R1–R6: Zero-Backoff Collinear Endpoints, the Repaired Conic Bridge, Additional $\Omega_4$ Rays, and the 6-Parameter Cubic-Phase FamilyDate: 17 October 2026
Scope: Exact computer-assisted global attractivity certificates, exact leave-one-out separation witnesses, and 6,125-shape audit results implementing all four recommendations on top of Attractive_Centers_Independent_Rays_2026-10-17.zip.
Building on the baseline established in Attractive_Centers_Independent_Rays_2026-10-17.zip—which comprises the 46 benchmark centers $\text{B46}$, the complete constant-coefficient Symmedian/Steiner domain $\Omega_{K, HK}$, the four earlier shape-dependent maps, and the six independent rays $R_1, \dots, R_6$—this extension establishes four rigorously certified enlargements of the triangle Heart:
Zero-Backoff ($\delta = 0$) Global Radial Endpoints $R_2^$ and $R_5^$:
By shifting the 4-parameter ansatz $Z = G + (f_0 + f_1 q)K + (g_0 + g_1 q)HK$ to collinear-centered coordinates $\bar{f} = f_0 + \frac{1}{4}f_1, \; \bar{g} = g_0 + \frac{1}{4}g_1$ with $\tilde{q} = \frac{1}{4} - q = 3zv$, the derivative-coupling perturbations $\Delta a(z, v)$ and $\Delta c(z, v)$ vanish identically on the collinear boundary $v = 0$. Consequently, the exact supremal radial thresholds along the seeds of $R_2 = (3/4, 23/10, -1, 6/5)$ and $R_5 = (9/20, 6/5, -8/5, 13/5)$ are the exact rational numbers:
\(\lambda_{R_2}^* = \frac{8960}{9081} \approx 0.9866754763, \qquad \lambda_{R_5}^* = \frac{3040}{3113} \approx 0.9765499518.\)
Using a 4-box collar decomposition that isolates the exact square collinear factor $c(z, 0) \ge 0$ and certifies the polynomial remainder $R(z, v) = (\det M - a(z,0)c(z,0))/v > 0$ on the collinear collar, both exact endpoints $R_2^$ and $R_5^$ are globally certified with zero backoff ($\delta = 0$) in 28 and 25 2D Bernstein leaves, respectively (CONIC_BRIDGE_CERTIFICATE.json).
Repaired Continuous Conic Bridge $\mathbf{c}{\text{bridge}}(t)$ Between $R_2^*$ and $R_5^*$ ($\delta = 0$):**
In probe_collinear_bridge.py, the linear numerator $(1-t)\mathbf{c}{R_2} + t\mathbf{c}{R_5}$ normalized by the collinear tangency factor $\lambda{\text{flat}}(t) = \frac{640(5t+14)}{1329t^2+2042t+9081}$ failed at the midpoint $t = 1/2$ on the isosceles slice $h = 1$ ($a(9/40, 53/160) = -58932991/2086850000 < 0$). Adding the collinear-invisible quadratic bow $10560\,t(1-t)(-1/4, 1, 3/10, -6/5)$ (which satisfies $\Delta\bar{f} = \Delta\bar{g} = 0$ and thus leaves the collinear boundary $v = 0$ untouched) yields the rational quadratic arc:
\(\mathbf{c}_{\text{bridge}}(t) = \frac{640(5t + 14)\bigl[(1 - t)\mathbf{c}_{R_2} + t\mathbf{c}_{R_5}\bigr] + 10560\,t(1 - t)\bigl(-1/4, \; 1, \; 3/10, \; -6/5\bigr)}{1329t^2 + 2042t + 9081}, \qquad t \in [0, 1].\)
Along this entire curve, $D(t)c(z, 0, t) = (120tz - 17t + 336z - 165)^2 \ge 0$ remains an exact square vanishing at $z_*(t) = \frac{17t+165}{24(5t+14)}$, while at the former midpoint counterexample $(t, z, v) = (1/2, 9/40, 53/160)$ both the diagonal and determinant become strictly positive ($a = 27673/417370 > 0, \; \det M = 84395890319/1741977169000 > 0$). The entire continuous curve $t \in [0, 1]$ is globally certified with **zero backoff ($\delta = 0$) in 78 exact 3D Bernstein leaves across 5 sub-domains (CONIC_BRIDGE_CERTIFICATE.json), and its support function $\max_{t \in [0,1]} n \cdot Z_{\text{bridge}}(t)$ is evaluated in closed form via quadratic stationary roots in conic_bridge_and_cubic_centers.py.
Four Additional Certified $\Omega_4$ Rays (R7–R10):
Four additional simple rational directions in the 4-parameter $(f_0, f_1, g_0, g_1)$ family are globally certified with radial brackets of width $3 \times 10^{-8}$ (NEW_OMEGA4_RAYS_CERTIFICATES.json), expanding the $q$-linear slice $\pi_q(\Omega_4)$ across previously uncaptured angular sectors.
The 6-Parameter Cubic-Phase Family (P1–P3):
Introducing the root-free cubic squared-side central moment $p := 36\sum_{i=1}^3(s_i - 1/3)^3 = (1 - 3z)(36z^2 - 24z + 1 + 36zv) \in [-1, 1]$ breaks the acute/obtuse phase degeneracy $\pm\cos(6\theta_0)$ of $q$-only models while keeping all entries of $3\operatorname{Sym} D_1 Z$ polynomial in $(z, v)$ and at most quadratic in $v$. Both the product-rule derivation and direct fixed-vertex Cartesian differentiation in $\mathbb{Q}[r, X, Y]$ agree with zero residual (CUBIC_PHASE_MATRIX.json), and three high-gain 6-parameter rays (P1, P2, P3) are globally certified with radial brackets of width $3 \times 10^{-8}$ (CUBIC_PHASE_CERTIFICATES.json).
| Label | Family | Seed $\mathbf{c}_0$ | Certified Radial Lower Bound $\lambda_-$ (or Exact $\lambda^*$) | Upper Non-Attractivity Bound $\lambda_+$ | Bernstein Leaves (diag, det / 5-task) |
|---|---|---|---|---|---|
R2_star |
$\Omega_4$ ($\delta=0$) | $(3/4, \; 23/10, \; -1, \; 6/5)$ | $8960/9081 \approx 0.98667548$ (exact $\lambda^*$) | $8960/9081$ (width $0$) | $28$ ($10, 1, 15, 1, 1$) |
R5_star |
$\Omega_4$ ($\delta=0$) | $(9/20, \; 6/5, \; -8/5, \; 13/5)$ | $3040/3113 \approx 0.97654995$ (exact $\lambda^*$) | $3040/3113$ (width $0$) | $25$ ($12, 1, 10, 1, 1$) |
Conic_Bridge |
$\Omega_4$ curve ($\delta=0$) | $\mathbf{c}_{\text{bridge}}(t), \; t \in [0, 1]$ | $\delta = 0$ on full interval $t \in [0, 1]$ | Exact collinear tangency $\forall t \in [0,1]$ | $78$ 3D leaves ($40, 1, 35, 1, 1$) |
Bridge_Mid |
$\Omega_4$ ($\delta=0$) | $(43/80, \; 2, \; -49/40, \; 8/5)$ | $42240/41737 \approx 1.01205166$ (exact $\lambda^*$) | $42240/41737$ (width $0$) | Slice $t=1/2$ of 3D certificate |
R7 |
$\Omega_4$ ray | $(3/5, \; 0, \; -29/20, \; 2)$ | $94423179/10^8 = 0.94423179$ | $94423182/10^8 = 0.94423182$ | $6, \; 26$ |
R8 |
$\Omega_4$ ray | $(1, \; 4/3, \; -1/2, \; -7/10)$ | $99579960/10^8 = 0.99579960$ | $99579963/10^8 = 0.99579963$ | $4, \; 24$ |
R9 |
$\Omega_4$ ray | $(0, \; -12/5, \; 8/5, \; -9/2)$ | $91081582/10^8 = 0.91081582$ | $91081585/10^8 = 0.91081585$ | $4, \; 43$ |
R10 |
$\Omega_4$ ray | $(1/2, \; -7/5, \; -2, \; 9)$ | $96617699/10^8 = 0.96617699$ | $96617702/10^8 = 0.96617702$ | $9, \; 30$ |
P1 |
$\Omega_6$ $(q,p)$ ray | $(3/5, \; 1/10, \; -3/5, \; -8/5, \; 3, \; 7/10)$ | $101367358/10^8 = 1.01367358$ | $101367361/10^8 = 1.01367361$ | $13, \; 71$ |
P2 |
$\Omega_6$ $(q,p)$ ray | $(1/5, \; -14/5, \; 1/5, \; 13/10, \; -7/5, \; -1/5)$ | $95069548/10^8 = 0.95069548$ | $95069551/10^8 = 0.95069551$ | $7, \; 29$ |
P3 |
$\Omega_6$ $(q,p)$ ray | $(4/5, \; 11/5, \; -1/5, \; -4/5, \; -4/5, \; 2/5)$ | $99773390/10^8 = 0.99773390$ | $99773393/10^8 = 0.99773393$ | $9, \; 28$ |
ENLARGEMENT_SEPARATIONS.json)Every new discrete addition (Bridge_Mid, R7–R10, P1–P3) has an exact rational leave-one-out separation witness proving that it lies strictly outside the convex hull of:
B46 models + rigorous interval-arithmetic enclosures of $M_-$ and $M_+$,Independent_Rays maps $R_1, \dots, R_6$ + the exact zero-backoff endpoints $R_2^, R_5^$,R7–R10 and P1–P3, the entire continuous conic bridge $\mathbf{c}_{\text{bridge}}([0, 1])$ (bounded via an exact 1D Bernstein positivity certificate on $t \in [0, 1]$),| Target Map | Witness Sides $(a, b, c)$ | Separating Normal $n = (n_B, n_C)$ | Exact Point Gap | Robust Bracket Gap | 6,125-Shape Individual Gain Count | 6,125-Shape Leave-One-Out Count |
|---|---|---|---|---|---|---|
Conic_Bridge ($t \in [0,1]$) |
$(12, 8, 6)$ (at $t=1/2$) | $(-8546/9569, \; -4305/9643)$ | $+9.030070 \times 10^{-4}$ | $+9.030031 \times 10^{-4}$ | $520 / 6125$ | $421 / 6125$ ($322$ for $t=1/2$ alone) |
R7 |
$(14, 11, 5)$ | $(-4869/5009, \; 1647/7067)$ | $+1.444020 \times 10^{-3}$ | $+1.444020 \times 10^{-3}$ | $328 / 6125$ | $300 / 6125$ |
R8 |
$(6, 5, 3)$ | $(6977/7896, \; -3220/6889)$ | $+2.867143 \times 10^{-3}$ | $+2.867143 \times 10^{-3}$ | $294 / 6125$ | $286 / 6125$ |
R9 |
$(9, 8, 7)$ | $(-1717/5889, \; -9215/9639)$ | $+3.551533 \times 10^{-3}$ | $+3.551532 \times 10^{-3}$ | $104 / 6125$ | $46 / 6125$ |
R10 |
$(9, 8, 7)$ | $(8620/9399, \; -3931/9922)$ | $+6.749631 \times 10^{-3}$ | $+6.749631 \times 10^{-3}$ | $82 / 6125$ | $82 / 6125$ |
P1 |
$(6, 5, 3)$ | $(-9069/9316, \; 1301/5701)$ | $+1.322587 \times 10^{-2}$ | $+1.322586 \times 10^{-2}$ | $659 / 6125$ | $647 / 6125$ |
P2 |
$(6, 5, 3)$ | $(-6609/9709, \; -6832/9331)$ | $+3.768958 \times 10^{-2}$ | $+3.768958 \times 10^{-2}$ | $271 / 6125$ | $263 / 6125$ |
P3 |
$(6, 5, 3)$ | $(9883/9908, \; -685/9742)$ | $+1.131836 \times 10^{-2}$ | $+1.131836 \times 10^{-2}$ | $585 / 6125$ | $531 / 6125$ |
Across all $6,125$ nondegenerate archived shapes (ENLARGEMENT_AUDIT_SUMMARY.json), the new additions strictly enlarge the Independent_Rays hull ($\text{B46} \cup \Omega_{K, HK} \cup \text{prev}_4 \cup {R_1, \dots, R_6}$) on $1,389 / 6,125$ shapes ($22.68\%$): $520$ shapes from the repaired conic bridge, $513$ shapes from the four new $\Omega_4$ rays R7–R10, and $1,007$ shapes from the three 6-parameter cubic-phase rays P1–P3.
Run the standalone exact proof replay (requires only Python standard library, SymPy, NumPy, and mpmath; do not use -O):
python3 verify_conic_bridge_and_extensions.py
python3 test_conic_bridge_and_extensions.py
To regenerate all certificates, separation proofs, figures, and the complete archive from scratch:
python3 certify_conic_bridge_and_extensions.py
python3 certify_enlargement_separations.py
python3 plot_heart_enlargements.py
python3 audit_heart_enlargements.py
python3 build_heart_enlargement_bundle.py