Prove exact two-call capacity for homogeneous query chains (no new κ)

research/homogeneous-chain-extraction/PROOF.md

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+# Exact two-call capacity of homogeneous query chains
+
+This is a conditional operator theorem, not a physical supplier. The three-field
+case is already in Cinna Davis's PR4698. The new result is the all-odd-length
+construction, a new attaining even-length construction, explicit linear-size
+boundaries, and a matching two-call bound
+with arbitrary restored dirty workspace in the restricted ring below.
+
+## Model and statement
+
+Work over A = Z[q]/(q^4-1), with p=q^2, and let B=Z[p]/(p^2-1).
+For the upper bound scalars may extend to any characteristic-zero field K,
+and free boundaries may be arbitrary invertible matrices over K[p]/(p^2-1).
+Thus the bound includes more scalar mixing than the construction needs.
+It does **not** include arbitrary physical address maps, independently selected
+column operations, or arbitrary Pauli operators beyond this global p. A physical
+substitution must satisfy Q^T=Q and Q^4=I; transpose fixes the formal q.
+
+Let n=2m+1 >= 3, and let J be the n-by-n upper shift. The two paid
+offers are U=I+qJ and its **actual** raw inverse transpose
+
+    U^-T = (I+qJ^T)^-1 = sum_{d=0}^{n-1} (-q J^T)^d.
+
+There are invertible B-valued boundaries L,E,D,R with
+
+    L U E D U^-T D R = diag(q I_(n-1), 1).
+
+They use 5m-2 scalar ADDs, m field swaps, and 3m signed-p field scales.
+ADD coefficients are only ±1 or ±p. These counts include **all** boundaries;
+there are no zero-only inputs. The last input is arbitrary and is restored.
+An ADD with coefficient p is implemented by p-conjugating its source, adding,
+and restoring the source, so no q query is implicit in a boundary. The counts
+are field macros, not a physical tape/runtime invoice.
+
+In this model any two calls to U or its inverse/transpose/raw-dual orientations,
+with arbitrary free boundaries and any number of restored dirty fields, can
+produce at most n-1 independent q jobs. The number of such jobs must be even.
+Consequently n-1 is optimal for every odd n. For even n we prove only the
+upper bound n-2, attained by the separate even construction below. The exact
+capacity for every n>=2 is therefore 2 floor((n-1)/2).
+
+## Odd-chain constructive factorization
+
+Index fields 0,...,2m. Define D=diag(1,p,1,p,...,1) and E=I-p(J^T)^2.
+Both have explicitly invertible B-valued circuits. Conjugation gives
+
+    D U^-T D = (I+q^-1 J^T)^-1.
+
+Since (I-a)(I+a)=I-a^2 for a=q^-1 J^T, multiplication by E gives
+
+    E D U^-T D = I-q^-1 J^T.
+
+This removes every long-root term, including odd powers beyond the quadratic
+term of the three-field predecessor; E itself has only p entries. Therefore
+
+    U E D U^-T D = E_(2m,2m) + q T,   T=J-pJ^T.
+
+The identity follows from JJ^T=diag(1,...,1,0). All operations are in A.
+
+Let u_(2k)=p^(m-k), u_(2k+1)=0. Then Tu=0 and u_(2m)=1.
+Take R to have columns e_0,...,e_(2m-1),u. It adds
+p^(m-k) times the arbitrary spectator to field 2k, for 0<=k<m.
+The first 2m inputs are unchanged otherwise. These are m reversible ADDs.
+
+Let W have columns T e_0,...,T e_(2m-1),e_(2m). Then
+
+    (E_(2m,2m)+qT) R = W diag(q I_(2m),1).
+
+The following explicit circuit is W^-1, so L=W^-1 and no determinant
+assumption or division is being hidden. On output vector y:
+
+1. For k=1,...,m-1, add p y_(2k-2) to y_(2k), in that order.
+2. For k=m-2,...,0, add p y_(2k+3) to y_(2k+1), in that order.
+3. Multiply every odd field by -p.
+4. Add p y_(2m-2) to the last field y_(2m).
+5. Swap each adjacent pair (0,1),(2,3),...,(2m-2,2m-1).
+
+For a vector Wz, the even-field prefix sums recover each z_(2k+1).
+The odd-field suffix sums, followed by -p, recover each z_(2k).
+The last ADD restores z_(2m), including its arbitrary initial value.
+This proves the circuit identity on every independent input column.
+
+E is emitted by adding -p times field i-2 to field i in descending order
+i=2m,...,2, using n-2 ADDs. L uses n-2 ADDs, m signed-p scales and m swaps.
+The two D boundaries use 2m p scales. With the m ADDs of R, the full inventory
+is (m+2(n-2), m, 3m) = (5m-2, m, 3m).
+
+## Even-chain construction
+
+Let n=2m>=4, gamma=p^(m-1), and E=I-p(J^T)^2. This time no D
+conjugation is needed: E U^-T=I-qJ^T. Define a weighted cyclic permutation M by
+
+    (Mx)_i = p^i x_(i+1) for 0<=i<n-1,
+    (Mx)_(n-1) = gamma x_0.
+
+All its weights are 1 or p, so it is invertible by field permutation and
+monomial maps. The special wrap weight gamma is essential; simply extending
+p^i to the last row fails when n is 2 modulo 4. Direct multiplication gives
+
+    U M E U^-T = F + q O,
+    F = E_(n-2,n-1) + gamma E_(n-1,0),
+    (Ox)_i = p^i (-x_i+p x_(i+2)) for 0<=i<n-2,
+    (Ox)_(n-2) = gamma x_0-x_(n-2),
+    (Ox)_(n-1) = 0.
+
+Define u_(2k)=p^k, u_(2k+1)=0, and v_(2k)=0,
+v_(2k+1)=p^(m-1-k). Then Ou=Ov=0, u_0=v_(n-1)=1,
+Fu=gamma e_(n-1), and Fv=e_(n-2).
+Take R with columns e_1,...,e_(n-2),u,v. It is an invertible source
+permutation followed by n-2 ADDs from the two arbitrary spectator fields.
+For the other n-2 columns F e_i=0. Consequently
+
+    (F+qO) R = W diag(q I_(n-2),I_2),
+    W = [O e_1,...,O e_(n-2),gamma e_(n-1),e_(n-2)].
+
+Here is an explicit reversible W^-1 circuit. Prefix-accumulate the even output
+fields y_0,y_2,...,y_(n-4), adding p times the previous accumulated even field.
+Suffix-accumulate the odd output fields y_1,y_3,...,y_(n-3), adding p times
+the next accumulated odd field. Add p y_(n-4) to y_(n-2) to restore the odd
+spectator. Multiply each of the first n-2 fields by p on even indices and -p
+on odd indices, and multiply the last field by gamma. Swap every adjacent
+pair, including the last pair. The query fields become their original order;
+the last pair becomes the two restored spectators in their original order.
+
+Thus with L=W^-1,
+
+    L U M E U^-T R = diag(q I_(n-2),I_2).
+
+The emitted circuit realizes the source permutation with n-2 swaps, M's
+cyclic shift with n-1 swaps, and the final order with m swaps. For
+eta=(m-1) mod 2 its full boundary inventory is
+
+    ADDs = 3n-7, swaps = 5m-3, signed-p scales = 3m-3+2 eta.
+
+There are n-2 ADDs each in R and E, and n-3 in L. M uses m-1+eta
+p scales and L uses n-2+eta. Identity scales are omitted. For n=2,
+conjugating U by the field swap S gives U^T, so S U S U^-T=I:
+two swaps suffice, attaining the capacity zero. No clean spectator is assumed.
+
+## Two-call upper bound with restored dirty workspace
+
+Let N=n+s include arbitrary dirty workspace, and pad J by a zero s-by-s block.
+Every oracle orientation reduces to I+epsilon q J_i, with J_i=J or J^T and
+epsilon=±1, up to invertible B-valued factors. For inverses use
+
+    (I+qJ_i)^-1 = (I-qJ_i)(I-pJ_i^2)^-1.
+
+The second factor is free and invertible because J_i is nilpotent.
+It can be absorbed into adjacent free boundaries. Thus an arbitrary two-call
+word has the form A_0 (I+epsilon q J_1) B_0 (I+delta q J_2) C_0,
+with A_0,B_0,C_0 invertible over K[p]/(p^2-1).
+
+Specialize p=1 but retain both q=1 and q=-1. Invertible free boundaries
+remain invertible at this component of K[p]/(p^2-1). The even coefficient is
+
+    A_0 (B_0 + epsilon delta J_1 B_0 J_2) C_0.
+
+Its rank is at least N-rank(J_1) = N-(n-1), by the rank inequality for
+an invertible matrix plus a rank-at-most-(n-1) perturbation. The desired
+diag(q I_r,I_(N-r)) has even coefficient rank N-r. Therefore r<=n-1.
+This proof includes every workspace field and its arbitrary input. If a free
+boundary or output permutation is absorbed into the desired endpoint, the
+same rank conclusion holds.
+
+Every padded oracle orientation has determinant 1 at both q=1 and q=-1,
+because its shift is nilpotent. Every free boundary has the same determinant
+at these two points, since p=1 at both. The endpoint determinant ratio is
+(-1)^r. Characteristic zero gives r even. The same statements hold for any
+mixture of q and q^-1 jobs, since q^-1=q at these points. For odd n our circuit
+attains r=n-1; for even n the resulting upper bound is r<=n-2, also attained above.
+
+This is a formal homogeneous-ring restriction. For the standard Gaussian
+C with eigenvalues 1,i, C^tensor(f) represents all four q eigenvalues only
+when f>=3. The construction holds at **every** positive width by substitution.
+The ring lower bound applies to a faithful representation, and consequently
+to any uniform family satisfying this restricted model at those larger widths.
+It is not a lower bound for every isolated small width or for all physical
+monomial/scalar circuits. For example at f=1, C=((1+i)/2)I+((1-i)/2)P
+is itself a unit of the scalar/P ring, so no such two-call bound follows there.
+
+## Conditional complete ledger and a direct-synthesis stop
+
+Suppose two provider calls have a common commuting transverse rank-b Gaussian
+on every chain field, using disjoint transverse axes, and the actual raw dual
+admits its true Pauli normalization. Suppose each call has autonomous helper
+target moment H_t and a **complete** paid moment P_t, with t=1-C. Then this
+controller has moment surplus
+
+    2 H_t + r(2b+1)^t + (n-r)(2b)^t - 2 P_t,
+    r = 2 floor((n-1)/2).
+
+Every provider charge and both orientations must be present in P_t. For
+E=P_1-H_1-nb, the rank surplus at C=0 is r-2E.
+In particular a positive rank surplus requires E<r/2 (E<m for odd n). This necessary threshold
+is not sufficient for a positive moment at a chosen C, much less for compiler,
+BIT and outer funding. A raw-dual orientation with different helpers needs its
+own target and paid terms instead of a symmetric factor of two.
+
+Directly synthesizing U by its n-1 unary q shears does not meet the allowance.
+Even favorably granting transverse cost exactly n b^t per invocation, its
+paid moment is n b^t+(n-1), before any further costs. For b>=1 and 0<=t<=1,
+subadditivity yields (2b)^t<=2b^t and (2b+1)^t<=2b^t+1, so the complete
+surplus is at most r-2(n-1), which is -(n-1) for odd n and -n for even n.
+At b=0 and t>0 the same expression is exact.
+Thus direct acquisition of each chain edge loses for every useful saving.
+Already-paid native observations or a changed joint provider would be needed.
+Their physical acquisition and restoration costs are **unknown** here.

research/homogeneous-chain-extraction/README.md

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+# Exact two-call capacity of homogeneous query chains
+
+**Conditional operator theorem, author-checked; no new kappa or native supplier.**
+For every n>=2, two calls to a homogeneous chain U_n=I+Q J_n and its actual raw
+inverse transpose can deliver exactly
+
+    r = 2 floor((n-1)/2)
+
+independent Q jobs while restoring the other arbitrary input fields. Every
+boundary uses only scalar mixing and the global monomial P=Q^2. For odd
+n=2m+1 the explicit circuit uses 5m-2 scalar ADDs, m swaps and 3m signed-P
+scales. A different weighted cyclic middle boundary attains n-2 jobs for every
+even n. Counts are field macros; physical movement and precision remain unpriced.
+
+| Chain fields | Q jobs from two calls | Restored spectators | Boundary ADDs | Swaps | Signed-P scales |
+|---:|---:|---:|---:|---:|---:|
+| 3 | 2 | 1 | 3 | 1 | 3 |
+| 4 | 2 | 2 | 5 | 7 | 5 |
+| 5 | 4 | 1 | 8 | 2 | 6 |
+| 6 | 4 | 2 | 11 | 12 | 6 |
+| 7 | 6 | 1 | 13 | 3 | 9 |
+
+This extends the three-field conditional identity in Cinna Davis's
+[PR4698](https://github.com/CrocSwap/integer-mult-bounds/pull/4698), whose frozen
+source/proof are pinned in [SOURCES.json](SOURCES.json). The proof also shows
+that no two-call word in the scalar/global-P ring can produce more jobs, even
+with arbitrary restored dirty workspace. It permits any inverse/transpose
+orientation and arbitrary invertible boundaries over that ring. Broader physical
+monomial/address operations and isolated small tensor widths are excluded from
+the upper bound. The constructive identities hold at every positive width.
+
+[PROOF.md](PROOF.md) gives the all-length constructions, restricted optimality
+proof, width qualification and complete conditional moment ledger.
+[verify.py](verify.py) checks all independent inputs symbolically over
+Z[q]/(q^4-1) for every length 2 through 32, every boundary count, both parity
+circuits, thirteen semantic mutations, 256 finite dirty-workspace rank fixtures,
+and the tensor-width qualification. These finite checks supplement the written
+proof; they do not prove physical realizability or certify multiplication.
+
+## Budget and expansion decision
+
+With n transverse rank-b main fields per provider, autonomous helper moment H_t
+and complete paid moment P_t, the two-call surplus is
+
+    2 H_t + r(2b+1)^t + (n-r)(2b)^t - 2 P_t.
+
+Writing E=P_1-H_1-nb, the C=0 rank surplus is r-2E. A new physical offer
+needs E<r/2 to pass this positive-rank test; the full moment at the proposed
+saving and every later cost still need checking. Directly building the chain
+from its n-1 unary queries loses at least 2(n-1)-r moment units for every
+0<=t<=1 with b>=1. Longer chains alone do not repair that bill.
+
+**Decision: retain the proved module; stop direct unary synthesis.** No native
+expansion is warranted until a named changed provider has a complete favorable
+budget. The retained-center source has heterogeneous charts and incoming helper
+noise, as recorded in [Discussion4840](https://github.com/CrocSwap/integer-mult-bounds/discussions/4840).
+No native homogeneous offer is obtained from its scalar Jordan chains.
+Acquisition, actual chart maps, dirty restoration and independent helper targets
+are unknown for the proposed longer chain. Physical movement/precision,
+completion, compiler, BIT and outer recovery remain unrun and unpriced.
+
+## Reproduce
+
+From the repository root, using only the Python standard library:
+
+```sh
+python3.12 -I -B research/homogeneous-chain-extraction/verify.py
+python3.12 -I -O -B research/homogeneous-chain-extraction/verify.py
+python3.12 -B scripts/research_package.py --package research/homogeneous-chain-extraction
+```
+
+[RESULT.json](RESULT.json) is deterministic and must match replay. No Lean,
+Docker, all-history suite or selected-frontier replay is required for this
+scoped theorem. The selected conditional kappa remains 0.00302015. See
+[TASK.json](TASK.json) for the exact base, original assignment, overlap boundary,
+claim, and explicit expansion gate. Upstream and frozen predecessors are unchanged.
+
+## Open next questions
+
+- Can an actual changed native provider supply even one five-field homogeneous
+  chain with excess E<2, arbitrary spectator/helper restoration and a positive
+  full moment? Begin with its literal source/target chart contract and complete
+  two-orientation invoice; stop on an unfavorable optimistic bill. No provider
+  meeting this allowance is identified here.
+- Can a named additional free physical monomial/address operation exceed this
+  ring-restricted capacity? Begin with one actual source-compatible operation
+  and replay the full endpoint on all independent inputs. The broader physical
+  operation model is outside the optimality proof and still needs a full bill.
+
+## Credit and scope
+
+Cinna Davis / 1337isnot1337 supplies PR4694's coupled retained-center extension,
+PR4698's helper normalization and conditional three-chain consumer, and the
+PR4440/4470 controller lineage. Retain Cao Yifan / intqwq, zxv, Jacob Alan
+Sussman, Ryanson Jonathan and all cited native/compiler/precision predecessors.
+Their actual native construction is not replayed or modified here. The new
+contribution is the all-length proof/circuits, restricted workspace bound,
+and explicit acquisition allowance. Prepared for Cinna Davis with substantial
+OpenAI Codex assistance. Apache-2.0. No independent human or proof-kernel
+acceptance is claimed.