Add deterministic Balan–Wang strong partial-results dossier

.github/workflows/balan-wang-partial-results-check.yml

@@ -0,0 +1,45 @@
+name: Balan-Wang partial-results certificate
+
+on:
+  pull_request:
+    paths:
+      - "phase_retrieval/balan_wang/**"
+      - ".github/workflows/balan-wang-partial-results-check.yml"
+  push:
+    branches:
+      - "agent/balan-wang-strong-partial-results"
+    paths:
+      - "phase_retrieval/balan_wang/**"
+      - ".github/workflows/balan-wang-partial-results-check.yml"
+
+permissions:
+  contents: read
+
+jobs:
+  exact-small-dimension-certificates:
+    runs-on: ubuntu-latest
+    timeout-minutes: 5
+
+    steps:
+      - name: Check out repository
+        uses: actions/checkout@v4
+
+      - name: Set up Python
+        uses: actions/setup-python@v5
+        with:
+          python-version: "3.12"
+
+      - name: Compile checker
+        run: >-
+          python -m py_compile
+          phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py
+
+      - name: Run exact certificates
+        run: >-
+          python
+          phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py
+
+      - name: Run with optimized bytecode
+        run: >-
+          python -O
+          phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py

README.md

@@ -14,13 +14,21 @@ record, audits, reproducibility material, and available formal verification.
   gap problem for \(G(n,1/2)\), together with supporting checks and a partial
   Lean 4 formalization.
 
+## Adjacent research notes
+
+- [Universal Balan–Wang phase-retrieval instability](phase_retrieval/balan_wang/)
+  — deterministic strong partial results, exact low-dimensional certificates,
+  and a quantitative totally-nonsingular-matrix obstruction. This note does
+  not claim a resolution of the universal conjecture.
+
 Submission-oriented source packages and compiled PDFs are collected in
 [arxiv_preprints](arxiv_preprints/).
 
-Both manuscripts are preprints and have not undergone external peer review.
-The repository makes the arguments and supporting material available for
-expert scrutiny; it does not assert an official change in the published
-status of either Erdős problem.
+Both Erdős manuscripts are preprints and have not undergone external peer
+review. The repository makes the arguments and supporting material available
+for expert scrutiny; it does not assert an official change in the published
+status of either Erdős problem. The adjacent research notes carry their own
+scope and status statements.
 
 ## Citation and license
 
@@ -33,4 +41,4 @@ changes were made. Scholarly users are also asked to cite the project using
 
 See [`LICENSE_SCOPE.md`](LICENSE_SCOPE.md) for the scope of the license and
 the exclusions for third-party literature, source scans, dependencies, and
-files carrying their own notices.
+files carrying their own notices.

phase_retrieval/balan_wang/README.md

@@ -0,0 +1,106 @@
+# Universal Balan–Wang instability at the critical threshold
+
+> **Status:** strong deterministic partial result. This directory does **not**
+> claim a proof or disproof of the universal conjecture.
+
+This dossier studies the Balan–Wang conjecture for a full-spark real matrix
+
+\[
+A\in\mathbb R^{(2M-1)\times M},
+\qquad
+\omega(A)=\min_{|S|=M}\sigma_{\min}(A_S),
+\qquad
+R(A)=\max_i\|a_i\|_2.
+\]
+
+The open problem asks whether universal constants `C > 0` and `0 < beta < 1`
+exist such that
+
+\[
+\omega(A)\le C R(A)\beta^M
+\]
+
+for every dimension and every full-spark critical frame.
+
+## Contents
+
+- [`proofs/STRONG_PARTIAL_RESULTS.md`](proofs/STRONG_PARTIAL_RESULTS.md) —
+  self-contained theorem-by-theorem write-up, independent proof-first and
+  counterexample-first cycles, merged obstruction, and adversarial audit.
+- [`experiments/verify_small_dimension_certificates.py`](experiments/verify_small_dimension_certificates.py)
+  — standard-library exact quadratic-surd checks for the algebraic
+  three-dimensional construction, the equal-norm Parseval comparison, and the
+  stated upper-bound constants.
+
+## Main conclusions
+
+1. The square-submatrix, projection-order-statistic, and row-to-hyperplane
+   incidence formulations are proved with exact constants.
+2. Rowwise normalization is monotone, and the general problem is equivalent at
+   exponential scale, up to a factor `sqrt(2M-1)`, to the Parseval problem.
+3. Every critical frame satisfies an explicit universal
+   `O(M^(-3/2))` upper bound. This is polynomial and therefore does not settle
+   the conjecture.
+4. Normalized real moment-curve/Vandermonde frames satisfy
+   \[
+   \omega(A)\le 16M R(A)4^{-M}.
+   \]
+5. A max-volume basis converts the unresolved universal problem into one
+   quantitative question about square submatrices of totally nonsingular
+   matrices whose minors are bounded by one.
+6. The exact two-dimensional value is
+   \[
+   \rho_2=1/\sqrt2.
+   \]
+7. The dossier proves the rigorous three-dimensional bracket
+   \[
+   \frac{\sqrt{105-40\sqrt5}}{11}
+   \le \rho_3
+   \le \sqrt{\frac{4-\sqrt6}{6}}.
+   \]
+8. The lower construction is strictly better than every equal-norm Parseval
+   frame in dimension three. Hence tightness is not a valid universal extremal
+   reduction.
+
+## Precise remaining obstruction
+
+It is enough, and is necessary up to polynomial factors, to prove the following
+quantitative totally-nonsingular statement:
+
+> There are universal `c > 0` and `p < infinity` such that every totally
+> nonsingular matrix
+> \[
+> C\in\mathbb R^{(M-1)\times M}
+> \]
+> whose every square minor has modulus at most one contains a square submatrix
+> `E` with
+> \[
+> \sigma_{\min}(E)\le M^p e^{-cM}.
+> \]
+
+No such deterministic estimate is proved here. No explicit infinite family
+with subexponential normalized stability is constructed either.
+
+## Reproduction
+
+From the repository root:
+
+```bash
+python phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py
+python -O phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py
+```
+
+The checker uses explicit exceptions rather than optimization-sensitive
+`assert` statements and has no third-party dependencies.
+
+## Literature boundary
+
+The note uses the original Balan–Wang formulation, the Liu–Wang work on decay
+of least singular values of submatrices, Shmalo's Gaussian critical-base
+result, and recent work connecting full-spark frames with totally nonsingular
+matrices. The literature search recorded in the proof note is targeted rather
+than exhaustive. No novelty or priority claim should be inferred without a
+separate specialist review.
+
+All claims in this directory are preprint-level research claims and have not
+undergone external peer review.