Prove the sharp fixed-product unitary-cycle conditioning optimum

research/unequal-gain-cycle-optimum/README.md

@@ -0,0 +1,221 @@
+# Equal gains uniquely minimize cycle conditioning at fixed product
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Let `W>=2`, `d>=1`, and let `U_0,...,U_(W-1)` be arbitrary complex unitary
+`d x d` matrices. For positive real gains `g_j` with geometric mean
+`r=(product_j g_j)^(1/W)` in `(0,1)`, define
+
+\[
+ (T_gx)_j=g_jU_jx_{j+1\bmod W},\qquad A_g=I+T_g.
+\]
+
+In the ordinary, unweighted Euclidean norm,
+
+\[
+ \boxed{\kappa_2(A_g)\ \geq\ \kappa_2(A_r),
+ \qquad\text{equality if and only if every }g_j=r.}
+\]
+
+Here `A_r` uses the **same unitary edges**, with every gain replaced by `r`.
+Thus unequal positive gains cannot improve conditioning, even before the
+unitary edges are optimized. Individual gains may exceed one.
+
+This is a **scoped algebraic proof**, with exact finite corroboration. It
+settles the fixed-product algebraic part of the nonuniform-gain question
+left by [PR #4880](https://github.com/CrocSwap/integer-mult-bounds/pull/4880).
+That predecessor proves the equal-gain optimum and is cited as a premise,
+not audited or replayed here. No new supplier, kappa or multiplication
+theorem is claimed.
+
+## Sharp fixed-product optimum
+
+Put `a=1+r^2`, `b=2r`, `c=cos(pi/W)`, and `H=U_0...U_(W-1)`.
+Combining the new comparison with #4880 gives
+
+\[
+ \min_{\substack{g_j>0,\ \prod g_j=r^W\\ U_j\text{ unitary}}}
+ \kappa_2(A_g)=
+ \begin{cases}
+ \sqrt{(a+bc)/(a-bc)},&W\text{ even},\\
+ (1+r)/\sqrt{a-bc},&W\text{ odd}.
+ \end{cases}
+\]
+
+Equality holds exactly when every gain equals `r` and
+`H=(-1)^(W+1) I_d`. The numerical formula and the ordered-product criterion
+on this equal-gain locus belong to #4880. The new step is that no nonuniform
+positive gain vector can attain or beat them. At three banks and product
+`1/8`, the exact optimum remains `sqrt(3)`; every unequal gain vector has a
+strictly larger condition number, whatever its unitary couplings.
+
+This fixes the product, rather than the arithmetic mean or a physical gain
+tariff. It does not compute the exact optimum for a prescribed unequal vector.
+
+## Proof of the strict comparison
+
+### Positive diagonal similarity preserves the spectrum
+
+Set `D_0=1` and `D_(j+1)=r D_j/g_j` for `0<=j<W-1`. Regard `D` as the block
+diagonal matrix with blocks `D_j I_d`. All blocks are positive. The product
+constraint gives `g_(W-1)=r D_(W-1)/D_0`, so the closing edge satisfies the
+same identity as every other edge. Direct substitution yields
+
+\[
+ T_g=D(rS)D^{-1},\qquad A_g=D(I+rS)D^{-1}=DA_rD^{-1},
+ \qquad (Sx)_j=U_jx_{j+1}.
+\]
+
+The operator `S` is unitary. Hence `A_r` is normal, every eigenvalue `z` of
+`T_g` has modulus `r`, and every eigenvalue `1+z` of `A_g` is nonzero.
+In particular all matrices in the statement are invertible, even when an
+individual gain exceeds one.
+
+Let `M` and `m` be the largest and smallest eigenvalue moduli of `A_g`.
+The normal matrix `A_r` has the same spectrum, so
+`kappa_2(A_r)=M/m`. The spectral-radius inequalities give
+
+\[
+ \|A_g\|_2\geq M,\qquad \|A_g^{-1}\|_2\geq 1/m.
+\]
+
+Their product proves the non-strict comparison. This uses similarity only
+to identify eigenvalues; it does not assume similarity preserves Euclidean
+singular values.
+
+### Equality forces all gains to agree
+
+Suppose the condition numbers agree. Each of the preceding inequalities has
+a positive right side, so equality of their product forces `||A_g||_2=M`.
+Choose a nonzero eigenvector `x` of `T_g` with eigenvalue `z` for which
+`lambda=1+z` has modulus `M`. Then `A_g x=lambda x` attains the operator
+norm, so
+
+\[
+ A_g^*A_gx=M^2x,
+ \qquad A_g^*x=\overline\lambda x,
+ \qquad T_g^*x=\overline z x.
+\]
+
+The middle identity follows by dividing the first by the nonzero `lambda`.
+Consequently `T_g^*T_gx=r^2x`. But direct block multiplication gives
+
+\[
+ T_g^*T_g=\operatorname{diag}
+ (g_{W-1}^2I_d,g_0^2I_d,\ldots,g_{W-2}^2I_d).
+\]
+
+Every block `x_j` is nonzero. Indeed `g_jU_jx_(j+1)=z x_j`, and `g_j>0`,
+`z!=0` and invertibility of every `U_j` propagate a zero block around the
+whole cycle, contradicting `x!=0`. The diagonal identity therefore forces
+`g_j^2=r^2` for every `j`. Positivity gives `g_j=r`. Conversely those gains
+make `A_g=A_r`, proving the equality statement.
+
+The corollary now follows from #4880's equal-gain theorem. Its attaining
+closing edge remains available, so the minimum is achieved in every `d`.
+The spectral-radius and norm-attainment tools are classical; no literature
+priority for those methods is asserted.
+
+## Exact finite evidence
+
+The standalone verifier constructs literal rational matrices, with no
+predecessor imports or scientific replays. Its fixed domain has
+`W in {2,3,4,5,7}`, `d in {1,2}`, `r in {1/3,1/2}`, two signed scalar edge
+products, and two rational orthogonal bank-gauge families. One matrix family
+has a noncentral ordered product. The dimension-two cases include
+noncommuting edges and nonmonomial rational orthogonal matrices.
+
+For each edge tuple it checks three gain profiles `g_j=r 2^(e_j)`:
+all exponents zero, `(1,-1,0,...)`, and `(2,0,...,-2)`. It checks exact
+diagonal similarity, both weighted-shift Gram diagonals, and normality of
+the uniform comparator. Independent rational positive-definiteness tests
+enclose both extreme eigenvalues of each literal Gram matrix. The resulting
+condition-ratio brackets are strictly separated for every nonuniform fixture.
+The global sharp formula is not numerically retested as the deliverable.
+
+The generated receipt enumerates 120 literal cycles, 80 strict gaps and 40
+uniform controls, including 54 cycles with noncommuting edges and 40 with
+a gain above one. Thirteen named boundary controls cover invalid gains,
+product, mean, bank count, shape and unitarity, a reversed similarity gauge,
+singular and indefinite pivots, and two changes of contract.
+
+In the same-spectrum exhibit, `A=[[1,1],[-1/4,1]]` has the uniform comparator's
+eigenvalues but a nonscalar Euclidean Gram matrix satisfying
+`det(tI-A^T A)=(16t^2-49t+25)/16`. In the changed-metric exhibit, its diagonal weighted
+bank norm makes it a scaled isometry. This shows why the unweighted Euclidean
+norm and the distinction between eigenvalues and singular values matter.
+
+The arbitrary-width, arbitrary-complex theorem rests on the written proof.
+Finite rational comparisons corroborate it within the enumerated domain.
+Receipts come from the verifier and bind all five package files by SHA256;
+none is hand-written or retained as a generated checkout output.
+
+## Reproduce
+
+From the repository root, using only the Python standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/unequal-gain-cycle-optimum/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/unequal-gain-cycle-optimum/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/unequal-gain-cycle-optimum
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/unequal-gain-cycle-optimum --check-source-history
+```
+
+Other hosts may substitute any CPython 3.12 interpreter. The direct marker is
+`PASS_UNEQUAL_GAIN_CYCLE_OPTIMUM`. Redirect stdout outside the checkout to retain
+the generated receipt. The registered normal and optimized commands run every
+fixture and control; scientific checks use explicit exceptions.
+
+The selected source-binding orientation passed on canonical main
+`de5ec2e7a6074068452f45b1a191b5c72832df4a`, priority revision 8, after its
+omitted inputs were materialized with the sparse helper. That command is
+metadata orientation, not a supplier replay. The original formal-interface
+card is explicitly changed to a written algebraic theorem rather than a
+proof-assistant formalization or predecessor audit. TASK.json records this
+scope and the [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19213514).
+
+## Attribution and limitations
+
+Credit Nick Petalas #4880 for the equal-gain theorem and open nonuniform
+question. Chris R / Clear1nsight #4842 supplies the complete-cascade context;
+hexapode #4818/#4819/#4826 supplies the proposed schedules. The retained
+construction lineage includes Dugongue #4743, Yiyang / Lyy-iiis #4710,
+lee-phys #4688 and Jacob Alan Sussman #4628. William Lawyer #852 studies
+balanced weighted-cycle inverses in the infinity norm under local weight caps;
+it is a different contract and is not replayed or contradicted here.
+All predecessor credit and source notices remain unchanged by citation.
+
+The theorem assumes scalar positive gains, unitary edge factors, one directed
+cycle and unweighted Euclidean bank norms, with geometric mean strictly below
+one. Nonunitary factors, zero gains, singular mean-one cases, changed graphs
+and other norms are excluded. No quantitative uniform gap away from equal
+gains or exact optimum for a prescribed unequal vector is supplied.
+
+Native emission of changed gains/couplings, arbitrary dirty-bank ownership and
+restoration, independent targets, normalization, intermediate amplification,
+complete caller costs, stock, compiler, physical tape, rounding and analytic
+recovery remain unpriced. A diagonal similarity is not a free physical
+normalization. No family construction expansion or selected-frontier change
+follows. Lean, Docker, predecessor scientific suites and the selected complete
+cold replay are unrun. Independent human and formal proof-kernel review are
+unclaimed.
+
+Power-of-ten inspection: no recursion; all scientific loops have fixed finite
+domains, matrices have at most 14 rows and each root bracket uses exactly 24
+iterations. Functions stay below 60 physical lines. Python fractions and
+collections allocate dynamically, a managed-runtime deviation; fixed rational
+inputs and bounded matrix sizes bound live data. The rejection helper's only
+indirect targets are the two named local functions `cycle` and
+`check_similarity`. Warning-strict ordinary and optimized replay are registered.
+No package verifier writes files or launches processes.
+
+## Open next questions
+
+- Can one physical three-bank native word realize the attaining closing
+  product while retaining arbitrary input banks and every independent target?
+  Start with one literal caller and its full paid invoice. Native compatibility,
+  ownership and restoration costs are the blockers.
+- For one prescribed nonuniform gain vector, what is the exact optimum over
+  unitary couplings? Start with a priced two- or three-bank vector and a full
+  singular-value calculation. The present fixed-product theorem gives strict
+  exclusion of the uniform benchmark, not that different optimum or its tariff.

research/unequal-gain-cycle-optimum/SOURCES.json

@@ -0,0 +1,24 @@
+{
+  "schema_version": 1,
+  "base_commit": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/unitary-cycle-optimal-conditioning/README.md",
+      "usage": "provenance",
+      "git_blob": "a2f36e8bf5a313ea919ad4d3f9ef054d574be58c",
+      "sha256": "de0d5bf48f56e7666d9da8cbfa3c37ee823bc62e1b3b450b713e409b9370a574"
+    }
+  ]
+}