Logarithmic γ₂ over Cardioid S*_℘ · Geometric Function Atlas

← all proofs

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 Logarithmic γ₂ over Cardioid S*_℘. 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 cardioid_exp log_gamma2 "1/4" --slack 0
Recheck: python3 -m gft.proofs recheck data/proofs/cardioid_exp__log_gamma2.json

{
 "status": "PROVED",
 "engine": "closed-form",
 "statement": "max over S*(cardioid_exp) of |log_gamma2| = 1/4  (proven exact, closed form)",
 "class": "cardioid_exp",
 "functional": "log_gamma2",
 "bound": "1/4",
 "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": "1/4",
  "value_float": 0.25,
  "bracket_lo": 0.25,
  "bracket_hi": 0.25,
  "extremal_gammas": [
   "0",
   "1"
  ],
  "extremal_omega": null,
  "decomposition": {
   "L": "A*e^{i t} + C  (single harmonic in t)",
   "A_at_rho1": "\\frac{1}{4} - \\frac{r_{0}^{2}}{4}",
   "C_at_rho1": "\\frac{r_{0}^{2}}{4}",
   "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": "1/4",
   "value_float": 0.25,
   "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 a one-parameter family of extremals (rho=1, every r0 in [0,1]; representatives omega=z and omega=z^2) \u2014 a positive-dimensional maximizer set a fine-slack branch-and-bound cannot close; the closed form settles it exactly.",
  "maximizer_dim": 1,
  "maximizer_desc": "the one-parameter family \\(\\gamma_0=r_0\\in[0,1],\\ \\gamma_1=e^{i\\theta}\\) (\\(\\rho=1\\))",
  "representatives": [
   {
    "r0": "1",
    "omega": "z",
    "gammas": [
     "1",
     "0"
    ]
   },
   {
    "r0": "0",
    "omega": "z^2",
    "gammas": [
     "0",
     "1"
    ]
   }
  ]
 }
}