Analy ic Ene gy Bound and he
Ze a–Cu a u e Limi :
A T anscenden al Lowe Bound in
Mul ica ie Phase Geome y
Bo a Ak a¸s & Cha GPT (co-au ho )
2025
Abs ac
We in oduce he concep o an Analy ic Ene gy Bound in mul ica ie phase ge-
ome y, a ising om he cu a u e hie a chy o analy ic phase ields. While classical
ene gy unc ionals a e limi ed by local me ic g adien s, analy ic con inua ion ex-
ends he ield cu a u e in o highe o de s, each weigh ed by an odd–ze a cons an
ζ(3), ζ(5), ζ(7), . . . . This yields a s ic ly posi i e ene gy loo : he o al ene gy
o an analy ic sys em can ne e anish, es ablishing a “ze a–cu a u e limi ” ha
couples geome y, analysis, and quan um phase dynamics. The p oposed heo em
shows ha e e y ζ(2k+1) e m adds a nonze o s i ness con ibu ion o he ene gy
densi y, linking numbe – heo e ic esidues o physical non–dissipa ion.
1 In oduc ion
Classical ield heo ies de ine ene gy h ough local g adien s o a po en ial o phase ield,
Eclass ∼ ∥∇Φ∥2.Howe e , when he phase ield Φ is analy ic on a mul ica ie mani old
Cn, i s highe –o de de i a i es encode a s uc u ed cu a u e hie a chy ha pe sis s
unde analy ic con inua ion. Each cu a u e o de co esponds o a deepe laye in he
phase–cone geome y, and he associa ed ene gy ecei es non i ial con ibu ions om
anscenden al cons an s eme ging in he Mellin–Ba nes expansion o he hype geome ic
ke nel:
p+1Fp(1) =
∞
X
k=0
Ak
(2k+ 1)! ζ(2k+ 1).
These ζ(2k+ 1) esidues gene a e wha can be iewed as analy ic cu a u e s i ness —
he inabili y o a phase mani old o ully la en unde me ic dissipa ion. As a esul ,
ene gy in analy ic mani olds admi s a nonze o lowe bound, analogous o a ze o–poin
ene gy, bu o igina ing om analy ic geome y a he han quan iza ion.
1
2 Analy ic Ene gy Bound
Fo mula ion
We de ine he analy ic ene gy unc ional
Eζ(m) =
m−2
X
k=1
α2m,2k+1 ζ(2k+ 1) ∥∇kΦ∥2,(1)
whe e ∇kΦ deno es he k– h o de cu a u e de i a i e o he phase ield Φ, and α2m,2k+1 >
0 a e analy ic con inua ion coe icien s de e mined by he pa i y o he ca ie o de C2m.
The ene gy bound ollows immedia ely:
Eζ(m)≥
m−2
X
k=1
α2m,2k+1 ζ(2k+ 1) ∥∇kΦ∥2>0.(2)
In e p e a ion
Because all coe icien s ζ(2k+ 1) >1, each cu a u e laye inc eases he minimal a ain-
able ene gy. Thus, e en in pe ec phase alignmen , he sys em e ains a esidual analy ic
oscilla ion, p e en ing comple e dissipa ion. This de ines he analy ic cu a u e loo :
Eclassical ≤Eζ(m).
Physically, his lowe bound plays he ole o an analy ic ze o–poin ene gy a ising om
he geome y o phase con inua ion.
[Ze a–Cu a u e Ene gy Bound] Le Φ be an analy ic phase ield on a smoo h mani old
Cnwi h cu a u e hie a chy {∇kΦ}m−2
k=1 . Assume:
(i) Φ ∈Hm(Ω) o compac Ω;
(ii) Each cu a u e ope a o Kk=∇†
k∇kis posi i e de ini e;
(iii) α2m,2k+1 >0 a e he Mellin–Ba nes esidue weigh s.
Then
Eζ(m) =
m−2
X
k=1
α2m,2k+1 ζ(2k+ 1) ⟨KkΦ,Φ⟩ ≥
m−2
X
k=1
α2m,2k+1 ζ(2k+ 1) ∥∇kΦ∥2>0.
Equali y holds only o he analy ically sa u a ed (maximally cohe en ) s a e o Φ.
3 Discussion
The heo em implies ha analy ic con inua ion ans o ms geome ic cu a u e in o an
a i hme ic ene gy hie a chy. Each odd–ze a cons an unc ions as a s i ness coe icien :
ζ(3) : p ima y analy ic igidi y (C6 egime),
ζ(5) : seconda y cu a u e s i ness (C8),
ζ(7) : hype bolic ex ension (C10),
ζ(9)+ : analy ic sa u a ion (C12 and beyond).
2
Consequen ly, he analy ic ene gy canno collapse o ze o: a minimal analy ic ib a ion
is in insic o all phase mani olds. This “ze a–cu a u e limi ” uni es analy ic numbe
heo y and physical ene gy bounds in o a single in a ian geome ic p inciple.
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