research: bind the fixed kernel entrance to native replay

research/kernel-entrance-native-boundary/PROOF.md

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+# The fixed entrance gauge and its normalization boundary
+
+This note concerns one finite frame word: the archived
+`bit-kernel-entrance-seed18` transformation of the PR #589 word. Adam Gall supplied
+the gauge, its kernel relation, sparse census, and cost saving. Nick Petalas
+supplied PR #3273's occurrence-parity normalizer. The new obligation is their
+source-bound composition with an explicit nonempty kernel declaration and the
+native record checks. It is not a physical owner, bank, or supplier theorem.
+
+## Algebra with arbitrary dirty helpers
+
+Let the unchanged raw A/scatter/B map have source/helper output `Uz` and target
+output `y + x + Ks`, where `z=(x,s)`. The raw source/helper output does not depend
+on targets. Original D adds `Ks`, and original C restores `U`. These facts are
+checked on all 661 formal scalar columns. The independent model derives `K`
+from raw A/scatter/B before comparing it to D; it does not infer the raw kernel
+relation from the constructor's test of D alone.
+
+For `p=444`, `q=470`, put `S=I+e_q e_p^T` on helper coordinates. The source-derived
+columns `K[:,p]` and `K[:,q]` are equal, with weight 55. Thus, over F2, `S²=I`
+and `KS` has zero column `p`, with its other columns unchanged. The candidate
+removes those 55 reads of `p` from D, performs `q+=p` before the raw word,
+executes original A/scatter/B and C, and performs `q+=p` again. Its target is
+
+`y + (KS)s + x + (KS)s = y+x`,
+
+and its source/helper endpoint is `S U⁻¹ U S z=z`. Every helper input is
+independent and arbitrary. Deleting the final shear leaves exactly
+`helper470 += input444`; the model records that explicit dirty-state witness.
+The endpoint proof and the predecessor's gain are retained obligations, not
+new discoveries.
+
+## Composition with occurrence normalization and an active alias
+
+Let `N` reduce each scatter slot's occurrence count modulo two. Let `G` be the
+fixed entrance transformation. `N` changes only `scatter`, while `G` changes
+only `start`, D, A, and C. Therefore `N(G(w))=G(N(w))` as complete words.
+The new model constructs both sides independently and binds the candidate to
+`G` of the exact normalized constructor input. Before applying `G`, the caller
+exports that normalized word through the pinned original exporter and requires
+all three generated artifact files to be byte-identical to the actual public
+CLI output. This supplies the explicit export bridge between the public
+execution and the word passed to the constructor.
+
+The slot parity identity permits equal helper values in different slots.
+Aliasing is substitution into an F2 identity; it does not require independent
+slot values. During scatter, targets are the only destinations and retained
+sources are helpers. Hence every retained helper is immutable throughout
+scatter, so serial COPY/ERASE windows read the same values as simultaneous
+symbolic snapshots.
+
+The active-alias fixture appends helper 472 as slot 9 and transfers the slot 0
+read at target 113 to three occurrences of slot 9. After `N`, slots 0 and 9
+both have genuine uses and the helper-level target map is unchanged. Each slot
+still incurs its own COPY/ERASE resource. A scalar equality cannot erase the
+extra rank-seven COPY charge. The independent interpreter checks every ERASE
+against its saved COPY value and the current source, in both scalar
+orientations; it does not execute physical address transforms.
+
+## Exact entrance geometry and invoice
+
+Frame 79 is the rational line generated by label 77,
+`(0,1,0,-1,1,0,1,0)`. The model derives its rank and a rank-seven annihilator by
+rational elimination, then checks their zero pairing. Both helpers' first raw
+contacts are at that frame. Starting `p` there removes one rank-one advance;
+moving `q` there for the new shear preserves its later dimension path.
+
+All source-word frame advances are independently checked for rational
+containment and increasing dimension. Their charges also telescope: the
+initial rank sum becomes 112 and the final sum remains 5,177, giving MOVE mass
+5,065. The nine retained frames each have rank seven. Thus the candidate has
+COPY mass 63 and raw mass 5,128; the active alias has COPY mass 70 and raw mass
+5,135. All values are calculated from source label rows and gate chronology,
+then compared with the exported/native invoice.
+
+The non-scatter ADD count is `23,778 - 55 + 2 = 23,725`, and normalized scatter
+has 527 ADDs, for 24,252 total. There are 2,910 MOVEs and 27,180 records for
+nine COPY/ERASE pairs, or 27,182 records for ten pairs. These are finite local
+bills. Retaining an old width-eight host and charging the unused rank-one
+axis cancels the local rank saving, as the predecessor explains.
+
+## Exact boundary and sharp limitation
+
+The declaration `(a=444, partners=[470], frame=79, dimension=1)` is bound to
+the actual entrance and the two shears with typed canonical JSON equality.
+The complete candidate is separately bound to the source-derived `G`.
+Original native selection checks alone do not prove this correspondence or
+the kernel identity. Changing only the declared partner to 223 tests that
+missing implication. The model derives its raw target response, verifies that
+it differs from the pivot's, and records a target witnessing the failure.
+Its native acceptance is execution evidence that must be reported separately
+from the model's rejection.
+
+The original public PR #3273 CLI retains its zero-helper-entrance guard and
+must reject the changed candidate. Applying `N` through that public CLI to
+the zero-entrance input, then constructing `G` under this separate contract,
+does not grant selected H2 admission. Nonempty kernel selection also does not
+make empty source-owner or donor inventories nonempty. Physical allocation,
+address transforms, reverse-frame realization, hosting, packed precision,
+outer assembly, and a new kappa remain outside this finite result.
+
+Prepared by Clear1nsight with substantial OpenAI assistance. No independent
+human review is claimed. Apache-2.0.