
    ziu                     P    d dl mZmZmZmZmZmZ d	dZde_         eddd      Z	y)
    )mpcoslogsqrtpie     c                     dt         _        t        j                          t        dd        t        d       t        d        t        d         t         j                  dt         j
                  z   z  z  }t        dt        |      d       t        d	t        |      d
d       t        j                  d      }g }g }t        ||dz         D ]b  }t        t
        dz   t        |      z  z
        }|d|z  t        |      z  z  }|j                  |       |j                  t        |             d t        d  d|        t        d       t        |      }	t        |      }
t        dt        |	      d
dt        |
      d
d       t        d       dt
        z   z  }t        dt        |      d       |
|	z
  |z  }t        dt        |      d
        fd}d}||z  }t        j                  d      }t        |      D ]-  }||z  }|dz   |z  } ||      } ||      }|||z   dz  |z  z  }/ t        d|        ||z   dz  }dt        |      z  }d|z  |z  |z  }t        d       t        d       t        dt        |      d
d| dt        |      d
       t        d|        t        d        t        d!||z
  dz    d"       t        d#t        |      d
       t        d$||z
  t        |      z  d%       t        d&       t        dd'      D ]  }|||z  z  }||k  st        t
        dz   t        |      z  z
        }t        t
        dz   t        |      z  z
        }t        t
        dz   t        |      z  z
        }t        t
        dz   t        |      z  z
        }t        d(| d)t        |      d*d+t        |      d,d-|d
d.|d
d/t        |      d
d.t        |      d
        t        dd        t        d0       t        d1|        t        d2|        t        d3t        ||z
                t        d        |||||d4S )5u{  
    Z(t) = Σ cos(π/4 - t*log(n)) / √n
    
    Collapsing discrete gap d between n using periodicity:
    cos(θ) = cos(π/4 - t*log(n)) repeats when t*log(n) changes by 2πk
    
    Period in n-space:
    t*log(n_next) - t*log(n) = 2πk
    log(n_next/n) = 2πk/t
    n_next/n = e^(2πk/t) = d
    
    So we collapse the sum into an integral over the repeating gaps.
    2   
zP================================================================================z8INTEGRAL COLLAPSE: Converting discrete sum to continuouszt =    u   
Period d = e^(2π/t) = z.6fu   → Every ~z.4fu   × increase in n, cos() repeatsr   r	      z
[1] Discrete Sum Z(z) = z
[2] Integral Collapse:z#   Integral bounds in x = log(n): [z, ]u'      Integrand: cos(π/4 - t*x) * e^(x/2)u"      One period in x: Δx = 2π/t = z   Number of periods: c                 b    t        t        dz  | z  z
        t        j                  | dz  z  z  S )Nr   r   )r   r   r   r   )xts    verificiation_collapse_gap.py	integrandz&Z_integral_collapse.<locals>.integrandH   s+    2a4!a%< 244AE?22    r
   u      ∫ over one period = z
[3] Integral Approximation:u2      Z ≈ 2 × n_periods × integral × avg(1/√n)u      Z ≈ 2 × u    × z   Z_integral = z!
[4] Gap Collapse Interpretation:u      Discrete: Σ over n = z termsu+      Continuous: ∫ collapsed by period d = z   Equivalent terms: ~z.2fz
[5] Periodicity Check:   z   n=u    → n'=z.1fu    (×z.3fz): arg=u    → z, cos=zRESULT COMPARISON:z  Z_discrete  = z  Z_integral = z  Error       = )
Z_discrete
Z_integralperiod_dperiod_x	n_periods)r   dpsmpfprintr   r   floatranger   r   r   appendabs)r   n_startn_enddr   n_values
cos_valuesncx_startx_endr   r   r   N_stepsdxintegral_one_periodix0x1f0f1avg_navg_inv_sqrt_nr   kn_checkarg1arg2c1c2s   `                              r   Z_integral_collapser=      s    BF
q	A	Bvh-	DF	VH	D* 	RUUQA	%eAhs^
45	Ka~%D
EF JHJ7EAI& $1q3q6z!"a!ed1go%
%(#	$ 
!!D
56 
$& 'lGJE	/gs/C2eElSVEWWX
YZ	35
 2vzH	.uXs.C
DE H,I	"5#3C"8
9:3 G	G	B&&)7^ 2V!er\r]r]R1}r112 
%&9%:
;<
 u_!Ee_NY!44~EJ	)+	>@	N5+C05H4IeTbNcdgMh
ij	ZL
)* 
.0	%ego&9%:&
AB	7a~
FG	"EGOuQx#?"D
EF 
$&1a[ ]Q!V$eACL 001DACL 001DRTAG,,-BRTAG,,-BE'(5>#*>d58C. Qc
%Szb	#eERTIVY?\ ]] 
Bvh-	 	ZL
)*	OJ<
()	Sj!89:
;<	VH !  r   r   g>٬D,@)r   r$   r%   N)r	   r
   )
mpmathr   r   r   r   r   r   r=   r   result r   r   <module>rA      s)    , ,CL 
	y!4	@r   