature priority is asserted. No new external theorem is imported here;40’s Kushilevitz-Mansour comparison and159’s two-paper primary review remain applicable. This ordinary research assessment does not increment the last dedicated review or satisfy the future review70 requirement.

Reader40’s spectral learning and determining-prefix proof are reused with a new registration endpoint. Assessment141’s adoption debt is avoided only because old mask meanings are preserved by an explicit transport, not because old rows are assumed free to scan. Assessment154’s ready-readout condition is met before composition. Assessment156’s final-rank warning remains substantive: the live rank restriction cannot be inferred from a small final rank. Replay slices, affine-fragment joins, persistent causal algebra, unary coherent algebra and native SAT controls stay unchanged. Entry129’s imported bounds retain their matching/positive-path contracts and growing alphabet.

Seven-surface disposition and next target

Source custody: only original proof, checker and review are added; no primary files are copied. Claim mapping: written extension109, no CLAIMS.yaml admission. Proof/evidence: one maintained reader41 and finite checker. Failures: late leaf repair, global queued callback, historical-basis accumulation and unsafe direction deletion are closed only for their stated representations/schedules. Frontier: temporary excess live rank with arbitrary intermediate keys; then heterogeneous target widths and independent joins. Reader synthesis: current research overview. Site: existing reader and homepage entry, same audience, parity in integration state.

Next test: allow incompatible provisional classifiers to coexist, then reconcile without rescanning an unbounded old-key population. A proposed finite interface must charge both recovery and late readout; a semantic final quotient is not sufficient. Do not reset to the already completed missing-quarter or Horn bridge.

Validation

Full make verify:1490 passed,1 skipped,5 failed in226.06s. These are the unchanged four custody-provenance failures and unavailable-lake frontier test; later make targets were not reached. The semantic checker and Python compilation passed. Final Site/scoped checks and publication parity are recorded in integration state. The next substantive result110 closes the six-result window and requires its reassessment; it is not automatically supplied by the next scheduled assessment.