Add deterministic Balan–Wang strong partial-results dossier

[SamPetkov]

Purpose

This PR adds a self-contained deterministic research dossier for the universal Balan–Wang phase-retrieval instability conjecture at the critical threshold N=2M-1.

It is deliberately a strong partial-result PR, not a claimed proof or disproof of the universal conjecture.

The work is independent of the Erdős 625 stack and is created directly from current main under phase_retrieval/balan_wang/.

Main mathematical results

The dossier proves and audits:

1. the exact square-submatrix reduction

   omega(A) = min_{|S|=M} sigma_min(A_S);
   

2. the exact projection-order-statistic formulation;
3. the negative second-moment identity and the incidence comparison

   D(A)/sqrt(M) <= omega(A) <= D(A);
   

4. monotonicity under rowwise normalization;
5. exponential-scale equivalence of the general and Parseval problems, with only a sqrt(2M-1) loss;
6. an explicit universal O(M^(-3/2)) upper bound from an M+2-row Gale argument;
7. an exact weighted global incidence identity;
8. the restricted moment-curve/Vandermonde theorem

   omega(A) <= 16 M R(A) 4^(-M);
   

9. a max-volume normal form reducing the unresolved universal problem to a quantitative theorem about square submatrices of totally nonsingular matrices whose minors are bounded by one.

Small-dimensional certificates

The note proves

rho_2 = 1/sqrt(2)

and the rigorous dimension-three bracket

sqrt(105-40 sqrt(5))/11
  <= rho_3
  <= sqrt((4-sqrt(6))/6).

The lower bound comes from an explicit five-row cyclic algebraic construction. Exactly five of the ten triples are active.

The PR also determines the exact optimum over equal-norm Parseval frames in dimension three:

(2/sqrt(3)) sin(pi/10)
  = (sqrt(5)-1)/(2 sqrt(3)).

The cyclic construction is strictly better, so equal-norm tight frames cannot be imposed as universal extremizers.

Precise remaining obstruction

The universal conjecture is reduced, up to polynomial factors, to the following quantitative TNS statement:

Every totally nonsingular C in R^((M-1) x M) whose square minors all have modulus at most one contains a square submatrix E with

sigma_min(E) <= M^p exp(-cM)

for universal c>0 and finite p.

This lemma is not proved here. No explicit infinite counterexample family with subexponential normalized stability is constructed.

Files

• phase_retrieval/balan_wang/README.md
• phase_retrieval/balan_wang/proofs/STRONG_PARTIAL_RESULTS.md
• phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py
• .github/workflows/balan-wang-partial-results-check.yml
• root README.md discoverability link

Exact verification

The standard-library checker uses exact arithmetic in Q(sqrt(5)) and Q(sqrt(6)). It verifies:

• both cyclic inner-product identities;
• both triple-orbit eigenvalue formulas;
• all ten triples and the 5+5 orbit count;
• positivity/full-spark certificates;
• the exact strict gap above the equal-norm Parseval optimum;
• the algebraic root giving the dimension-three upper constant;
• the prefactor absorption used in the moment-curve statement.

Local validation passed under both modes:

python phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py
python -O phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py

Selected output:

BALAN-WANG SMALL-DIMENSION CERTIFICATES: PASS
  triple patterns: {('d', 'p', 'p'): 5, ('d', 'd', 'p'): 5}
  equal-norm Parseval M=3 value: 0.356822089773...
  cyclic algebraic lower value: 0.358570173636...
  universal M=3 upper value: 0.508348675814...

The focused workflow repeats compilation, normal execution, and optimized execution.

Literature and claim boundary

The note records the original Balan–Wang formulation, Liu–Wang's polynomial-decay context, Shmalo's 2026 Gaussian critical base 1/4, and the 2026 full-spark/TNS literature. The search is targeted rather than exhaustive; no novelty or priority claim is made without a separate specialist review.

Recommended review order

1. audit the three exact reformulations and row/complement conventions;
2. check the M+2-row Gale bound;
3. check the Chebyshev/Vandermonde determinant estimate;
4. audit the max-volume/TNS equivalence;
5. review the rho_3 upper-root extremization argument;
6. run the exact certificate workflow.

Keep this PR draft until the mathematical arguments have received independent expert review.