Research: PR #3189's eighteenth-position QQ16 error witness survives the local native core

research/qq16-native-noise-lift/README.md

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+# The QQ16 eighteenth-position witness survives the local native core
+
+**Finite native error-channel certificate.** The B1,2 child-error witness in
+[Tekalign's PR3189](https://github.com/CrocSwap/integer-mult-bounds/pull/3189)
+survives the literal retained native core, including its changing frames and
+occupied residual adapters. It is not an artifact of replacing those maps by
+identities. This answers PR3189's second **Open next question** for one local
+512-coordinate payload factor. It does not supply a rounded recursive child,
+a global producer accuracy bound, or an affordable consumer.
+
+All event numbers below are zero-based in the expanded `B_1_2` program of the
+unchanged [joint-query certificate](../joint-query/certificate.json.gz).
+The retained backward core starts at event 1864. Its suffix after the named
+injection contains no cover maps or new native-line wrappers.
+`ONE` is the real vector of 512 ones. The complete Gaussian-integer injection
+is recorded explicitly in `RESULT.json`, so it can also be used without relying
+on its frame-based description.
+
+| Item | Exact value |
+| --- | --- |
+| Injection | After child 13489: `['climb','A',344,226,236,1]` |
+| Error direction | `w = T_236 ONE`, source frame basis `[416,75,19]` |
+| Components of w | Real and imaginary entries in `{-1,0,1}`; squared Hilbert norm 512 |
+| Attaining cut | After 26472: `L93 += -L103/8`, local frame 1 |
+| Error at that cut | `(-81921/262144) ONE` |
+| Largest component denominator exponent at completed events | 18, first attained at this cut |
+| Largest absolute component in this error trace | `81921/4` |
+| Final data error | `L75 = -ONE/2`, `L93 = ONE/2`; all other data fields zero |
+| Final helper error support | L337, L338, L339, L341, L343, L346 |
+| Final private error | All eight fields zero |
+
+The helper defect is material: this error channel does **not** restore borrowed
+state. The six helper squared norms are respectively `32,128,128,32,128,128`.
+Their full component histograms and vector hashes are in `RESULT.json`; for
+example the final coordinate zero of L337 is `i/2`. The originally perturbed
+L344 itself has zero final error.
+
+## What this proves about a grid
+
+Let the ideal and perturbed computations agree until this completed child.
+Insert the additive defect `h w`, for any positive grid quantum h, then apply
+the subsequent native maps and scalar operations exactly. Linearity makes the
+error at each cut h times the checked trace, regardless of the underlying
+ideal data and dirty-helper values. This is a completed-child defect model,
+not a claim that a particular child or rounding algorithm produces this error.
+
+The injection is on the Gaussian component lattice `h Z[i]`. At the attaining
+cut, a real component expressed in units of `h/2^17` is `-81921/2`, which is
+not an integer. Thus 17 fractional guard positions cannot represent even this
+one channel exactly. If the ideal component at this cut is on the working
+grid, the perturbed component is off it; rounding that component to the grid
+incurs absolute error at least `h/2^18`. The exact error trace is on
+`h/2^18 Z[i]` at every
+**completed program event**, so 18 is necessary and sufficient for this
+particular direction and these cuts.
+
+This is a lower-bound witness for a more general native error-lattice contract,
+not a sharp upper bound for every child direction. It does not bound the
+intermediate arithmetic inside a child recipe. The final data and helper
+errors need only one fractional position, so an endpoint-only grid check would
+miss the intermediate requirement.
+
+## Evidence and reproduction
+
+From the repository root, standard-library Python 3.12:
+
+```sh
+python3 -Werror -I -B scripts/research_package.py \
+  --package research/qq16-native-noise-lift
+```
+
+Expected markers:
+
+```text
+PASS_QQ16_NATIVE_NOISE_LIFT guard=18 coefficient=-81921/262144 data_fields=2 helpers=6 private=0
+PASS_QQ16_NATIVE_NOISE_FORWARD guard=18 complete_trace_and_endpoint=matched
+PASS_QQ16_NATIVE_NOISE_DUAL coefficient=-81921/262144 checks=4 coordinate=0
+```
+
+Nine behavioral test groups and the registered-package PASS follow. The
+primary replay is read-only; `verify.py --write PATH` exclusively creates an
+absent receipt. It authenticates four live inputs and three archive members.
+The regenerated receipt compares every proof-relevant field, including the
+explicit injected vector, all endpoint vectors' canonical hashes, every
+changed-field cut's trace digest, source event bytes and the scalar-projection
+positive control.
+
+The primary interpreter executes the 13,529-event local suffix on Gaussian
+dyadic vectors. Zero operands may skip a linear native kernel because that
+kernel maps zero to zero; no cost saving is attributed to this shortcut.
+There are 16 child occurrences, nine adapter occurrences, four residual
+adapter occurrences and 584 scalar occurrences with nonzero changed outputs.
+The other suffix events are still visited and authenticated.
+
+`independent.py` imports neither the primary checker nor predecessor executable
+code. Its sparse Gaussian `Fraction` engine independently executes the complete
+single-channel suffix, matching the full changed-field trace digest, the 18-bit
+guard, all 512 attaining components and every endpoint vector. It also applies
+backward adjoints of the literal monomial/line recipes. Starting from coordinate
+zero of L93 at the attaining cut, it derives the response to the explicit
+witness input. Three additional dual calculations check terminal L75, L93 and
+the occupied residual output L337. This is independent arithmetic coverage of
+one channel, not an all-child census or an independent review of native semantics.
+
+Controls reject changed global phase, an unframed injection, injection before
+instead of after the child, a changed attaining ADD sign, a residual-adapter
+orientation error, source corruption, changed injection bytes and unknown
+events. Zero and signed dyadic scaling are positive controls. A nonzero complex
+adjoint identity tests the independently implemented kernel order; duplicate
+JSON keys and boolean/integer receipt confusion are also checked.
+
+## Scope, costs and expansion decision
+
+The certificate concerns the actual **local retained core recipes** at one
+nine-axis payload factor, after a single independent completed-child defect.
+It checks local physical component values, including nonzero residual outputs.
+It does not execute the earlier B1 cover sweeps or validate a global owner
+placement, 73-axis embedding, tensor-depth extension or selected supplier.
+Those are inherited or open interfaces, and no payload-only replacement for
+a cover action is made here: no cover map remains in the studied suffix.
+
+The ideal native recipes and their access interpretation are inherited from
+the reviewed QQ16/native work. The numerical replay checks their exact linear
+response, not their implementation at the retained recursive tariff.
+Actual child-return rounding and correlations, errors within recipes, dirty
+input bounds, fixed-format overflow and a receiver-compatible grid remain
+unresolved. Nor does this certificate price storage, movement, cleanup,
+hosting, the F17 consumer or a complete recursive invoice. These unknown costs
+are not zero. [PR2965's conditional output-error transfer](../qq16-producer-defect-transfer/README.md)
+is unchanged; its producer accuracy and restoration premises are not supplied.
+
+Family decision: **narrow and stop** after this one handoff. No new search,
+controller implementation, selected-frontier change or kappa improvement is
+claimed. The selected cold replay, complete predecessor suites, all 7,004
+projected error columns and a formal proof kernel were not run. The result is
+author-checked with a separately implemented arithmetic check; no independent
+human review or hosted CI pass is claimed.
+
+## Open next questions
+
+Can an actual child implementation exclude or correctly price this error
+channel? Begin with the named `226 -> 236` child on L344, its returned-value
+and defect lattices, and the following receiver operations. A returned-value
+grid and a norm bound alone do not establish defect-lattice closure. This
+requires a specific child implementation and dirty-input/grid contract.
+
+Can the all-cut local guarantee be extended to the actual global caller's
+physical grid while restoring every borrowed helper? Bind the retained-core
+embedding and owner placement, then propagate its actual errors and charge
+native arithmetic, movement, storage and F17 consumption together. The local
+channel certificate does not establish an affordable full producer/consumer.
+See `TASK.json` for the structured handoff and stop rules.
+
+## Sources and credit
+
+Base main: `463113d675af7410ac91d53a4fed51ec6494eb9a`, priority revision 6.
+The claim is in discussion #2727. `SOURCES.json` pins the live certificate,
+selected archive, native primitive evaluator and PR3189 receipt. Exact
+predecessor heads are recorded in `TASK.json`:
+
+- Tekalign (`tekuuu`), PR3189 `70537595e46444c8a63455523d9851a5efe55385`:
+  the child-error lattice criterion, projected witness and explicit native-lift handoff.
+- dcharb78, PR3181 `80fc72d14546088e9fce3c06dc12bc67270da0af`:
+  ideal scalar interior/grid and adjoint work motivating the audited implication.
+- Tekalign, PR3017 `0f493f4c264fbc70ee14364b68af260d858c143c`:
+  exact Gaussian-dyadic primitives and local native execution conventions.
+- Ross Richardson, PR3067 `e1e28b4070ae52efa5a9d0aa826bc588be7206f2`:
+  physical F0 execution/rounding methodology. The new primary event dispatcher
+  adapts the PR3017/3067 conventions; it is not an independent native implementation.
+- Nick Petalas, PR2965 `526f2c242202b31b15e09a7502dfb8ff757b2bad`;
+  Arham Tahir, PR2995/3036; Ali Romman and the joint-query team; Sagar Thakrar,
+  DLamdata, ywz65, starship-s, Douglas Colkitt and the retained selected/native lineage.
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+New work is Apache-2.0. No predecessor bytes or notices are changed.
+
+The checks use managed Python integers and collections, fixed 712 fields and
+512 coordinates, bounded serialized input sizes and explicit replay timeouts.
+They are research verifiers, not fixed-storage or constant-time machine programs.