DOI: 10.5281/zenodo.18079441
title: FUNt — Golden Python Play Model (φ⁷ Edition) now with -  Schrödinger boundary solver
title: Schrödinger boundary solver

description: note - Python Version Updated 12/28/25

 note - Python Version Updated 11/17/25


The FUNt Golden Python Play Model  7.2 (φ⁷ Edition) -

  - Student-safe Schrödinger boundary solver (default)#  - A/B/C Golden module preserved#  - φ-HRB boundaries hidden behind safe presets (A2, B3, C3)#  - Minimal label added for clarity (no new menus)# - Single-line model label (requested):# - “Model: 1D finite quantum well (Schrödinger) with student-safe generalized boundaries.”

A wide family of mathematical constants, reveal a unified behavior,

when examined through the lens of geometric resonance.

Many of these constants, 

— including φ, φ⁷, silver, supersilver, metallic means, plastic, supergolden, 

and custom bosonic inputs.

— all universal constants, collapse into the same universal geometric behavior when expressed through the Golden-Resonance Spiral.

This behavior is shown through a reproducible Python tool (v7.2) optimized for tablets and low-compute (mobile) devices, revealing a structural equivalence across constants traditionally treated as distinct.

The purpose of this paper is to:

Document the discovery.

Present  the underlying algebra.

Provide  a public reference version suitable for; Zenodo DOI archiving.

 

1. INTRODUCTION

Most mathematical constants appear isolated 

— each with its own history, origin, and symbolic niche. 

Yet when examined through the lens of geometric resonance, many of these constants reveal a unified behavior.

The FUNt Golden Python Play Model v7.2 emerged from an engineer’s intuition, refined through computational visualization: 'Pattern Recognition'

'If nature uses a pattern once, it will use it again wherever alignment is possible.'

Colab notebook Python Code. (Copy to paste in Colab)

# ============================================================# FUNt Golden Python Play Model v7.2 — Minimal Label Edition# Author: M.K. Nowlin (Omm) • Helper: Oasis## Purpose:#  - Student-safe Schrödinger boundary solver (default)#  - A/B/C Golden module preserved#  - φ-HRB boundaries hidden behind safe presets (A2, B3, C3)#  - Minimal label added for clarity (no new menus)## Single-line model label (requested):# “Model: 1D finite quantum well (Schrödinger) with student-safe generalized boundaries.”# ============================================================

import os, json, mathimport numpy as npimport matplotlib.pyplot as plt

OUTDIR = "/outputs"os.makedirs(OUTDIR, exist_ok=True)

# ============================================================# CORE CONSTANTS# ============================================================phi = (1 + 5**0.5)/2phi7 = phi**7fractaile_D = 1.618

silver = 1 + np.sqrt(2)supersilver = (1 + np.sqrt(2))**2supergolden = 1.4655712318767687plastic = 1.3247179572447458sacred_default = 1.7320508075688772

ħ = 1.0m_default = 1.0V0_default = 0.0

def metallic_mean(n:int):    return (n + np.sqrt(n*n + 4))/2

def log_phi(x):    return np.log(x)/np.log(phi)

# ============================================================# GOLDEN FAMILY + SPIRAL# ============================================================

def golden_family(n_window=(-7,8)):    fam = {}    for n in range(n_window[0], n_window[1]):        fam[f"G[{n}]"] = (phi**7)**(n/7)    return fam

def plot_spiral(value, label=""):    theta = np.linspace(0, 2*np.pi, 600)    r = value ** (theta/(2*np.pi))    fig = plt.figure(figsize=(6,6))    ax = fig.add_subplot(111, projection='polar')    ax.plot(theta, r, linewidth=2, label=label)    ax.set_title("Golden Resonance Spiral")    ax.legend()    fig.savefig(os.path.join(OUTDIR, "golden_spiral.png"), dpi=150)    return fig

# ============================================================# HRB BOUNDARY SYSTEM (hidden, student-safe presets)# ============================================================

def Z_HRB(Z_base:float, m:int, theta_deg:float):    return Z_base * (phi**m) * math.cos(math.radians(theta_deg))

def dispersion_F(k:float, L:float, kappa:float, Z0:float, ZL:float):    if k == 0:        cos_kL = 1 - (k*L)**2/2        sin_kL = k*L    else:        cos_kL = math.cos(k*L)        sin_kL = math.sin(k*L)

    term1 = cos_kL    term2 = ((Z0 + ZL)/(2*kappa*(k if k!=0 else 1))) * sin_kL    term3 = (Z0*ZL)/(kappa**2 * (k**2 if k!=0 else 1e-30)) * (1 - cos_kL)    return term1 + term2 + term3

def _bisection(F, a, b, args, max_iter=60, tol=1e-10):    Fa = F(a,*args); Fb = F(b,*args)    if np.isnan(Fa) or np.isnan(Fb): return None    if Fa == 0: return a    if Fb == 0: return b    if Fa*Fb > 0: return None    left, right = a, b    for _ in range(max_iter):        mid = (left + right)/2        Fm = F(mid,*args)        if abs(Fm) < tol: return mid        if Fa*Fm <= 0:            right = mid        else:            left = mid; Fa = Fm    return (left + right)/2

def scan_roots(F, k_min, k_max, args, samples=12000):    ks = np.linspace(k_min, k_max, samples+1)    vals = np.array([F(k,*args) for k in ks])    roots = []    for i in range(len(ks)-1):        a, b = ks[i], ks[i+1]        Fa, Fb = vals[i], vals[i+1]        if np.isnan(Fa) or np.isnan(Fb): continue        if Fa == 0:            roots.append(a)        elif Fa*Fb < 0:            r = _bisection(F,a,b,args)            if r is not None and (len(roots)==0 or abs(r-roots[-1])>1e-6):                roots.append(r)    return np.array(roots), ks, vals

def solve_boundary_spectrum(    L=1.0, kappa=1.0,    Z0_base=1.0, m0=0, theta0=60.0,    ZL_base=1.0, mL=0, thetaL=30.0,    k_min=0.001, k_max=80.0,    show_plot=True, save_plot=True):    Z0 = Z_HRB(Z0_base, m0, theta0)    ZL = Z_HRB(ZL_base, mL, thetaL)    args = (L, kappa, Z0, ZL)

    roots, ks, Fs = scan_roots(dispersion_F, k_min, k_max, args)

    fig = plt.figure(figsize=(8,5))    ax = fig.add_subplot(111)    ax.plot(ks, Fs, linewidth=1)    if len(roots)>0:        ax.scatter(roots, np.zeros_like(roots), s=10)    ax.axhline(0, linestyle="--", linewidth=0.8)    ax.set_title("Boundary Dispersion F(k)")    ax.set_xlabel("k")    ax.set_ylabel("F(k)")    fig.tight_layout()    if save_plot: fig.savefig(os.path.join(OUTDIR,"boundary_dispersion.png"),dpi=150)    if show_plot: plt.show(fig)    else: plt.close(fig)

    eigen_table = []    if len(roots) > 0:        k1 = roots[0]        for i,kn in enumerate(roots, start=1):            eigen_table.append({                "n": i,                "k_n": float(kn),                "phi_log_rel": float(log_phi(kn/k1))            })

    payload = {        "params":{            "L":L,"kappa":kappa,            "Z0_base":Z0_base,"m0":m0,"theta0_deg":theta0,            "ZL_base":ZL_base,"mL":mL,"thetaL_deg":thetaL        },        "phi":float(phi),        "phi7":float(phi7),        "fractaile_D":float(fractaile_D),        "eigen":eigen_table    }

    with open(os.path.join(OUTDIR,"boundary_eigs.json"),"w") as f:        json.dump(payload,f,indent=2)

    return roots, payload

# ============================================================# SCHRÖDINGER WRAPPER (DEFAULT MODE)# ============================================================

def schrodinger_energies_from_k(ks, ħ=ħ, m=m_default, V0=V0_default):    return (ħ**2)*(ks**2)/(2*m) + V0

# HRB presets → student-safeHRB_PRESETS = {    "A2": dict(theta=0.0,  m=0, Z_base=1.0),    "B3": dict(theta=30.0, m=0, Z_base=1.0),    "C3": dict(theta=60.0, m=0, Z_base=1.0),}

def run_schrodinger_block(    L=1.0, m=m_default, ħ_val=ħ, V0=V0_default,    left_preset="C3", right_preset="B3",    k_min=0.001, k_max=80.0, show_plot=True):    print("Model: 1D finite quantum well (Schrödinger) with student-safe generalized boundaries.\n")

    roots, payload = solve_boundary_spectrum(        L=L, kappa=1.0,        Z0_base=1.0, m0=HRB_PRESETS[left_preset]["m"],  theta0=HRB_PRESETS[left_preset]["theta"],        ZL_base=1.0, mL=HRB_PRESETS[right_preset]["m"], thetaL=HRB_PRESETS[right_preset]["theta"],        k_min=k_min, k_max=k_max, show_plot=show_plot, save_plot=True    )

    En = schrodinger_energies_from_k(roots, ħ=ħ_val, m=m, V0=V0)

    # Save with label    spectrum = {        "model_label": "1D Schrödinger finite well with student-safe generalized boundaries",        "ħ":ħ_val, "m":m, "V0":V0,        "k_modes":[float(x) for x in roots],        "E_modes":[float(x) for x in En]    }    with open(os.path.join(OUTDIR,"schrodinger_spectrum.json"),"w") as f:        json.dump(spectrum,f,indent=2)

    # Energy ladder plot    if len(En)>0:        fig = plt.figure(figsize=(6,5))        ax = fig.add_subplot(111)        ax.plot(np.arange(1,len(En)+1), En, marker='o')        ax.set_xlabel("n")        ax.set_ylabel("E_n")        ax.set_title("Schrödinger Spectrum\n1D finite well with student-safe boundaries")        fig.tight_layout()        fig.savefig(os.path.join(OUTDIR,"schrodinger_spectrum.png"),dpi=150)        if show_plot: plt.show(fig)        else: plt.close(fig)

    print("Saved: boundary_dispersion.png, boundary_eigs.json, schrodinger_spectrum.json/.png\n")    return roots, En, payload

# ============================================================# RESOLVE FOR GOLDEN PLAY# ============================================================

OPTIONS = {    "Golden φ": phi,    "φ^7": phi7,    "D (Fractaile)": fractaile_D,    "Silver δ": silver,    "Supersilver ς": supersilver,    "Supergolden ψ": supergolden,    "Plastic ρ": plastic,    "Metallic mean φ_n": "metallic",    "Sacred": "sacred",    "Boson": "boson",}

def resolve(name, metallic_n=1, boson=1.0, sacred=sacred_default):    base = OPTIONS[name]    if base=="metallic": return metallic_mean(metallic_n)    if base=="boson":    return boson    if base=="sacred":   return sacred    return base

# ============================================================# GOLDEN BLOCK# ============================================================

def run_golden_block(A_name="Golden φ", B_name="φ^7", C_name="(none)",                     metallic_n=1, boson=1.0, sacred=sacred_default,                     save_summary=True, do_spiral=True):

    A = resolve(A_name, metallic_n, boson, sacred)    B = resolve(B_name, metallic_n, boson, sacred)    C = None if C_name=="(none)" else resolve(C_name, metallic_n, boson, sacred)

    print(f"A = {A}")    print(f"B = {B}")    if C is not None: print(f"C = {C}")

    if do_spiral:        figA = plot_spiral(A, label=A_name)        plt.show(figA)

    fam = golden_family()    print("\nGolden Family G[k]:")    for k,v in fam.items():        print(f"{k:>5} = {v:.9f}")

    if save_summary:        summary = {            "A":A,"B":B,"C":C,            "phi":phi,"phi7":phi7,            "fractaile_D":fractaile_D,            "Metallic_n":metallic_n,            "Boson":boson,            "Sacred":sacred,        }        with open(os.path.join(OUTDIR,"summary.json"),"w") as f:            json.dump(summary,f,indent=2)        print("\nSaved /outputs/summary.json")

# ============================================================# MAIN (RUN BOTH BLOCKS)# ============================================================

if __name__ == "__main__":    run_golden_block(        A_name="Golden φ", B_name="φ^7", C_name="(none)",        metallic_n=1, boson=1.0, sacred=sacred_default,        save_summary=True, do_spiral=True    )

    run_schrodinger_block(        L=1.0, m=1.0, ħ_val=1.0, V0=0.0,        left_preset="C3", right_preset="B3",        k_min=0.001, k_max=80.0, show_plot=True    )

 

 
description: Final 'corrected' Model of v5.0 uploaded