KADE Packing Diagnostic

What this tool does

KADE Packing Diagnostic is a structural diagnostic for finite-dimensional complex Grassmannian line packings. Given a packing Φ of n unit vectors in ℂ^d, it computes a compact fingerprint of structural indicators that go beyond the worst pairwise coherence μ. The tool runs entirely in the browser. No data is uploaded anywhere.

Quantities computed

• μ — coherence, computed by five independent float64 kernels (naive, Kahan compensated, pairwise divide-and-conquer, Gram-triangular with reverse accumulation, and double-double TwoProduct+TwoSum at roughly 31 extra bits of precision) and cross-checked. The maximum disagreement is reported.
• μ / Welch — slack against the Welch–Rankin bound. Near 1.0 indicates a near-Welch (typically near-ETF) configuration; large values indicate room above the dyadic algebraic floor.
• Saturated graph G_sat: pairs with μ − |⟨v_i, v_j⟩| < 10⁻⁸. Reported as density and as a histogram across 20 bins from 0 to μ.
• Cliff D₁: the ratio gap[n_sat] / gap[n_sat−1] of sorted gaps to μ. Large cliff (≫ 1) signals a smooth-rigid basin: the saturated set is separated by orders of magnitude from the rest.
• kade base: the minimum of |det G_T| over all d-subsets T of the n vectors. The worst d-tuple is a smallest rank-deficient subframe. Standard pair-based analysis cannot see this.
• kade spectral: λ_max / λ_min on the worst d-tuple's Gram. Reveals how close the worst d-tuple is to true rank-deficiency at d − 1.
• kade couple: max/median of pairwise coherences inside the worst d-tuple. Distinguishes uniform-trio morphology (ratio ≈ 1) from couple-plus-witnesses morphology (ratio ≫ 1).
• Tightness gap: spread of frame-operator eigenvalues λ_max(VV*) − λ_min(VV*). Zero within ε means tight frame.
• Bargmann phase test on K₃ cliques in the saturated graph: angles of triple products ⟨v_a, v_b⟩⟨v_b, v_c⟩⟨v_c, v_a⟩. Discrete distribution suggests algebraic structure; continuous suggests numerical optimisation.
• Reference bounds: Welch–Rankin (always), orthoplex (valid when n > d²), Levenstein-2 (valid when n > d(d+1)/2). The active bound is the maximum of those that apply.
• Field type: a check on max |Im(v)| over all vector components — flags real-only frames (all imaginary parts < 10⁻¹²) where Bargmann phases collapse to {0, π}.
• Duplicate detection: identifies vectors v_i ≈ v_j within 10⁻¹⁰ L¹ component distance, which usually indicates a submission error.
• Holder comparison: when (d, n) appears in the embedded Game of Sloanes snapshot, the holder file is fetched live from the upstream repository and compared metric-by-metric.
• Parsed vectors: displayed with 16-digit complex components, copy-paste-able for downstream verification.

Why each thing

Why coherence μ

The defining quantity of the Grassmannian packing problem. The optimal projective packing in ℂ^d is the configuration minimising μ = max_{i < j} |⟨v_i, v_j⟩|. Applications include compressed sensing, digital fingerprinting, quantum state tomography, and multiple description coding.

Why μ / Welch and the cliff D₁

The Welch–Rankin bound is universal for any (d, n) with n > d. The ratio μ / Welch reports how close the configuration is to the dyadic floor. The cliff D₁ goes further: it measures whether the basin is smooth-rigid (large cliff means refinement of any single pair hits saturated pair constraints from many other pairs simultaneously) or still descent-feasible (small cliff means smooth-max optimisation has room to move).

Why kade — the worst d-tuple

Coherence is a function of pairs. Two configurations with identical μ and identical saturated graph can have radically different triplet / quartet / d-tuple geometries. The minimum |det G_T| across all d-subsets exposes near-rank-deficient configurations that are invisible to pair-based analysis. This is the methodological pivot of the kade toolkit: reframing the basin analysis from k = 2 to k = d. The output triplet, plus the internal pair couple ratio, plus the spectral ratio λ_max / λ_min, together characterise the morphology of the basin floor in a way the worst pair alone cannot.

Why Bargmann phase and tightness gap

Bargmann triple products on K₃ cliques are projective invariants — they do not depend on phase choices of individual vectors. A discrete phase distribution (a small finite set of distinct values) signals algebraic structure such as an equiangular tight frame or a real frame. A continuous phase distribution signals a numerically optimised packing without algebraic regularity. The frame-operator tightness gap λ_max(VV*) − λ_min(VV*) is the standard test for tightness: zero (within numerical noise) means VV* is a multiple of the identity.

Why holder comparison

A single number out of context is hard to interpret. Side-by-side comparison with the current Game of Sloanes holder for the same (d, n) immediately shows whether a submitted packing improves on the existing best, ties it (which is usually noise or coincidence), or falls short. Every metric is computed by the same code path on both packings, so the comparison is byte-exact within float64.

Rigorous background

Bounds and inequalities cited above

Welch, L. R. "Lower bounds on the maximum cross correlation of signals." IEEE Trans. Inform. Theory 20.3 (1974): 397–399.

Levenshtein, V. I. "Designs as maximum codes in polynomial metric spaces." Acta Appl. Math. 29.1–2 (1992): 1–82.

Bukh, B., and Cox, C. "Nearly orthogonal vectors and small antipodal spherical codes." Israel J. Math. 238 (2020).

Conway, J. H., Hardin, R. H., and Sloane, N. J. A. "Packing lines, planes, etc.: packings in Grassmannian spaces." Experiment. Math. 5.2 (1996): 139–159.

Rankin, R. A. "The closest packing of spherical caps in n dimensions." Proc. Glasgow Math. Assoc. 2 (1955): 139–144.

Strohmer, T., and Heath, R. W. Jr. "Grassmannian frames with applications to coding and communication." Appl. Comput. Harmon. Anal. 14.3 (2003): 257–275.

Frame theory and equiangular tight frames

Fickus, M., and Mixon, D. G. "Tables of the existence of equiangular tight frames." arXiv:1504.00253 (2015–2016).

Casazza, P. G., Fickus, M., Mixon, D. G., Wang, Y., and Zhou, Z. "Constructing tight fusion frames." Appl. Comput. Harmon. Anal. 30.2 (2011): 175–187.

Bodmann, B. G., and Haas, J. "Frame potentials and the geometry of frames." J. Fourier Anal. Appl. 21.6 (2015): 1344–1383.

Benedetto, J. J., and Fickus, M. "Finite normalized tight frames." Adv. Comput. Math. 18.2 (2003): 357–385.

Bargmann invariants and projective geometry

Bargmann, V. "Note on Wigner's theorem on symmetry operations." J. Math. Phys. 5.7 (1964): 862–868.

Simon, R., et al. "Operational Bargmann invariants and projective measurements." Phys. Rev. Lett. 104.20 (2010).

Game of Sloanes problem and leaderboard

Jasper, J., King, E. J., and Mixon, D. G. "Game of Sloanes: Best known packings in complex projective space." SPIE Proc. 11138, Wavelets and Sparsity XVIII (2019). arXiv:1907.07848.

Game of Sloanes repository: github.com/gnikylime/GameofSloanes. Curated by John Jasper, Emily J. King, and Dustin G. Mixon.

Sphere packing and projective designs

Cohn, H., and Kumar, A. "Universally optimal distribution of points on spheres." J. Amer. Math. Soc. 20.1 (2007): 99–148.

Renes, J. M., Blume-Kohout, R., Scott, A. J., and Caves, C. M. "Symmetric informationally complete quantum measurements." J. Math. Phys. 45.6 (2004): 2171–2180.

What this tool does NOT claim

• It does not claim global optimality of any packing it analyses.
• It does not claim that the structural classes referenced (Welch-saturated tight algebraic, generic non-tight numerical, near-Welch non-tight numerical) are exhaustive of all possible Grassmannian basin morphologies.
• It does not claim that small cliff implies a packing is attackable, or that large cliff implies it is not; cliff is one of several diagnostic indicators.
• It does not claim that the kade toolkit is complete. Additional invariants (persistent homology of the coherence graph, higher-order p-frame potentials, erasure-robustness) remain to be integrated.
• It does not claim authority over the field; it is the output of an independent investigator collaborating with an AI model, and is offered as a diagnostic instrument to anyone working on or studying Grassmannian frame packings.
• It does not claim that μ computed in float64 is sufficient for a Game of Sloanes record submission at the 8th-decimal byte-exact rule. Quad-precision verification (gcc __float128 or Python mpmath at ≥ 30 digits) is required for record submissions.
• It does not claim novelty for individual components. μ, the Welch bound, the frame operator, Gram subdeterminants as RIP δ_d quantities, and Bargmann invariants all have precedents in the cited literature. The integrated cross-cell application is what the kade toolkit organises.

Who this is useful for

• Researchers in frame theory who want to diagnose packings on their own disk and obtain a structural fingerprint in seconds.
• Students learning the geometry of Grassmannian configurations who want to see what lies beyond the worst pair.
• Anyone preparing a Game of Sloanes submission who wants a pre-submission sanity check (with the caveat that the float64 disclaimer above applies).
• Practitioners in compressed sensing, MIMO precoding, quantum tomography, or multiple description coding who use Grassmannian frames in applications and want to characterise the configurations they work with.

The tool does not replace mathematical reasoning, quad-precision verification, or peer review. It is one diagnostic instrument among others.

Authorship

Independent investigator: Rafael Amichis Luengo (RAL), Madrid · tretoef@gmail.com.

The kade diagnostic framework was developed in an iterative collaboration between RAL and Claude (Anthropic), across a series of empirical cross-cell investigations on Grassmannian frame packings during 2026. The HTML tool is open source and runs locally in the user's browser; no data is transmitted off-device by the Calculator. The Screener tab and the holder-comparison feature make optional outbound requests to raw.githubusercontent.com to retrieve public packing files from the Game of Sloanes repository.

Source code & repository

Public repository (source code, license, citation, issue tracker): github.com/tretoef-estrella/KADE.

Bug reports, feature requests, and pull requests are welcome via the repository's issue tracker. The tool is licensed under CC BY-NC 4.0 (free for academic and personal use; commercial use requires permission — contact tretoef@gmail.com). Citation formats (BibTeX, APA, MLA, Chicago, IEEE, CFF) are in CITATION.md in the repository.