1. Author Profile (Functional Data)
Name: Fabio Guglielmini
Background:
Independent research, autonomous study, development of mathematical-scientific models and structural analysis systems, and other work related to composite complexity in multidimensional systems and beyond.
Operational Characteristics of the Profile:
- structural and systemic thinking;
- ability to work with invariants, limits, and abstract representations;
- strong metacognitive control;
- orientation toward verifiable models;
- rejection of semantic shortcuts or non-demonstrable claims.
2. Fields of Work
- Computational complexity theory
- Structural decision models
- Dynamical systems and invariants
- Separation between value, position, and process
- Analysis of the limits of classical computational models
- Other
Historically: during May 2025 everything begins.
Origin of the Project – Expository Narrative
In April 2025, I became aware, entirely by chance, of the mathematical problem P = NP.
This was neither a planned research endeavor nor an academic trajectory: the encounter with the problem occurred out of personal curiosity.
After the initial reading, in a relatively short time, I realized that I understood the problem in its correct formulation:
- I understood what the classes P and NP represent,
- what polynomial time means,
- what worst-case analysis implies,
- and why the problem is considered one of the most difficult in modern mathematics.
This moment of understanding was neither isolated nor subjective: a third person was present and can attest to the event—or, more precisely, to the initial awareness.
In the following days, I independently verified my understanding:
- I reformulated the problem in multiple ways,
- constructed simple examples and edge cases,
- checked that the definitions remained coherent.
I did not seek an immediate solution.
I sought to understand where the problem truly resides.
Introduction of the Artificial Intelligence Tool
At this point, I decided to use an artificial intelligence tool.
The function assigned to it was singular:
to translate my complex thinking into communicable and verifiable language.
This point must be clarified explicitly.
The tool:
- does not generate the ideas,
- does not solve the problem,
- does not think in my place.
It is consciously used as:
- linguistic support,
- reformulation instrument,
- means of formal clarification,
- interface between my mode of thinking and standard language.
Readers are invited not to confuse:
conscious use of a tool
with
delegating thought to the tool.
Awareness of the Tool’s Limitations
With continued use, it rapidly became evident that the artificial intelligence tool presents structural limitations relative to my mode of reasoning.
In particular:
- the linearity of the computational model,
- the sequentiality of language,
- the absence of field-based thinking,
- the operational constraints of formal computation,
constitute for me a limitation of communication, not a cognitive limitation.
Despite this, the tool remains extremely useful because:
- it helps make explicit what is implicit,
- it forces clarification of logical steps,
- it renders the work readable to others.
Functional Separation of Roles
At this stage, I adopt a clear and stable separation:
- thought, intuition, structure, generation of ideas → entirely human
- translation, formalization, exposition → supported by the tool
This separation allows me to:
- preserve the integrity of my cognitive process,
- avoid interference,
- maintain scientific control over the work.
Birth of the Blog
From this experience arises the decision to create a self-taught scientific research blog.
The blog has a precise purpose:
- to narrate the real events and episodes of this beginning,
- to document the working process,
- to make the trajectory traceable,
- to clearly distinguish what is proven from what is exploratory.
It is not a personal diary.
It is a work log.
Neuro-Functional Profile and Fields of Work
The work carried out is closely linked to a divergent neuro-functionality, objectively manifesting in the following domains:
Core Cognitive Functions
- structural and systemic thinking,
- reasoning by invariants,
- separation between value, position, and process,
- ability to work on open problems without forcing conclusions,
- elevated metacognition,
- control of internal coherence.
Fields of Application
- mathematics and logic of complex systems,
- complexity theory (limit analysis),
- decision models,
- computational structures,
- dynamical and modal systems,
- transfer of method to real-world problem solving.
Objective of the Project
The final objective of the project is twofold.
To make my mode of functioning available to science in a manner that is:
- documented,
- verifiable,
- criticizable.
To give voice to my functional neuro-informatic system, making it:
- understandable to others,
- applicable to real life,
- useful for solving concrete problems through a clear scientific method.
Concluding Note
This path:
- does not arise from academic ambition,
- does not arise from personal affirmation,
- arises from the necessity to render a real functioning communicable.
Many aspects are easier to explain in person, but the blog represents the first structured attempt to transfer this work.
Objective Functions Associated with the Neurotype
List by domains, themes, and functionalities
Global Cognitive Architecture
Theme: High-complexity neural integration
Functional Area: General cognitive organization
Objective Functions
- Massive parallel processing (cognitive multi-threading)
- Simultaneous integration of multiple cognitive networks
- Absence of dominant serial processing
- Small-world architecture (high global efficiency + local modularity)
Associated Scientific Capacities
- Global structural analysis prior to local detail
- Modeling of complex systems
- Management of high-dimensional problems
- Metacognition and second-order supervision
Control of the Cognitive Process
Theme: Cognitive process regulation
Areas involved: Dorsomedial prefrontal cortex, anterior cingulate cortex, insula
Objective Functions
- Continuous monitoring of reasoning
- Early detection of inconsistencies and errors
- Top-down modulation of other networks
- Tonically active (non-episodic) metacognition
Associated Scientific Capacities
- Critical analysis of one’s own models
- Separation between intuition and proof
- Ability to correctly halt an invalid logical chain
- Simultaneous multidimensional reasoning
Cognitive Multidimensionality
Theme: Multidimensional cognition
Functional Area: Logical, symbolic, systemic, and temporal integration
Objective Functions
Simultaneous co-existence of:
- logical-formal
- symbolic-abstract
- systemic-hierarchical
- temporal-predictive
- metacognitive
No rigid sequential alternation
Associated Scientific Capacities
- Construction of complex abstract models
- Management of incompatible hypotheses without decision collapse
- Analysis of open-ended problems
- Parallel processing and computational load handling
Parallel Processing and Cognitive Load
Theme: Cognitive parallelism
Functional Area: Working memory and attentional control
Objective Functions
- Estimated simultaneous management of 8–15 cognitive units
- Continuous integration without sharp switching
- Performance limits linked to attention, not structure
Associated Scientific Capacities
- Multi-parametric analysis
- Simultaneous comparison of scenarios
- Work on systems with many interdependent variables
Cognitive State Superposition
Theme: Overlapping cognitive states
Functional Area: Flexibility and ambiguity tolerance
Objective Functions
- Maintenance of overlapping cognitive states
- Deferred decision collapse
- Parallel evaluation of incompatible solutions
Associated Scientific Capacities
- Work on conjectures and unresolved problems
- Structured exploratory thinking
- Controlled (non-chaotic) creativity
Hyper-Conscious Analytical Control
Theme: Logical precision and control
Functional Area: Analytical monitoring
Objective Functions
- Explicit articulation of implicit assumptions
- Conscious control of every logical step
- Preventive interception of errors
Associated Scientific Capacities
- Production of clean proofs
- Analysis of formal limits
- Rejection of unjustified simplifications
Conceptual Representation and Language
Theme: Language as structural space
Functional Area: Conceptual formalization
Objective Functions
- Topological and relational mental representations
- Concepts as nodes, relations as vectors
- High semantic density per sentence unit
Associated Scientific Capacities
- Mathematical formalization
- Use of language as a modeling tool
- Translation across different conceptual levels
Nonlinear Dynamics and Adaptation
Theme: Complex cognitive adaptation
Functional Area: Dynamic networks
Objective Functions
- Context sensitivity
- Rapid priority reorganization
- Global system stability
- Integration of inputs as parameters rather than commands
Associated Scientific Capacities
- Rapid learning
- Deep conceptual restructuring
- Resistance to simple biases
Embodied Cognition and Sensorimotor Regulation
Theme: Mind-body integration
Functional Area: Somatomotor and cerebellar systems
Objective Functions
- Integration of proprioceptive and interoceptive signals
- Anchoring abstraction to bodily feedback
- Temporal predictive optimization (cerebellar role)
Associated Scientific Capacities
- Improved decision calibration
- Reduction of model–reality disconnection
- Practical application of theoretical models
Advanced Predictive Functioning
Theme: Predictive processing
Functional Area: Generation and updating of internal models
Objective Functions
- Continuous hypothesis generation
- Constant comparison with input
- Implicit Bayesian updating
- Multi-level, multi-domain prediction
Associated Scientific Capacities
- Anticipation of future states
- Predictive modeling
- Analysis of dynamic systems
Conditions of Maximum Effectiveness
Theme: Optimal cognitive environment
Objective Functions
Maximum efficiency in problems that are:
- complex
- open
- multidimensional
Performance decline in:
- repetitive tasks
- hyper-normative environments
- low informational complexity
Ideal Application Domains
- scientific research
- theoretical modeling
- complex systems
- non-standard problems
Statistical Rarity of the Profile
Theme: Population positioning
Objective Data
- Co-occurrence of all functions: even rarer
- Many capacities > +2 standard deviations
- Estimated prevalence: < 2–3%
Final Functional Synthesis
The described profile corresponds to a highly integrated neural system, metacognitively supervised, with strong functional parallelism, nonlinear adaptive dynamics, and advanced capacities for abstract modeling, prediction, and cognitive control.
Case Study 0:
P vs NP
4.1 Correct Problem Formulation
The P vs NP problem is considered in its classical formulation:
- deterministic computation;
- polynomial time in the worst case;
- absence of admissible adaptive dynamics.
4.2 Central Object Introduced
For an instance x and a subset of variables S, define:
RS(x)={assegnamenti parziali estendibili a soluzione globale}
This representation is:
- exact;
- non-heuristic;
- semantically complete.
4.3 Proven Properties
- Monotonicity under restriction: S⊆T⇒RT(x)↾S⊆RS(x)
- Decision equivalence: x∈L⟺R[n](x)=∅
4.4 Identified Critical Point
The P vs NP problem reduces to the question:
∃p(n) polinomiale: ∀x,S, ∣RS(x)∣≤p(n)
This constitutes the actual frontier of the state of the art.
4.5 Epistemic Result
- The identified properties are necessary.
- They are not sufficient to prove P=NP.
- No forcing argument is proposed.
5. Separation of Paradigms
5.1 Classical Paradigm (Deterministic)
- static;
- discrete;
- worst-case;
- syntactic;
- devoid of semantics or dynamics.
This paradigm excludes by definition:
- adaptation;
- feedback;
- geometric structure;
- convergence processes.
5.2 Declared Alternative Paradigm
An alternative model is introduced based on:
- dynamic states;
- coherent binary composition;
- monotone functionals;
- distinction between value and position;
- persistence of modes.
This paradigm does not claim to resolve P vs NP.
6. Dynamic Theorem (Alternative Paradigm)
A theorem of the following type is established:
- the process value decreases monotonically;
- the value tends to zero;
- the state does not collapse;
- a residual invariant dynamic persists.
This scheme is formally analogous to Ricci/Perelman-type methods.
7. Current State of the Work
- Complete understanding of P vs NP up to its structural limit.
- Construction of mathematically correct objects.
- Proof of valid lemmas.
- Foundation of a coherent alternative paradigm.
Not produced:
- a proof of P = NP;
- a proof of P ≠ NP.
8. Multilevel Reading Structure
- Level 1: Linear conceptual description.
- Level 2: Technical definitions and lemmas.
- Level 3: Analogies with physical systems and flows.
- Level 4: Research program and limits.
- Level 5: Advanced structural thinking.
- Level 6: Institutional and academic evaluation.
Each content layer is clearly labeled.
9. Project Objective
- clarify where shortcuts end;
- make real constraints explicit;
- construct only what is defensible;
- promote collaboration on formal grounds.
10. Collaborations
Collaborations are open with:
- researchers;
- mathematicians;
- physicists;
- engineers;
- structurally oriented professionals.
Focus areas:
- technical work;
- formal comparison;
- development of verifiable models.
11. Contact
Website: fabioguglielmini.com