IC–IIa: Fo mal Consolida ion o he
In o ma ional Dynamic Calculus in
he Theo y o In o ma ional
Eme gence (TIE)
Adol o J. Céspedes Jiménez
ORCID: 0009-0003-3026-7611
Uni e si y o G oningen – Facul y o A s
Abs ac
This documen o malizes he s able e sion o he In o ma ional Calculus II (IC–II), he
dynamic compu a ional laye o he Theo y o In o ma ional Eme gence (TIE). IC–IIa
p o ides he inalized no a ion, ope a o s, dynamical ules, cohe ence unc ions,
in o ma ional de i a i es, and ansi ion laws linking in e nal con igu a ions (Iₛ),
ma ix- ield con igu a ions (Iₘ), and eme gen pe spec i es. I also es ablishes he o mal
b idge o TIE–Dialog, he empi ical implemen a ion o IC–II o he measu emen o
con e sa ional cohe ence.
This consolida ed e sion se es as he ounda ion o IC–III and he opological phase o he
In o ma ional Calculus.
This e sion ( 1) p o ides he ma hema ically consolida ed o mula ion o IC–II as used in
TIE–Dialog. Fu u e e sions will expand he opological ea men in oduced in IC–III.
1. In oduc ion
IC–II de ines how in o ma ional sys ems e ol e, synch onize, di e ge, and sel - epai . I
p o ides:
• a dynamic cohe ence unc ion 𝒞ₜ
• ope a o s o esonance, con as , and usion
• ules o in o ma ional s abiliza ion and epai
• a minimal di e en ial ∂ᵢ cap u ing pe spec i al change
• and a gene al ansi ion cycle Iₛ → Iₘ → Iₛ′
IC–IIa consolida es hese componen s in o a single, in e nally consis en and empi ically
es able o mal s uc u e.
2. Co e En i ies and No a ion
In e nal con igu a ion (Iₛ)
S a e ec o o he sys em a ime .
Ma ix- ield con igu a ion (Iₘ)
In o ma ion p ojec ed om he b oade ield/ma ix a ime ; ep esen s cons ain s,
con ex , and en i onmen al s uc u e.
Cohe ence (𝒞ₜ)
Deg ee o dynamic alignmen be ween Iₛ and Iₘ, modula ed by ola ili y and asynch onous
in e -sys emic di e ence:
𝒞ₜ = (sim(Iₛ, Iₘ), ∂ᵢIₛ, Δₐₛyₙc)
High cohe ence implies ha Iₛ and Iₘ e ol e in nea -synch ony wi h minimal empo al o se .
In o ma ional di e en ial (∂ᵢ)
Ra e o change o Iₛ ac oss he pe spec i al imeline:
∂ᵢIₛ( ) = Iₛ( ) − Iₛ( −1)
Repai ope a o (ℛ)
Minimal in e en ion ha es o es cohe ence a e b eakdown.
Resonance (⊗)
No malized alignmen be ween wo con igu a ions.
Fusion (⊕)
S able combina ion o wo con igu a ions.
Con as (⊖)
Di e ence be ween con igu a ions.
3. In o ma ional Ope a o s
3.1 Fusion (⊕)
Iₐ ⊕ Iᵦ = η·Iₐ + (1−η)·Iᵦ
3.2 Con as (⊖)
Iₐ ⊖ Iᵦ = Iₐ − Iᵦ
3.3 Resonance (⊗)
Iₐ ⊗ Iᵦ = cos(Iₐ, Iᵦ)
4. Tempo al Dynamics
4.1 In o ma ional Di e en ial (∂ᵢ)
∂ᵢIₛ( ) = Iₛ( ) − Iₛ( −1)
Measu es in e nal ola ili y.
4.2 Ex ended Dynamic Cohe ence Func ion (𝒞ₜ)
𝒞ₜ = σ( α·cos(Iₛ( ), Iₘ( +δ)) − β·|∂ᵢIₛ( )| − γ·Δₐₛyₙc( ) )
Whe e:
• cos(Iₛ, Iₘ) → seman ic alignmen
• ∂ᵢIₛ → ola ili y penal y
• Δₐₛyₙc → empo al-pe spec i al misalignmen
• δ → in insic pe spec i al o se
• σ(x) = 1 / (1 + e^(−x))
• α, β, γ ∈ ℝ⁺
Cohe ence becomes he join minimiza ion o seman ic misalignmen , in e nal ola ili y, and
asynch ony.
4.3 Asynch onous In e -Sys em Di e ence (Δₐₛyₙc)
Δₐₛyₙc( ) = ∥ Iₛ( ) − Iₘ( +δ) ∥
• δ ∈ ℝ⁺ models pe spec i al o se
• δ = 0 eco e s synch onous cohe ence
• δ > 0 cap u es s agge ed in o ma ional lows
Δₐₛyₙc e lec s misma ches be ween:
• lexical s composi ional la encies
• p edic ion s in eg a ion cycles
• u n- aking and epai
• pe spec i al eme gence
IC–IIa adop s Δₐₛyₙc as a p imi i e dimension o cohe ence.
4.4 Dimensionali y-D i en Synch oniza ion
The a e a which a sys em educes asynch onous di e ence depends on i s in o ma ional
dimensionali y (Dₛ):
dΔₐₛyₙc( ) / d = −k(Dₛ)·Δₐₛyₙc( )
Whe e k(Dₛ) is mono onically inc easing.
In e p e a ion:
• Low-dimensional sys ems → agile cohe ence
• High-dimensional sys ems → obus synch oniza ion capaci y
Dimensionali y de e mines he sys em’s abili y o in eg a e dispe sed and empo ally
displaced lows.
4.5 The Ma ix as a Nega i e Sys em
The ma ix- ield con igu a ion Iₘ ac s as a nega i e sys em:
• Iₛ( ) → posi i e p ojec ion
• Iₘ( +δ) → nega i e s abiliza ion
App oxima ion:
Iₘ ≈ − ∂Iₛ/∂ + con ex ual s uc u e
In e p e a ion:
Cohe ence eme ges om a posi i e–nega i e in e play:
• he sys em gene a es s a es
• he ma ix cons ains, coun e balances, and s abilizes hem
This p e en s unaway di e gence and ancho s pe spec i al cohe ence.
5. Repai and S abili y
5.1 B eakdown condi ion
𝒞ₜ < Φ_low
5.2 Law o Minimal Repai
ΔIₛ = a gmin Δ | 𝒞ₜ₊₁ − 𝒞ₜ | subjec o 𝒞ₜ₊₁ > 𝒞ₜ
5.3 S abiliza ion zone
Φ_low ≤ 𝒞ₜ ≤ Φ_high
6. The T iadic T ansi ion Cycle
Iₛ( ) → Iₘ( +δ) → cohe ence check → epai (i needed) → Iₛ′( +1)
Fo mal upda e:
Iₛ′( +1) = Iₛ( ) ⊕ g(Iₘ( ), ℛ( ))
7. Minimal Pseudocode
o in ange(1, T - del a): # asummed del a >= 0
# in o ma ional di e en ial (∂ᵢ I_s)
dI = I_s[ ] - I_s[ -1]
dI_no m = no m(dI)
# seman ic simila i y + asynch onous di e ence
sim = cosine(I_s[ ], I_m[ + del a])
Del a_async = no m(I_s[ ] - I_m[ + del a])
# ex ended cohe ence unc ion
C_ = sigmoid(alpha * sim - be a * dI_no m - gamma * Del a_async)
# Law o Minimal Repai
i C_ < Phi_low:
I_s[ ] = epai (I_s[ ], I_m[ + del a])
s o e(C_ )
8. Empi ical Implemen a ion: TIE–Dialog
TIE–Dialog ins an ia es IC–IIa by:
• using seman ic embeddings as Iₛ and Iₘ
• compu ing 𝒞ₜ u n-by- u n
• de ec ing b eakdown (𝒞ₜ < Φ_low)
• iden i ying epai s (minimal inc eases)
• compu ing Δₐₛyₙc( )
• acking 𝒞ᵢ pe pa icipan
• gene a ing ull analy ical epo s
IC–IIa becomes ully empi ical and es able.
9. Nume ical Example
Gi en:
Iₛ = (0.6, 0.8)
Iₘ = (0.5, 0.7)
Iₛ(p e ) = (0.7, 0.6)
sim = cos(Iₛ, Iₘ) = 0.992
∂ᵢIₛ = (−0.1, 0.2) → |∂ᵢIₛ| = 0.224
𝒞ₜ = σ( 1.2·0.992 − 0.8·0.224 )
I 𝒞ₜ < Φ_low → epai igge ed.
10. Figu e (Concep ual)
Iₛ( ) → Iₘ( +δ) → cohe ence → epai (i needed) → Iₛ′( +1)
11. Conclusion
IC–IIa consolida es he dynamic laye o he TIE, p o iding a comple e, s able, and
ep oducible o mal basis o modeling cohe ence, up u e, epai , and pe spec i al
e olu ion.
I addi ionally inco po a es he asynch onous in e -sys emic di e ence Δₐₛyₙc, g ounding
cohe ence in he minimiza ion o empo al desynch oniza ion be ween in o ma ional lows.
This aligns IC–IIa wi h neu ocogni i e e idence and p o ides a uni ied mechanism o
synch oniza ion ac oss na u al and a i icial sys ems.
Re e ences
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F amewo k o Cohe ence and Repai Dynamics in he Theo y o In o ma ional
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F is on, K. (2010). The ee-ene gy p inciple: A uni ied b ain heo y? Na u e Re iews
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