Prove optimal conditioning of unitary observation cycles (no new κ)

research/unitary-cycle-optimal-conditioning/README.md

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+# Best conditioning of a unitary-coupled observation cycle
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+For any number of banks `W>=2`, internal dimension `d>=1`, and real
+`0<r<1`, let
+
+\[
+ (Sx)_j=U_jx_{j+1\bmod W},\qquad A=I+rS,
+\]
+
+where every `U_j` is unitary. Put `a=1+r²`, `b=2r`, and `c=cos(π/W)`.
+The exact minimum Euclidean condition number over all the couplings is
+
+\[
+ \boxed{\min\kappa_2(A)=
+ \begin{cases}
+ \sqrt{(a+bc)/(a-bc)},&W\text{ even},\\
+ (1+r)/\sqrt{a-bc},&W\text{ odd}.
+ \end{cases}}
+\]
+
+Equality holds exactly when the ordered product
+`H=U_0 U_1 ... U_(W-1)` is `-I` for even `W`, or `I` for odd `W`.
+This includes every finite internal dimension, arbitrary complex unitary
+couplings, arbitrary dirty input vectors, and all `0<r<1`.
+
+At `W=3,r=1/2`, the optimum is **√3**. The real rational matrix
+
+\[
+ A=\begin{pmatrix}1&1/2&0\\0&1&1/2\\1/2&0&1\end{pmatrix}
+\]
+
+attains it. Its Gram eigenvalues are `9/4,3/4,3/4`.
+At `W=2`, choosing one edge `I` and the other `-I` gives condition number
+one for every `r`. The new contribution is an exact optimum, equality
+classification and attaining changed interface. It is a **scoped algebraic
+proof with exact finite corroboration**, not a supplier or multiplication theorem.
+The spectral method is classical; no literature-priority claim is intended.
+
+## Why this question
+
+Chris R / Clear1nsight's [PR #4842](https://github.com/CrocSwap/integer-mult-bounds/pull/4842),
+head `acd3384454083d4eb673343eb7762877d43e5394`, establishes condition number
+3 for the existing cyclic observation matrices in hexapode's
+[#4818](https://github.com/CrocSwap/integer-mult-bounds/pull/4818),
+[#4819](https://github.com/CrocSwap/integer-mult-bounds/pull/4819) and
+[#4826](https://github.com/CrocSwap/integer-mult-bounds/pull/4826).
+That source result is cited, not rechecked here. This package changes the
+couplings and asks for the best possible conditioning in the resulting
+algebraic class. The smaller constant does not apply to the unchanged sources.
+
+The retained construction lineage includes Dugongue #4743, Yiyang / Lyy-iiis
+#4710, lee-phys #4688, and Jacob Alan Sussman #4628. Their source notices and
+credit are retained by citation. No predecessor code, scientific receipts or
+unmerged Git objects are needed by this verifier. The separate two-bank port
+conditioning result #4828 studies a different matrix and is not this target.
+
+A bankwise unitary change of basis only conjugates `H`. It cannot turn an
+existing noncentral product into the attaining scalar product. Thus the result
+identifies a changed-interface design target, rather than a free reencoding of
+the existing observations. There is no paid-call or κ improvement claimed.
+
+## Proof
+
+### The ordered cycle product controls the whole spectrum
+
+The map `S` is unitary. The eigenvalue equation `Sx=z x` gives
+`x_(j+1)=z U_j* x_j`, and closure gives `H x_0=z^W x_0`.
+For every eigenvalue `h` of `H`, each of its `W` distinct roots `z` is an
+eigenvalue of `S`, with the multiplicity of `h`. These eigenvectors span the
+whole `Wd`-dimensional space. Equivalently, successively changing bank bases
+makes every edge except the closing edge the identity, leaving `H` there.
+
+Since `A` is normal, its squared singular values are exactly
+
+\[
+ a+b\cos\big((\theta+2\pi k)/W\big),\quad k=0,\ldots,W-1,
+ \qquad e^{i\theta}\in\operatorname{spec}(H).
+\]
+
+All are positive because `a-b=(1-r)²>0`. The full condition number is at
+least the condition number within any one such root orbit. We minimize one
+orbit first, then characterize when every orbit can attain the minimum.
+
+### Even number of banks
+
+The orbit is closed under negation. Let `t` be the angular distance of its
+nearest point to `+1`, with `0<=t<=δ=π/W`. Its extreme cosines are
+`cos(t)` and `-cos(t)`. Its squared condition number is
+
+\[
+ R(t)=\frac{a+b\cos t}{a-b\cos t}.
+\]
+
+It decreases strictly on this interval away from zero and is minimized
+uniquely at `t=δ`. The root orbit then satisfies `z^W=-1`. The minimum is
+`(a+bc)/(a-bc)`. This reasoning also covers `W=2`, where `c=0`.
+
+### Odd number of banks
+
+Here `W>=3`. With the same `t` and `δ`, the distance of the nearest orbit
+point to `-1` is `δ-t`. The squared condition number is
+
+\[
+ R(t)=\frac{a+b\cos t}{a-b\cos(\delta-t)}.
+\]
+
+The sign of its derivative is the sign of
+
+\[
+ a\{\sin(\delta-t)-\sin t\}+b\sin\delta
+ =2a\cos(\delta/2)\sin(\delta/2-t)+b\sin\delta.
+\]
+
+The last expression is strictly decreasing on `[0,δ]`, is positive at zero,
+and negative at `δ`, since `a>b>0`. Thus the sole interior stationary point
+is a maximum, and the minimum lies at an endpoint. The endpoint values are
+
+\[
+ R(0)=\frac{a+b}{a-bc},\qquad R(\delta)=\frac{a+bc}{a-b}.
+\]
+
+Their denominators are positive, and
+`(a+bc)(a-bc)-(a+b)(a-b)=b²(1-c²)>0`.
+Therefore `R(δ)>R(0)` and zero is the unique minimizer. Its root orbit has
+`z^W=1`, giving the stated odd formula.
+
+### Attainment and equality for arbitrary dimension
+
+If the complete matrix attains the lower bound, every individual root orbit
+must do so too: each orbit's largest-to-smallest ratio lies between one and
+the full ratio, and is already bounded below by the optimum. The unique
+one-orbit minimizers force every eigenvalue of `H` to be `-1` for even `W`,
+or `+1` for odd `W`. A unitary matrix with only that eigenvalue is the
+corresponding scalar matrix. Conversely that scalar `H` makes all the root
+orbits identical, so it attains the bound.
+
+For any prescribed first `W-1` unitary edges, choose
+
+\[
+ U_{W-1}=(U_0\cdots U_{W-2})^*\,\sigma I,
+ \qquad \sigma=(-1)^{W+1}.
+\]
+
+Then `H=σI` and the optimum is attained. This is an explicit algebraic
+construction; it does not assert that a source native port can implement the
+changed closing edge at its existing price.
+
+## Reproduction and finite evidence
+
+From the repository root, with the Python standard library only:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/unitary-cycle-optimal-conditioning/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/unitary-cycle-optimal-conditioning/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/unitary-cycle-optimal-conditioning
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/unitary-cycle-optimal-conditioning --check-source-history
+```
+
+Any CPython 3.12 executable can replace the displayed interpreter. The direct
+verifier computes complete rational Gram characteristic polynomials from
+literal bank matrices, independently of the root-orbit formula. It compares
+them with Chebyshev predictions, checks exact attaining fixtures, exercises
+noncommuting edges and nonmonomial rational orthogonal edges, and rejects
+boundary mutations. Its generated JSON receipt binds all package sources by
+SHA256. Use `--receipt PATH` to save it outside the checkout.
+The exact enumeration checks 756 scalar cycles, including 378 attaining
+products, and 576 two-dimensional cycles, including 72 attainers. Forty-two
+noncommuting rational gauge fixtures and eleven boundary controls pass. The
+all-size and arbitrary-complex claims depend on the written proof, rather
+than the finite fixtures. See `TASK.json` for the exact verification domain.
+
+## Scope, budget and limitations
+
+The original `complex-rank-shape-decoder` card is changed to a local conditioning
+result for a specified observation interface. Current source binding, rather
+than the card's historical packing baseline, is the orientation control. The
+source-bound positive control passed on base
+`624de9dc66858af0945e5f8a0942488d62280d43`, priority revision 7.
+The independent inquiry is the changed-coupling optimum; no predecessor audit
+is its deliverable. Live PRs, recent claims, local reservations, merged searches
+and distinctive main-tree README searches found no same optimum in their stated
+coverage. Sparse notes, unsearched diffs and unpublished work limit coverage.
+[Advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19134008).
+
+This result assumes equal edge magnitudes `r`, Euclidean bank norms and unitary
+edge factors. It does not optimize unequal gains, bank-dependent weighting,
+nonunitary couplings, or intermediate execution norms. The norm formula applies
+to the completed observation matrix only. Source-compatible native emission,
+dirty-bank ownership/restoration, full caller costs, stock, compiler, physical
+storage/tape, rounding and analytic recovery remain open and unpriced. Lean,
+Docker, predecessor scientific replays and selected-frontier replay are unrun.
+Independent human and formal proof-kernel review are unclaimed.
+
+Power-of-ten inspection: no recursion; every computation is bounded by fixed
+fixture limits `W<=13,d<=2` and matrix size at most 26. Functions stay below 60
+lines. Explicit exceptions enforce scientific checks under optimized Python.
+Python's rational arithmetic and collection allocation are a managed-runtime
+deviation from fixed steady-state allocation; finite fixture sizes bound live
+matrix storage, while exact coefficient sizes are determined by the fixed
+rational inputs. There is no external runtime input except an optional receipt
+path. Warning-strict ordinary and optimized runs are registered.
+
+## Open next questions
+
+- Can one actual native port change its closing coupling to the attaining
+  product while retaining all independent targets and restoring dirty state?
+  Start with a literal three-bank word and its complete paid invoice. A
+  source-compatible realization is missing; the algebra alone does not provide it.
+- Does a nonuniform gain profile improve the complete interface after paying
+  its normalization and precision costs? First fix those physical prices and
+  one concrete gain vector. Equal-gain spectral normality does not settle that
+  different problem, and no expansion is justified without that ledger.

research/unitary-cycle-optimal-conditioning/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    }
+  ]
+}

research/unitary-cycle-optimal-conditioning/TASK.json

@@ -0,0 +1,159 @@
+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 7,
+  "base_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a decoder deliver the local paid mass in fewer, larger-rank children (block frame climbs instead of rank-1 to rank-3 steps)?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/complex-decoder-field-extension-seed44/README.md",
+      "research/complex-frame-pair-decoders/README.md",
+      "research/selected-k8-unrestricted-ends/README.md",
+      "research/eisenstein-reduction-copy-floor/README.md"
+    ],
+    "methods": [
+      "Find one exact small decoding identity whose frame path climbs in blocks, and price its complete per-call profile.",
+      "Bound the best effective child rank achievable for one fixed support alphabet, including copies, frames and restoration."
+    ],
+    "acceptance_gate": "A per-call local profile, with all copied information, frames and restoration counted, whose ceiling exceeds the selected word's by at least 5%; masked or overcomplete supports must change an assumption of the eight-term all-pairs bound. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin. Ideal budget: rank-16 delivery is 2.31x (levers.py).",
+    "bottleneck_link": "Rank-1 children are about half of recursive cost; the selected word's effective child rank is about 3.9. Larger-rank delivery lowers recursive cost per unit of paid mass.",
+    "allowed_variations": [
+      "One masked, overcomplete or block decoder family with exact labels and supports.",
+      "A scoped lower bound on effective rank for one alphabet.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Repacking idle remainders of an unchanged word (covered by the packing ceiling).",
+      "Repeating the eight-term all-pairs search inside its proved scope.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Searching below C=40 in the #2017 Eisenstein binary scalar-rectangle or fixed-support block-linear additive pairing model; specify a changed premise instead. This bound does not cover every complex decoder.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "rank shape",
+      "block climb",
+      "masked decoder",
+      "overcomplete",
+      "effective rank"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact minimum Euclidean condition number of I+rS over all unitary-coupled W-bank cycles, including equality cases and an attaining changed three-bank interface.",
+  "scope_change_reason": "The card is narrowed to an original algebraic conditioning step for changed cyclic observation interfaces, following the distinct complete-cascade setting in Clear1nsight PR4842. No paid-profile gain or predecessor audit is the deliverable.",
+  "bottleneck_link": "A changed complete observation interface needs a precision budget. The exact optimum distinguishes achievable conditioning by changing couplings from free bankwise basis changes; actual native cost remains unknown.",
+  "budget_minutes": 60,
+  "stop_rule": "Ship the first proved optimum with bounded exact corroboration; do not expand native synthesis or optimize unequal gains.",
+  "outcome": "completed",
+  "evidence": "README proof for every W>=2,d>=1,0<r<1; exact Gram characteristic polynomials and finite optimum corroboration in verify.py.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can an actual native closing port realize the attaining ordered product?",
+      "why_it_matters": "Only a source-compatible interface with funded restoration can use the conditioning gain.",
+      "first_step": "Emit one complete three-bank word and retain every dirty input and independent target.",
+      "blocked_by": "Native implementation and complete paid invoice are unknown."
+    },
+    {
+      "question": "Can unequal edge gains improve a physically priced complete interface?",
+      "why_it_matters": "Equal-gain normality does not settle nonuniform conditioning or normalization costs.",
+      "first_step": "Fix one actual gain vector and charge normalization and precision before optimization.",
+      "blocked_by": "No concrete physical gain tariff or complete interface ledger is supplied."
+    }
+  ],
+  "verification_domain": {
+    "scalar": {
+      "banks": [
+        2,
+        3,
+        4,
+        5,
+        6,
+        7
+      ],
+      "gains": [
+        "1/3",
+        "1/2",
+        "3/4"
+      ],
+      "edges": "all signed scalar edge tuples"
+    },
+    "matrix": {
+      "banks": [
+        2,
+        3
+      ],
+      "gain": "1/2",
+      "edges": "all tuples from the eight real signed-permutation 2x2 matrices"
+    },
+    "gauge": {
+      "banks": [
+        2,
+        3,
+        4,
+        5,
+        6,
+        7,
+        13
+      ],
+      "gains": [
+        "1/3",
+        "1/2"
+      ],
+      "internal_dimension": 2,
+      "products": [
+        "parity-optimal scalar",
+        "rotation with cosine 3/5",
+        "reflection"
+      ],
+      "method": "noncommuting rational orthogonal bank bases"
+    },
+    "optimality_checks": "Exact extremal-root brackets for nonattainers at W=2,3; exact scalar or quadratic Gram identities for attainers."
+  },
+  "predecessors": [
+    {
+      "pr": 4842,
+      "author": "Chris R / Clear1nsight",
+      "head": "acd3384454083d4eb673343eb7762877d43e5394",
+      "use": "Cited context for complete observation cycles; not a runtime source or audit deliverable."
+    },
+    {
+      "pr": 4818,
+      "author": "hexapode",
+      "head": "fd29d065bbe12f5be5af02fdf5a8b4d86dda3381",
+      "use": "Cited schedule context only."
+    },
+    {
+      "pr": 4819,
+      "author": "hexapode",
+      "head": "a7b42c2f8ffe5a18d305ca4dda5729d2f1d72ea8",
+      "use": "Cited schedule context only."
+    },
+    {
+      "pr": 4826,
+      "author": "hexapode",
+      "head": "d7b53d3992e658f0dd49d0207ba33bbc39272bac",
+      "use": "Cited schedule context only."
+    }
+  ],
+  "overlap": {
+    "canonical_main": "624de9dc66858af0945e5f8a0942488d62280d43",
+    "priority_revision": 7,
+    "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19134008",
+    "coverage": "Live open and recent account PR lists; candidate PR titles/bodies; latest handoffs and active Discussions; local reservations; distinctive main README and merged-title/body searches. Sparse local notes, arbitrary diffs and unpublished work are not exhaustive."
+  }
+}