Carlos Tomas Grahm
Carlos Tomas Grahm is an independent mathematical researcher whose work has ranged from inverse constitutive modeling of highly nonlinear physical systems to prime structure and computational complexity. His current research includes a proposed resolution of P versus NP that is undergoing independent technical review.
01 / EDITORIAL PORTRAIT
Photograph forthcoming
01 / CURRENT WORK
A proposed resolution of P versus NP
The current project approaches P versus NP through canonical Subset Sum and an information-based lower-bound framework.
Rather than beginning with a particular algorithm, the argument asks a more basic question: what are all the possible ways a candidate subset sum can be established as unequal to a target?
The framework classifies those possibilities, identifies the information cost at the terminal cases, and constructs families of instances intended to force those costs simultaneously.
The unusual feature of the approach is its conceptual simplicity. The central ingredients are elementary ordering, counting, and information constraints rather than extensive new machinery.
The work is currently undergoing independent review by multiple PhD-level readers. The purpose of the present public effort is not to treat the result as established in advance, but to make the argument visible enough to receive serious scrutiny.
Status: Proposed result — active independent review.
Current objective: Broader mathematical scrutiny and formal evaluation.
02 / THE IDEA IN PLAIN LANGUAGE
What is different about the approach?
A NO instance of Subset Sum contains many candidate sums that must somehow be distinguished from the target.
The project asks whether those candidates can be eliminated collectively through a genuine shared relation, or whether the computation ultimately has to account for them separately.
The proposed lower bound comes from identifying the circumstances in which the usual many-at-once shortcuts fail. The resulting architecture reduces the problem to a small number of information mechanisms rather than a catalogue of particular algorithms.
03 / SELECTED RESEARCH
Selected research
COMPLEXITY THEORY
P versus NP through Subset Sum
A proposed lower-bound framework based on an exhaustive classification of how inequality between candidate subset sums and a target can be established.
Status: active independent review.
NUMBER THEORY
Prime Avoidance in Reflected Sieve Sets
This work studies Goldbach-type structure through a mirror map: given an even number 2n, a prime p on one side of the center is reflected to 2n − p.
Instead of claiming a proof of Goldbach’s conjecture, the paper reformulates failure as a rigid arithmetic condition: every local prime would have to be captured by a center-dependent congruence obstruction. A single prime that escapes all relevant obstructions has a prime reflection and therefore gives a Goldbach pair.
This changes the perspective from counting representations globally to asking whether a local prime set can be totally covered by arithmetic obstruction classes.
APPLIED MATHEMATICS / BIOMEDICAL ENGINEERING
Recovering the mechanics of a highly nonlinear physical system
Earlier work in biomedical engineering focused on constitutive modeling of arterial tissue — a highly nonlinear composite biological material.
Working within an inverse framework, the project reconstructed strain-energy behavior from experimentally measured pressure, geometry, and load rather than simply fitting a conventional forward model to a narrow response regime.
In the arterial application, the resulting model remained within approximately experimental error across the measured regimes, while prior constitutive approaches could diverge substantially in portions of the response range.
Carlos Tomas describes the conceptual result as a breakthrough in macroscopic theory of everything: the idea that sufficiently general constitutive structure can recover the governing behavior of complex macroscopic systems from observable response data.
“We can now recover the strain-energy function of a material from its data, within experimental error, everywhere — and nothing in the method is specific to the material.”
“A breakthrough in macroscopic theory of everything” is Carlos Tomas’s description of the broader theoretical significance, not the formal title of the paper.
04 / A COMMON THREAD
Different problems. A recurring instinct.
The subjects look unrelated: arterial mechanics, reflected primes, and computational complexity.
The common instinct is structural.
Instead of adding more machinery to a difficult problem, ask which relations are actually necessary, which alternatives exhaust the possibilities, and what happens when the system is represented in the right variables.
In applied mechanics, that meant recovering a constitutive structure from experimental response.
In reflected-prime work, it meant replacing an absence-of-representations problem with a concrete obstruction-covering problem.
In the current complexity work, it means classifying the information by which candidate sums can be separated from a target.
The research program is therefore broad in subject but relatively consistent in method: reduce complicated phenomena to the smallest structural question that still controls the outcome.
05 / BACKGROUND
From grant-funded theory to independent research
Carlos Tomas began research in a university environment spanning biomedical engineering and applied mathematics. As an undergraduate, he held an unusually substantial grant-funded research position that supported year-round theoretical work rather than occasional undergraduate assistance.
His work centered on the mathematical mechanics of arteries and constitutive theory. That experience established a research style centered on inverse problems, structural reduction, and the search for models that explain rather than merely interpolate.
Faced with publication and development timelines measured in years, he chose not to move directly into the conventional graduate-school pipeline and instead continued research independently.
What began as that decision eventually became a broader mathematical program spanning geometry, number theory, applied mathematics, and computational complexity.
02 / AT THE WORKSTATION — PHOTOGRAPH FORTHCOMING
06 / PAPERS
Selected papers and notes
Number theory / reflected prime structures. Public PDF.
Applied mathematics / biomechanics / constitutive modeling. Public PDF.
P versus NP / Subset Sum lower-bound project
Complexity theory. Current manuscript undergoing review.
07 / MEDIA
For media
Carlos Tomas is available for conversations about P versus NP, independent mathematical research, complexity theory, applied mathematics, the process of reviewing unconventional results, and the path from biomedical modeling to foundational mathematics.
What would a conceptually simple P versus NP proof look like?
Why take a major mathematical claim public before review is complete?
How does independent research get evaluated without an institutional platform?
What can arterial mechanics teach someone about mathematical structure?
Why reformulate an open problem instead of attacking it in its conventional form?
What does it mean to search for “macroscopic” universal structure in physical systems?
Download short bio — forthcoming
Download headshot — forthcoming
08 / CONTACT
Research and media inquiries
For media, podcast, research-review, or academic inquiries:
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