binary-convolution-theory 1.0.0

Binary Convolution Theory: A Structural Approach to Perfect Numbers

Binary Convolution Theory (BCT) - Verification Code

Computational verification code for the paper:

Binary Convolution Theory: A Structural Approach to Perfect Numbers
Masamichi Iizumi, Tamaki Iizumi
INTEGERS 26 (2026)

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Overview

Binary Convolution Theory (BCT) provides a novel framework for studying the multiplicative structure of integers through their binary representations. This repository contains the verification code for all theorems presented in the paper.

Key Concepts

• Binary Convolution Height H(a,b): Maximum value in the pre-carry convolution of bin(a) and bin(b)
• BCT-Perfect Numbers: Integers where H(a,b) = 1 for all non-trivial factorizations
• The Main Insight: BCT-perfectness (structural balance) appears incompatible with classical perfectness (σ(n) = 2n) for odd numbers

Installation

git clone https://github.com/miosync-masa/BinaryConvolutionTheory.git
cd BinaryConvolutionTheory
pip install -e .

Repository Structure

BinaryConvolutionTheory/
├── README.md
├── setup.py
├── src/
│   ├── core/
│   │   ├── __init__.py
│   │   ├── binary_utils.py           # Binary representation utilities
│   │   └── bct_invariants.py         # BCT invariants (H, C, L, etc.)
│   └── theorems/
│       ├── __init__.py
│       ├── thm1_2_upper_bound.py           # Theorems 1-2: Upper Bounds
│       ├── thm3_equality_condition.py      # Theorem 3: Equality Condition
│       ├── thm4_5_mersenne.py              # Theorems 4-5: Mersenne Properties
│       ├── thm6_fermat_resonance.py        # Theorem 6: Fermat Minimum Resonance
│       ├── thm7_prop5_sweep.py             # Theorem 7 & Prop 5: Carry Schedules
│       ├── thm8_even_perfect.py            # Theorem 8: Even Perfect Numbers
│       ├── lemma4_thm9_fermat.py           # Lemma 4 & Theorem 9: Fermat Primes
│       ├── thm10_structure.py              # Theorem 10: BCT-Perfect Structure
│       ├── thm11_conjecture1_abundance.py  # Theorem 11(b) & Conjecture 1
│       └── remaining_lemmas_theorems.py    # Lemmas 1-3, Thm 11(a), 12, 13

Usage

from src.core.binary_utils import bin_seq, bin_str, popcount, is_fermat
from src.core.bct_invariants import H, C, L, is_bct_perfect, sigma, abundance_ratio

# Binary Convolution Height
print(H(3, 5))   # 1 (orthogonal)
print(H(7, 7))   # 3 (Mersenne self-convolution)

# Check BCT-perfectness
print(is_bct_perfect(15))  # True (3 × 5)
print(is_bct_perfect(21))  # False (3 × 7)

# Even perfect numbers
print(is_bct_perfect(28))  # True
print(abundance_ratio(28)) # 2.0 (perfect!)

Verified Theorems

Lemma 1 (Counting Lemma for Sumsets)

For S ⊂ {0, ..., L-1} with |S| = w:

max_k r_S(k) ≥ ⌈w² / (2L-1)⌉

Verified: 99,998 integers (n < 10⁵), 0 violations

Lemma 2 (Power of Two Orthogonality)

H(2^a, m) = 1 for any a ≥ 1 and m ≥ 1

Verified: 20,000 cases (a ≤ 20, m ≤ 1000), all H = 1

Lemma 3 (BCT-Perfectness Inheritance)

n = 2^a · m is BCT-perfect ⟺ m is BCT-perfect

Verified: 8,769 composite numbers (n < 10⁴), 0 violations

Lemma 4 (Characterization of Sparse Odd Primes)

An odd prime p has popcount(p) = 2 ⟺ p is a Fermat prime

Verified: All odd primes p < 10⁶ with popcount = 2
Found: Exactly 5 such primes: F₀=3, F₁=5, F₂=17, F₃=257, F₄=65537

Theorem 1 (Upper Bound)

For any factorization n = a × b:

H(a, b) ≤ min(popcount(a), popcount(b))

Verified: 483,533 factorizations (n ≤ 10⁵), 0 violations

Theorem 2 (Self-Convolution Upper Bound)

H(n²) := H(n, n) ≤ popcount(n)

Verified: 99,999 integers (n ≤ 10⁵), 0 violations

Theorem 3 (Equality Condition)

H(n²) = popcount(n) ⟺ the 1-bits of bin(n) are centrally symmetric

Verified: 999,999 integers (n < 10⁶), 100% biconditional
Notable: Maximum gap = 6 at n = 807,743 (unique!)

Theorems 4-5 (Mersenne Properties)

For Mersenne numbers M_k = 2^k - 1:

H(M_k²) = k = popcount(M_k)     (Theorem 4)
C(M_k²) = (k-1)²                 (Theorem 5)

Verified: k ∈ [2, 20], exact match

Theorem 6 (Fermat Minimum Resonance)

For Fermat numbers F_k = 2^(2^k) + 1:

H(F_k²) = 2  (constant, independent of k)

Verified: k ∈ [0, 7], all H = 2

Key Contrast:

Type	k=2	k=3	k=4	Pattern
Mersenne	2	3	4	H = k (grows!)
Fermat	2	2	2	H = 2 (constant!)

Theorem 7 (Single-Sweep Normalization)

L(a, b) = 1 for all factorizations (sequential LSB→MSB model)

Verified: 16,723 factorizations (n ≤ 5,000), all L = 1

Proposition 5 (Arbitrarily Long Parallel Chains)

For m = (2^k + 1)/3 with odd k ≥ 3:

H(3, m) = 2, but L_par(3, m) = k - 1

Verified: odd k ∈ [3, 21], all match
Key Insight: L_par can grow arbitrarily while H stays at 2!

Theorem 8 (Even Perfect Number Structure)

For even perfect P = 2^(p-1) × (2^p - 1) where 2^p - 1 is Mersenne prime:

H(P) = 1
bin(P) = 1^p 0^(p-1)

Verified: First 7 even perfect numbers, all H = 1

p	P	Binary Pattern	H	σ/P
2	6	110	1	2.00
3	28	11100	1	2.00
5	496	111110000	1	2.00
7	8128	1111111000000	1	2.00

Theorem 9 (Fermat Prime Orthogonality)

All Fermat primes F_i, F_j are pairwise binary orthogonal: H(F_i, F_j) = 1 for i ≠ j

Verified: All 10 pairs of F₀, F₁, F₂, F₃, F₄, all H = 1

Structural Reason: F_k has 1-bits at positions {0, 2^k}, so sumsets are always disjoint.

Theorem 10 (Structure of BCT-Perfect Odd Numbers)

BCT-perfect odd composites are constrained to specific structural types:

Type	Count	Percentage	Max σ/n	Example
p × q	487	93.30%	1.6000	15 = 3 × 5
p × q × r	18	3.45%	1.6941	255 = 3 × 5 × 17
p² × q	7	1.34%	1.6508	63 = 3² × 7
p³	3	0.57%	1.4815	27 = 3³
Other	7	1.34%	1.7007	65535 = 3×5×17×257

Verified: 522 BCT-perfect odd composites (n < 10⁵)

Theorem 11 (Abundance Bounds)

(a) General bound: For odd squarefree semiprimes n = pq:

σ(n)/n ≤ 8/5 = 1.6

Verified: 4,371 semiprimes, maximum at (p,q) = (3,5)

(b) Computational observation: For all BCT-perfect odd composites n < 10⁵:

σ(n)/n < 1.71

Verified: 522 BCT-perfect odds, max = 1.7007 at n = 65535

Theorem 12 (Semiprime Obstruction)

An odd perfect number cannot be a squarefree semiprime pq

Verified: Follows from Theorem 11(a) since 1.6 < 2

Theorem 13 (BCT Obstruction for Odd Perfect Numbers)

No odd BCT-perfect composite n < 10⁶ satisfies σ(n)/n = 2

Verified: 2,017 BCT-perfect odd composites (n < 10⁶), all σ/n < 1.71

Conjecture 1 (BCT-Perfect Odd Abundance Bound)

For all BCT-perfect odd composite n: σ(n)/n < 2

Verified: n < 10⁶, max σ/n = 1.7007
Gap from perfection: 0.2993

🌟 IMPLICATION: If Conjecture 1 holds for all n, then:

{Odd perfect numbers} ⊆ {BCT-imperfect numbers}

Complete Verification Summary

Item	Range Verified	Result
Lemma 1	n < 10⁵	✅ No violations
Lemma 2	a ≤ 20, m ≤ 1000	✅ All H = 1
Lemma 3	n < 10⁴	✅ Inheritance holds
Lemma 4	p < 10⁶	✅ All sparse primes are Fermat
Theorem 1-2	n ≤ 10⁵	✅ No counterexamples
Theorem 3	n < 10⁶	✅ 100% biconditional
Theorem 4-5	k ∈ [2, 20]	✅ Exact match
Theorem 6	k ∈ [0, 7]	✅ All H = 2
Theorem 7	n ≤ 5,000	✅ All L = 1
Proposition 5	odd k ∈ [3, 21]	✅ L_par = k - 1
Theorem 8	First 7 even perfects	✅ All H = 1
Theorem 9	F₀ through F₄	✅ All pairs orthogonal
Theorem 10	n < 10⁵	✅ Structure matches Table 1
Theorem 11(a)	p, q < 500	✅ σ/n ≤ 1.6
Theorem 11(b)	n < 10⁵	✅ σ/n < 1.71
Theorem 12	—	✅ Logical consequence
Theorem 13	n < 10⁶	✅ No BCT-perfect perfect
Conjecture 1	n < 10⁶	✅ σ/n < 2

Running Verification

Each theorem module can be run standalone:

# Run all verifications
python -m theorems.thm1_2_upper_bound
python -m theorems.thm3_equality_condition
python -m theorems.thm4_5_mersenne
python -m theorems.thm6_fermat_resonance
python -m theorems.thm7_prop5_sweep
python -m theorems.thm8_even_perfect
python -m theorems.lemma4_thm9_fermat
python -m theorems.thm10_structure
python -m theorems.thm11_conjecture1_abundance
python -m theorems.remaining_lemmas_theorems

Core API

Binary Utilities (src/core/binary_utils.py)

bin_seq(n)       # LSB-first bit sequence [b₀, b₁, ...]
bin_str(n)       # MSB-first string for display
popcount(n)      # Number of 1-bits (Hamming weight)
bit_positions(n) # Set of positions with 1-bits
is_centrally_symmetric(n)  # Check bit symmetry
is_mersenne(n)   # Check if Mersenne number
is_fermat(n)     # Check if Fermat number

BCT Invariants (src/core/bct_invariants.py)

H(a, b)           # Binary Convolution Height
C(a, b)           # Total Carry Count
L(a, b)           # Chain Length (sequential)
L_parallel(a, b)  # Chain Length (parallel)
is_bct_perfect(n) # Check BCT-perfectness
sigma(n)          # Sum of divisors σ(n)
abundance_ratio(n) # σ(n)/n

Implementation Notes

Bit Ordering Convention

• Computation: LSB-first (index 0 = least significant bit)
• Display: MSB-first (standard binary notation)

This matches the paper's convention where:

bin(n) = (b₀, b₁, ..., bₖ) where n = Σ bᵢ · 2ⁱ

Verification Ranges

All verifications use ranges matching or exceeding those in the paper's Table 4.

Citation

@article{iizumi2026bct,
  title={Binary Convolution Theory: A Structural Approach to Perfect Numbers},
  author={Iizumi, Masamichi and Iizumi, Tamaki},
  journal={INTEGERS},
  volume={26},
  year={2026}
}

License

MIT License - see LICENSE for details.

Author

Masamichi Iizumi
Tamaki Iizumi
Miosync, Inc., Tokyo, Japan
[e-mail]

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