Rule out one mixed affine helper for the QQ16 interface

research/qq16-affine-helper-mix-seed98c45/PROOF.md

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+# One mixed affine QQ16 helper cannot recover the lost input
+
+## Statement and scope
+
+Fix the source-bound QQ16 `F0/B1` pair, retained ordinary schedule, and
+standard observations (selected outputs, completed-X descendants, and all five
+forward-query histories). On a finite regular cover `G`, let
+`B_s : Q^(96|G|) -> Q^(8|G|)` be the actual ordinary-source query operators
+for `s=0,1,2`.
+
+Allow exactly one additional eight-coordinate helper, immediately before
+`F0`, with response
+
+```text
+H' = H + B0 X_U + (M lifted cover-wise) B2 X_U + c,
+```
+
+where `M` is any rational 8x8 matrix, `H` is an arbitrary but common dirty
+pre-state, and `c` is any fixed 8-vector lifted identically at each cover
+location. For every `M`, `H`, and `c`, the augmented observation map is
+noninjective. More precisely,
+`dim B2(ker[B0+M B2; B1]) >= 32|G|/7`. This is an information obstruction,
+not a native cost or new `kappa` result.
+
+## Pinned rank input
+
+The exact three-query rank bound used here is the source-checked result in
+[PR #2104](https://github.com/CrocSwap/integer-mult-bounds/pull/2104), head
+`4bdc574da394a39f955e8cf4637075649c43a632`, exact base
+`0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`. Its `PROOF.md` SHA-256 is
+`9c2ad4600d65dd3354db4bae5ac78d13fedf2da6e500224fafaeed596846c887`.
+That proof pins the QQ16 source/certificate and checks, for each pair
+`(0,1),(0,2),(1,2)`, that the 96 relative route labels are distinct and the
+cross-Gram identity coefficient is `192 I_8`; it also checks the actual
+ordinary column norms and Gram. The local source-specific verifier from that
+exact PR head was run read-only against this base with Python 3.14. It printed
+`PASS_QQ16_EXTRA_QUERY_RESIDUAL rank_stack>=144|G|/7 missing_A>=ceil(32|G|/7) no_new_kappa`.
+At the last successful hosted-status check at 19:52Z, PR #2104 was open and
+its checks were pending. A later status refresh hit the GitHub API rate limit,
+so its current hosted status is unverified.
+
+For `N=|G|`, those exact route facts imply
+
+```text
+B_s B_s* = 48 I_(8N),
+tr((B_s B_s*)^2) = 18432 N,
+tr(B_s B_t* B_t B_s*) = 1536 N  (s != t).
+```
+For `S=[B0;B1;B2]` and `K=S* S`,
+
+```text
+tr(K)   = 1152 N,
+tr(K^2) = 64512 N,
+rank(S) >= tr(K)^2/tr(K^2) = (144/7) N.                (1)
+```
+The rank bound is Cauchy--Schwarz on the nonzero eigenvalues of `K`. The
+current package uses this exact predecessor lemma; it does not rederive the
+route labels.
+
+## All-matrix kernel argument
+
+Let `A_M = B0 + (M lifted cover-wise) B2` and `T_M=[A_M;B1]`. Both query
+blocks have `8N` rows, so `rank(T_M) <= 16N`. Appending `B2` recovers exactly
+the row space of `S`, since
+
+```text
+A_M = B0 + M B2,       B0 = A_M - M B2,
+row([A_M;B1;B2]) = row([B0;B1;B2]).
+```
+This holds for every rational `M`, including singular `M`. Therefore (1)
+gives
+
+```text
+rank([T_M;B2]) - rank(T_M)
+  >= (144/7)N - 16N = (32/7)N > 0.                     (2)
+```
+The left side is `dim B2(ker T_M)`. Rationality of all maps means the
+rational nullspace spans the real nullspace. Thus some rational `h` satisfies
+
+```text
+B1 h = 0,       B0 h + M B2 h = 0,       B2 h != 0.   (3)
+```
+Clearing denominators preserves (3).
+
+Use the same `H,c` for inputs `h` and `0`. Equation (3) makes their new
+helper records equal. The affine offset and dirty pre-state cancel in the
+difference.
+
+## Existing selected-value witness
+
+For any `h` satisfying (3), use the source-pinned QQ16 input compensation on
+pair 0:
+
+```text
+X_U = h,                             Y_U = 0,
+X_A = -P_(0,A) B0 h - P_(2,A) B2 h,  Y_A = -X_A,
+X_T = P_(0,T) ell_(0,A) P_(2,A) B2 h, Y_T = -X_T.
+```
+
+The formulas are linear in `h`. The selected pair sums vanish; the ordered
+`F0/B1` equations use only `B1h=0`, `B2h!=0`, and the pinned two-sided
+selected-section identities. They leave a nonzero initial selected value
+`X_(0,T)` but zero selected outputs, completed-X descendants and all five
+forward histories, as detailed in
+`research/joint-query/evidence/QQ16-SELECTED-WITNESS.md`. By (3), the mixed
+affine helper also agrees with the zero input. No decoder from these
+observations can restore every original selected input.
+
+The zero-cost relaxation grants every potentially relevant expense for free:
+
+| Expense | Favorable charge |
+|---|---:|
+| Source access and query arithmetic | 0 |
+| Helper storage and output hosting | 0 |
+| Helper update, inverse, and dirty-state restoration | 0 |
+| Frame and address transitions | 0 |
+| Local calls and recursive child operations | 0 |
+| Recursive precision | 0 |
+
+It still fails injectivity, so there is no implementable cost or moment gain
+to invoice for this interface.
+
+## Limits
+
+The theorem allows all rational 8x8 mixers and fixed offsets on one helper.
+It keeps the actual `F0/B1` maps, ordinary tail, observation set and source
+cut fixed. It does not cover a second independent helper, an input-dependent
+offset, another source field, a changed first pair or tail, or a later read of
+other ordinary banks. `32|G|/7` is an information dimension, not a paid
+recursive-child rank.
+
+`verify.py` checks the rank/row-space corollary on exact rational controls and
+checks the exact trace arithmetic. It does not replace the pinned route
+verification of (1) or replay the full inherited native trajectory.

research/qq16-affine-helper-mix-seed98c45/README.md

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+# QQ16 mixed affine helper obstruction
+
+**Scoped negative result; no supplier or kappa claim.** One added 8-coordinate
+affine helper `H + B0 X_U + M B2 X_U + c` cannot restore all original
+selected inputs under the fixed QQ16 `F0/B1` pair and retained schedule, for
+any rational 8x8 mixer `M`. The proof converts the exact three-query rank
+bound from PR #2104 into an all-mixer kernel bound. The helper and every
+operation needed to implement or restore it are granted free, yet the
+observations remain noninjective.
+
+## Reproduce
+
+From the repository root:
+
+```sh
+python3 -I -O -B research/qq16-affine-helper-mix-seed98c45/verify.py
+python3 scripts/research_package.py --package research/qq16-affine-helper-mix-seed98c45 --check-only
+```
+
+Expected direct result: `PASS_QQ16_AFFINE_HELPER_MIX cases=4 positives=4 controls=3 no_new_kappa`.
+Expected metadata result: `PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)`.
+Registered replay uses Python 3.12; hosted CI is the runtime check if that
+interpreter is unavailable locally.
+
+The exact route lemma is pinned to PR #2104 head
+`4bdc574da394a39f955e8cf4637075649c43a632`, `PROOF.md` SHA-256
+`9c2ad4600d65dd3354db4bae5ac78d13fedf2da6e500224fafaeed596846c887`.
+Its source-specific verifier ran locally on Python 3.14 against the exact
+base. At the last successful status check at 19:52Z, that PR was open with
+hosted checks pending; the later refresh hit the GitHub API rate limit, so
+current hosted status is unverified. Full argument and scope limits are in
+[PROOF.md](PROOF.md); original source and selected witness remain unchanged.
+
+## Attribution and limits
+
+Credit Ali Romman and PR #1969 for the QQ16 maps and selected witness; Cinna
+Davis PR #2104 for the source-bound three-query rank result; Cinna Davis PR
+#2079 for the B2-only observation this result extends. Adam Gall PR #2093 and
+#2086 address related but different observation/logger cases. Prepared with
+substantial OpenAI Codex assistance. No independent human or formal-kernel
+review is claimed.
+
+## Open next questions
+
+- **Can a second independent ordinary-to-selected record restore injectivity?**
+  The row-count cap used here would no longer apply. Specify the second source
+  and compute the joint kernel before native synthesis.
+- **Can one changed ordinary-tail map retain the erased selected value?**
+  Derive its inverse and observations before expanding the controller.

research/qq16-affine-helper-mix-seed98c45/RESULT.json

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+{
+  "dimension_bound": "dim B2(ker[B0+M B2;B1]) >= 32|G|/7 for every rational 8x8 M",
+  "no_kappa_change": true,
+  "outcome": "negative-result",
+  "remaining_limits": [
+    "Uses the exact route-rank theorem in open PR #2104; its hosted checks were pending at 19:52Z and the later status refresh failed on API rate limit.",
+    "No native helper implementation, complete physical cost, supplier admission, precision proof, or global controller."
+  ],
+  "reproduction": "python3 -I -O -B research/qq16-affine-helper-mix-seed98c45/verify.py",
+  "schema_version": 1,
+  "source_rank_predecessor": {
+    "git_blob": "388bca8797d90d385d984ad8bfa72af20cf80123",
+    "head": "4bdc574da394a39f955e8cf4637075649c43a632",
+    "local_check": "Downloaded exact-head verifier run against pinned base on Python 3.14; PASS_QQ16_EXTRA_QUERY_RESIDUAL.",
+    "name": "QQ16 three-query route-rank proof",
+    "path": "research/qq16-extra-query-residual/PROOF.md",
+    "sha256": "9c2ad4600d65dd3354db4bae5ac78d13fedf2da6e500224fafaeed596846c887",
+    "status_at_claim": "open; hosted checks pending",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2104",
+    "verifier_path": "research/qq16-extra-query-residual/verify.py",
+    "verifier_sha256": "68e3c1c8c044a9b7ae245b189ef2154b6cfe6c96faca87349cd336ebf731fd2e"
+  },
+  "status": "SCOPED_NEGATIVE_RESULT"
+}

research/qq16-affine-helper-mix-seed98c45/SOURCES.json

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