Research: isolate orthogonal remainder overlays (no new κ)

research/pr3190-distinct-axis-basis-isolation/README.md

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+# Isolation for orthogonal simultaneous remainder transfers
+
+**Scoped negative result; no new κ and no selected-result change.** This package
+closes one part of the simultaneous-transfer question raised by open PR
+[#3296](https://github.com/CrocSwap/integer-mult-bounds/pull/3296). It proves a
+conditional operator obstruction for any number of pairwise-orthogonal moved
+axes, and independently reproduces the exact singleton facts used by that
+argument. PR [#3311](https://github.com/CrocSwap/integer-mult-bounds/pull/3311)
+independently replays a different, port-zero literal witness; this package
+covers the all-port singleton domain and the orthogonal-axis implication.
+
+## Exact scope and result
+
+The source universe is fixed at main commit
+`463113d675af7410ac91d53a4fed51ec6494eb9a`: the authenticated 2,717,280-row
+PR3148 fill, the 112 port recipes, the fixed chronology, allocation and template
+inputs listed in [SOURCES.json](SOURCES.json), owner replica zero, cover identity,
+and PR3187's exact eight-stream Gaussian-integer replay. The proposed per-axis
+rule is transcribed in `declared_rule` in [verify.py](verify.py) from PR #3190 at
+head `5e4a3235169edddd033889da4e9f47635ff50c3e`: a moved axis belongs to its
+donor remainder, donor and receiver differ, and the axis is orthogonal to the
+receiver's current remainder axes, every stage entry/exit X/Y translation, the
+terminal Y translation, and the donor and receiver terminal translations.
+The resulting child ranks must not exceed 51 and each child's axes must remain
+orthonormal.
+
+The enumerated domain deliberately relaxes PR #3190's construction restrictions:
+it admits any receiver with a nonempty full axis list, including a hosted
+receiver, and does not forbid a child from being both donor and receiver in a
+bundle. Thus it is a superset for the commuting-rule audit, not an assertion
+that every enumerated overlay satisfies the native controller or hosting
+contract. Excluding operator-preserving overlays in this superset is stronger
+than excluding only the narrower PR #3190 move class.
+
+Across all 112 ports, the verifier enumerates 9,384 candidate single-axis
+transfers from a hosted child with a nonempty remainder to another child whose
+full axis list is nonempty (the receiver's residual may be empty). Of those,
+5,575 fail the declared rule. The other 3,809 pass the
+rule and final rank/frame checks. At each such transfer's own moved-axis
+Fourier character, the unchanged baseline is the exact scalar matrix and the
+single transfer changes the exact 8-by-8 Gaussian-integer matrix. The stored
+[RESULT.json](RESULT.json) contains every per-port count and the pair-control's
+baseline, singleton and bundle matrices at both moved-axis characters, including
+the exact changed entries. This is finite source-bound evidence for those
+singleton premises.
+
+## Isolation lemma
+
+Fix any one of the 112 ports and its source-bound schedule. Let
+`T_i = (d_i, r_i, v_i)` be a nonempty finite collection of distinct
+single-axis transfers from the enumerated singleton domain. Suppose their moved
+axes are pairwise orthogonal over `F_2`, each move uses its donor axis at most
+once, and the combined final assignment satisfies the rank and within-child
+orthonormality checks. For each `i`, let `M_i(v_i)` be the exact replay after
+only `T_i`, evaluated at character `v_i`; let `M_B(v_i)` be the replay after
+the whole bundle `B`, at the same character.
+
+Then, for every `i`,
+
+```text
+M_B(v_i) = M_i(v_i) != M_empty(v_i).
+```
+
+For the equality, inspect an idle-cover-sweep phase in the source replay. A
+residual axis `w` contributes `dot(character, w)` to its child's phase exponent;
+this rule is implemented in PR3187's unchanged
+[algebra.py](../rank7-idle-star-stream-obstruction/algebra.py). Set the
+character to `v_i`. The selected move changes the residual membership of
+`v_i`, with `dot(v_i, v_i) = 1`. Every other moved axis `v_j` contributes zero
+because `dot(v_i, v_j) = 0`. Thus all other transfers leave every idle-sweep
+phase unchanged at this character. Hosted rows, translations, bridges, local
+sweeps, terminal events and chronology are fixed, so the full matrix replay is
+exactly the singleton replay. The singleton census proves the inequality for
+each possible `T_i`; hence the bundle cannot preserve the operator.
+
+The orthogonality premise also keeps each component's receiver-axis check true
+while the bundle is applied: other transferred axes added to or removed from a
+receiver have zero dot product with it. Distinct donor axes remain available,
+and only the combined rank/frame check adds a bundle-level condition.
+
+This argument quantifies over arbitrary finite bundle size within the stated
+source and legality conditions; it does not enumerate bundles. The verifier
+also finds a legal port-zero pair, `4→0` with axis `2^32` and `14→0` with axis
+`2^72`, and replays the bundle. At each moved-axis character, the bundle matrix
+equals the corresponding singleton matrix. Both baselines are `i I_8`; the
+singletons differ from them in 2 and 5 matrix entries respectively, so this pair
+also supplies an explicit joint-change example. RESULT.json records the exact
+baseline, singleton and bundle matrices. The proof, rather than that fixture,
+establishes the all-finite-bundle implication.
+
+## Open next questions
+
+This result does **not** exclude a simultaneous bundle containing nonorthogonal
+moved axes (including repeated copies of the same vector), an overlay whose
+components are not individually in the enumerated singleton domain, another
+owner replica, a changed host schedule, or a repaired/rescheduled exchange. It
+does not prove that all of PR #3190's proposed gains fail under every repair, and
+it does not establish any general κ bound. Those are separate questions.
+
+The next concrete step is to identify the smallest legal simultaneous
+hosted-remainder overlay in this fixed schedule with two nonorthogonal moved
+axes, then test for a character that isolates it or a written obstruction that
+does not require orthogonality. Other replicas and schedule repairs remain
+separate scope expansions.
+
+## Reproduction and provenance
+
+The verifier imports PR3187's authenticated source loader and exact event
+semantics unchanged. Its new work reconstructs the owner-zero rows for all 112
+ports, transcribes the declared rule, enumerates every scoped singleton, and
+proves the character-isolation implication above. This is a new binding and
+algebraic implication, **not** an independent reimplementation of the physical
+operator model. Inputs, Git blobs and SHA256 hashes are pinned in
+[SOURCES.json](SOURCES.json); all live inputs are listed in [VERIFY.json](VERIFY.json).
+
+From the repository root, with Python 3.12:
+
+```sh
+python3.12 scripts/research_package.py --package research/pr3190-distinct-axis-basis-isolation
+python3 scripts/research_package.py --package research/pr3190-distinct-axis-basis-isolation --check-only
+```
+
+`verify.py --write` regenerates RESULT.json after a deliberate source change;
+ordinary verification compares its freshly computed result byte-semantically
+to the committed receipt. The registered replay passed with Python 3.12.15; a
+direct replay also passed with Python 3.14.8. The verifier uses only the Python
+standard library.
+
+The exact replay model and source loader come from Nick Petalas's PR #3187 and
+retain its PR3133 notice and credits through the pinned vendor files. The
+audited remainder-transfer rule is the one stated by Rohan Arun's open PR #3190
+at the head above. New proof and verifier: Cinna Davis, with substantial
+OpenAI Codex assistance. No human or proof-kernel review is claimed.