OpenLLM-France/wikimedia huggingface dataset:
# Orthogonal complement/C/4+7i/Complex and real/Exercise

We consider the complex numbers $\mathbb {C}$ with the standard inner product $\left\langle -,-\right\rangle$.
a) Determine for the complex linear subspace of $\mathbb {C}$ generated by $4+7{\mathrm {i} }$ the orthogonal complement with respect to $\left\langle -,-\right\rangle$.
b) Determine for the complex linear subspace of $\mathbb {C}$ generated by $4+7{\mathrm {i} }$ the orthogonal complement with respect to the real part of $\left\langle -,-\right\rangle$ (that is, the corresponding real inner product).
c) Determine for the real linear subspace of $\mathbb {C}$ generated by $4+7{\mathrm {i} }$ the orthogonal complement with respect to of the real part of $\left\langle -,-\right\rangle$.
# Polynomials/1 to x^3/Unit interval/Orthonormalization/Exercise

Apply Gram-Schmidt orthonormalization to the linearly independent polynomial functions
    $1,x,x^{2},x^{3}\in V={\left\{f:[0,1]\rightarrow \mathbb {R} \mid f{\text{ continuous}}\right\}}$
with respect to the inner product described in example.
# MDLD/orthogonal complements

Start=orthogonal complements See=orthogonal complement MDLD/orthogonal complement Target=Vector space with inner product/Orthogonal complement/Definition
# Orthogonal complement/(-2,8,9)/Exercise

Determine the orthogonal complement of the linear subspace in $\mathbb {R} ^{3}$ generated by ${\begin{pmatrix}-2\\8\\9\end{pmatrix}}$.
# Orthogonal complement/(5,8,-3,9) (6,2,0,3)/Exercise

Determine the orthogonal complement of the linear subspace in $\mathbb {R} ^{4}$ generated by ${\begin{pmatrix}5\\8\\-3\\9\end{pmatrix}}$ and ${\begin{pmatrix}6\\2\\0\\3\end{pmatrix}}$.
# Orthonormalization/C^2/2/Exercise

Apply Gram-Schmidt orthonormalization to the basis
    ${\begin{pmatrix}5-2{\mathrm {i} }\\3-3{\mathrm {i} }\end{pmatrix}},\,{\begin{pmatrix}6-{\mathrm {i} }\\-2+4{\mathrm {i} }\end{pmatrix}}$
of $\mathbb {C} ^{2}$.
# Orthogonal projection/R^3/Plane/2/Exercise

Determine the value of the vector ${\begin{pmatrix}7\\6\\8\end{pmatrix}}\in \mathbb {R} ^{3}$ under the orthogonal projection to the linear subspace generated by ${\begin{pmatrix}2\\2\\5\end{pmatrix}}$ and ${\begin{pmatrix}4\\-1\\3\end{pmatrix}}$.
# Orthogonal basis/K/Subfamily/Orthogonal complement/Exercise

Let $V$ be a vector space over ${\mathbb {K} }$, endowed with an inner product $\left\langle -,-\right\rangle$. Let $v_{i}$, $i\in I$, denote an orthogonal basis of $V$. For every subset $J\subseteq I$, denote the linear subspace generated by $v_{i}$, $i\in J$, by $U_{J}$. Show that the orthogonal complement of $U_{J}$ equals $U_{I\setminus J}$.
# Orthogonal complement/(8,3,-6,-4) (4,-2,7,5)/Exercise

Determine the orthogonal complement of the linear subspace in $\mathbb {R} ^{4}$ generated by ${\begin{pmatrix}8\\3\\-6\\-4\end{pmatrix}}$ and ${\begin{pmatrix}4\\-2\\7\\5\end{pmatrix}}$.
# Orthogonal projection/R^3/Plane/1/Exercise

Determine the value of the vector ${\begin{pmatrix}3\\4\\-2\end{pmatrix}}\in \mathbb {R} ^{3}$ under the orthogonal projection to the linear subspace generated by ${\begin{pmatrix}3\\0\\6\end{pmatrix}}$ and ${\begin{pmatrix}1\\7\\1\end{pmatrix}}$.
# The Idea Incubator

Many useful products originated from university research work. These include seat belts, Gatorade, the Internet, and Google. This product concept incubator provides an opportunity for the people of Wikiversity to conceive, describe, improve, and develop important new product concepts.
Consumers can describe products they would like to see here. Creative inventors can propose product concepts. Innovators can refine product concepts into requirements statements and design specifications. Entrepreneurs can then develop and launch products inspired by these specifications.
Please add your product ideas to this list.
* Universal Authentication, One ID.
* Blockchain-Secured Power of Attorney
* Mebot—A chatbot informed by your life and personality
* A Human Rights Exhibition
* Virtual Nations
* Reforming Social Media Platforms
* Global Land Trust
* Improving Constituent Assignments to Representatives
* Wiki Mural and The Big History Timeline
* Motivational Interviewing Chatbot
* Socrates, the chatbot
* The Real Good Navigator
* Improved Social Operating Systems
* An Encyclopedia of Ought
* The Wisdom Playground
* Carbon Dividend as Carbon Offset
* Crowdsourcing Weather Stations
* Eyelevel Video Camera
* Smart HVAC Systems
* User Agreement Summary Service
* Blood Type Certification
* Men’s Swimsuit Design
# Properties of Boolean functions/soft/integer

Properties of Boolean functions :
| hard | soft |
| binary | binary |
| integer | integer |
Soft properties depend on the arity of the BF. Integer properties have consecutive non-negative integers as values.

## softened
### weight

the number of true places in the truth table of length 2$^{arity}$ (hard weight is a fraction)

### nonlinearity

extent to which a BF is not linear (hard nonlinearity is a fraction)

### gravity and depth

2$^{gravity}$ and 2$^{depth}$ are the cardinalities of legion and cohort. Equivalently for the softened properties. Only the result for the contradiction differs from hard gravity and depth.

## consul (binary Walsh spectrum)

The Walsh spectrum is the product of a truth table with a Walsh matrix. The binary Walsh spectrum is the product of a truth table with a binary Walsh matrix, using F$_2$ operations (mod 2). It is always a Walsh function, and shall be called consul. The term is also used for the integer denoting the Walsh function.
One could refine the consul with a sign. That signed consul would be the twin prefect. So the (unsigned) consul is the Walsh index of the twin prefect. (See an overview for arity 3.)
The consul weight is the binary weight of the consul integer. (E.g. consul 6 has consul weight 2.)
| sharp family with consuls 0...7 | blunt family with consul 2 (compare 4-ary family with consul 6) |
| 3-ary families with Walsh spectra (integers) and consuls (red backgrounds on the right) ||

## patron

The patron of a BF is the XOR of itself and its twin. It is a noble. (3-ary images)

## praetor

XOR of left and right half of the truth table. (3-ary images)

## quaestor

XOR of left and reversed right half of the truth table (i.e. of the coordinates in the Hasse matrix) (3-ary images)
# Studies of Boolean functions

These articles about Boolean functions use a similar style and terminology.  They are created by Watchduck a.k.a. Tilman Piesk.
* Studies of Boolean functions/terminology
* Studies of Boolean functions/examples
* Discrete helpers/boolf
* properties
* sequences
* linear
* gender
* mentors
* legion and cohort
* permutations
* principalities and dominions
* families
* nonlinearity
* seals
* Tesseract and 16-cell faces

## Zhegalkin indices

* Algebraic normal form
* Zhegalkin matrix, Zhegalkin twins
* Noble Boolean functions
* Gentle sets of Boolean functions
* Linear and noble Boolean functions
* Smallest Zhegalkin index

## Commons

Some topics exist mainly in the form of categories on Commons. Articles here should follow.
| 3-ary (4) | 4-ary (20) | great principalities with entries | on diagonal (related to patron dominions) |
| 3-ary (4) | 4-ary (22) | great principalities with entries | in top row (related to patron principalities) |
* 2-ary     3-ary     4-ary         great prefects in hypercube matrix
* 2-ary     3-ary     4-ary         principalities and dominions in hypercube matrix
* Quadrant colors in hypercube matrices (image set)
* 3-ary Boolean functions in octeract matrix; faction clusters
* 3-ary Boolean functions; graphs of faction clusters
* Walsh spectra of 3-ary Boolean functions; families

## old

These old pages require some updates.
* 3-ary Boolean functions
* 4-ary Boolean functions
# Triangles Chestnut and Chinkapin

* Pascal's triangle with multiplied columns

## Triangles Chestnut$_{(Drop)}$

🌊    triangle Chestnut    row sums ~Grass :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 1 · 2 2 | 1 · 0 0 | | | | | 2 |
| 2 | 1 · 2 2 | 2 · 0 0 | 1 · 2 2 | | | | 4 |
| 3 | 1 · 2 2 | 3 · 0 0 | 3 · 2 6 | 1 · 8 8 | | | 16 |
| 4 | 1 · 2 2 | 4 · 0 0 | 6 · 2 12 | 4 · 8 32 | 1 · 210 210 | | 256 |
| 5 | 1 · 2 2 | 5 · 0 0 | 10 · 2 20 | 10 · 8 80 | 5 · 210 1050 | 1 · 64384 64384 | 65536 |
💧    triangle ChestnutDrop    row sums ~GrassDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 0 | 1 · 0 0 | | | | | 0 |
| 2 | 0 | 1 · 0 0 | 1 · 2 2 | | | | 2 |
| 3 | 0 | 1 · 0 0 | 2 · 2 4 | 1 · 8 8 | | | 12 |
| 4 | 0 | 1 · 0 0 | 3 · 2 6 | 3 · 8 24 | 1 · 210 210 | | 240 |
| 5 | 0 | 1 · 0 0 | 4 · 2 8 | 6 · 8 48 | 4 · 210 840 | 1 · 64384 64384 | 65280 |

## Triangles Chinkapin$_{(Drop)}$

🌊    triangle Chinkapin    row sums ~Grass :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 0 0 | | | | | | 0 |
| 1 | 1 · 0 0 | 1 · 2 2 | | | | | 2 |
| 2 | 1 · 0 0 | 2 · 2 4 | 1 · 0 0 | | | | 4 |
| 3 | 1 · 0 0 | 3 · 2 6 | 3 · 0 0 | 1 · 10 10 | | | 16 |
| 4 | 1 · 0 0 | 4 · 2 8 | 6 · 0 0 | 4 · 10 40 | 1 · 208 208 | | 256 |
| 5 | 1 · 0 0 | 5 · 2 10 | 10 · 0 0 | 10 · 10 100 | 5 · 208 1040 | 1 · 64386 64386 | 65536 |
💧    triangle ChinkapinDrop    row sums ~GrassDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 0 0 | | | | | | 0 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 1 · 2 2 | 1 · 0 0 | | | | 2 |
| 3 | 0 | 1 · 2 2 | 2 · 0 0 | 1 · 10 10 | | | 12 |
| 4 | 0 | 1 · 2 2 | 3 · 0 0 | 3 · 10 30 | 1 · 208 208 | | 240 |
| 5 | 0 | 1 · 2 2 | 4 · 0 0 | 6 · 10 60 | 4 · 208 832 | 1 · 64386 64386 | 65280 |
# Sequence Thyme

families :
| introverted: Mint | blunt: Balm |
| extroverted: DubSage | sharp: Sage |
| total: Thyme ||
${\text{Thyme}}(n)={\frac {2^{2^{n}}+(2^{2^{n-1}})(2^{n}-1)}{2^{n}}}$

## Sequences Thyme$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Thyme (A000231) | 2 | 3 | 7 | 46 | 4336 | 134281216 | 288230380379570176 | 2658455991569831764110243006194384896 |
| 💧 ThymeDrop | 2 | 1 | 4 | 39 | 4290 | 134276880 | 288230380245288960 | 2658455991569831763822012625814814720 |
row sums of Olive$_{(Drop)}$ :
🌊    triangle Olive (A227725)    row sums Thyme (A000231) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 1 · 2 2 | 1 · 1 1 | | | | | 3 |
| 2 | 1 · 2 2 | 3 · 1 3 | 1 · 2 2 | | | | 7 |
| 3 | 1 · 2 2 | 7 · 1 7 | 7 · 2 14 | 1 · 23 23 | | | 46 |
| 4 | 1 · 2 2 | 15 · 1 15 | 35 · 2 70 | 15 · 23 345 | 1 · 3904 3904 | | 4336 |
| 5 | 1 · 2 2 | 31 · 1 31 | 155 · 2 310 | 155 · 23 3565 | 31 · 3904 121024 | 1 · 134156284 134156284 | 134281216 |
💧    triangle OliveDrop    row sums ThymeDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 0 | 1 · 1 1 | | | | | 1 |
| 2 | 0 | 2 · 1 2 | 1 · 2 2 | | | | 4 |
| 3 | 0 | 4 · 1 4 | 6 · 2 12 | 1 · 23 23 | | | 39 |
| 4 | 0 | 8 · 1 8 | 28 · 2 56 | 14 · 23 322 | 1 · 3904 3904 | | 4290 |
| 5 | 0 | 16 · 1 16 | 120 · 2 240 | 140 · 23 3220 | 30 · 3904 117120 | 1 · 134156284 134156284 | 134276880 |
# Sequence Sage

families :
| introverted: Mint | blunt: Balm |
| extroverted: DubSage | sharp: Sage |
| total: Thyme ||
${\text{Sage}}(n)={\frac {2^{2^{n}}}{2^{n{\color {Red}+1}}}}~=~2^{(2^{n}-n{\color {Red}-1})}$  ${\text{DubSage}}(n)={\frac {2^{2^{n}}}{2^{n}}}~=~2^{(2^{n}-n)}$

## Sequences Sage$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Eulerian numbers (A000295) | 0 | 0 | 1 | 4 | 11 | 26 | 57 | 120 |
| 🌊 Sage (~A016031) | 1 | 1 | 2 | 16 | 2048 | 67108864 | 144115188075855872 | 1329227995784915872903807060280344576 |
| 💧 SageDrop | 1 | 0 | 1 | 14 | 2032 | 67106816 | 144115188008747008 | 1329227995784915872759691872204488704 |

## Sequences DubSage$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 DubSage | 2 | 2 | 4 | 32 | 4096 | 134217728 | 288230376151711744 | 2658455991569831745807614120560689152 |
| 💧 DubSageDrop | 2 | 0 | 2 | 28 | 4064 | 134213632 | 288230376017494016 | 2658455991569831745519383744408977408 |

## Sequences TornSage$_{(Drop)}$

${\text{for}}~n\geq 2:~{\text{TornSage}}(n)={\frac {{\text{Sage}}(n)}{2}}$
| | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 TornSage (~A076688) | 1 | 1 | 8 | 1024 | 33554432 | 72057594037927936 | 664613997892457936451903530140172288 |
| 💧 TornSageDrop | 1 | 0 | 7 | 1016 | 33553408 | 72057594004373504 | 664613997892457936379845936102244352 |
Compare Clove.
# Sequence Clove

families :
| introverted: Mint | blunt: Balm |
| extroverted: DubSage | sharp: Sage |
| total: Thyme ||

## Sequences Clove$_{(Drop)}$

Clove = Balm − TornSage
| | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Balm | 2 | 5 | 30 | 2288 | 67172352 | 144115192303714304 | 1329227995784915891206435945914040320 |
| TornSage | 1 | 1 | 8 | 1024 | 33554432 | 72057594037927936 | 664613997892457936451903530140172288 |
| 🌊 Clove | 1 | 4 | 22 | 1264 | 33617920 | 72057598265786368 | 664613997892457954754532415773868032 |
| 💧 CloveDrop | 1 | 3 | 18 | 1242 | 33616656 | 72057598232168448 | 664613997892457954682474817508081664 |
# Properties of Boolean functions/hard

Properties of Boolean functions :
| hard | soft |
| binary | binary |
| integer | integer |
Hard properties can be assigned to a BF, without referencing its arity.

## weight

Weight is the quotient of true and all places of the truth table. It is usually shown as an integer for a given arity (soft weight). This triangle shows the numbers of BF by weight.

## nonlinearity

Nonlinearity is the extent to which a BF is not linear. It is usually shown as an integer for a given arity (soft nonlinearity).

## atoms

Atoms or atomvals are the relevant arguments of the BF. They correspond to the circles of an Euler diagram. The number of atoms is the valency. I this project the letters A, B, C... correspond to atoms 0, 1, 2... E.g. $B\land C$ has atoms 1 and 2.

## root

The root of a BF created by replacing its atoms with the set {0, ..., valency−1}. When it is its own root, the BF is dense.

## equivalence classes

based on input negation and permutation: family (negation), faction (permutation), clan (both) (Typically represented by Smallest Zhegalkin index.)
extended by complement: super-family, super-faction, super-clan (Families and clans can be self-complementary. Factions can not.)
(Further extension by half-complement leads to ultra-famlies and -clans, which are soft properties.)

## prefect

The prefect is a way to assign each BF to a linear BF. A linear is assigned to itself. (3-ary images)

## legion and cohort

These properties assign every almost every BF to a finite set of integers. The result for the contradiction is infinite. Soft equivalents have been defined to avoid this problem. The cardinalities of the results are gravity and depth.

## complement

all places of the truth table negated, e.g. 0001 and 1110 (LSB of the Zhegalkin index negated) The complement of a set of BF is the set of complements. (Families and clans can be self-complementary. Factions can not.)

## reverse

truth table reversed, e.g. 0001 and 1000

## dual

complement of the reverse, e.g. 0001 and 0111
# Sequence Mint

families :
| introverted: Mint | blunt: Balm |
| extroverted: DubSage | sharp: Sage |
| total: Thyme ||
${\text{Mint}}(n)={\frac {(2^{2^{n-1}})(2^{n}-1)}{2^{n}}}$

## Sequences Mint$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Mint (A001320) | 0 | 1 | 3 | 14 | 240 | 63488 | 4227858432 | 18302628885633695744 |
| 💧 MintDrop | 0 | 1 | 2 | 11 | 226 | 63248 | 4227794944 | 18302628881405837312 |
# Bully Metric Length Apan

Bully Metric Main Page
The length apan is by definition the distance light travels in vacuum in 30.55 femtoseconds. SI prefixes have the same meaning and conventions when used with Bully lengths as they have when used with standard SI units. However, the word "length" may be dropped when the context is clear. For example, the statement: "An American football field has ten mega-length-apan (10 Mla) between the goal lines", should be shortened to "An American football field has ten megapan (10 Mla) between the goal lines". See Table 1 below for the list of SI prefixes used with Bully lengths. In addition to SI prefixes, the table also identifies the smallest meaningful length (Planck length), largest meaningful length (Observable Universe radius), and a few other lengths for comparison.
Table 1: The length prefix table :
| Prefix | Prefix | Prefix | Symbols |
| Name | Symbol | Base 10 | Length |
| Observable Universe Radius | Observable Universe Radius | Observable Universe Radius | $48\,Qla$ |
| quetta | Q | 10$^{30}$ | Qla |
| ronna | R | 10$^{27}$ | Rla |
| yotta | Y | 10$^{24}$ | Yla |
| zetta | Z | 10$^{21}$ | Zla |
| exa | E | 10$^{18}$ | Ela |
| Earth-Sun Distance | Earth-Sun Distance | Earth-Sun Distance | $16.33\,Pla$ |
| peta | P | 10$^{15}$ | Pla |
| tera | T | 10$^{12}$ | Tla |
| giga | G | 10$^9$ | Gla |
| American Football Field Length | American Football Field Length | American Football Field Length | $10\,Mla$ |
| mega | M | 10$^6$ | Mla |
| kilo | k | 10$^3$ | kla |
| — | — | 10$^0$ | la |
| milli | m | 10$^{−3}$ | mla |
| micro | μ | 10$^{−6}$ | μla |
| nano | n | 10$^{−9}$ | nla |
| The Proton Radius | The Proton Radius | The Proton Radius | $91.716\,pla$ |
| pico | p | 10$^{−12}$ | pla |
| femto | f | 10$^{−15}$ | fla |
| atto | a | 10$^{−18}$ | ala |
| zepto | z | 10$^{−21}$ | zla |
| yocto | y | 10$^{−24}$ | yla |
| ronto | r | 10$^{−27}$ | rla |
| quecto | q | 10$^{−30}$ | qla |
| Minimum value (Planck Length) | Minimum value (Planck Length) | Minimum value (Planck Length) | ${\frac {qla}{0.566660}}$ |

# The Bully Constants

A surprising number of earth's physical constants can be approximated using various algebraic combinations of the following three numbers (click here to learn more) with Bully Metric units.
One can efficiently approximate the Earth's sidereal year and tropical year to eight digits, and approximate the Earth's radius (r ≈ 6371), Schwarzschild radius (R), standard gravitational parameter (μ = MG ≈ 3.984e14), and a typical gravitational acceleration on earth's surface (g ≈ 9.813 ).
${1\,Sidereal\,Year}={1.033\,Zta}={31,558,150\,s}$
${1\,Tropical\,Year}=(1.033-0.00004)\,Zta-2\,s=31,556,926\,s$
$r_{earth}\approx {\frac {1\,Tla}{\sqrt {2\times 1.033}}}=6371\,km$
$R_{earth}\approx {\frac {kla}{1.033}}\approx {8.866\,mm}$
${\mu }_{earth}\approx {\frac {1}{2\times 1.033}}\,{\frac {kla^{3}}{kta^{2}}}\approx {398,400,000,000,000\,{\frac {m^{3}}{s^{2}}}}$
$g_{earth}\approx 1{\frac {Zla}{Zta^{2}}}\approx {9.813\,{\frac {m}{s^{2}}}}$
# Wikiphilosophers/Consciousness

Consciousness, at its simplest, is awareness of internal and external existence. But what exactly is consciousness? Where is it located? And how did it emerge? Will we lose our consciousness after death?
Read users' philosophical views below.
* S. Perquin argues that consciousness is a universal and eternal phenomenon, not reducible to matter or energy, implying that it exists everywhere in the universe and persists beyond physical death. Read more...
* Elías believes that a person's consciousness isn't limited to him but can be shared/experienced by/with other people. People can share thoughts, though we don't how to do so. Being a person isn't just having a body and a mind. A person can have to minds as seen...Read more
* Mars Sterling Turner puts Logical Tautologies together to prove the existence of an Eternal, Omnipresent, Omnipotent Consciousness.
# Chapter 6: The Future of Cosmic Influx Theory
## Introduction

As the Cosmic Influx Theory (CIT) continues to develop, several key areas require further exploration and verification. The ultimate goal of CIT is to test its predictions against observational data and explore how it may contribute to a broader unification of physics.
This chapter explores:
* Future observational tests for CIT.
* How CIT could unify electromagnetism and gravity.
* The role of AI-human collaboration in advancing new scientific theories.

## 6.1 Experimental and Observational Tests for CIT

CIT makes specific predictions about planetary structuring, gravity, and mass-energy influx. Some of the key areas where it can be tested include:
1. Exoplanet surveys
2. Gravitational constant
3. Plate tectonics and planetary growth
4. Protoplanetary disk structures

## 6.2 CIT and the Unification of Physics

One of the most ambitious goals of CIT is to contribute to the unification of fundamental forces. Current physics separates: - Gravity (described by General Relativity). - Electromagnetism (described by Quantum Field Theory).
CIT proposes that: 1. Gravitational effects may be linked to energy influx mechanisms. 2. The role of VRMS and Lorentz transformation could bridge the gap between electromagnetism and gravity. 3. The vacuum energy field may serve as a common foundation for both forces.
This opens the possibility of a new framework for understanding fundamental interactions.

## 6.3 The Role of AI-Human Collaboration in Science

A unique aspect of CIT’s development is its reliance on AI-assisted research and human intuition. Scientific progress is increasingly shaped by: - AI-driven data analysis (pattern recognition, statistical modeling). - Human creativity and theoretical reasoning (hypothesis formulation, conceptual breakthroughs).
CIT highlights the importance of balancing AI automation with human scientific intuition. Future scientific advancements may depend on:
* AI-supported verification of non-mainstream theories.
* Combining AI modeling with observational data.
* Ensuring ethical decision-making in scientific progress.
By embracing AI-human collaboration, CIT can serve as an example of how new scientific ideas emerge and evolve.

## Summary

This chapter introduced:
* How CIT can be tested through exoplanet surveys, gravitational measurements, and planetary geology.
* The potential for CIT to contribute to a unified theory of physics.
* The role of AI-assisted research in scientific discovery.
With this, the Cosmic Influx Theory (CIT) Wikiversity page is complete, providing an open-access resource for future research, discussion, and verification.
# Sequence Lotus

| | valency gravity | strength depth |
| triangle | Willow | Poplar |
| diagonal | Lotus | Lily |
| difference | Orchid | Oxalis |
Lotus are the row sums of the weight triangle A163353. See here.

## Sequences Lotus$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Lotus (A000371) | 2 | 2 | 10 | 218 | 64594 | 4294642034 | 18446744047940725978 | 340282366920938463334247399005993378250 |
| 💧 LotusDrop | 2 | 0 | 8 | 208 | 64376 | 4294577440 | 18446744043646083944 | 340282366920938463315800654958052652272 |
half :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 HalfLotus (A003465) | 1 | 1 | 5 | 109 | 32297 | 2147321017 | 9223372023970362989 | 170141183460469231667123699502996689125 |
| 💧 HalfLotusDrop | 1 | 0 | 4 | 104 | 32188 | 2147288720 | 9223372021823041972 | 170141183460469231657900327479026326136 |
row sums of SignedCedar :
triangle SignedCedar    row sums Lotus (A000371)   🌊 :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 1 · 2 −2 | 1 · 4 4 | | | | | 2 |
| 2 | 1 · 2 2 | 2 · 4 −8 | 1 · 16 16 | | | | 10 |
| 3 | 1 · 2 −2 | 3 · 4 12 | 3 · 16 −48 | 1 · 256 256 | | | 218 |
| 4 | 1 · 2 2 | 4 · 4 −16 | 6 · 16 96 | 4 · 256 −1024 | 1 · 65536 65536 | | 64594 |
| 5 | 1 · 2 −2 | 5 · 4 20 | 10 · 16 −160 | 10 · 256 2560 | 5 · 65536 −327680 | 1 · 4294967296 4294967296 | 4294642034 |
LotusDrop   💧 :
| 0 | 2 |
| 1 | 0 |
| 2 | 8 |
| 3 | 208 |
| 4 | 64376 |
| 5 | 4294577440 |
half :
triangle HalfSignedCedar    row sums HalfLotus (A003465)   🌊 :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 1 · 1 −1 | 1 · 2 2 | | | | | 1 |
| 2 | 1 · 1 1 | 2 · 2 −4 | 1 · 8 8 | | | | 5 |
| 3 | 1 · 1 −1 | 3 · 2 6 | 3 · 8 −24 | 1 · 128 128 | | | 109 |
| 4 | 1 · 1 1 | 4 · 2 −8 | 6 · 8 48 | 4 · 128 −512 | 1 · 32768 32768 | | 32297 |
| 5 | 1 · 1 −1 | 5 · 2 10 | 10 · 8 −80 | 10 · 128 1280 | 5 · 32768 −163840 | 1 · 2147483648 2147483648 | 2147321017 |
HalfLotusDrop   💧 :
| 0 | 1 |
| 1 | 0 |
| 2 | 4 |
| 3 | 104 |
| 4 | 32188 |
| 5 | 2147288720 |

## Sequences SmallLotus$_{(Drop)}$

SmallLotus differs from Lotus only in the first entry. SmallLotus$_{(Drop)}$ are the left columns of SmallTurnedWillow$_{(Drop)}$.
| | 0 | 1 | 2 | 3 | 4 |
| 🌊 SmallLotus | 1 | 2 | 10 | 218 | 64594 |
| 💧 SmallLotusDrop | 1 | 1 | 8 | 208 | 64376 |
# Cross product/Computation/R/2/Exercise

Compute the cross product
    ${\begin{pmatrix}3\\5\\2\end{pmatrix}}\times {\begin{pmatrix}-4\\6\\-5\end{pmatrix}}$
in $\mathbb {R} ^{3}$.