Inverse coeff. |A₃| over Bean S*_𝔅 · Geometric Function Atlas

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Certificate under review. This legacy or incomplete artifact is retained for audit, but cannot carry public proof wording until its v1 replay contract validates.

Stored legacy claim for Inverse coeff. |A₃| over Bean S*_𝔅. Its proof wording is withheld pending a valid replayable v1 certificate.

What this rests on

The certificate reduces to these classical lemmas - each checkable by hand, independently of the engine.

1. Schur parametrisation is exact

The Schur recursion maps the closed polydisc \(\overline{\mathbb D}^{m}\) exactly onto the admissible coefficient vectors \((a_2,\dots,a_{m+1})\) of the class. Ranging the Schur parameters over \(\overline{\mathbb D}^{m}\) therefore sweeps every member's coefficients — nothing admissible is missed and nothing spurious is added.

Schur 1917; Foias–Frazho; Simon, OPUC. · schur-onto-coefficient-body
2. Rotation normalisation

The functional is weight-homogeneous under \(f(z)\mapsto e^{-i\theta}f(e^{i\theta}z)\), so its maximum is attained on the rotation-normalised slice \((\gamma_0\ge 0\) real\()\). This removes one real dimension and is verifiable by hand.

Elementary weight homogeneity. · rotation-reduction
3. Sharp in closed form (single-harmonic reduction)

For the Fekete–Szegő functional the reduced problem has one free angle, and the functional is affine in it: L = A·e^{iθ} + C with A, C real. So max over θ of |L| = |A| + |C| exactly (triangle equality, attained by the right phase), and the remaining maximum over the two radii is an exact, closed-form box maximum. The certified bracket closes to a point — no slack, no second-variation needed. This is the classical Keogh–Merkes argument, mechanized.

Keogh & Merkes (1969), single-angle Schur reduction. · fekete-szego-single-harmonic

In the literature

This value over \(S^*(\varphi)\) is fixed by a published general theorem - Fekete–Szegő, \(|\gamma_2|\) and \(|A_3|\) by Ma & Minda (1994); the second Hankel determinant \(H_2(2)=(B_1/2)^2\) by Lee–Ravichandran–Supramaniam (2013). The registry re-certifies it mechanically - it is not a new result. See provenance & novelty.

closed-form engine m = 2 Schur params · closed-form proof (no box partition)

Reproduce & raw certificate

Reproduce: python3 -m gft.proofs prove bean_tanh inv_a3 "7/16" --slack 0
Recheck: python3 -m gft.proofs recheck data/proofs/bean_tanh__inv_a3.json

{
 "status": "PROVED",
 "engine": "closed-form",
 "statement": "max over S*(bean_tanh) of |inv_a3| = 7/16  (proven exact, closed form)",
 "class": "bean_tanh",
 "functional": "inv_a3",
 "bound": "7/16",
 "slack": 0.0,
 "m": 2,
 "proof": "closed-form-single-harmonic (m=2; positive-dimensional maximizer set; B&B does not close at fine slack)",
 "lemmas": [
  "schur-onto-coefficient-body",
  "rotation-reduction",
  "fekete-szego-single-harmonic"
 ],
 "failure": null,
 "sharp": {
  "proven": true,
  "method": "m2-closed-form",
  "value_exact": "7/16",
  "value_float": 0.4375,
  "bracket_lo": 0.4375,
  "bracket_hi": 0.4375,
  "extremal_gammas": [
   "1",
   "0"
  ],
  "extremal_omega": null,
  "decomposition": {
   "L": "A*e^{i t} + C  (single harmonic in t)",
   "A_at_rho1": "\\frac{r_{0}^{2}}{4} - \\frac{1}{4}",
   "C_at_rho1": "\\frac{7 r_{0}^{2}}{16}",
   "angular_max": "max_t |L| = |A| + |C|  (triangle equality, sharp)"
  },
  "reference": "Single-angle Schur reduction + triangle equality in the phase: the m=2 coefficient body has one free angle, so |L| = |A e^{i t} + C| maximizes to |A| + |C| exactly.",
  "candidate": {
   "value_exact": "7/16",
   "value_float": 0.4375,
   "method": "m2-closed-form"
  },
  "exact": true,
  "note": "Proven exact (sharp). The Fekete-Szego functional reduces to a single harmonic in the one free angle, so max_t|L| = |A|+|C| exactly (triangle equality), and the sharp constant is the exact box maximum \u2014 the certified bracket closes to a point, no slack. The maximum is attained on 1 distinct extremals \u2014 a positive-dimensional maximizer set a fine-slack branch-and-bound cannot close; the closed form settles it exactly.",
  "maximizer_dim": 0,
  "maximizer_desc": "1 distinct extremals (omega=z)",
  "representatives": [
   {
    "r0": "1",
    "omega": "z",
    "gammas": [
     "1",
     "0"
    ]
   }
  ]
 }
}