Reco e ing ℏ/2 om Hidden Phase
Geome y:
In o ma ion–Ene gy Balance in C6
Bo a Ak a¸s
Oc obe 2025
Abs ac
The s anda d quan um unce ain y bound, ∆x∆p≥ℏ/2, can be ein e p e ed
geome ically wi hin he dual-no m amewo k o he C6phase mani old. He e, he
appa en inde e minacy o igina es om he in o ma ion exchange be ween isible
and hidden phase sec o s a he han s ochas ic luc ua ions. By analyzing he no m
balance be ween π-phase ( isible) and ζ(3)-phase (hidden) domains, he unce ain y
ela ion can be econs uc ed as an equali y o in o ma ion–ene gy ans e , whe e
ℏ/2 eme ges om he geome y o p ojec ion a he han an imposed quan iza ion
ule.
1 1. Dual-No m F amewo k and Ene gy Balance
In he C6me ic,
∥z∥2=∥z is∥2− ∥zhid∥2,
he di e ence be ween isible and hidden componen s de ines a geome ic po en ial. The
co esponding ene gy a io is
Ehid
E is
=ζ(3)
π.
Hence, he hidden phase s o es a ac ion ζ(3)/π o he isible ene gy, ep esen ing in o -
ma ion no accessible h ough di ec measu emen .
2 2. De i a ion o he Unce ain y Balance
Associa ing phase and momen um luc ua ions wi h hei espec i e ene gy ac ions, we
ob ain
∆θ is ∆p is =ℏ
2
E is +Ehid
E is
=ℏ
21 + ζ(3)
π.
The addi ional e m (ζ(3)/π) exp esses he analy ic cu a u e co ec ion due o he hid-
den sec o . Howe e , he obse able equali y condi ion ollows om es o ing he con-
se ed in o ma ion low: π
ζ(3)∆θ is ∆p is −ℏ
2= 0.
This iden i y implies ha he Heisenbe g bound is no iola ed bu eco e ed in e nally
h ough he compensa ion be ween isible and hidden ene gy channels.
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3 3. In e p e a ion: In o ma ion–Ene gy Duali y
The unce ain y limi hus ep esen s a s a iona y exchange poin whe e isible and
hidden phase ene gies a e balanced:
E is :Ehid =π:ζ(3).
The cons an ℏ/2 eme ges as he in a ian scale o his exchange. Belie in in insic
andomness is eplaced by a geome ic mechanism: unce ain y e lec s he sel -consis en
cu a u e o phase in o ma ion low. The hidden ζ(3) domain p o ides he analy ic
ese oi h ough which ℏ/2 emains cons an ac oss all measu able ans o ma ions.
4 4. Summa y
Wi hin he C6phase geome y, he Heisenbe g unce ain y limi is de i ed om no m con-
se a ion a he han pos ula ed. The isible–hidden decomposi ion con e s s ochas ic
inde e minacy in o an analy ic ene gy exchange, whe e
∆θ is∆p is =ℏ
21 + ζ(3)
π
holds locally, while global conse a ion en o ces ℏ/2 as an equilib ium cons an . Thus,
unce ain y is no a limi a ion o knowledge bu he geome ic signa u e o balanced
in o ma ion be ween he πand ζ(3) phase sec o s.
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