research: certify local-first two-ADD obstruction on staged544

research/staged544-local-two-add/.gitattributes

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+* -text

research/staged544-local-two-add/.gitignore

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research/staged544-local-two-add/PROOF.md

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+# Local-first two-ADD retiming obstruction
+
+## Statement and fixed contract
+
+Fix either literal PR3734 provider retained in `inputs/`. A first existing ADD
+may change its common binary frame in place, or move across exactly one adjacent
+ADD or nonzero SCALE and change that frame. A second existing ADD may then move
+any distance inside its allowed scalar corridor and choose any common binary
+frame. Both the intermediate and final words must satisfy the fixed boundaries.
+Every other gate, gate frame, entrance, target/centre cut, native warp, and exit
+is unchanged. Each move stays within A after its prelude, B between native warp
+cuts, or restore. ADD crossings use the sufficient condition
+`destination(g) != source(h)` and `destination(h) != source(g)`; SCALE crossings
+use exact conjugation. These are the PR3886 move rules, not every possible
+semantics-preserving rewrite. The same ADD may be chosen twice.
+
+**Theorem.** Every word in this class has new-minus-old raw rank moment at least
+zero for every real `0 < a < 1`. The raw moment is `sum n_r r^(1-a)`. This is a
+finite source-bound theorem conditional on the explicitly stated word/frame
+contract, supported by the exact census below. It is not a universal two-gate
+barrier or supplier admission.
+
+The first move need not be neutral. The census also includes every locally
+losing first move in this radius-one class. The second move has no radius bound.
+A simultaneous edit which needs an illegal intermediate, a longer first move,
+a moved SCALE, extra/deleted gates, warp/phase crossings, or another changed
+boundary is outside the statement.
+
+## 1. Reused scalar and single-move facts
+
+The elementary matrix `I+cE_ds` commutes with `I+bE_uv` under the displayed
+sufficient condition. Moving it right across a scale q at d multiplies c by q;
+a scale at s divides c by q. Moving left reverses these factors. The source
+scales are nonzero powers of two, so coefficients remain dyadic. These are
+identities on arbitrary inputs, including dirty helpers. Cleanup is defined as
+the literal inverse of the edited forward programme, with the inherited buffer
+substitution; it does not require zero dirty state.
+
+The PR3886 trace, move applier, complete single-move census, and sign criterion
+are reused verbatim. The primary replay regenerates all 1,776,295 single-move
+cut/dimension cases separately for each provider and compares the entire frozen
+receipt, rather than trusting its old PASS flag. That census proves every
+single-move delta is nonnegative for all real `0<a<1` in its fixed-check
+relaxation. This package adds the following reductions and the actual paired
+census; it does not claim the scalar identity or sign theorem as new work.
+
+## 2. Reduction for an arbitrary changed first frame
+
+If the second move selects the first moved ADD again, all other instructions
+are still in original order with original frames. Removing that ADD from either
+word leaves exactly the same sequence. Legal crossings compose, scale factors
+telescope, and the final word is an original single-ADD candidate.
+
+If the second ADD has no register in common with the first moved ADD, their
+frame changes affect disjoint registers. The first ADD cannot be a scalar
+barrier for the second. Removing it preserves the order and frames of every
+instruction relevant to the second's price and crossing restrictions. Its
+second-move delta is therefore an original single-move delta. The final delta
+is the sum of two original nonnegative deltas. This argument remains valid if
+their temporal intervals overlap or the moves are in different phases.
+
+It suffices to enumerate second ADDs sharing at least one register with the
+first, in all three phases, excluding the first ADD itself. The programme does
+this without a distance cutoff. For the changed first frame it enumerates
+actual subspaces, not just dimensions: those subspaces can change the next
+move's geometry. The full census also compares every such intermediate invoice
+with the separately retained frame auditor.
+
+## 3. A fixed-frame adjacent swap creates just one new gap
+
+Suppose the first move swaps adjacent gates g,h without changing either frame.
+If they share a register, the two frames on that register are equal. Indeed,
+nesting in the original and swapped orders gives both inclusions. If they are
+disjoint, their frame actions affect different registers. Consequently the
+complete frame state before both and after both is unchanged. The same is true
+after removing any third ADD q: deleting q can change the preceding frame but
+does not change either fixed gate frame.
+
+If q is one of g,h, the final order is obtainable by a single original ADD move.
+If a crossed gate is a SCALE, it cannot itself be chosen as the second ADD.
+If q is a third gate, its old removal cost is unchanged. An insertion outside
+the unique interior gap meets the same frame constraints as in the original
+word. It also crosses the same relevant gates, either both members of the pair
+or neither; their order does not change the sufficient crossing conditions or
+commuting scalar scale factors. Thus only the interior gap in h,g order can
+have a genuinely new price. Moves in another phase or warp corridor reduce to
+the original census as well.
+
+Even at that gap, q must touch both g and h to require a new check. If it touches
+only h, the identical price/frame occurs after both in the original g,h order;
+if it touches only g, it occurs before both. If it touches neither, either side
+works. The additional gate crossed in that comparison is disjoint from q and
+cannot be a barrier. Duplicate descriptions of an unchanged placement must be
+normalised: insertion immediately after the original q is the same position
+as immediately before it once q is removed. The unpruned control implements
+this address convention explicitly.
+
+The fast bridge census therefore tests third ADDs touching both members of
+each legal fixed-frame adjacent pair, at its one new gap. It computes removal
+costs from the original trace and insertion constraints after the literal
+swap. A separate direct adjacent-pair enumerator checks the first-pair set.
+Controls compare the fast formula against full intermediate traces, include all
+four nonzero bridge cases per provider, and test omitted non-bridge cases
+without the pruning predicate. These finite controls supplement this reduction;
+they do not replace its proof.
+
+## 4. Complete finite frames and prices
+
+For either moved ADD, with preceding frames P1,P2 and next fixed constraints
+Q1,Q2 after removal, its possible common frame F satisfies
+`L=P1+P2 <= F <= H=Q1 intersect Q2`. If L is not contained in H there is no
+candidate. Otherwise every dimension between `dim L` and `dim H` occurs.
+On one register, removal contributes
+
+    f(q-p)-f(u-p)-f(q-u),       f(r)=r^(1-a), f(0)=0,
+
+and insertion at dimension d contributes
+
+    f(d-p)+f(q-d)-f(q-p).
+
+The second price depends only on d, so enumerating dimensions covers every
+second frame without sampling bases. The first frame is enumerated as a full
+subspace because it changes the second interval. One-vector extensions enumerate
+all such subspaces inductively. Every actual first interval has quotient rank
+at most three; the implementation raises an error, rather than silently
+truncating, if its declared resource bound of four is exceeded. An independent
+quotient-basis/subset-XOR enumeration checks all 3,118 source intervals across
+both providers.
+
+As in PR3886, a fixed check is relaxed to a virtual climb for pricing. This
+charge is zero on the original and every genuinely legal candidate. Native-map
+entrances are explicit checks, and no move crosses them. Centre descents are
+separately billed without overwriting the main forward frame. Intermediate
+words are checked against the actual boundaries; second candidates form a
+possibly larger relaxation. Non-improvement throughout that relaxation implies
+the stated result for legal final words.
+
+## 5. Exact all-exponent conclusion and accounting scope
+
+For each combined new-minus-old histogram d, define
+`C_k=sum_r d_r min(r,k)` and `S_j=sum_{k<=j} C_k`. All emitted deltas have zero
+rank mass and nonnegative S_j. PR3886's summation-by-parts identity writes their
+moment difference as
+
+    sum_j S_j (theta_j-theta_(j+1)),
+    theta_k=2f(k)-f(k-1)-f(k+1),
+
+with the final theta set to zero only for this finite identity. The real-power
+curvatures are positive and decreasing on `0<a<1`; the triangular integral
+proof, including its convergent endpoint at k=1, is retained in the predecessor.
+Every coefficient in this sum is nonnegative. Every nonzero delta is strictly
+losing; zero is exactly neutral. The 49 combined types are retained with their
+multiplicities, caps, and prefixes. No floating-point grid supplies the proof.
+
+An unchanged positive normaliser and common added bill preserve this ordering.
+Extra nonnegative charges cannot turn it into a saving. Helper widths, fixed
+entrances/exits, and buffer identities are unchanged by the class. This does
+not establish uniform coefficient/precision costs, native Clifford identities,
+hosting, or outer admission for every edited word. A programme which also
+reduces some previously common physical bill would be a different comparison.
+There is no new kappa, all-size multiplication theorem, human peer review, or
+formal-kernel proof in this package.

research/staged544-local-two-add/README.md

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+# A local first move does not unlock a second ADD saving on staged544
+
+This follows the neutral-first handoff from **CrocSwap PR3886** and fork PR9.
+It proves a broader, but still precisely bounded, negative result on both exact
+PR3734 staged544 providers. A first ADD may change its frame in place or cross
+one adjacent scalar gate, with any legal first frame. A second ADD may then move
+arbitrarily far within its allowed scalar corridor with any common frame.
+Neither a neutral nor a locally losing first move enables a strictly better raw
+rank moment for any real `0<a<1` in this class.
+
+This is not an arbitrary two-ADD impossibility result. The first displacement
+is limited to one neighbour, the intermediate word must be legal, and other
+frames and all caller boundaries stay fixed. Longer first moves, simultaneous
+changes requiring an illegal intermediate, warp crossings, and changed
+interfaces remain open. No new circuit saving, supplier, or kappa is claimed.
+
+## What was checked
+
+For each provider the exact first-state census contains 4,709 distinct
+fixed-frame adjacent swaps, 49 neutral and 616 losing adjacent changes with a
+changed first frame, and 29 neutral and 353 losing stationary frame changes.
+There are **5,756 nontrivial first-move records**. The identity first move is already
+covered by the regenerated predecessor single-move census.
+
+The new interaction and single-new-gap reductions avoid an unstructured square
+of the old census. Fixed-frame swaps require 14,693 new-gap cut/dimension checks.
+The 1,047 changed-frame first states require 11,903,025 interacting second
+cut/dimension checks. The total is **11,917,718 reduced candidate prices per
+provider**, yielding **49 signed histogram types**, all exactly certified as
+non-improving. These are price cases, not a count of distinct complete circuits.
+All actual first subspaces are enumerated; second dimensions suffice because
+their price depends only on dimension. No other first states are silently
+represented by one arbitrary frame orientation.
+
+`PROOF.md` states the contract and proves the reductions. The compact
+`RESULT-base.json` and `RESULT-extra.json` retain every resulting signed type,
+multiplicity, capped sum, and sign prefix, together with source and first-state
+hashes. The standalone verifier regenerates them without a solver or network.
+It also regenerates the entire PR3886 single-move premise and checks both source
+words with the retained arbitrary-dirty interpreter in both orientations.
+
+## Reproduce
+
+Python 3.12 is the registered environment; the standalone scientific replay
+also passes on CPython 3.13.5, Linux x86_64, standard library only.
+
+```sh
+python3 -B research/staged544-local-two-add/verify.py --provider base
+python3 -B research/staged544-local-two-add/verify.py --provider extra
+python3 -B research/staged544-local-two-add/test_checks.py
+python3 -O -B research/staged544-local-two-add/test_checks.py
+python3 -B scripts/research_package.py --package research/staged544-local-two-add
+```
+
+Ordinary execution is read-only, authenticates local source snapshots before
+loading retained code, and compares the complete regenerated receipt.
+`--emit /an/external/path.json` explicitly regenerates an external receipt; it
+refuses to overwrite files inside this package. The registered full census is
+bounded separately from the smaller controls. Hosted execution and metadata
+checks are reported independently of the original scientific execution.
+
+Controls independently enumerate all 3,118 first-frame intervals by quotient
+bases and subset XOR, compare 26 fast bridge cases with full intermediate
+traces, and check 3,804 unpruned non-bridge prices. The sample includes every
+nonzero fast-bridge case. An actual disabled bridge predicate is detected.
+There are 48 exact matrix checks of scale conjugation, source corruption and
+resource-bound controls, literal undo checks, and scalar/frame/barrier mutants.
+The credited Jeremy Kothe PR2957 counterfactual rollback/recovery still produces
+`2+3 -> 1+4` and is rejected by a non-improvement certificate. Wrong-scalar
+controls rederive inverse cleanup before testing semantics, so stale cleanup
+is not the cause of rejection. This control is not a new current improvement.
+
+## Boundary and cost decision
+
+The declared milestone is a scoped obstruction, not an implementation admission.
+Fixed helper widths, buffer identities, entrances/exits, and unchanged gates are
+part of the class. Elementary-matrix identities establish scalar equality on
+arbitrary dirty state; inverse cleanup is rederived for literal controls.
+Frame invoices are fully reconstructed. Each changed-frame intermediate also
+passes the separately retained frame auditor. The negative raw comparison needs
+no new allocation, and cannot fund added nonnegative cost under unchanged
+normalisers and common bills.
+
+Native Clifford phase, full physical allocation, uniform coefficients/stock,
+precision, all-size compiler, whole supplier admission, and outer/BIT/analytic
+chains were not freshly replayed here. Unknown costs are not zero. In particular,
+this is not a claim that every candidate has been admitted as a complete
+physical supplier. The expansion decision is **narrow this search away from a
+local first displacement**. Repeating the same first-move class with more seeds
+cannot alter this exact result.
+
+## Open next questions
+
+Can a first ADD moving across at least two neighbours, with a concrete changed
+intermediate frame, unlock a profitable second move? First freeze one such
+literal witness and its complete local invoice. This is the nearest escape from
+the proved radius-one premise; long-range exploratory samples in this session
+were incomplete and are not a barrier certificate.
+
+Can a simultaneous two-frame rewrite, or a changed entrance/exit contract,
+realise PR3863's `2+2 -> 1+3` target without an individually legal intermediate?
+First provide the exact before/after word and dirty restoration contract, then
+price all changed children and consumer costs. This is distinct from choosing
+compatible-looking ranks from an aggregate histogram. Native and physical
+admission remain separate blockers even after a favourable local identity.
+
+## Sources and credit
+
+Scientific base is canonical main `20ecf2891d80b5acf18d980b2164fb44ea7faa90`,
+priority revision 6. Neither the selected certificate nor frozen packages are
+modified. `SOURCES.json` pins every local snapshot by byte count, SHA256, and Git
+blob. PR3734/dataisfire supplies the words at
+`fa041037a3374eaa08feeca0b26a5d6893194147`. PR3886/Chris R supplies the reused
+trace, single-move proof, sign criterion, and applier at
+`359fe0fe77d29544972fd22ee674085ed5a3b4d0`. Dugongue's PR3863 supplies the
+paid-profile target; Jeremy Kothe's PR2957 supplies the existing swap control.
+PR2979 and the staged-entry/retiming lineage retain their construction credit.
+
+Sagar Thakrar's auditor and roger-sato's semantic adapter are retained unchanged,
+including Arham Tahir's frame-contract attribution. They are not newly
+independent interpreters. New reductions, paired census, and controls are by
+Chris R / Clear1nsight with substantial OpenAI assistance. Apache-2.0 and all
+notices in `vendor/SOURCE-CREDITS.md` apply. There is no independent human or
+formal-kernel review claim.
+
+The official overlap helper refreshed 630 open PRs and 2,544 Discussion records
+with complete pagination at 2026-10-19 03:39 UTC; local-note coverage was limited
+to the exported inputs. PR3890's width-price region is a different contract.
+The upstream claim comment was denied with HTTP403; no posted upstream claim is
+asserted. Temporary fork-only input-export infrastructure is excluded from this
+research PR. Exact execution, exploration limits, and publication checkpoints
+are recorded in `VALIDATION.json`.