Algebraic Operations¶

TriadicGPT provides eight algebraic operations over prime-factor signatures. All are O(1) via bitwise arithmetic.

Cosine Similarity vs Prime Algebra¶

Operation	Cosine Similarity	Prime Algebra
"Does A contain all features of B?"	Approximate	`phi(A) % phi(B) == 0`
"What features do A and B share?"	Not possible	`gcd(phi(A), phi(B))`
"Combine features of A and B"	Not possible	`lcm(phi(A), phi(B))`
"A is to B as C is to ?"	Approximate	Exact factor transfer
Explainability	Black box	Full factor decomposition

The Eight Operations¶

Subsumption¶

Does concept A contain all semantic features of concept B?

\[A \subseteq B \iff \Phi(A) \bmod \Phi(B) = 0\]

from triadic_head.algebra import TriadicValidator

validator = TriadicValidator()
print(validator.subsumes(phi_dog, phi_animal))
# --> True (dog contains all features of animal)

Composition¶

Create a new concept with all features of both A and B:

\[A \cup B = \text{lcm}(\Phi(A), \Phi(B))\]

composed = validator.compose(phi_king, phi_female)
# --> Integer containing all factors of king AND female

Intersection¶

Find the features shared by A and B:

\[A \cap B = \gcd(\Phi(A), \Phi(B))\]

shared = validator.intersection(phi_king, phi_queen)
# --> Integer whose factors are the features they share

Difference¶

Features present in A but not in B:

\[A \setminus B = \Phi(A) / \gcd(\Phi(A), \Phi(B))\]

unique_to_king = validator.difference(phi_king, phi_queen)
# --> Factors that king has but queen lacks

Symmetric Difference¶

Features present in exactly one of A or B:

\[A \triangle B\]

sym_diff = validator.symmetric_difference(phi_king, phi_queen)
# --> Factors unique to either king or queen (but not both)

Gap Analysis¶

Complete breakdown of shared and unique features:

gap = validator.gap_analysis(phi_king, phi_queen)
# --> {
#   'shared': gcd,
#   'only_in_a': factors unique to king,
#   'only_in_b': factors unique to queen,
#   'a_contains_b': bool,
#   'b_contains_a': bool
# }

Analogy Prediction¶

Solve A:B :: C:? via algebraic factor transfer:

predicted = validator.analogy_predict(phi_king, phi_queen, phi_man)
# --> Predicted prime composite for "woman"

Analogy Verification¶

Verify whether D is a valid solution to A:B :: C:D:

is_valid = validator.analogy_verify(phi_king, phi_queen, phi_man, phi_woman)
# --> True/False

Using with TriadicWrapper¶

The compare method on TriadicWrapper provides a high-level interface:

from triadic_head import TriadicWrapper

model = TriadicWrapper("gpt2", n_bits=64, align_mode="infonce")
sigs = model.encode(["king", "queen", "man", "woman"])

# Compare returns similarity + shared factors + subsumption
result = model.compare("king", "queen")
print(result)
# {'similarity': 0.89, 'shared_factors': [2, 3, 5, ...],
#  'subsumes': False, 'subsumed_by': False}

Performance¶

All operations are implemented via bitwise arithmetic, achieving 5--78x speedup over naive prime factorization:

Operation	Bitwise	Prime arithmetic
Subsumption	AND + compare	mod
Composition	OR	lcm
Intersection	AND	gcd
Difference	AND NOT	div + gcd