[petalas]

Summary

Scoped algebraic construction and exact optimality proof for completed partial external-source echoes. This answers the normalized-block part of PR4860's second open question. Every partial receiver update is complete when observed; arbitrary borrowed workspace is restored at return.

H_i(D,W) = (D + L_i W, W), with independent arbitrary L_i
forward/inverse prices c_i^+, c_i^-; injection prices a,b > 0

exact price = sum_i(c_i^+ + c_i^-) + ceil(s/2)(a+b)

An alternating forward/inverse schedule attains the lower bound for every length and every positive price tuple over characteristic-zero fields. The first strict improvement uses six calls for two completed updates, compared with eight calls for two separately restored echoes. It removes two source injections and retains every child call. The proof allows arbitrary label interleaving, repetition and integer source shifts.

Only research/completed-partial-echo/ is added. This contribution was produced by Nick Petalas. Credit Nick Petalas / PR4860 for the question, Ross Richardson / PR1287 for cyclic batching and complete dirty echoes, Sagar Thakrar / PR1227, mightyOil / PR1203, Adam Gall / PR1205, Hanson Sun / PR1488, Cinna Davis / PR1391 and their retained restoration lineage. Predecessor contracts are cited premises, without replay or audit of their results.

Evidence

Before, the normalized completed-observation boundary has no sharp source-injection tariff in PR4860. After, PROOF.md gives a matching lower bound and explicit universal schedule, including the boundary that prevents the saving when workspace must be restored at every observation.

Local CPython 3.12.15 on macOS arm64 passes:

• 1,600 exact weighted schedule cases through 64 blocks, including rational and unequal direction prices.
• 448 literal matrix fixtures, all 2,328 X/W/D basis columns and 6,294 complete output observations.
• Eight independently implemented bounded shortest-word searches, including unrestricted repeated labels and nonbinary source shifts; the unit-price minima are four and six.
• Seventeen semantic and input rejection controls. Normal and optimized generated receipts agree byte for byte; direct commands took 0.30 and 0.31 seconds, below their 120-second limits. These are local replay durations, without a speedup claim.
• Registered metadata/pins and both replay commands; in-memory compilation and whitespace checks; exact committed bytes, executable modes and declared source ancestry for all eight package/input files.
• Official committed export integrity and both declared inputs; normal and optimized read-only Git-free replay with an empty PATH; all nine exported files retain bytes and modes. Corrupting either predecessor text input rejects in both modes. Export receipts equal the local receipts.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/completed-partial-echo/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/completed-partial-echo/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/completed-partial-echo
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/completed-partial-echo --check-source-history --check-modes

Substitute another CPython 3.12 path elsewhere. Expected markers are PASS_COMPLETED_PARTIAL_ECHO, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier emits a generated receipt bound to all six package files and both predecessor text inputs by SHA256; no handwritten receipt is used.

Limitations

The theorem requires a uniform word over independently arbitrary linear maps and indivisible source-preserving blocks. The observer sees D only and permits W_initial+X at odd checkpoints. If W must be restored at each checkpoint, the exact injection price is s(a+b), so the saving disappears. Direct source queries, opened child bodies, map-dependent schedules, feedback, extra data fields, retained query records and adaptive observations have different contracts.

Every block's complete normalization, literal inverse and internal scratch costs remain in its stated price. Observation execution and physical routing are separate. No actual selected caller, native/precision/ownership/tape implementation, complete compiler or outer funding is supplied. The five inherited framework assumptions remain. No new kappa or selected-frontier change. The universal theorem rests on the written proof; independent human and formal proof-kernel review, a selected cold replay, Lean and Docker are unrun.

Open next questions

• Bind one actual cyclic-equivariant caller that permits shifted borrowed workspace at a completed output observation. Start with the complete normalized block, its literal inverse and the observer's read set, then price all source movement.
• Determine the optimum when cyclic normalization and injections may interleave inside the blocks. That needs a new literal child alphabet and complete observation contract; this proof treats the blocks as indivisible.

Related open work

No other open PR computing this completed-observation injection tariff was found in the refreshed 361-PR body snapshot and live lists. Sparse local notes, unpublished work and arbitrary historical diffs remain coverage limits.

• Physical operation frames with deferred readouts: conditional κ = 1.11192082577e-4 #126 changes physical operation frames and deferred readouts on a frozen scalar DAG while retaining its source injections. This PR prices a uniform normalized-block observation schedule; it does not instantiate or check that physical network or compare its numerical certificate.
• Prove the exact weighted tariff for multilinear dirty-input queries (no κ) #6393 prices Boolean multilinear evaluation through dirty input rows with one final output. This PR requires ordered complete observations of independently labelled linear blocks; it does not recompute that cube tariff.

Merged #2020/#5968 have fixed binary drifting children and partial receiver observations. #3417 explicitly excludes intermediate observations. Their numerical results are not recomputed here.

The completion refresh inspected all twelve open account PRs created since the CLAIM, including this PR. None of the other eleven computes this completed-observation injection tariff. No shared-instance numerical disagreement was found, and no additional computations audited another contribution.

CI notes

Both hosted checks passed at head cab3df8a0e45d3ba72a898c062afe7bcc2a294cc: plan and replay (research/completed-partial-echo), in run 41528694810. The completed checks were read once. Run metadata confirms conclusion success and the matching head. Hosted replay certifies the registered finite checks and bindings; the universal theorem remains a written proof outside a formal kernel. All registered commands and publication/export checks also passed locally.

Coordination: #1166 (comment)

Merge Danger

Door: two-way. The contribution is an isolated research package.

Blast Radius: package. Shared code, frozen source inputs and the selected record are unchanged.