diff --git a/.github/workflows/balan-wang-partial-results-check.yml b/.github/workflows/balan-wang-partial-results-check.yml new file mode 100644 index 00000000..33aa3187 --- /dev/null +++ b/.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 diff --git a/README.md b/README.md index 41ade075..a36855b6 100644 --- a/README.md +++ b/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. \ No newline at end of file diff --git a/phase_retrieval/balan_wang/README.md b/phase_retrieval/balan_wang/README.md new file mode 100644 index 00000000..653f16ef --- /dev/null +++ b/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. \ No newline at end of file diff --git a/phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py b/phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py new file mode 100644 index 00000000..a37797f9 --- /dev/null +++ b/phase_retrieval/balan_wang/experiments/verify_small_dimension_certificates.py @@ -0,0 +1,279 @@ +#!/usr/bin/env python3 +"""Exact algebraic checks for the Balan--Wang partial-results dossier. + +The script uses only the Python standard library. It verifies identities in +Q(sqrt(5)) and Q(sqrt(6)) exactly, enumerates every three-row subset of the +five-row cyclic construction, and checks the strict comparison with the +optimal equal-norm Parseval value in dimension three. + +It is a regression layer for the displayed certificates, not a proof of global +optimality of the cyclic construction or of the universal conjecture. +""" + +from __future__ import annotations + +from collections import Counter +from dataclasses import dataclass +from decimal import Decimal, getcontext +from fractions import Fraction +from itertools import combinations +from typing import Union + + +Scalar = Union[int, Fraction, "QuadraticSurd"] + + +@dataclass(frozen=True) +class QuadraticSurd: + """An exact element a + b*sqrt(d), with rational a,b and squarefree d.""" + + d: int + a: Fraction = Fraction(0) + b: Fraction = Fraction(0) + + @classmethod + def rational(cls, d: int, value: Union[int, Fraction]) -> "QuadraticSurd": + return cls(d=d, a=Fraction(value), b=Fraction(0)) + + def _coerce(self, other: Scalar) -> "QuadraticSurd": + if isinstance(other, QuadraticSurd): + if other.d != self.d: + raise TypeError(f"incompatible quadratic fields: {self.d} and {other.d}") + return other + return QuadraticSurd.rational(self.d, Fraction(other)) + + def __add__(self, other: Scalar) -> "QuadraticSurd": + rhs = self._coerce(other) + return QuadraticSurd(self.d, self.a + rhs.a, self.b + rhs.b) + + def __radd__(self, other: Scalar) -> "QuadraticSurd": + return self + other + + def __sub__(self, other: Scalar) -> "QuadraticSurd": + rhs = self._coerce(other) + return QuadraticSurd(self.d, self.a - rhs.a, self.b - rhs.b) + + def __rsub__(self, other: Scalar) -> "QuadraticSurd": + return self._coerce(other) - self + + def __neg__(self) -> "QuadraticSurd": + return QuadraticSurd(self.d, -self.a, -self.b) + + def __mul__(self, other: Scalar) -> "QuadraticSurd": + rhs = self._coerce(other) + return QuadraticSurd( + self.d, + self.a * rhs.a + self.d * self.b * rhs.b, + self.a * rhs.b + self.b * rhs.a, + ) + + def __rmul__(self, other: Scalar) -> "QuadraticSurd": + return self * other + + def __truediv__(self, other: Union[int, Fraction]) -> "QuadraticSurd": + denominator = Fraction(other) + if denominator == 0: + raise ZeroDivisionError("division by zero") + return QuadraticSurd(self.d, self.a / denominator, self.b / denominator) + + def __pow__(self, exponent: int) -> "QuadraticSurd": + if exponent < 0: + raise ValueError("negative powers are not needed by this checker") + result = QuadraticSurd.rational(self.d, 1) + base = self + power = exponent + while power: + if power & 1: + result = result * base + base = base * base + power //= 2 + return result + + def sign(self) -> int: + """Return the exact sign of a+b*sqrt(d).""" + + if self.a == 0 and self.b == 0: + return 0 + if self.b == 0: + return 1 if self.a > 0 else -1 + if self.a == 0: + return 1 if self.b > 0 else -1 + if self.a > 0 and self.b > 0: + return 1 + if self.a < 0 and self.b < 0: + return -1 + + rational_square = self.a * self.a + radical_square = self.d * self.b * self.b + if rational_square == radical_square: + return 0 + + if self.a > 0 and self.b < 0: + return 1 if rational_square > radical_square else -1 + if self.a < 0 and self.b > 0: + return 1 if radical_square > rational_square else -1 + raise RuntimeError("unreachable sign case") + + def to_decimal(self) -> Decimal: + return ( + Decimal(self.a.numerator) / Decimal(self.a.denominator) + + (Decimal(self.b.numerator) / Decimal(self.b.denominator)) + * Decimal(self.d).sqrt() + ) + + def __str__(self) -> str: + return f"({self.a}) + ({self.b})*sqrt({self.d})" + + +def require(condition: bool, message: str) -> None: + """Optimization-independent regression gate.""" + + if not condition: + raise RuntimeError(message) + + +def require_equal(left: object, right: object, message: str) -> None: + if left != right: + raise RuntimeError(f"{message}: {left!r} != {right!r}") + + +def q5(a: Union[int, Fraction] = 0, b: Union[int, Fraction] = 0) -> QuadraticSurd: + return QuadraticSurd(5, Fraction(a), Fraction(b)) + + +def q6(a: Union[int, Fraction] = 0, b: Union[int, Fraction] = 0) -> QuadraticSurd: + return QuadraticSurd(6, Fraction(a), Fraction(b)) + + +def pair_type(i: int, j: int) -> str: + distance = min((i - j) % 5, (j - i) % 5) + if distance == 1: + return "p" + if distance == 2: + return "d" + raise RuntimeError(f"invalid distinct pair {i}, {j}") + + +def verify_rho2_certificate() -> None: + # Two unit rows at projective angle pi/3 have Gram eigenvalues 1 +/- 1/2. + cosine = Fraction(1, 2) + least_gram_eigenvalue = 1 - cosine + require_equal(least_gram_eigenvalue, Fraction(1, 2), "rho_2 Gram value") + + +def verify_cyclic_rho3_certificate() -> Counter[tuple[str, str, str]]: + one = q5(1) + h = q5(Fraction(48, 121), Fraction(-1, 121)) + p = q5(Fraction(57, 242), Fraction(-39, 242)) + d_value = q5(Fraction(31, 121), Fraction(17, 121)) + + cos_144 = q5(Fraction(-1, 4), Fraction(-1, 4)) + cos_72 = q5(Fraction(-1, 4), Fraction(1, 4)) + + require(h.sign() > 0, "h must be positive") + require((one - h).sign() > 0, "h must be smaller than one") + require_equal(h + (one - h) * cos_144, p, "inner product p") + require_equal(h + (one - h) * cos_72, d_value, "inner product d") + + lambda_a = one - d_value + expected_lambda_a = q5(Fraction(90, 121), Fraction(-17, 121)) + require_equal(lambda_a, expected_lambda_a, "type-A least eigenvalue") + + lambda_b = q5(Fraction(105, 121), Fraction(-40, 121)) + radical = 2 + p - 2 * lambda_b + require(radical.sign() > 0, "reconstructed square root must be positive") + require_equal( + radical * radical, + p * p + 8 * d_value * d_value, + "type-B radical identity", + ) + require(lambda_b.sign() > 0, "cyclic lower Gram eigenvalue must be positive") + require((lambda_a - lambda_b).sign() > 0, "type B must be the active orbit") + + patterns: Counter[tuple[str, str, str]] = Counter() + for triple in combinations(range(5), 3): + labels = tuple( + sorted(pair_type(i, j) for i, j in combinations(triple, 2)) + ) + patterns[labels] += 1 + + require_equal( + patterns, + Counter({("d", "p", "p"): 5, ("d", "d", "p"): 5}), + "dihedral triple-orbit count", + ) + + parseval_squared = q5(Fraction(1, 2), Fraction(-1, 6)) + # (3-sqrt(5))/6 = 1/2 - sqrt(5)/6. + expected_gap = q5(Fraction(267, 726), Fraction(-119, 726)) + require_equal(lambda_b - parseval_squared, expected_gap, "Parseval gap identity") + require(expected_gap.sign() > 0, "cyclic construction must beat Parseval value") + require_equal(267 * 267 - 5 * 119 * 119, 484, "integer positivity witness") + + return patterns + + +def verify_rho3_upper_constant() -> None: + # r=(4-sqrt(6))/6=2/3-sqrt(6)/6 is the smallest positive root + # of q(x)=10x^3-30x^2+25x-125/27 at equal positive eigenvalues 5/3. + r = q6(Fraction(2, 3), Fraction(-1, 6)) + polynomial_value = ( + 10 * (r**3) + - 30 * (r**2) + + 25 * r + - q6(Fraction(125, 27)) + ) + require_equal(polynomial_value, q6(0), "rho_3 upper-root identity") + require(r.sign() > 0, "rho_3 upper squared constant must be positive") + + +def verify_auxiliary_numeric_ordering() -> tuple[Decimal, Decimal, Decimal]: + getcontext().prec = 80 + + cyclic_squared = q5(Fraction(105, 121), Fraction(-40, 121)).to_decimal() + parseval_squared = q5(Fraction(1, 2), Fraction(-1, 6)).to_decimal() + upper_squared = q6(Fraction(2, 3), Fraction(-1, 6)).to_decimal() + + require( + Decimal(0) < parseval_squared < cyclic_squared < upper_squared < Decimal(1), + "numeric ordering of dimension-three squared constants", + ) + + return ( + parseval_squared.sqrt(), + cyclic_squared.sqrt(), + upper_squared.sqrt(), + ) + + +def verify_simple_prefactor_inequality() -> None: + # The moment-curve estimate 16 M 4^{-M} is absorbed by + # 32 (1/3)^M because M(3/4)^M <= 2 for M>=2. + value = Fraction(2) * Fraction(3, 4) ** 2 + require(value <= 2, "prefactor inequality at M=2") + for dimension in range(2, 200): + current = Fraction(dimension) * Fraction(3, 4) ** dimension + next_value = Fraction(dimension + 1) * Fraction(3, 4) ** (dimension + 1) + if dimension >= 3: + require(next_value <= current, "prefactor sequence must decrease from M=3") + require(current <= 2, f"prefactor inequality failed at M={dimension}") + + +def main() -> None: + verify_rho2_certificate() + patterns = verify_cyclic_rho3_certificate() + verify_rho3_upper_constant() + verify_simple_prefactor_inequality() + parseval_value, cyclic_value, upper_value = verify_auxiliary_numeric_ordering() + + print("BALAN-WANG SMALL-DIMENSION CERTIFICATES: PASS") + print(" rho_2 = 1/sqrt(2)") + print(f" triple patterns: {dict(patterns)}") + print(f" equal-norm Parseval M=3 value: {parseval_value}") + print(f" cyclic algebraic lower value: {cyclic_value}") + print(f" universal M=3 upper value: {upper_value}") + print(" cyclic lower value is strictly above the Parseval optimum") + + +if __name__ == "__main__": + main() diff --git a/phase_retrieval/balan_wang/proofs/STRONG_PARTIAL_RESULTS.md b/phase_retrieval/balan_wang/proofs/STRONG_PARTIAL_RESULTS.md new file mode 100644 index 00000000..f3aa3fcc --- /dev/null +++ b/phase_retrieval/balan_wang/proofs/STRONG_PARTIAL_RESULTS.md @@ -0,0 +1,1288 @@ +# Deterministic Balan–Wang instability at `N = 2M-1` + +## Strong partial results, exact reductions, and the remaining TNS obstruction + +**Status, 25 July 2026.** This note does not resolve the universal conjecture. +It records the strongest deterministic package obtained from two logically +independent cycles: a proof-first cycle and a counterexample-first cycle. The +cycles are merged only after each has produced a concrete theorem or a precise +obstruction. + +The claims below are self-contained research claims pending external review. +The literature discussion is deliberately conservative and makes no novelty or +priority assertion. + +--- + +## 1. Setup and conjecture + +Let `M >= 2`, `N = 2M-1`, and let + +\[ +A\in\mathbb R^{N\times M} +\] + +have rows `a_1^T,\ldots,a_N^T`. For `S\subseteq[N]`, let `A_S` denote the +row submatrix. Assume that `A` is **full spark**, meaning that every `M` rows +are linearly independent. + +Define + +\[ +R(A)=\max_i\|a_i\|_2 +\] + +and, in the original complement convention, + +\[ +\omega(A)= +\min_{\substack{S\subseteq[N]\\ +\operatorname{rank}(A_{S^c})0` and `0= 1/4`; it does not prove the deterministic upper bound. + +--- + +# Part I. Mandatory exact reductions + +## 2. Reduction to square row submatrices + +### Proposition 2.1 + +For every full-spark `A\in\mathbb R^{(2M-1)\times M}`, + +\[ +\boxed{\omega(A)=\min_{\substack{S\subseteq[2M-1]\\|S|=M}} +\sigma_{\min}(A_S).} +\tag{2.1} +\] + +### Proof + +If `rank(A_{S^c})0`. Define + +\[ +\widetilde a_i=a_i/r_i. +\] + +For every subset `S`, + +\[ +\widetilde A_S^T\widetilde A_S-A_S^TA_S +=\sum_{i\in S}(r_i^{-2}-1)a_ia_i^T\succeq0. +\] + +Consequently + +\[ +\omega(\widetilde A)\ge\omega(A). +\] + +Hence + +\[ +\boxed{ +\rho_M=\max\{\omega(A):A\text{ full spark and }\|a_i\|_2=1 +\text{ for every }i\}.} +\tag{5.1} +\] + +The maximum exists. The product of unit spheres is compact, `omega` is +continuous, and every non-full-spark configuration has `omega=0`, while +full-spark configurations with positive `omega` exist. + +This proves that unit row norms may be imposed at an extremizer. It does not +prove tightness. + +--- + +## 6. Parseval frames are exponentially equivalent to general frames + +Define + +\[ +\rho_M^{\mathrm P}= +\sup\left\{ +\frac{\omega(U)}{R(U)}: +U^TU=I_M,\ U\in\mathbb R^{(2M-1)\times M} +\text{ full spark} +\right\}. +\] + +### Proposition 6.1 + +\[ +\boxed{ +\rho_M^{\mathrm P}\le\rho_M +\le\sqrt{2M-1}\,\rho_M^{\mathrm P}.} +\tag{6.1} +\] + +In particular, + +\[ +\limsup_{M\to\infty}(\rho_M^{\mathrm P})^{1/M}=\beta_*. +\] + +### Proof + +The first inequality follows by rescaling a Parseval frame by `R(U)^{-1}`. + +For the second, normalize `R(A)=1`, let + +\[ +G=A^TA, +\qquad +U=AG^{-1/2}. +\] + +Then `U^TU=I_M` and `A_S=U_SG^{1/2}`. For every `S`, + +\[ +\sigma_{\min}(U_SG^{1/2}) +\le\|G^{1/2}\|\,\sigma_{\min}(U_S). +\] + +Therefore + +\[ +\omega(A)\le\sqrt{\lambda_{\max}(G)}\,\omega(U). +\] + +A Parseval frame has `R(U)\le1`, because `UU^T` is an orthogonal projection. +Moreover, + +\[ +\lambda_{\max}(G)\le\operatorname{tr}G +=\sum_i\|a_i\|_2^2\le2M-1. +\] + +Taking the supremum gives (6.1). `\square` + +Thus an exponential theorem for all Parseval frames would settle the universal +problem with the same exponential base, up to a polynomial loss. + +--- + +# Cycle A — proof-first + +## 7. A universal polynomial bound + +### Theorem 7.1 — `M+2`-row Gale bound + +For every `M>=3`, + +\[ +\boxed{ +\rho_M\le +\frac{2\sin(\pi/(M+2))} +{\sqrt{M+2-2\sin^2(\pi/(M+2))}}.} +\tag{7.1} +\] + +Consequently, + +\[ +\boxed{\rho_M\le(2\pi+o(1))M^{-3/2}.} +\tag{7.2} +\] + +This is a polynomial theorem and therefore does not resolve (BW). + +### Proof + +Choose any `n=M+2` rows of a unit-row critical frame and call the resulting +matrix `B`. Let + +\[ +U\in\mathbb R^{n\times2} +\] + +have orthonormal columns spanning `ker(B^T)`. Every two rows of `U` are +independent. Indeed, if `T` is a two-set and `U_Tt=0` for nonzero `t`, then +`z=Ut` is nonzero, belongs to `ker(B^T)`, and is supported on `S=T^c`; hence +`B_S^Tz_S=0`, contradicting the full spark of the `M` rows in `S`. + +Write the rows of `U` as + +\[ +u_i=r_iv_i, +\qquad r_i>0, +\qquad v_i\in S^1. +\] + +Order their projective directions cyclically in an interval of length `pi`. +Let the successive gaps be `alpha_i>0`, with + +\[ +\sum_{i=1}^{n}\alpha_i=\pi. +\] + +Put `m_i=min(r_i,r_{i+1})`. Since `U^TU=I_2`, + +\[ +\sum_i r_i^2=2, +\qquad +\sum_i m_i\le\sum_i r_i\le\sqrt{2n}. +\] + +Concavity of sine gives + +\[ +\sum_i\sin\alpha_i\le n\sin(\pi/n). +\] + +Cauchy–Schwarz implies + +\[ +\min_i m_i\sin\alpha_i +\le +\left(\frac1n\sum_i\sqrt{m_i\sin\alpha_i}\right)^2 +\le +\frac{\sum_i m_i}{n}\frac{\sum_i\sin\alpha_i}{n} +\le +\sqrt{\frac2n}\sin\frac\pi n. +\] + +Choose an adjacent pair `T` attaining this bound. Testing `U_T` on a unit +vector perpendicular to the larger of the two row directions gives + +\[ +\sigma_{\min}(U_T) +\le\delta_n, +\qquad +\delta_n:=\sqrt{\frac2n}\sin\frac\pi n. +\] + +Choose `t\in S^1` with `\|U_Tt\|\le\delta_n`, and put `z=Ut`. Then +`z\in\ker(B^T)` and `\|z\|=1`. For `S=T^c`, + +\[ +B_S^Tz_S=-B_T^Tz_T. +\] + +Since `\|z_S\|\ge\sqrt{1-\delta_n^2}` and +`\|B_T^T\|\le\sqrt2 R(B)=\sqrt2`, + +\[ +\sigma_{\min}(B_S) +\le +\frac{\sqrt2\,\delta_n}{\sqrt{1-\delta_n^2}}. +\] + +Substituting `n=M+2` proves (7.1). `\square` + +This excludes any counterexample family with normalized stability +asymptotically larger than order `M^{-3/2}`. It does not exclude polynomially +smaller families, which would still disprove the exponential conjecture. + +--- + +## 8. A global weighted incidence identity + +For `|T|=M-1`, put + +\[ +v_T^2=\det(A_TA_T^T) +\] + +and let `u_T` be a unit normal to `span(A_T)`. Let `G=A^TA`. + +For `i\notin T`, + +\[ +\det(A_{T\cup\{i\}})^2 +=v_T^2|\langle a_i,u_T\rangle|^2. +\] + +Double-counting incidences and applying Cauchy–Binet gives the exact identity + +\[ +\boxed{ +M\det G +=\sum_{|T|=M-1}v_T^2\,u_T^TGu_T.} +\tag{8.1} +\] + +Also, + +\[ +\sum_{|T|=M-1}v_T^2=e_{M-1}(G). +\] + +Since every incidence distance is at least `D(A)`, and there are `M` rows +outside each `T`, + +\[ +u_T^TGu_T +=\sum_{i\notin T}|\langle a_i,u_T\rangle|^2 +\ge M D(A)^2. +\] + +Hence + +\[ +D(A)^2\le\frac{\det G}{e_{M-1}(G)} +=\frac1{\operatorname{tr}(G^{-1})}. +\] + +Using + +\[ +\operatorname{tr}(G)\operatorname{tr}(G^{-1})\ge M^2 +\] + +and `tr(G)\le(2M-1)R(A)^2`, one obtains + +\[ +\boxed{ +D(A)^2\le\frac{2M-1}{M^2}R(A)^2.} +\tag{8.2} +\] + +This identity explains why a bare second-moment or Cauchy–Binet argument stops +at polynomial scale: it controls a weighted quadratic average rather than an +exponentially small extreme incidence. + +--- + +## 9. Exponential decay for real moment-curve frames + +### Theorem 9.1 — normalized Vandermonde frames + +Let `t_1,\ldots,t_{2M-1}` be distinct points of `[-1,1]`, and define + +\[ +v(t)=(1,t,\ldots,t^{M-1}), +\qquad +a(t)=\frac{v(t)}{\|v(t)\|_2}. +\] + +Let `A` have rows `a(t_i)^T`. Then `A` is full spark and + +\[ +\boxed{ +\omega(A) +\le M\,2^{-2(M-1)^2/M} +\le16M\,4^{-M}.} +\tag{9.1} +\] + +For arbitrary row scalings along the same projective directions, + +\[ +\boxed{\omega(A)\le16M R(A)4^{-M}.} +\tag{9.2} +\] + +Thus this restricted class has upper exponential base at most `1/4`. + +### Proof + +One of the intervals `[-1,0)` and `[0,1]` contains at least `M` nodes. Choose +`s_1,\ldots,s_M` in that interval and define + +\[ +q(t)=\prod_{j=1}^{M}(t-s_j). +\] + +For each `i`, let + +\[ +P_i(t)=\frac{q(t)}{t-s_i} +\] + +and let `c_i` be its coefficient vector in the monomial basis. Since `P_i` is +monic, `\|c_i\|_2\ge1`. Put `x_i=c_i/\|c_i\|_2`. On the selected submatrix, +all but one coordinates of `A_Sx_i` vanish, and + +\[ +|\langle a(s_i),x_i\rangle| +=\frac{|q'(s_i)|}{\|v(s_i)\|_2\|c_i\|_2} +\le|q'(s_i)|. +\] + +Therefore + +\[ +\sigma_{\min}(A_S)\le\min_i|q'(s_i)|. +\] + +Let + +\[ +V=\prod_{i **Quantitative TNS lemma.** There exist universal `c>0` and `p such that every totally nonsingular +> \[ +> C\in\mathbb R^{(M-1)\times M} +> \] +> whose square minors all have modulus at most one contains a square submatrix +> `E` satisfying +> \[ +> \boxed{\sigma_{\min}(E)\le M^p e^{-cM}.} +> \tag{10.5} +> \] + +Sufficiency follows from (10.4). Conversely, apply a hypothetical universal +frame theorem to `F(C)`. Since `R(F(C))\le\sqrt M`, Proposition 10.1 yields +(10.5) with only polynomial losses. + +This is the precise deterministic obstruction isolated by Cycle A. + +--- + +## 11. Why the remaining proof routes stop + +### 11.1 Hyperplane-arrangement volume + +The number of sign cells is exponential, so one cell has exponentially small +spherical volume. In dimension `M-1`, converting volume to a linear scale takes +an `(M-1)`st root. An `exp(-cM)` volume estimate therefore yields only a +constant-scale radius. Small volume also does not, by itself, imply small +inradius. + +### 11.2 Plücker coordinates without conditioning control + +Cauchy–Binet controls + +\[ +\sum_{|S|=M}\det(A_S)^2, +\] + +but + +\[ +|\det(A_S)|=\prod_{j=1}^{M}\sigma_j(A_S). +\] + +An exponentially small determinant may be distributed across all singular +values and give only constant-scale information after taking an `M`th root. +The max-volume/TNS reduction is the point at which the Plücker information is +converted into an explicit conditioning problem. + +### 11.3 Gale/Naimark recursion + +For a Parseval `U\in\mathbb R^{(2M-1)\times M}` and a Naimark complement +`V\in\mathbb R^{(2M-1)\times(M-1)}`, complementary square submatrices satisfy + +\[ +\sigma_{\min}(U_S)=\sigma_{\min}(V_{S^c}). +\] + +However, `V` has `2M-1=2(M-1)+1` rows, two more than the lower-dimensional +critical count. This row-count mismatch blocks the direct induction. + +### 11.4 Polynomial products + +The polynomial method succeeds on the moment curve because Chebyshev +polynomials give coefficient control and a sharp Vandermonde-product estimate. +For arbitrary hyperplane normals, no comparable deterministic coefficient +bound was obtained. + +--- + +# Cycle B — counterexample-first + +## 12. Exact solution in dimension two + +### Theorem 12.1 + +\[ +\boxed{\rho_2=\frac1{\sqrt2}.} +\tag{12.1} +\] + +Up to row permutations, row sign changes, and right multiplication by an +orthogonal matrix, the unique extremal projective configuration consists of +three equally spaced lines. + +### Proof + +By rowwise normalization, take the three rows to be unit vectors. For two +projective lines with acute angle `alpha`, + +\[ +\sigma_{\min}=\sqrt{1-|\cos\alpha|} +=\sqrt2\sin(\alpha/2). +\] + +Three points on a projective circle of total length `pi` contain a pair at +projective distance at most `pi/3`. Therefore + +\[ +\omega(A)\le\sqrt2\sin(\pi/6)=1/\sqrt2. +\] + +The lines at angles `0`, `pi/3`, and `2pi/3` attain equality. Equality in the +spacing argument forces all three gaps to be `pi/3`; decreasing any row norm +strictly decreases the least singular value of a pair containing that row. +`\square` + +--- + +## 13. An exact algebraic lower construction for `rho_3` + +Set + +\[ +h=\frac{48-\sqrt5}{121}. +\] + +For `k=0,\ldots,4`, define + +\[ +a_k= +\left( +\sqrt{1-h}\cos\frac{4\pi k}{5}, +\sqrt{1-h}\sin\frac{4\pi k}{5}, +\sqrt h +\right). +\tag{13.1} +\] + +All rows have norm one. Their two possible off-diagonal inner products are + +\[ +p=\frac{57-39\sqrt5}{242}, +\qquad + d=\frac{31+17\sqrt5}{121}. +\tag{13.2} +\] + +There are two dihedral orbits of triples. + +For triples with off-diagonal pattern `(p,p,d)`, the smallest Gram eigenvalue +is + +\[ +1-d=\frac{90-17\sqrt5}{121}. +\] + +For triples with pattern `(p,d,d)`, the smallest Gram eigenvalue is + +\[ +1+\frac p2-\frac12\sqrt{p^2+8d^2} +=\frac{105-40\sqrt5}{121}. +\] + +Both values are positive, so the frame is full spark. Consequently + +\[ +\boxed{ +\rho_3\ge +\frac{\sqrt{105-40\sqrt5}}{11} +=0.358570173636\ldots.} +\tag{13.3} +\] + +Exactly five of the ten triples are active. The exact identities and orbit +count are independently checked by the accompanying standard-library script. +Global optimality is not claimed. + +--- + +## 14. A rigorous universal upper bound for `rho_3` + +### Theorem 14.1 + +\[ +\boxed{ +\rho_3\le\sqrt{\frac{4-\sqrt6}{6}} +=0.5083486758\ldots.} +\tag{14.1} +\] + +Hence + +\[ +\boxed{ +0.3585701736\ldots +\le\rho_3 +\le0.5083486758\ldots.} +\tag{14.2} +\] + +### Proof + +By rowwise normalization, let `A\in\mathbb R^{5\times3}` have unit rows. Put +`K=AA^T`. Its three positive eigenvalues `mu_1,mu_2,mu_3` satisfy + +\[ +\mu_1+\mu_2+\mu_3=\operatorname{tr}K=5. +\] + +The characteristic polynomial is + +\[ +p_K(x)=x^2\prod_{j=1}^{3}(x-\mu_j). +\] + +The principal-minor derivative identity gives + +\[ +\frac{p_K''(x)}2 +=\sum_{|S|=3}\det(xI_3-K_S). +\tag{14.3} +\] + +Let `r` be the smallest positive zero of `p_K''`. If every `K_S` had smallest +eigenvalue greater than `r`, then every determinant on the right-hand side of +(14.3) would be strictly negative, contradicting `p_K''(r)=0`. Therefore some +triple satisfies + +\[ +\lambda_{\min}(K_S)\le r. +\tag{14.4} +\] + +Write `e_2,e_3` for the elementary symmetric polynomials of the positive +eigenvalues. Then + +\[ +q_{\mu}(x):=\frac{p_K''(x)}2 +=10x^3-30x^2+3e_2x-e_3. +\tag{14.5} +\] + +It remains to maximize the smallest positive root `r(\mu)` over +`mu_i>0`, `sum mu_i=5`. Strict interlacing places this root in +`(0,min_i mu_i)`, so it is simple and tends to zero at the boundary of the +simplex. Hence a maximum is attained in the interior and the root is a smooth +function there. + +At an interior constrained extremum, implicit differentiation and Lagrange +multipliers give equality of the three partial derivatives of `q_mu` with +respect to `mu_i`. Subtracting the `i` and `j` equations yields + +\[ +(\mu_i-\mu_j)(3r-\mu_k)=0, +\qquad\{i,j,k\}=\{1,2,3\}. +\tag{14.6} +\] + +If the three eigenvalues are distinct, (14.6) forces all three to equal `3r`, +a contradiction. If exactly two are equal, say `mu_1=mu_2=a` and `mu_3=b`, +then (14.6) gives `r=a/3`. Substituting `b=5-2a` and `r=a/3` into (14.5) +gives + +\[ +\frac{a^2(45-17a)}{27}=0, +\] + +so `a=45/17` and `b=-5/17`, impossible. Thus the only interior extremum has + +\[ +\mu_1=\mu_2=\mu_3=5/3. +\] + +At this point + +\[ +q_{\mu}(x)=\frac5{27}(3x-5)(18x^2-24x+5), +\] + +whose smallest positive root is + +\[ +r=\frac{4-\sqrt6}{6}. +\] + +Combining this with (14.4) and taking square roots proves (14.1). `\square` + +The bracket remains too wide to determine `rho_3` exactly. + +--- + +## 15. Equal-norm Parseval frames are not universal extremizers + +### Proposition 15.1 — exact dimension-three Parseval value + +Among full-spark equal-norm Parseval frames +`U\in\mathbb R^{5\times3}`, + +\[ +\boxed{ +\sup\frac{\omega(U)}{R(U)} +=\frac2{\sqrt3}\sin\frac\pi{10} +=\frac{\sqrt5-1}{2\sqrt3} +=0.3568220898\ldots.} +\tag{15.1} +\] + +### Proof + +Let `V\in\mathbb R^{5\times2}` be a Naimark complement. Then + +\[ +\|u_i\|^2=3/5, +\qquad +\|v_i\|^2=2/5. +\] + +For complementary sets `|S|=3`, `|T|=2`, the cosine-sine decomposition gives + +\[ +\sigma_{\min}(U_S)=\sigma_{\min}(V_T). +\] + +Five projective lines in `R^2` contain a pair at projective distance at most +`pi/5`. Thus + +\[ +\min_{|T|=2}\sigma_{\min}(V_T) +\le\sqrt{\frac25}\,\sqrt2\sin\frac\pi{10}. +\] + +Dividing by `R(U)=sqrt(3/5)` proves the upper bound. Five equally spaced +projective lines form an equal-norm Parseval frame in `R^2`; their Naimark +complement attains equality. `\square` + +The squared gap between the cyclic construction and the Parseval optimum is + +\[ +\frac{105-40\sqrt5}{121}-\frac{3-\sqrt5}{6} +=\frac{267-119\sqrt5}{726}>0, +\] + +because + +\[ +267^2-5\cdot119^2=484>0. +\] + +Therefore + +\[ +\boxed{ +\text{no global maximizer for }\rho_3 +\text{ can be an equal-norm tight frame}.} +\tag{15.2} +\] + +This is a concrete refutation of a blanket tight-frame extremizer reduction. + +--- + +## 16. Counterexample mechanisms that fail + +### 16.1 Near-block perturbations + +Suppose an `M`-row submatrix of `A_0` is singular and every corresponding row +is perturbed by Euclidean norm at most `epsilon`. Testing on a unit null vector +of the unperturbed submatrix gives + +\[ +\boxed{\sigma_{\min}(A_S)\le\sqrt M\,\epsilon.} +\tag{16.1} +\] + +Thus the small generic coupling used to make a block construction full spark +immediately creates a bad mixed subset. + +### 16.2 Highly nonuniform row norms + +For every row `a_i`, choose an `M`-subset containing it. Since +`\sigma_{\min}(B)=\sigma_{\min}(B^T)`, testing `B^T` on the coordinate vector +corresponding to this row gives + +\[ +\boxed{\omega(A)\le\min_i\|a_i\|_2.} +\tag{16.2} +\] + +Tiny rows cannot be shielded by the remaining rows. + +### 16.3 Plücker-flat constructions + +Comparable maximal minors control products of singular values, not individual +least singular values. Without an independent upper bound on the other +singular values, determinant flatness is not a conditioning certificate. + +### Cycle-B conclusion + +No explicit infinite family satisfying + +\[ +\omega(A_M)/R(A_M)\ge e^{-o(M)} +\] + +was found. The cyclic `M=3` construction is a finite-dimensional lower bound, +not an asymptotic counterexample. + +--- + +# Part III. Merge of the cycles + +## 17. Common bottleneck + +Cycle A shows that the conjecture would follow from the quantitative TNS lemma, +or equivalently at exponential scale from the Parseval case. + +Cycle B shows that the most immediate escape mechanisms—equal-norm tight +extremizers, tiny-row shielding, near-block perturbations, and Plücker-flat +bases—do not provide a counterexample. + +The central missing deterministic statement can be expressed in three +polynomially equivalent forms. + +### Incidence form + +There are universal `c>0` and `p