Research: all-depth QQ16 cover-map parity floor at B1 cut

research/qq16-map-parity-floor-seed-ed361f3c/LICENSE

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research/qq16-map-parity-floor-seed-ed361f3c/PR-BODY.md

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+## Scope and result
+
+**Scoped all-depth obstruction; no new kappa.** At the exact post-`B_1_2` QQ16 cut, the eight `Y_A` output frames each have 18 odd Z4 terms, while the existing private fields `C0..C7` return in the zero frame. Every registered cover map applies an invertible binary owner transform and adds only coefficient 2. Modulo two, the payload shift vanishes and the owner transform preserves odd-support size. Therefore every finite composition of cover maps leaves all eight frames at odd-support weight 18; this rules out depth-one, depth-two, depth-three, and longer map-only normalization.
+
+In the declared one-term native-line model, one rank-one correction changes at most one parity bit. Reaching the private zero frames requires at least 18 corrections per output, or **144 rank-one line calls** total. This exceeds by 72 times the favorable budget of at most two extra rank-one-equivalent operations after the rank-eight output charge. The bill also includes 288 associated full-cover map operations, eight scalar copies priced at 32 real primitives, and up to eight terminal clear primitives (zero additional clears if the existing next core follows). Further map implementation, hosting, source routing, and precision charges cannot lower the line floor.
+
+This extends the released one-map / one-term result in PR #2197 with an all-depth invariant. It is limited to the pinned eight frames, registered invertible cover maps, and one-term Z4 corrections. Alternate field layouts and multi-term native operations remain open. No full controller, supplier admission, or selected-bound change is claimed.
+
+## Exact evidence and reproduction
+
+The source-pinned verifier reconstructs the eight port frames from the native recipes, checks all 3,650 owner maps for rank 73, applies every map to every port frame (29,200 checks), and validates a two-map positive control. Negative controls demonstrate that an odd coefficient shift could cancel a parity bit and that a singular owner map violates the invariant's hypothesis.
+
+```sh
+python3 -I -B research/qq16-map-parity-floor-seed-ed361f3c/verify.py
+python3 scripts/research_package.py --package research/qq16-map-parity-floor-seed-ed361f3c --check-only
+```
+
+Both commands pass locally under CPython 3.14.7. The registered package replay requests Python 3.12, which is unavailable locally.
+
+## Pins and predecessor attribution
+
+- Seed `ed361f3c876dd81e664556c9f95bd36e`; `recursive-interfaces`, `joint-query-controller`, priority revision 4.
+- Scientific base `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`.
+- Selected-result SHA-256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+- Catalogue SHA-256 `d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7`.
+- Map/frame input SHA-256 `a5ed85fac5ebf3c091482a113ab0d351eb5a7892fe2841a35e16017ec1ff49f7`.
+- Predecessor [PR #2197](https://github.com/CrocSwap/integer-mult-bounds/pull/2197), head `0a6f487ddff27cde0369e225f9ccb328ca2d8043`, identifies the one-map obstruction and releases the depth-two/three question. This package proves the parity floor for every composition depth using the map law in the selected source certificate.
+- [PR #2174](https://github.com/CrocSwap/integer-mult-bounds/pull/2174) supplies the conditional two-query decoder; merged [PR #1969](https://github.com/CrocSwap/integer-mult-bounds/pull/1969) supplies retained QQ16 map and private-field conventions.
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance in scope selection, proof development, exact verifier implementation, and packaging. Apache-2.0.

research/qq16-map-parity-floor-seed-ed361f3c/PROOF.md

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+# All-depth parity obstruction for QQ16 cover-map normalization
+
+## Exact object
+
+Use the pinned QQ16 certificate at the post-`B_1_2` cut. The eight output frames are `f117`, `f124`, `f137`, `f144`, `f153`, `f163`, `f174`, and `f228`; the existing private fields `C0..C7` return in frame `f0`. The frame registry and each `Y_A` port label are bound to the certificate SHA-256 recorded in `SOURCES.json`.
+
+A frame is a finite function `a: F2^73 -> Z4`, represented by its nonzero `line_terms_Z4`. Define
+
+```text
+odd_support(a) = { x : a(x) is 1 or 3 }.
+```
+
+The exact source frames each have 18 terms, all with odd coefficient, so `|odd_support|=18`. The private target frame is zero, so its odd-support size is 0.
+
+## Lemma: cover maps preserve odd-support size
+
+A registered cover map has the pinned action
+
+```text
+M(g,delta) f(d,z1,...,z_lambda)
+    = f(d*g, z1 XOR d*delta, ..., z_lambda XOR d*delta).
+```
+
+On frame axes this applies the invertible owner map `g`; the payload translation adds coefficient 2 on its translated axes. Reducing coefficients modulo 2 removes that translation. Thus the odd support after one map is exactly the image of the original odd support under `g`. Since `g` is invertible, this map preserves support cardinality.
+
+The claim composes: after any finite sequence of registered cover maps, every one of the eight frames still has odd-support size 18. In particular, no depth-one, depth-two, depth-three, or longer cover-map sequence alone can normalize one of these frames to `f0`.
+
+The verifier checks all 3,650 registered owner matrices for full rank 73, reconstructs the eight source frames from their pinned port recipes, and checks all 29,200 map/frame actions. These finite checks bind the stated inputs. The all-depth conclusion follows from the preceding invariant, not from a bounded path census.
+
+## One-term line cost
+
+In the declared one-term native-line model, one rank-one line correction changes one Z4 frame coefficient by `+1` or `-1`. Modulo 2 this toggles at most one support bit, so it can decrease odd-support size by at most one. Each port therefore needs at least 18 such calls to reach the zero frame. Across eight ports the floor is
+
+```text
+8 * 18 = 144 rank-one line calls.
+```
+
+The inherited wrapper associates two full-cover map operations with each line call, for 288 map operations. Copying the eight observed scalar values into existing private fields costs eight scalar ADDs, priced at 32 real scalar primitives. If the cut is terminal, clearing eight fields adds eight scalar clear primitives; when the existing next core runs, its clears perform this cleanup with no additional clear charge. Any separate map route only adds operations and cannot lower the line floor.
+
+The favorable rank-eight output screen in Discussion #2074 leaves at most two additional rank-one-equivalent calls. The lower bound 144 exceeds that allowance by a factor of 72 before source-routing implementation, global hosting, or recursive precision costs. This rules out the multi-map normalization route within the stated one-term line interface.
+
+## Scope
+
+This is a structural obstruction for the eight pinned QQ16 frames, the registered invertible cover-map action, and the one-term Z4 line correction model. It does not rule out a different private-field layout, a native operation that changes multiple frame coefficients in one call, or a different controller. It supplies no physical supplier, selected-bound, or kappa result.