Prove QQ16 shear plus an old-chart read still loses selected inputs

research/qq16-shear-old-chart-kernel/README.md

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+# The QQ16 shear still loses selected inputs with a dense old-chart record
+
+Grant the paired A-to-T shear between F0 and B1 and retain the complete
+ordinary old-chart query `q0 = B0 X_U^initial`. These changes still do not
+restore all original selected inputs. For a finite regular cover of size
+`N`, at least `ceil(32N/7)` dimensions of selected-input differences remain
+invisible under the observation contract below.
+
+This is a scoped linear-algebra obstruction, not a new supplier or κ.
+It answers the dense-observation handoff in [PR #2326](https://github.com/CrocSwap/integer-mult-bounds/pull/2326).
+It does not claim that every dense query fails. The new step is a graph
+intersection for the changed first pair; the three-query rank estimate
+and ordinary-tail argument are credited predecessor results.
+
+## Exact scope
+
+Use the conserved F0 and B_1_2 maps from
+[the retained QQ16 package](../joint-query/README.md), with its A, T and U
+port sets, selected sections, owner action and ordinary tail. All arrays
+are arbitrary over the rationals. Restrict the counterexamples to fields
+constant in payload coordinates, but allow independent cover-owner values.
+This preserves the actual owner action; it does not replace routes by a
+nonfaithful quotient.
+
+After the complete F0, insert the logical paired shear from
+[PR #2216](https://github.com/CrocSwap/integer-mult-bounds/pull/2216):
+
+```text
+X_T <- X_T + C X_A
+Y_T <- Y_T - C X_A.
+```
+
+Here `C` is the identity pairing from the eight ordered A fields to the
+eight ordered T fields. No ordinary field changes. The proof also works
+for any fixed invertible rational-linear `C` between those selected spaces.
+
+The observations are:
+
+- `q0 = B0 X_U^initial`, retained without loss;
+- every selected output immediately after B1 and at all later cuts;
+- all five forward-query histories;
+- all final completed-X descendants under the retained endpoint typing.
+
+A separately retained transient F0 A/T output, a later ordinary read and
+the final former-Y hole are **not** observations. Later homogeneous
+selected-only linear controls are allowed. The observation record does
+not feed back into ordinary state. The final decoder may be nonlinear:
+two identical observation tuples still cannot have different answers.
+
+Grant the shear, read, storage, native frame conversions, address work,
+cleanup, hosting and precision zero cost. Recovery fails even in this
+favorable relaxation, so no construction or moment claim is made. In
+particular, this does not adopt PR #2216's bare-scalar native pricing.
+
+## Ordered kernel construction
+
+Write `B_s = R_s iota_U` and `K_1 = K_(1,U)`, so `B1 K_1 = 12 I`.
+Let `P0A = P_(0,A)`, `P0T = P_(0,T)` and `P1T = P_(1,T)`.
+Every selected section and its selected query are two-sided inverses.
+Define the following ordered maps from query coordinates:
+
+```text
+L = ell_(1,T) C P0A
+M = (ell_(2,A) + ell_(2,T) C) P0A.
+```
+
+`L` is invertible. Take a query vector `c` and an ordinary vector `w`
+satisfying
+
+```text
+B0 w = 0,
+B1 w = -11 L c,
+B2 w = M c.                                           (1)
+```
+
+Only pair 0 is initially nonzero:
+
+```text
+X_U = w,       Y_U = 0,
+X_A = -P0A c,  Y_A = P0A c,
+X_T = P0T c,   Y_T = -P0T c.                           (2)
+```
+
+The other pairs and borrowed helpers may be set to zero: this is an
+allowed subspace of an arbitrary-dirty-input contract. Both selected pair
+sums vanish. The initial selected defect is injective in `c`.
+
+The F0 query is `f0 = B0 w - c + c = 0`. Its exact output is
+
+```text
+X_U = Y_U = w,
+X_A = P0A c,  Y_A = -P0A c,
+X_T = Y_T = 0.
+```
+
+After the shear, `X_T = C P0A c` and `Y_T = -C P0A c`.
+Set `a = L c`. The future query at B1 is
+
+```text
+q2 = R2 Y = B2 w - M c = 0.
+```
+
+The systematic logger `E2` therefore gives `u_A = P_(2,A) q2 = 0`,
+`u_U = w` and `u_T = -C P0A c = -P1T a`. Consequently
+
+```text
+b = R1 u = B1 w - a = -12a.
+```
+
+The conserved backward equations from the
+[ordered QQ16 proof](../joint-query/evidence/JOINT-QUERY-TRANSFER.md) give
+
+```text
+Y'_U = w + K_1 a = h,
+Y'_A = 0,
+Y'_T = u_T - P1T b - 11 P1T a = 0,
+X'_U = w - w = 0,
+X'_A = X'_T = 0.                                     (3)
+```
+
+No operators have been commuted. The output (3) is exactly the
+[retained witness's post-B1 form](../joint-query/evidence/QQ16-SELECTED-WITNESS.md),
+with an arbitrary ordinary value `h`. That tail argument does not require
+`B1 h = 0`: the first positive bridge puts `-h` in Y1, F2 has zero X
+source, and the negative bridge removes `-h` before B3. Later active
+pairs are zero. The terminal moves `h` to named final X0 with the
+former-Y native type. Every selected output, forward history and
+completed-X descendant is zero. Homogeneous selected-only controls keep
+the selected state zero.
+
+Finally `q0 = B0 w = 0`. Thus every solution of (1) gives the same
+allowed observations as the zero input, despite the selected defect (2).
+
+## Why nonzero solutions exist on the actual cover
+
+Let `S = [B0; B1; B2]`, a map into a `24N`-dimensional query space,
+and let
+
+```text
+H = { (0, -11 L c, M c) : c in Q^(8N) }.
+```
+
+Because `L` is invertible, `dim H = 8N`, and every vector of
+`im S intersect H` determines a unique `c`. The elementary intersection
+inequality gives
+
+```text
+dim(im S intersect H) >= rank(S) + 8N - 24N
+                         = rank(S) - 16N.             (4)
+```
+
+Every vector in this intersection has a rational preimage `w`, so (4)
+counts actual invisible selected defects, not just unconstrained query
+values. The zero element is the only one with `c = 0`.
+
+The second-moment argument in [PR #2104](https://github.com/CrocSwap/integer-mult-bounds/pull/2104),
+rebound to the retained packet by [PR #2153](https://github.com/CrocSwap/integer-mult-bounds/pull/2153),
+gives `rank(S) >= 144N/7`. Its finite premises are rechecked here:
+
+```text
+sum_(p in U) r_p r_p^T = 48 I_8,   ||r_p||^2 = 4,
+R_A R_A^T = R_T R_T^T = 4 I_8,
+g_(s,p) g_(t,p)^-1 distinct for all 96 p, each s<t in {0,1,2}.
+```
+
+A nonidentity regular permutation has trace zero. Hence, for `K=S* S`,
+
+```text
+tr K = 1152 N,
+tr K^2 = 64512 N,
+rank S >= (tr K)^2 / tr K^2 = 144N/7.
+```
+
+Combining this with (4) proves the stated `ceil(32N/7)` lower bound.
+The dimension is across the full owner array, not a per-owner native
+rank tariff. Clearing rational denominators and scaling yields dyadic
+inputs of arbitrarily small nonzero magnitude. This proves existence;
+it does not give a compact actual-cover vector or a precision bound.
+
+## Reproduce and verification boundary
+
+From the repository root, with Python 3.12 and the standard library:
+
+```sh
+python3.12 scripts/research_package.py --package research/qq16-shear-old-chart-kernel
+python3.12 -I -O -B research/qq16-shear-old-chart-kernel/verify.py
+```
+
+The checker binds the packet and proof inputs through [SOURCES.json](SOURCES.json),
+checks the actual three-chart route distinctions and Gram identities,
+and exercises (1)--(3) with exact rational noncommuting selected-section
+fixtures. It also checks the retained formal-h tail and deliberate
+failures. Finite fixtures do not prove the quantified intersection
+argument; the proof is written above.
+
+The registered normal and optimized replays passed on CPython 3.12.13:
+three sets of 96 relative routes, four rational basis fixtures, 17 tail
+events and six rejection-control groups. A Git-free relocated optimized
+replay preserved all 11 copied files; corrupting a copied source was
+rejected by its digest check. Separate Astra/max agent reviews found no
+blocking mathematical/spec or verifier-quality issue. These are agent
+reviews, not independent human review.
+
+No pending executable is imported. The packet's reviewed scalar/native
+binding is inherited, not cold-replayed. There is no full supplier replay,
+proof-assistant check or independent human review. The stronger later
+moment estimates are not needed or reverified here. The selected record
+and all frozen sources remain unchanged.
+
+## Open next questions
+
+Can a changed first-pair interface make a *different* eight-coordinate
+dense record sufficient? This matters because detecting a known witness
+is weaker than closing the complete kernel. Start by writing its ordered
+analogue of (1) and intersecting its query graph with `im S`. Actual
+native access, retention and a favorable complete bill remain blockers
+even if that algebra succeeds.
+
+Can one recover the privileged post-B1 selected cut used by
+[PR #2174](https://github.com/CrocSwap/integer-mult-bounds/pull/2174)
+from legal final records? Its two-record theorem is a different positive
+contract. The first step is an ordered reconstruction of that cut; the
+current obstruction neither supplies nor contradicts it.
+
+## Provenance and credit
+
+Base: `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`, priority revision 4.
+The accepted chooser nonce is `7dcbcbed93c608a2c89c3e8330a31a30`;
+`joint-query-controller` is a refined scope on the recursive-interfaces track.
+[Task metadata](TASK.json) records the claim, overlap and stopping rule.
+
+- Ali Romman, PR #1969 and the [retained notices](../joint-query/NOTICE.md):
+  local maps, selected sections, owner action and ordinary tail.
+- Ross Richardson, PR #2104 `4bdc574da394a39f955e8cf4637075649c43a632`,
+  and commdata2338, PR #2153 `5646398dcdc01769df83b939c6b3eb5ecb4d5ebd`:
+  the three-query rank method and source-binding checks.
+- Cinna Davis, PR #2216 `b80e9ba43367fb8b20877a97da1cb677c34f9264`:
+  the paired shear and its bounded witness-separation result;
+  PR #2122 `7045301a8c15eb4408e3e9d80969c9d50a914cde`: the mixed-helper
+  obstruction for the unchanged first pair. The new graph argument here
+  accounts for the intervening shear rather than reusing that trajectory.
+- commdata2338, PR #2326 `fb7d64cd4a51c49bc56920305b0aca64011a420c`:
+  the coordinate-access obstruction and dense-query handoff;
+  PR #2174 `f79470a7931f3943360c10d15b428e0e4e479c22`: the adjacent
+  two-record recovery theorem and recorded-cut boundary.
+- Adam Gall, PR #2093 `8b21a900eb520e4f11ea177b310f66316f8e8983`:
+  the common-axis witness family. The active coordinate-threshold and
+  whole-owner separation investigations concern that different family.
+
+Prepared by @starship-s with substantial Hermes and OpenAI Codex assistance.
+New material is Apache-2.0. No predecessor result is claimed as new.

research/qq16-shear-old-chart-kernel/SOURCES.json

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+{
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+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
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+    },
+    {
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+      "usage": "provenance",
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+}