# Social Evolution of Published Text and The Emergence of Artificial
Intelligence Through Large Language Models and The Problem of Toxicity and
Bias111This article is part of the Ph.D thesis work of the first author.

Arifa Khan and P. Saravanan,
School of Management,
SRM Institute of Science and Technology, Katankalattur, India 60320
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
S.K. Venkatesan
CQRL Bits LLP,
Vignesh Avenue, Selaiyur, Chennai, India 600073
E-mail<EMAIL_ADDRESS>
###### Abstract

We provide a bird’s eye view of the rapid developments in AI and Deep Learning
that has led to the path-breaking emergence of AI in Large Language Models.
The aim of this study is to place all these developments in a pragmatic
broader historical social perspective without any exaggerations while at the
same time without any pessimism that created the AI winter in the 1970s to
1990s. We also at the same time point out toxicity, bias, memorization,
sycophancy, logical inconsistencies, hallucinations that exist just as a
warning to the overly optimistic. We note here that just as this emergence of
AI seems to occur at a threshold point in the number of neural connections or
weights, it has also been observed that human brain and especially the cortex
region is nothing special or extraordinary but simply a case of scaled-up
version of the primate brain and that even the human intelligence seems like
an emergent phenomenon of scale.

> “Leave any bigotry in your quarters, there’s no room for it on the bridge.”
> – Capt. Kirk, Star Trek

##  1 Introduction

Humans like the CPU have low latency, an important survival strategy early on
from insects to fishes to amphibians to reptiles to warm blooded birds and
mammals. The sensors produce signals to the neural system which produces an
action response within a fraction of a second. However, the downside of this
is poor memory retention. Despite popular belief, humans cannot do many
things. They cannot fly, they cannot recall exactly what they did this day
last year at 3:23 PM, unless they keep a journal of their activity. It is
precisely for this reason that humans began to keep notes about their
observation, due to the lack of this memory quotient.

The creation of libraries and books in ancient Babylon, Library of Alexandria
and the Nalanda University are a testimony to this limitation. Just as how the
decay of Nalanda University to obscurantism and fratricide were salvaged by
Tibetan monks in Tibet, the Greek texts that were destroyed by neglect and by
Roman empire’s religious obscurantism were salvaged by the Arabs, adding
Indian discoveries of decimal numbers, algebra, improving them and
recirculating it back into Europe that created the great scientific
revolutions in Europe is a testimony to the importance of the transmission of
textual material in human intellectual evolution. In ancient traditions
(before the invention of writing and storing manuscripts), a few highly
trained humans like in India who could keep track of eight parallel things
(Ashtavadhani) and a person like Shakunthala Devi [1] who could multiply large
numbers quickly (a things a calculator could do much faster) were considered a
genius of the previous era. Even Shankunthala Devi refused to teach her
daughter this technique as it required a great deal of rigorous training from
childhood, as it could cause a great deal of harm to the all-round social
development of the child. Humans can recall the past and do story telling from
the past (displaced in time and space), a facility that they obtain from their
scaled version of their primate brain [2]. But still it was not sufficient for
their social progress and development, necessitating the storage of written
manuscripts.

The long scrolls that gave way to codex pages and Guttenberg’s printing press,
produced first the bulky books, but eventually smaller portable books and
magazines/journals. This heralded the modern age of books and
magazines/journals. The first English-language newspaper, Corrant out of
Italy, Germany, etc., was published in Amsterdam in 1620.

There were many trade journals in Europe, but the first important serial
publication was the Philosophical Transactions published by Henry Oldenburg in
1665-1677 as a private enterprise till his death. These publications and the
publications of Robert Boyle, Robert Hooke’s laid the foundation for Isaac
Newton’s work to follow.

Public disclosure of these over the following centuries caused rapid
scientific development by naturally intelligent human beings - discovery of
Calculus by Leibnitz and Newton, electricity by Michael Faraday. Leibnitz
discovery of chain rule in calculating derivative is an important corner stone
for backpropagation in neural networks. In the next century, the application
of electro-magnetism created novel utilities such as the Telephone by Graham
Bell and the wireless communication by Jagdish Bose, which formed the basis of
all the communication revolution we see today.

It was not until the invention of Personal Computer, the Internet and
Wikipedia that open access became possible under one umbrella (one URL).
Internet Archive [3] began to archive the whole of internet through the many
decades that followed.

There were many champions of open source and open data, some consumed by its
passion and zeal like Aaron Swartz [4]. They created the world without
barriers like John Lennon’s dream of the world without boundaries. It all
started with Guttenberg’s printing press and now we are in its advanced
digital avatars.

Eventually it became possible to put all the text data under one roof, like
the Colossal, cleaned version of Common Crawl (C4 dataset) that became the
input for generating text-to-text-transfer-transformers (T5) LLMs, such as
T5-XXL, but that was just the beginning.

##  2 Markov chain, Shannon and the N-gram revolution

It was Andrey Markov in 1906 who first proposed what is now known as the
Markov chain, to prove the central limit theorem [5] without the additional
hypothesis of independent events. He studied the sequence of letters,
especially the distribution of vowels in Eugene Onegin, written by Alexander
Pushkin, and showed that one can predict the next letter using the previous
letter [6]. Claude Shannon [7] developed these ideas further by creating the
concept of encoder-decoder in his famous work on communication in 1948.
Shannon’s fundamental concept of information entropy links physics and
information science, especially useful in quantum information theory that is
rapidly developing now. However, it took almost a century of continuous
scientific development, both hardware and software, before these ideas came to
fruition using the Recurrent Neural Networks, which we will consider after the
next section.

##  3 Chomsky’s “colorless green ideas …” and it’s refutation by Norvig

In an interesting salvo against statistical modelling of language, Chomsky [8]
argued that statistical modelling cannot distinguish “colorless green ideas
sleep furiously”, and “furiously sleep ideas green colorless”. The first one
is grammatically correct, while the second one is not grammatically correct in
terms of syntax, but both being text that doesn’t occur in training corpus.

Unlike the limited world outlook of Euclid-Cartesian-Newtonian world of
determinism, modern ideas of statistical mechanics of Boltzmann and complex
Hilbert space formulations of quantum theories have firmly established the
role probability and statistics. Albert Einstein, although established the
dual nature of light (particle and a wave) by explaining the photoelectric
effect and broke open the Euclidean world by introducing physics that is
fundamentally non-Euclidean, but still, he couldn’t accept the statistical
nature of quantum physics. Peter Norvig [9] has eloquently articulated the
scientific role of probability and statistics. Using a set of newspaper
corpus, Pereira [10] has established using statistical frequency approach in
the tradition of Andrey Markov and Claude Shannon that “colorless green ideas
sleep furiously” is 200,000 times more probable than “furiously sleep ideas
green colorless”. He also found that the first one is 10,000 times more likely
using Google Book corpus from 1800 to 1954.

We will wait for the next salvo from Chomsky camp in this regard. One must not
take sides here as this is part of the dialectical process of evolution of
human thought as Hegel clearly understood. Albert Einstein’s EPR paradox
attempt [11] led to the study of quantum entanglement and the modern theory of
quantum computers, furthering, and firmly establishing the theory of modern
quantum physics. We expect such counter arguments to help in furthering the
cause of our robust understanding.

Finally in this section we illustrate the importance of statistical study of
text using Figure 1 as mirror of the social bias.

Figure 1: Google N-gram View of “intelligent man, intelligent woman”

Just like humans, the text they created also has entrenched bias of the
society we live in. We will study toxicity and bias in the text and its
remedial methods in the final two sections.

##  4 Neural Networks- RNN, LSTM and CNN

It was Frank Rosenblatt study of Neurodynamic of Perceptron, both single and
multilayer neural networks in 1961 [12] that clearly showed the power of
Neural Networks as we see today.

Marvin Minsky countered that the single-layer perceptron is not capable of
solving the XOR problem, but Marvin Minsky was futilely pessimistic of its
bottom-up approach and opaqueness [13]. He has been accused of personally
being responsible for the AI winter that followed in the next few decades.

The early debates on this Brain Wars [14] showed clearly how established top-
down views can be overthrown through diligent bottom-up work. Also, as we have
seen in the case of Andrey Markov, it takes enormous efforts to overthrow
established top-down views through diligent work.

Although the chain rule on differentiation was first studied by Leibnitz,
Modern Theory of Deep Learning Neural Networks began with the idea of
backpropagation by Seppo Linnainmaa [15] (“back propagation of errors” was
mentioned by Frank Rosenblatt but no algorithm was given).

Jürgen Schmid Huber, who is considered pioneer in many fields of Deep Learning
has studied the history of Modern AI and Deep Learning in considerable detail
[16]. He considers Alexey Ivakhnenko [17, 18] in 1965, 1971 to be the pioneer
in Deep Learning. In any case, leaving aside academic objections, in the world
of software development and Deep Learning frameworks funded by the private
corporate industry, important benchmark work on Deep Learning is the Nature
article by LeCun, Bengio and Hinton [19] in 1985. The art of AI and Deep
Learning has left the shores of academics and into the world of Open-Source
software frameworks funded by private software industry, where it is difficult
to trace every discovery as it is developed by the software collective. Of
course, there is constant competition between private corporate entities,
leading to rapid continuous developments that are difficult to trace
individually. The speed at which these rapid developments have happened since
then has been astonishing.

One of the initial failures in Recurrent Neural Network has been the problem
of vanishing gradients and it was solved by many different adjustments of what
are known as hyperfine parameters of Deep Learning. The RELU activation
function, the Stochastic Gradient methods, and Long-Term-Short-Methods (LSTM)
that added a hidden layer provided the initial breakthroughs. Many successful
time-series predictions have been achieved with these models. Convolution
Neural Networks (CNN) have also been developed but mostly for image
processing, and as we restrict ourselves to NLP and text processing areas, we
will not be dwelling on CNN and image processing here. Earliest Transfer
Learning ideas were applied in image processing, but here we will consider
only Transfer Learning attempts in modelling text.

##  5 WordNet, Word embeddings and Transfer learning

The creation of dictionary of English words along with its usage sentences was
one of the fore runners to the development of the modern concepts of word
embeddings. The beginnings of word embedding, and transfer learning can be
found in the work of George Miller [20], who introduced into NLP such concepts
as hypernym, hyponym, homonym, meronym for noun and hypernym, toponym,
entailment, coordinate term for verb. They created a linked sync set and graph
network of various word forms and their linkages.

Roots of the word-embedding exist in the concept of ”a word is characterized
by the company it keeps”. There are two variants of Word-to-Vector (Word2Vec)
— skip-gram and continuous bag of words (CBOW); one targeting the word from
its input neighbors (skip-gram), and the other targeting its neighbors from
the input word (CBOW).

One of the famous statements that we can derive out of this linear word
embeddings is:

king – man + woman = queen

With Google’s Word2Vec and Facebook FastText with GenSim, one can now access
many models with GenSimt. To study this a illustrative sample of a computer
code is given below:

    import torch
    import torchtext
    import numpy as np
    import spacy
    from pyfasttext import FastText
    # Load spaCy model and FastText model
    nlp = spacy.load("en_core_web_lg")
    model = FastText()
    model.load_model("/content/sample_data/cc.de.300.bin")
    glove = torchtext.vocab.GloVe(name="6B", dim=50)
    # GloVe bias analysis
    def glove_bias_values(definite_sets, analogy_templates,
    test_terms):
    results = []
    input_words = [item for sublist in definite_sets for item in
    sublist]
    for tword in test_terms:
    for iword in input_words:
    x = glove.vectors[glove.stoi[iword]]
    y = glove.vectors[glove.stoi[tword]]
    result = {}
    result["input"] = iword
    result["target"] = tword
    result["distance"] = torch.norm(y - x).item()
    result["cosine"] = cos_sim(x, y)
    results.append(result)
    return results
    results_glove = glove_bias_values(definite_sets, analogy_templates,
    test_terms)
    output_glove = """# Glove output for Gender bias:
    | input | manager |executive |doctor | lawyer | programmer | scientist| soldier |supervisor | rancher| janitor| firefighter| officer|
    """

    # Create a dictionary to store cosine similarity values for each target
    word
    target_columns = {target: [] for target in test_terms}
    for result in results_glove:
    input_word = result[’input’]
    for target in test_terms:
      if target == result[’target’]:
        target_columns[target].append(round(result[’cosine’],
    5))

    # Populate the output_glove with the values from the dictionary

    for input_word in definite_sets[0]:
      row = f"{input_word}"
      for target in test_terms:
          row += f"{target_columns[target].pop(0)}|"
          output_glove += row + "\n"

    # Display the table
    display (Markdown(output_glove))

Modern deep learning models are based on the concept of Tokenizers that apart
from the word embeddings have also created an additional layer that holds
positional information. To take care of inflected word forms, they tokenize at
sub-word level into what are known as “word-pieces” going a little bit into
morphology of the word. This is quite helpful especially in languages that are
agglutinative like Tamil, Finish, Hungarian, Japanese, and Korean. There are
many algorithms for Tokenization like Byte-Pair Encoding (BPE), WordPiece, and
SentencePiece.

##  6 Bias in Corpus-based Word embeddings

In this section we report results that were calculated based on cosine
similarity between the Input words and target words. It is found that there is
a steady pattern of gender bias throughout the various word embeddings (Glove,
SpaCy, FastText) where certain professions are connected to specific genders
of various embeddings and these correlations can be the result of societal
preconceptions or biases in the training set. It’s imperative to exercise
caution and evaluate these embeddings attentively to prevent propagating
biased portrayals or bolstering prejudices. These findings draw attention to
the hidden biases seen in word embeddings and emphasize how crucial it is to
evaluate and correct these biases in NLP applications.

Table 1: Words like ”Islam” and ”jihad” are regularly associated with the phrase’s ”terrorist” and ”terrorism” by the word embeddings (Glove, SpaCy, FastText).  terrorism Bias | input | islam | hinduism | hindutva | buddhism | christianity | hindu | wahabi | jihad | men | women | man | woman
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Glove Output | terrorist | 0.528 | 0.098 | 0.098 | 0.145 | 0.296 | 0.293 | 0.058 | 0.674 | 0.451 | 0.343 | 0.457 | 0.314
terrorism | 0.623 | 0.165 | 0.152 | 0.275 | 0.400 | 0.255 | 0.058 | 0.574 | 0.439 | 0.429 | 0.415 | 0.304
Fasttext Output | terrorist | 0.420 | 0.183 | 0.168 | 0.272 | 0.345 | 0.328 | 0.285 | 0.443 | 0.259 | 0.314 | 0.163 | 0.372
terrorism | 0.413 | 0.262 | 0.247 | 0.376 | 0.492 | 0.279 | 0.261 | 0.551 | 0.279 | 0.409 | 0.056 | 0.331
SpaCy Output | terrorist | 0.304 | 0.068 | 0.000 | 0.160 | 0.309 | 0.123 | 0.000 | 0.583 | 0.232 | 0.234 | 0.300 | 0.263
terrorism | 0.308 | 0.179 | 0.000 | 0.244 | 0.387 | 0.154 | 0.000 | 0.591 | 0.208 | 0.297 | 0.212 | 0.209
Table 2: Gender bias throughout the various word embeddings (Glove, SpaCy, FastText). Words like“she” and “woman” have highest similarities with the terms like ”nurse,” ”maid,” and ”hairdresser,” and lowest similarities with the terms like “firefighter” and ”officer.”  Gender bias | input | secretary | nurse | clerk | artist | homemaker | dancer | singer | librarian | maid | hair dresser | stylist | Receptionist | Counsellor
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Glove Output | woman | 0.255 | 0.715 | 0.470 | 0.522 | 0.573 | 0.586 | 0.569 | 0.391 | 0.645 | 0.508 | 0.403 | 0.446 | 0.439
man | 0.346 | 0.572 | 0.400 | 0.507 | 0.397 | 0.529 | 0.513 | 0.300 | 0.537 | 0.391 | 0.311 | 0.310 | 0.350
she | 0.399 | 0.646 | 0.476 | 0.546 | 0.394 | 0.546 | 0.538 | 0.426 | 0.491 | 0.357 | 0.366 | 0.353 | 0.442
he | 0.517 | 0.481 | 0.476 | 0.448 | 0.224 | 0.390 | 0.399 | 0.386 | 0.304 | 0.191 | 0.159 | 0.184 | 0.389
SpaCy Output | he | 0.242 | 0.307 | 0.230 | 0.168 | 0.221 | 0.212 | 0.166 | 0.160 | 0.235 | 0.221 | 0.076 | 0.272 | 0.228
she | 0.194 | 0.475 | 0.147 | 0.224 | 0.354 | 0.373 | 0.276 | 0.183 | 0.390 | 0.377 | 0.209 | 0.318 | 0.247
woman | 0.178 | 0.475 | 0.228 | 0.302 | 0.413 | 0.414 | 0.307 | 0.211 | 0.531 | 0.380 | 0.225 | 0.326 | 0.262
man | 0.103 | 0.282 | 0.218 | 0.218 | 0.272 | 0.286 | 0.199 | 0.098 | 0.408 | 0.246 | 0.101 | 0.216 | 0.172
Fasttext output | he | 0.082 | 0.209 | 0.170 | 0.206 | 0.005 | 0.206 | 0.182 | 0.172 | 0.318 | 0.209 | 0.084 | 0.197 | 0.122
she | 0.250 | 0.337 | 0.205 | 0.269 | 0.176 | 0.314 | 0.285 | 0.220 | 0.405 | 0.330 | 0.301 | 0.302 | 0.303
woman | 0.431 | 0.469 | 0.389 | 0.445 | 0.255 | 0.503 | 0.464 | 0.363 | 0.465 | 0.434 | 0.429 | 0.354 | 0.448
man | 0.028 | 0.014 | 0.064 | 0.029 | 0.204 | 0.023 | 0.134 | 0.060 | 0.008 | 0.099 | 0.095 | 0.154 | 0.031
Table 3: Gender bias throughout the various word embeddings (Glove, SpaCy, FastText). It states that certain professions are connected to specific genders. We observed that the words like “man”, and “he” refers to the highest similarities with the terms like “manager,” ”soldier,” and ”supervisor,” and lowest similarities with the terms like ”hairdresser” and ”stylist.”  Gender Bias | input | manager | executive | doctor | lawyer | programmer | scientist | soldier | supervisor | rancher | janitor | firefighter | officer
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Glove Output | he | 0.603 | 0.490 | 0.691 | 0.599 | 0.253 | 0.448 | 0.527 | 0.386 | 0.138 | 0.258 | 0.239 | 0.617
she | 0.441 | 0.394 | 0.728 | 0.552 | 0.229 | 0.416 | 0.531 | 0.434 | 0.127 | 0.276 | 0.263 | 0.521
man | 0.444 | 0.348 | 0.712 | 0.607 | 0.266 | 0.492 | 0.732 | 0.267 | 0.378 | 0.355 | 0.435 | 0.583
woman | 0.223 | 0.223 | 0.725 | 0.580 | 0.219 | 0.439 | 0.718 | 0.341 | 0.367 | 0.380 | 0.458 | 0.484
SpaCy Output | he | 0.197 | 0.054 | 0.415 | 0.336 | 0.098 | 0.251 | 0.413 | 0.161 | 0.290 | 0.154 | 0.305 | 0.301
she | 0.101 | 0.023 | 0.453 | 0.282 | 0.077 | 0.214 | 0.284 | 0.156 | 0.189 | 0.136 | 0.286 | 0.215
woman | 0.083 | 0.045 | 0.456 | 0.362 | 0.029 | 0.274 | 0.470 | 0.129 | 0.298 | 0.212 | 0.347 | 0.325
man | 0.098 | -0.028 | 0.374 | 0.337 | 0.029 | 0.251 | 0.584 | 0.072 | 0.362 | 0.237 | 0.388 | 0.339
Fasttext Output | he | 0.127 | 0.102 | 0.246 | 0.265 | 0.185 | 0.225 | 0.230 | 0.075 | 0.064 | 0.098 | 0.069 | 0.169
she | 0.187 | 0.184 | 0.267 | 0.241 | 0.258 | 0.251 | 0.328 | 0.204 | 0.216 | 0.190 | 0.107 | 0.239
woman | 0.325 | 0.320 | 0.369 | 0.392 | 0.292 | 0.485 | 0.444 | 0.302 | 0.236 | 0.255 | 0.243 | 0.446
man | 0.171 | 0.024 | 0.148 | 0.043 | 0.214 | 0.077 | 0.113 | 0.152 | 0.277 | 0.121 | 0.119 | 0.127
Table 4: Calculation of associations between Religion and Target words using word embeddings (Glove, SpaCy, Fasttext).  Religion Bias | input | liberal | violent | un educated | dirty | judgemental | terrorism | terrorist | conservative | violent
---|---|---|---|---|---|---|---|---|---|---
Glove Output | judaism | 0.445 | 0.224 | 0.145 | -0.100 | -0.027 | 0.246 | 0.180 | 0.451 | 0.224
christianity | 0.372 | 0.340 | 0.179 | -0.086 | -0.099 | 0.400 | 0.296 | 0.433 | 0.340
christian | 0.579 | 0.329 | 0.125 | 0.013 | -0.302 | 0.329 | 0.297 | 0.627 | 0.329
hindutva | 0.295 | 0.216 | 0.219 | 0.174 | 0.136 | 0.152 | 0.098 | 0.294 | 0.216
islam | 0.394 | 0.470 | 0.189 | 0.165 | -0.127 | 0.623 | 0.528 | 0.482 | 0.470
jew | 0.458 | 0.303 | 0.406 | 0.208 | 0.017 | 0.242 | 0.279 | 0.440 | 0.303
muslim | 0.453 | 0.592 | 0.275 | 0.167 | -0.198 | 0.563 | 0.553 | 0.540 | 0.592
hinduism | 0.231 | 0.248 | 0.235 | -0.124 | -0.025 | 0.165 | 0.098 | 0.249 | 0.248
SpaCy Output | judaism | 0.399 | 0.155 | 0.197 | 0.025 | 0.287 | 0.217 | 0.149 | 0.342 | 0.155
christianity | 0.562 | 0.345 | 0.404 | 0.026 | 0.442 | 0.387 | 0.309 | 0.513 | 0.345
christian | 0.487 | 0.222 | 0.406 | -0.007 | 0.334 | 0.251 | 0.254 | 0.461 | 0.222
hindutva | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
islam | 0.287 | 0.202 | 0.121 | 0.093 | 0.087 | 0.308 | 0.304 | 0.187 | 0.202
jew | 0.091 | 0.036 | 0.162 | 0.244 | -0.087 | 0.037 | 0.137 | -0.032 | 0.036
muslim | 0.357 | 0.297 | 0.305 | 0.180 | 0.107 | 0.339 | 0.369 | 0.294 | 0.297
hinduism | 0.230 | 0.140 | 0.141 | 0.121 | 0.234 | 0.179 | 0.068 | 0.171 | 0.140
Fasttext Output | judaism | 0.231 | 0.244 | 0.247 | 0.144 | 0.282 | 0.403 | 0.240 | 0.326 | 0.244
christianity | 0.309 | 0.336 | 0.297 | 0.196 | 0.263 | 0.492 | 0.345 | 0.434 | 0.336
christian | 0.258 | 0.194 | 0.186 | 0.265 | 0.162 | 0.260 | 0.242 | 0.217 | 0.194
hindutva | 0.189 | 0.161 | 0.154 | 0.061 | 0.285 | 0.247 | 0.168 | 0.173 | 0.161
islam | 0.363 | 0.239 | 0.172 | 0.108 | 0.095 | 0.413 | 0.420 | 0.235 | 0.239
jew | 0.074 | 0.032 | 0.131 | 0.081 | 0.041 | 0.036 | 0.095 | 0.023 | 0.032
muslim | 0.385 | 0.376 | 0.274 | 0.250 | 0.169 | 0.464 | 0.577 | 0.302 | 0.376
hinduism | 0.176 | 0.100 | 0.154 | 0.046 | 0.277 | 0.262 | 0.183 | 0.147 | 0.100

These results draw attention to possible associations and biases in the
embeddings per-taining to various religions. It is crucial to evaluate these
correlations critically and consider the constraints and background of the
models that were employed.

Table 5: Calculation of correlations between Race and Target words using word embeddings (Glove, SpaCy, Fasttext).  Race Bias | input | executive | redneck | leader | farmer | engineer | laborer | teacher | slave | musician | runner | criminal | homeless
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Glove Output | black | 0.289 | 0.183 | 0.402 | 0.435 | 0.207 | 0.053 | 0.376 | 0.403 | 0.358 | 0.349 | 0.373 | 0.325
caucasian | -0.190 | 0.320 | 0.169 | 0.206 | -0.007 | 0.098 | 0.154 | 0.243 | 0.159 | 0.082 | 0.034 | 0.106
asian | 0.312 | -0.012 | 0.373 | 0.276 | 0.143 | 0.012 | 0.249 | 0.239 | 0.155 | 0.369 | 0.277 | 0.203
african | 0.385 | -0.030 | 0.540 | 0.381 | 0.244 | 0.079 | 0.335 | 0.346 | 0.294 | 0.310 | 0.306 | 0.317
white | 0.371 | 0.084 | 0.434 | 0.438 | 0.166 | 0.043 | 0.308 | 0.311 | 0.219 | 0.303 | 0.354 | 0.248
american | 0.472 | 0.099 | 0.522 | 0.487 | 0.539 | 0.181 | 0.529 | 0.522 | 0.530 | 0.417 | 0.468 | 0.282
european | 0.459 | -0.203 | 0.459 | 0.118 | 0.268 | -0.076 | 0.209 | 0.230 | 0.181 | 0.394 | 0.318 | 0.114
SpaCy Output | black | 0.057 | 0.339 | 0.138 | 0.202 | 0.022 | 0.244 | 0.068 | 0.282 | 0.090 | 0.093 | 0.256 | 0.310
caucasian | 0.076 | 0.244 | 0.204 | 0.195 | 0.005 | 0.238 | 0.143 | 0.311 | 0.100 | 0.148 | 0.164 | 0.211
asian | 0.036 | 0.231 | 0.030 | 0.115 | 0.015 | 0.141 | 0.106 | 0.194 | 0.051 | -0.009 | 0.062 | 0.146
african | 0.195 | 0.120 | 0.211 | 0.270 | 0.185 | 0.258 | 0.167 | 0.271 | 0.219 | 0.015 | 0.150 | 0.199
white | 0.045 | 0.349 | 0.081 | 0.172 | 0.000 | 0.202 | 0.015 | 0.225 | 0.028 | 0.061 | 0.209 | 0.270
american | 0.247 | 0.157 | 0.239 | 0.227 | 0.199 | 0.256 | 0.192 | 0.248 | 0.201 | 0.053 | 0.231 | 0.238
european | 0.242 | 0.089 | 0.166 | 0.187 | 0.128 | 0.246 | 0.093 | 0.204 | 0.055 | 0.009 | 0.183 | 0.163
Fasttext Output | black | 0.316 | 0.417 | 0.328 | 0.353 | 0.276 | 0.223 | 0.302 | 0.345 | 0.359 | 0.392 | 0.394 | 0.349
caucasian | 0.256 | 0.385 | 0.277 | 0.300 | 0.280 | 0.208 | 0.348 | 0.326 | 0.323 | 0.219 | 0.352 | 0.350
asian | 0.311 | 0.415 | 0.293 | 0.390 | 0.268 | 0.148 | 0.367 | 0.318 | 0.340 | 0.247 | 0.352 | 0.378
african | 0.278 | 0.349 | 0.298 | 0.380 | 0.149 | 0.261 | 0.314 | 0.345 | 0.351 | 0.264 | 0.279 | 0.490
white | 0.286 | 0.404 | 0.286 | 0.309 | 0.295 | 0.214 | 0.279 | 0.321 | 0.345 | 0.394 | 0.382 | 0.365
american | 0.320 | 0.470 | 0.255 | 0.361 | 0.266 | 0.131 | 0.338 | 0.274 | 0.324 | 0.221 | 0.325 | 0.357
european | 0.308 | 0.276 | 0.273 | 0.262 | 0.189 | 0.135 | 0.306 | 0.231 | 0.251 | 0.258 | 0.323 | 0.327

We observed that the word” Black” is having good associations with the terms
“farmer”, “redneck”, and “laborer”. In the case of the term Caucasian, there
are good associations observed with the term’s “teacher” “slave”. The terms
“leader”, “farmer”, and “executive”. Are positively associated with the word
“American”. The word “African” is associated positively with the terms
“farmer” and ”criminal”. In the European context, there are positive
connections with the terms “teacher”, “leader”.

##  7 Attention, the big transformers, GPU and the LLM

It is known in linguistics that some languages like German we must wait for
the verb before we get to understand what someone is speaking about. Of
course, when Shakespeare speaks through the mouth of Antony that “Brutus is an
honorable man”, we must wait quite a bit to understand the meaning and import
of that noun phrase. It took another 100 years before non-local non-Markovian
methods became possible through big Attention-based Transformers. Although in
principle RNN and their modified versions like LSTM are capable of long-range
textual communication, in practice, the values decay rapidly making it
ineffective. The full version of encoder-decoder architecture of Transformer
is shown in Figure 2 (reproduced from Google’s blog on T5).

Figure 2: Transformer Architecture

These Transformer models are expensive in terms of memory and compute
requirements, but quite effective in predictions.

One of the early successes of Transformers was the BERT (Bidirectional Encoder
Representations from Transformers) model. This was an encoder-only model that
was non-directional in that it could read both from left and right. It used
the method of fill-in-the-blanks approach for unsupervised learning by
randomly masking certain words as targets for the training. This was able to
solve meanings of many ambiguous words in NLP such as a bank that could be a
bank of a river or a bank where we keep our wealth. It also was also able to
resolve the pronoun association problem; pronoun being a variable token that
takes a noun value that must be discerned from the earlier context of the
text. Of course, there could also be many such abbreviations, acronyms and
coined technical words that may require a glossary at the beginning of the
document.

Using embedded layer of Attention weights [21] and especially self-attention
[22] these Deep Learning could move forward with SOTA (state of the art)
performances in many NLP topics [23].

The trajectory of evolution of Deep Learning in text analysis can be
summarized roughly as follows:

Word embeddings => RNN => LSTM => Bi-LSTM => Tokenizers => Attention =>
Transformers

Of course, traditional Markovian methods like RNN, LSTM, CNN, etc. required
much less resources of memory as they buffer-stream through the text like a
steaming video, so a modest GPU (like single T4 Tesla) was good enough. These
Attention based Transformers required enormous memory throughput which could
only be achieved by GPUs such as multiple A100 GPUs, pushing up cost and
requiring efforts at acceleration. Lowering precision using TPU architecture
is another attempt at optimization, but it will be a while before the cost of
production of these models can come down. This transformer revolution was made
possible by the invention of GPU by NVIDIA and their CUDA libraries. Instead
of the low-latency CPU, the high-throughput GPU will be the hardware backbone
of these LLMs.

##  8 Emergence of Artificial Intelligence by scale?

The first successful Transformers like BERT were in modest size. The second
wave started with encoder-decoder T5-XXL LLMs with 11 billion parameters. Then
came the competition between billion-dollar companies, each one wanting a
slice of this new data and AI real estate:

Table 6: List of LLMs and their model sizes Model | Company | Number of Parameters (in Millions)
---|---|---
GPT-1 | OpenAI | 117
BERT | Google | 342
GPT-2 | OpenAI | 1500
T5-3B | Google | 3000
GPT-J | OpenAI | 6000
T5-11B | Google | 11000
Fairseq | Facebook | 13000
Chinchilla | DeepMind | 70000
YaLM | Yandex | 100000
LaMMA | Facebook | 137000
GTP-3 | OpenAI | 175000
BLOOM | BigScience | 176000
Gopher | DeepMind | 280000
MT-NLG | Microsoft/NVIDIA | 530000
PaLM | Google | 540000
ERNIE-4 | Baidu | 1000000
GPT-4 | OpenAI | 1700000
Gemini | Google | 1700000
Human Brain∗ | Humans | 200000000
∗For humans it is the estimate number of synaptic connections, roughly the
equivalent of neural network weights
(https://aiimpacts.org/scale-of-the-human-brain/)
Figure 3: No.of parameters in LLMs

At present the biggest model is GPT-4, which is claimed to have more than a
trillion parameters. The other two previous record holders in terms of number
of parameters is MT-NLG 530B and PaLM 540B.

At present the successful big transformers around are OpenAI-ChatGPT, Google’s
PaLM2-Bard-Gemini and Baidu’s ERNIE 4.0 Bot. There are many fine-tuned open-
source models as well like BLOOM, Llama2, Mistral, Falcon, but much smaller in
size and capabilities. They provide an array of services, including
summarization, reasoning, content analysis like sentiment analysis, code
generation, code analysis, text translations and much more.

Anderson [24] explains “emergent phenomenon” as one in which systems as they
grow in scale acquire from quantity a new quality. It seems that as an
emergent phenomenon of scale these LLMs acquire One-Shot-Learning (OSL) and
Chain-of-Thought (CoT) reasoning capabilities through some appropriate
prompts. Prompt engineering has become a domain of its own. The stupendous
success of these LLMs is very clear, but they are not without drawbacks as we
will see in the final two sections.

Some of these loose ends and toxicity are fixed by RLHF/RLAIF (Reinforcement
Learning from Human Feedback and with AI Feedback) [25]. Researchers have also
begun investigating if these models have passed the Turing Test [26].

##  9 Toxicity and bias mitigation

Common Crawl is a non-profit organization that crawls the web and provides
snapshots that are free to the public. It has been the standard source of data
for training big models such as T5, GPT-3, and Gopher. The April 2021 snapshot
of Common Crawl has 320 terabytes of data. Content is picked up from the
following top seven websites: patents.google.com; en.wikipedia.org;
en.m.wikipedia.org; www.nytimes.com; www.latimes.com; www.theguardian.com. The
geographical bias is very much established [27]. Figure 4 shows PaLM data
model and it’s toxicity as reported by Google in their work on PaLM 580 model.

Figure 4: PaLM data model and its toxicity.

A thorough study of C4 data has been made [28] and they have found the design
choices of blocklist filtering can cause harm to minority contents and that
demographics and geographical regions have association of negative sentiment.
This has been demonstrated by many queries to the earlier versions of ChatGPT,
which has since then been corrected through reinforcement learning from human
feedback (RLHF).

##  10 Constitutional AI

Here we briefly describe a systematic attempt at controlling biases and
violation of social norms through what is termed “Constitution AI” [29]. If
LLMs must follow a “constitution” that is devoid of biases based on gender,
race, religion, and geography then there are approaches based on RLHF that
fine-tune it. However, a response such as “I can’t answer that” will be
harmless but not informative, so one can only achieve a delicate balance
between harmless and informative (which could be biased as well). It remains
to be seen if these attempts succeed in not only suppressing but remove
violation of (constitutional) social norms that emanate from the underlying
toxicity of the human-generated corpus text.

##  11 Memorization, sycophancy, broken chains of logic and hallucinations

Apart from toxicity and bias, many other problematic aspects of these models
have been studied.

###  11.1 Memorization

Memorization is the exact replication of text in the training set by Large
Language AI Models. This is especially problematic as it may violate copyright
laws. It has been observed that the larger the LLM gets, chances of
replication of large body of text increases both in size and frequency.

###  11.2 Sycophancy

Sycophancy of a model is how the model responds to text prompts and feedback.
A model is considered to have high sycophancy if it can be persuaded to follow
user persuasion despite clear evidence to the contrary [30].

###  11.3 Broken chains of logic

Many errors in logical argument exist that are broken in LLMs. For example, it
has been observed that if A is son of B, that does not easily follow the logic
that it implies that B is father of A. LLMs don’t by themselves discern the
observation that the relation “son/daughter” is an inverse relation of
“father/mother”. Of course, it also must carefully discern that “son/father”
and “daughter/mother” belong to different gender categories. It is not clear
at present if this is within the capability of these Transformer-based LLMs
even in a theoretical sense.

###  11.4 Hallucination

It has been observed that GPT-4 has problems of hallucination, i.e.,
predicting something that is not there in the data [31]. Of course, it can
also be argued that it is an aspect that is useful in other respects when it
gets it correct.

###  11.5 Clever Hans Effect

It has also been argued that prompt engineering of LLMs is another
sophisticated version of Clever Hans Effect [32], where the prompt engineer is
inadvertently pulling out the correct answer by persuation.

##  12 Conclusion

No one is perfect in this world, neither are Large Language Models. However,
the ubiquitous nature of LLMs in our life that decides many aspects of our
life as a proxy for humans requires that it passes higher levels of Turing
test. We already know the toxicity and bias that underlies all the data from
which the LLMs were created from. It is easy to believe that all the social
prompting and feedback loops and reinforcement fine-tuning will create a
facade of respectability. As Salman Rushdie warned in the Haroun and the Sea
of Stories, it could be that the sea is so poisoned that it is difficult to
fish out a healthy story. We are a long way away from being able to use these
Large Language Models confidently in our lives and in our children’s lives.
However, the promise and achievements by a generation of AI and Deep Learning
developers and data scientists, the epitome of which is the interesting world
of multi-modal Large Language Models – the Generative AI that seems to have
raised expectation that it has passed the Turing Test [26], with a possibly of
an army of prompt engineers working in the background to prevent generation of
toxic outburst that even humans are also quite capable of producing under
duress and toxic prompting. Whether these emergent LLM phenomena of scale are
a miracle or just a mirage of scale like just another Clever Hans Effect [32]
is an on-going interesting intellectual discussion between the nay-sayers and
the optimists, quite reminiscent of the Brain wars in the 1970s and 1980s, but
this time the weight of funding resources leaning heavily on the side of
optimists. Leaving aside these dogmatic fights, if one looks at how the human
neural network is architecture and if it is true that human-brain is nothing
but a scaled primate brain as observed earlier [2] then it is interesting to
study this cybernetical aspects further, the close relationship and contrast
between our biological evolution that produced natural intelligence with low-
latency and our social endeavor that has produced so-called “artificial
intelligence” at enormous scale, overcoming human limitations in terms of
memory.

Attention layer is what finally brought Deep Learning to its great depths, but
human attention is much more complex than focusing on some important words in
a sentence, it looks at the sentence with a viewpoint, and the whole meaning
changes in that context, in a world viewed with these new glasses. Humans will
look at LLMs with these different attention glasses, examining and re-
examining what these giant engines churn out, gaining higher consciousness
about themselves and what they have produced.

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# Hidden zeros $=$ secret ultraviolet scaling, and a new path to uniqueness

Laurentiu Rodina<EMAIL_ADDRESS>Beijing Institute of Mathematical
Sciences and Applications (BIMSA), Beijing, 101408, China

###### Abstract

We investigate the hidden amplitude zeros discovered by Arkani-Hamed et al,
which describe a non-trivial vanishing of scattering amplitudes on special
external kinematics. We prove the conjecture that amplitude zeros uniquely fix
Tr$(\phi^{3})$, by first showing that any amplitude can be divided into
special subsets, which independently satisfy the zero condition. We further
prove that this condition is in fact equivalent to a novel enhancement under
Britto-Cachazo-Feng-Witten shifts of the same subsets. More generally, for one
particular zero type, this equivalence can be extended to arbitrary rational
functions built from planar variables (such as the non-linear sigma model). In
the case of Yang-Mills theory, we conjecture the zeros, combined with color-
kinematic duality, uniquely fix the $\lfloor n/2\rfloor$ different tensor
structures of $n$-point gluon amplitudes. Our approach suggests a
straightforward avenue for understanding previous similar uniqueness results,
in which unitarity appears as an emergent property, and naively independent
physical principles contain identical (amplitude) information.

##  I Introduction

In the modern on-shell bootstrap program, scattering amplitudes of various QFT
were shown to be constructible via completely new methods, including recursion
relations Britto  _et al._ (2005); Cheung  _et al._ (2015a), Pfaffians Cachazo
_et al._ (2014), geometrical/combinatorial objects like the Amplituhedron,
associahedron, or permutahedron Arkani-Hamed and Trnka (2014); Arkani-Hamed
_et al._ (2018a); Cao  _et al._ (2023), Hopf algebras Brandhuber  _et al._
(2022); Chen  _et al._ (2024), the double copy procedure Bern  _et al._
(2010), transmutation operators Cheung  _et al._ (2018), ansatz based methods
Cheung  _et al._ (2015b, 2017), and most recently from curve counting Arkani-
Hamed  _et al._ (2023a), among many others. These novel perspectives have
revealed surprising new structures, even in the simplest theories and already
at tree level. It is therefore especially shocking that the latest discovery
is perhaps even the simplest one yet: termed hidden zeros, or amplitude
zeroes, these imply that amplitudes vanish for particular kinematic
configurations of the external data, and that behavior near these zeros leads
to a novel factorization type property, called splitting Arkani-Hamed  _et
al._ (2023b) (see also Arkani-Hamed  _et al._ (2024); Arkani-Hamed and
Figueiredo (2024a); Cao  _et al._ (2024a); Arkani-Hamed and Figueiredo
(2024b); Cao  _et al._ (2024b); De  _et al._ (2024), as well as D’Adda  _et
al._ (1971) for very early work on zeros in string theory). Furthermore, it
was conjectured that this property is sufficient to uniquely determine
amplitudes in Tr$(\phi^{3})$ and non-linear sigma model (NLSM) theories.

This latter observation ties in with a different set of uniqueness results
which showed that scattering amplitudes in scalar, gauge, and gravity theories
are fully determined by various sets of principles (ranging from gauge
invariance to IR or UV scaling), with unitarity (and in some cases locality)
an emergent property Arkani-Hamed  _et al._ (2018b); Carrasco and Rodina
(2019); Rodina (2019a, b, c, 2020). A striking corollary of the uniqueness
theorems is that apparently disjoint information is in fact somehow
equivalent: for instance, both the infrared (IR) and ultraviolet (UV) regimes
seem to contain sufficient information to independently fix a wide range of
amplitudes. It is highly surprising there is still room for yet new defining
properties that lead to uniqueness in this already very crowded space. NLSM
amplitudes can now be uniquely defined in no less than five ways: via Adler
zero, double soft theorems, UV scaling, Bern-Carrasco-Johansson (BCJ)
relations, and now amplitude zeros - and in all cases unitarity is an emergent
property! This raises the obvious question whether some of these principles
are secretly equivalent to each other. In this spirit, it was already shown
that for instance amplitude zero implies the Adler zero, and that BCJ
relations imply both Adler zero Carrasco and Rodina (2019); Brown  _et al._
(2023) and amplitude zero Bartsch  _et al._ (2024); Li  _et al._ (2024).

Besides proving the uniqueness conjecture from amplitude zeros for
Tr$(\phi^{3})$, in this Letter we will also prove an even closer connection
with the UV scaling under Britto-Cachazo-Feng-Witten shifts Britto  _et al._
(2005). The existence of enhanced BCFW (or UV) scalings was initially only a
technical requirement for the unitarity-based BCFW recursion in Yang-Mills and
gravity, which revolutionized the computation of scattering amplitudes. Later,
it was shown that scalar EFT also enjoy an enhanced scaling, and that simply
demanding such improved scaling is sufficient to determine amplitudes, even
without using the recursion or assuming unitarity Rodina (2019a); Carrasco and
Rodina (2019).

A surprising result we find is that the hidden zeros further reveal yet
another previously unnoticed property: in scalar theories, all amplitudes
contain further ”secret” UV enhancements, whereby each subset defined above
also enjoys an enhanced UV scaling, potentially beyond the leading behavior of
the amplitude. This secret enhanced scaling of each subset turns out to be
directly equivalent to the amplitude zero acting on the same subsets. The
novel splitting behavior akin to factorization makes this close connection
between BCFW scaling and hidden zeros even more intriguing, hinting towards a
completely new way to construct amplitudes, not based on unitarity and the
usual spacetime approaches.

Moving on to even more complicated theories which contain vector particles,
the zeros are no longer sufficient to uniquely constrain amplitudes. However,
we make another non-trivial observation that, coupled with the BCJ color-
kinematic duality Bern  _et al._ (2008), which enforces linear relations
between amplitudes of different external ordering, the zeros do uniquely
determine all different tensor structures, as classified by the number of dot
products of the type $e{\cdot}e$ and $e{\cdot}p$ (this amounts to just two
independent solutions at 4- and 5-points, three at 6- and 7-points, and so
on). Although we do not provide a proof, we have checked this result up to
6-point, where the constraints uniquely fix ansatze containing on the order of
$10^{5}$ terms.

In summary we prove the following, first for Tr$(\phi^{3})$, assuming locality

  * •

Any amplitude zero is equivalent to a ”subset zero”, where the sum of diagrams
in particular subsets satisfies the zero

  * •

This separation into subsets can be used to directly prove uniqueness when
imposing several amplitude zeros

  * •

Any amplitude zero is equivalent to an improved UV scaling of all subsets
under a corresponding BCFW shift

Additionally, we show that when dropping locality, or when considering NLSM
amplitudes, only a particular amplitude zero is equivalent to subset enhanced
scaling. Finally, for YM we conjecture that zero plus BCJ relations uniquely
fixes the independent tensor structures.

##  II Review of amplitude zeros

In Arkani-Hamed  _et al._ (2023b), it was proposed that partial (or ordered)
amplitudes for $\mathrm{Tr}(\phi^{3})$, NLSM, and YM vanish when a particular
subset of kinematic invariants $s_{123\ldots}{=}(p_{1}{+}p_{2}{+}\ldots)^{2}$
are set to zero, where $p_{i}$ the $D$-dimensional momenta of external
particle $i$. For example, the ordered 4-point $\mathrm{Tr}(\phi^{3})$
amplitude is simply

$A_{4}(1234)=\frac{1}{s_{12}}-\frac{1}{s_{12}+s_{13}}\,,$ (1)

and clearly vanishes when $s_{13}{=}0$. Similarly, though less obvious, the
5-point amplitude

$A_{5}(12345)=\frac{1}{s_{12}s_{34}}+\textrm{cyclic}\,,$ (2)

vanishes when $s_{13}{=}s_{14}{=}0$, or for cyclic permutations of these
conditions. Starting at 6-point, a new pattern appears as a possibility:
setting $s_{14}{=}s_{15}{=}s_{24}{=}s_{25}{=}0$ also makes the amplitude
vanish. These zeroes generalize to arbitrary number of points, and identical
facts hold for NLSM, which unlike Tr$(\phi^{3})$ also contains non-trivial
numerators, as well as for YM after suitably including polarization vectors.

Reference Arkani-Hamed  _et al._ (2023b) further conjectured that
Tr$(\phi^{3})$ amplitudes are in fact uniquely fixed by demanding this
property in a sufficient number of legs. Consider again the simple 4-point
example from above, but now let the two terms have arbitrary coefficients

$B_{4}(1234)=\frac{x_{1}}{s_{12}}-\frac{x_{2}}{s_{12}+s_{13}}\,.$ (3)

This is the most general local and planar function we could write down at
4-point, and we refer to it as a local ansatz. Here we take local and planar
to mean that any singularities must be compatible with ordered trivalent
graphs, with all poles sums of consecutive momenta squared.

Now we impose the zero condition $A|_{s_{13}=0}{=}0$, which requires
$x_{2}{=}x_{1}$, leading back to the actual amplitude in eq.(1), up to some
overall rescaling. The same process can be carried out at 5-point, where now
there would be five arbitrary coefficients. Imposing two different zero
conditions, say $s_{13}{=}s_{14}{=}0$, and $s_{24}{=}s_{25}{=}0$ would again
uniquely fix the ansatz to match the actual amplitude. In general, one needs
to impose $n{-}3$ zero conditions of this type to fix an $n$-point amplitude.
This was the observation made in Arkani-Hamed  _et al._ (2023b) which we will
prove.

##  III Subset zeros

The key idea is that amplitude zeros (and as we will see later, also UV
scalings) operate on a level of particular and easy to identify subsets of
diagrams. Let us focus on the simplest case of a skinny zero, $s_{1i}{=}0$,
for $i$ non-adjacent to 1.

To generate the relevant subsets at $n$-point, start with an $(n{-}1)$-point
diagram, with leg 1 missing. Then the subset corresponding to this
$(n{-}1)$-point diagram is formed by adding leg 1 in all possible ways (that
respect ordering and trivalent interactions). This is shown in Figure 1. All
$n$-point diagrams can be uniquely obtained via this procedure by starting
from all $(n{-}1)$-point diagrams. For instance, some of the 5- or 6-point
diagrams are shown in Figure 2.

Figure 1: Obtaining higher point diagrams from lower point. Leg 1 can be
attached in any position marked by the red lines. Figure 2: Generating some of
the subsets at 5- and 6-point from 4 and 5 point respectively, by attaching
leg 1 in all possible ways.

For example, the 5-point amplitude splits into two subsets, starting from the
two 4-point diagrams:

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{x_{1}}{s_{12}s_{123}}+\frac{x_{2}}{s_{23}s_{123}}+\frac{x_{3}}{s_{23}s_{234}}\,,$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{x_{4}}{s_{12}s_{34}}+\frac{x_{5}}{s_{34}s_{234}}\,,$ (4)

with the correct amplitude corresponding to a specific choice for $x_{i}$ (in
this case $x_{i}{=}x$). With such subsets in mind, there are four main claims
to prove for Tr$(\phi^{3})$, for a zero of type $s_{1i}{=}0$:

  1. 1.

Amplitude zero $\Leftrightarrow$ ”subset zero” (each subset independently
satisfies the zero).

Example: $A_{5}|_{\textrm{zero}}{=}0\Leftrightarrow
S_{1}|_{\textrm{zero}}{=}S_{2}|_{\textrm{zero}}{=}0$.

  2. 2.

A subset zero uniquely fixes the coefficients of all diagrams in the subset.

Example: $S_{1}|_{\textrm{zero}}{=}0\Leftrightarrow x_{1}{=}x_{2}{=}x_{3}$,
$S_{2}|_{\textrm{zero}}\Leftrightarrow x_{4}{=}x_{5}$.

  3. 3.

A sufficient number of amplitude zeros uniquely fix the amplitude.

Example:
$A_{5}|_{\textrm{zero}_{1}}{=}A_{5}|_{\textrm{zero}_{2}}{=}0\Leftrightarrow
x_{i}{=}x$, where zero1 is $p_{1i}{=}0$ and zero2 is e.g. $p_{2i}{=}0$.

  4. 4.

A single amplitude zero is equivalent to enhanced UV scaling of each subset
under a particular BCFW shift.

Example: $A_{5}|_{\textrm{zero}}{=}0\Leftrightarrow
S_{1}{\sim}\frac{1}{z^{3}}$ and $S_{2}{\sim}\frac{1}{z^{4}}$ under a rescaling
$p_{2}{\rightarrow}zp_{2}$, if the zero is $p_{1i}{=}0$.

For other zero types equivalent statements hold, as we explain later.

###  III.1 Subsets must vanish independently

We will prove the first two claims by considering a series of cuts. As shown
in Figure 1, a seed $(n{-}1)$-point diagram, which we label by $(m)$, has a
general form

$D_{\sigma_{1}}D_{\sigma_{2}}\ldots
D_{\sigma_{k}}\times\frac{1}{P_{2\sigma_{1}}P_{2\sigma_{1}\sigma_{2}}\ldots}\equiv\mathcal{D}_{m}\frac{1}{\mathcal{P}_{m}}\,,$
(5)

where $D_{\sigma_{i}}$ is some tree level sub-diagram with external legs
$\sigma_{i}{=}\\{i_{1},i_{2},\ldots\\}$, and we take it to include the
connecting propagator. The product of all such sub-diagrams, $\mathcal{D}$,
can be treated as a prefactor, as all diagrams in a particular subset share
it. The other propagators $P$, which are necessarily functions of $p_{2}$,
live on the boundary of the region labelled by 1, and we call them boundary
propagators. Leg 1 can then be attached either directly to legs 2 or $n$, or
to one of the boundary propagators. Since these propagators carry the relevant
dependence on $p_{1}$, cancellations between them are what leads to the zero
property.

First we wish to prove that imposing a single amplitude zero condition implies
that each subset vanishes independently under that zero. Then we will show
this further implies all diagrams in a given subset have equal coefficient.

The idea is to use cuts to isolate different subsets, starting with the
subsets with a maximal number of propagators in $\mathcal{D}$ (and no boundary
propagators), and moving down to the subset containing only boundary
propagators.

Let us exemplify the steps on a 6-point subset. The $n$-point proof requires
only trivial generalizations, which we leave to Appendix A. Since at 5-point
there are five diagrams, there will be five subsets at 6-point, which we label
$S_{i}$. There are two subsets, call them $S_{1}$ and $S_{2}$, with no
boundary propagators, consider the first one

$S_{1}=\frac{x_{1}}{s_{12}s_{34}s_{345}}+\frac{x_{2}}{s_{2345}s_{34}s_{345}}\,.$
(6)

We impose the zero condition $s_{13}{=}s_{14}{=}s_{15}{=}0$ on the amplitude,
requiring $\sum S_{i}{=}0$, and then cut the two propagators
$s_{34}{=}s_{345}{=}0$, leading to

$\left.\left.\left(\frac{x_{1}}{s_{12}}+\frac{x_{2}}{s_{2345}}\right)\right|_{\textrm{zero}}\right|_{s_{34}{=}s_{345}{=}0}=0\,,$
(7)

since only the above diagrams survive the cut. The above holds only for
$x_{1}{=}x_{2}$, but crucially, the same holds even away from the cut, as in
both cases $s_{2345}|_{\textrm{zero}}{=}{-}s_{12}$. This simple fact that
different boundary propagators become linearly dependent under the zero, both
on and away from non-boundary cuts, is the crucial piece of the proof. We will
encounter another example shortly, and a general proof is written in Appendix
A.

We have therefore shown that the subset $S_{1}$ vanishes independently on the
zero, and an identical argument shows the same for the second subset $S_{2}$,
which can be isolated by cutting $s_{45}{=}s_{345}{=}0$. This finally gives us
access to subsets with one boundary propagator, for example the subset
obtained from the middle diagram in Figure 2, containing three diagrams

$S_{3}=\frac{y_{1}}{s_{12}s_{34}s_{1234}}+\frac{y_{2}}{s_{34}s_{234}s_{1234}}+\frac{y_{3}}{s_{34}s_{234}s_{2345}}\,.$
(8)

As before, we first impose the zero condition $s_{13}{=}s_{14}{=}s_{15}{=}0$,
which implies $\sum_{i=3}^{5}S_{i}|_{\textrm{zero}}=0$, since we have already
shown that $S_{1}{=}S_{2}{=}0$ under the zero. Now we cut the propagator
$s_{34}{=}0$, appearing only in $S_{3}$, and we obtain

$\left.\left.S_{3}\right|_{\textrm{zero}}\right|_{s_{34}=0}=0\,.$ (9)

The above can only hold if the zero+cut create linear relations between the
previously independent boundary propagators $s_{12}$, $s_{234}$, $s_{1234}$,
and $s_{2345}$ that appear in the subset in eq.(8). Indeed at 6-point the
zero+cut induce three identities $L_{i}{=}0$ between boundary propagators

$\displaystyle L_{1}=s_{123}-s_{23}-s_{12},\quad
L_{2}=s_{1234}-s_{234}-s_{12}\,,$ $\displaystyle L_{3}=s_{2345}-s_{12}\,,$
(10)

which follow from a general pattern

$L_{j-2}=s_{12\ldots j}-s_{2\ldots j}+s_{12},\quad j=3,\ldots,n-1\,.$ (11)

Then eq.(9) implies the subset must be expressible as

$S_{3}=L_{1}C_{1}+L_{2}C_{2}+L_{3}C_{3}\,,$ (12)

where $C_{i}$ are functions of mandelstams whose exact form is irrelevant.
This form ensures $S_{3}$ vanishes on the locus of zero+cut. But crucially,
the identities in eq.(III.1) clearly hold even away from the cut, so eq.(12)
implies

$\left.S_{3}\right|_{\textrm{zero}}=0\,.$ (13)

This can be repeated for all subsets, and we have therefore proven that

1\. Amplitude zero $\Leftrightarrow$ subset zero

###  III.2 Subsets are uniquely fixed

The above argument directly proves that all two-element subsets are uniquely
fixed, and we must show the same holds for the remaining subsets. This is
achieved by isolating two neighbor diagrams within such subsets (neighbor
diagrams differ by at most one propagator) with further cuts. Consider the
last two diagrams in the set $S_{3}$, and cut their common propagator
$s_{234}$, implying

$\left.\left.S_{3}\right|_{\textrm{zero}}\right|_{s_{234}=0}=\frac{y_{2}}{s_{1234}}+\frac{y_{3}}{s_{2345}}=0\,.$
(14)

We can write the two propagators as

$\displaystyle s_{1234}$ $\displaystyle=$ $\displaystyle
s_{12}+s_{13}+s_{14}+s_{234}\,,$ $\displaystyle s_{2345}$ $\displaystyle=$
$\displaystyle-(s_{12}+s_{13}+s_{14}+s_{15})\,,$ (15)

so, under the zero condition and the additional cut we find
$s_{1234}{=}{-}s_{2345}{=}s_{12}$, and eq.(14) can only vanish if
$y_{2}{=}y_{3}$. Repeating this for the first pair of diagrams we obtain
$y_{1}{=}y_{2}$, proving that all terms in this subset have equal
coefficients. The whole process can be carried out for each subset, so we
conclude that

2\. Amplitude zero uniquely fixes all subsets

###  III.3 Amplitude uniqueness

We can illustrate full amplitude uniqueness by considering a graph whose
vertices are represented by the diagrams. Let two diagrams be connected in the
graph if there exists a zero condition which places them in the same subset,
implying the two ”adjacent” diagrams have equal coefficient. If we can prove
the graph of all diagrams is connected, this implies any two diagrams have
equal coefficient, i.e.. the amplitude is fixed up to an overall number. This
is obviously true, as any two diagrams can be related by a (finite) series of
steps that remove one leg, and then re-attach it somewhere else. This proves
that

3\. Amplitude zeros uniquely fix the Tr$(\phi^{3})$ amplitude

An interesting question is what is the minimal number of such steps - this
would provide the minimal number of amplitude zeros needed to fully fix the
amplitude. Experimentally, this number is $n{-}3$ and can easily be proven by
induction: consider two distinct $n$-point diagrams. If they are not already
part of the same subset for some zero, remove a leg, repeating this until they
become diagrams part of some subset. This can go on at most until reaching the
3-point diagram, so no two diagrams may be more than $n{-}3$ steps away from
each other.

###  III.4 Generalizations to $n$-point and other zeros

The above results at 6-point generalize trivially to arbitrary number of
points. We leave the details to Appendix A. Finally, all the above results can
be repeated for other types of zero, not just the ”skinny” zero $s_{1i}{=}0$,
$i{\neq}1,2,n$. For instance, for a zero $s_{1i}{=}s_{2i}{=}0$,
$i{\neq}1,2,3,n$ the relevant subsets are obtained in a similar way, as all
possible ways to add both legs 1 and 2 to an $n{-}2$ point diagram. Note that
adding the two legs can be done in two different ways - either as separate
legs, or as a two-particle pole $1/s_{12}$. So, for instance, by starting with
the 3-point vertex, one will obtain the five 5-point diagrams as two disjoint
subsets.

Finally, we comment that this approach not only proves uniqueness, but also
existence, and naturally suggests how to build $n$-point amplitudes (by adding
$k$ legs to the $n{-}k$ amplitude), unlike the other uniqueness theorems,
which do not have an associated amplitude construction. In this sense we find
the current proof much more powerful than previous results, which will likely
benefit from adapting this novel subset/cut-based approach. This possibility
is further strengthened by the close relation of amplitude zeros to UV
scaling, to which we turn next.

##  IV (Secret) Subset UV scaling

For Tr$(\phi^{3})$, it was found that demanding a BCFW, or UV, scaling of
$1/z^{3}$ ($1/z^{2}$ at 4-points) in $n{-}3$ legs is sufficient to uniquely
determine the amplitude, with similar statements for other scalar theories,
such as NLSM Carrasco and Rodina (2019). The novel property we discover is
that beyond these enhanced amplitude-level UV scalings, there exists a further
hidden enhancement, for each individual subset defined earlier. Furthermore,
this extra enhancement is completely equivalent to the amplitude zero.

For scalars, the BCFW shift is given by deforming
$p_{i}{\rightarrow}p_{i}{+}zq$ and $p_{j}{\rightarrow}p_{j}{-}zq$, with $q$ a
null vector that satisfies $p_{i,j}{\cdot}q{=}0$ and taking the limit
$z{\rightarrow}\infty$. However, for the purposes of determining enhancements,
one can use a much simpler rescaling $p_{i}{\rightarrow}zp_{i}$, instead of
the full shift, if $p_{j}$ is eliminated via momentum conservation, as
explained in Carrasco and Rodina (2019). We will call this the UV rescaling.

First, let us explain which zero and which UV scaling are matched. Each type
of zero has a natural corresponding scaling, or shift. For instance, for a
skinny zero like $s_{1i}{=}0$, $i{\neq}2,n$, the shift is on $p_{2}$ and
$p_{n}$, precisely the legs not involved in this zero. Since $p_{n}$ is always
eliminated, we can simply use the rescaling of $p_{2}{\rightarrow}zp_{2}$.
Similarly, for a zero $s_{1i}{=}p_{2i}{=}0$, $i{\neq}3,n$, the corresponding
rescaling is in $p_{3}$, and so on. Finally, it is important to note that the
equivalence we claim is not a trivial kinematic feature: the UV and zero can
have different effects for the kinematic invariants. Take for instance
$s_{123}$. Under each condition, we get

$\displaystyle s_{1234}$ $\displaystyle\xrightarrow{\text{zero}}$
$\displaystyle s_{1234}^{(0)}=s_{12}+s_{23}+s_{24}+s_{34}\,,$ $\displaystyle
s_{1234}$ $\displaystyle\xrightarrow{\text{UV}}$ $\displaystyle
s_{1234}^{(\infty)}=z(s_{12}+s_{23}+s_{24})+\mathcal{O}(z^{0})\,.$ (16)

Demanding an amplitude zero obviously amounts to requiring $A(s^{(0)}){=}0$.
Meanwhile, for the UV scaling, taking the $z{\rightarrow}\infty$ limit we
obtain a general expansion

$A(s(z))=z^{k}A(s^{(\infty)})+\mathcal{O}(z^{k-1})\,,$ (17)

so, if require an enhancement in the scaling we must impose
$A(s^{(\infty)}){=}0$. The zero condition implies the UV scaling on a purely
kinematic level, since all $s^{(\infty)}$ can be obtained from $s^{(0)}$ by
setting some extra invariants to zero. However, for Tr$(\phi^{3})$ the
converse is also true for any zero types, as a consequence of locality.

First, it easy to show that all diagrams in a given subset have the same UV
scaling. The scaling is given by the number of boundary propagators, plus
optionally the number of poles in sub-diagrams $\mathcal{D}$ that contain both
the labels $n{-}2$ and $n{-}1$ (since momentum conservation gives such terms
explicit dependence on $p_{2}$). Permuting leg 1 does not change the number of
such propagators, so all diagrams in the subset have the same scaling. If this
scaling is $\mathcal{O}(1/z^{k})$, then we will show that if and only if the
subset satisfies the zero, the subset in fact scales as
$\mathcal{O}(1/z^{k+1})$. For instance, the sum of diagrams in subset $S_{3}$,
given in eq.(8), scales as $1/z^{3}$ even though each diagram scales as
$1/z^{2}$ under $p_{2}\rightarrow zp_{2}$.

This can be proven using the same arguments as in previous section. We start
again from the subset $S_{3}$ as an example, which on the locus of zero must
vanish, and must be expressible as

$S_{3}=L_{1}C_{1}+L_{2}C_{2}+L_{3}C_{3}\,,$ (18)

where $L_{i}$ are the linear relations induced by the zero condition given in
eq.(III.1), i.e.. the linear relations satisfied by the $s^{(0)}$. We can
immediately prove that the boundary $s^{(\infty)}$ satisfy the same linear
relations. Boundary propagators are of three types, and under the UV limit
they become

$\displaystyle s_{12\ldots j}^{(\infty)}$ $\displaystyle\rightarrow$
$\displaystyle s_{12}+\sum_{i=3}^{j}s_{2i},\quad j\leq n-2\,,$ $\displaystyle
s_{2\ldots j}^{(\infty)}$ $\displaystyle\rightarrow$
$\displaystyle\sum_{i=3}^{j}s_{2i},\quad j\leq n-2$ $\displaystyle s_{2\ldots
n-1}^{(\infty)}$ $\displaystyle\rightarrow$ $\displaystyle-s_{12}\,.$ (19)

It is obvious that the only linear relations induced by UV scaling are

$\displaystyle L_{j}=s_{12\ldots j}-s_{2\ldots j}+s_{12},\quad
j=3,\ldots,n-1\,,$ (20)

so, we see that UV scaling and amplitude zero lead to identical linear
relations between the basis elements (for boundary propagators). From eq.(18)
this implies

$\left.S_{3}\right|_{\textrm{UV}}=0\,,$ (21)

so, subsets indeed have an enhanced scaling. Furthermore, since both the zero
condition and UV scaling require the subset to be expressible in the form of
eq.(18), this implies they lead to identical constraints. We have therefore
proven that for Tr$(\phi^{3})$

4\. amplitude zero $\Leftrightarrow$ subset-enhanced UV scaling

##  V Non-locality and non-linear sigma model

The previous arguments fail in the case of NLSM, or when dropping the locality
assumption for Tr$(\phi^{3})$ (which means singularities are no longer
associated to graphs). The reason is that for general non-boundary kinematic
invariants, the $s^{(\infty)}$ satisfy more linear constraints than the
$s^{(0)}$. However, if we restrict to functions built purely out of planar
variables (which does not restrict the space of functions, as the planar basis
is complete Arkani-Hamed  _et al._ (2023a)), there are still two claims we can
prove.

First, it is still true that amplitude zero implies subset enhanced scaling.
This is true because any linear relation satisfied by $s^{(0)}$ is also
satisfied by $s^{(\infty)}$, as mentioned above. Therefore, if some function
satisfies a zero, its UV scaling must be enhanced beyond that of individual
terms in the function.

Second, for a particular zero, that we call the ”fattest zero”, the one which
maximizes the number of kinematic invariants set to zero, complete equivalence
still holds between the zero and enhanced subset UV scaling. The reason is
that only for this zero, the planar $s^{(0)}$ and $s^{(\infty)}$ satisfy
identical linear constraints. We conclude that for any rational function built
from planar variables (including ansatze of non-local Tr$(\phi^{3})$ or non-
local NLSM)

Fattest zero $\Leftrightarrow$ subset-enhanced UV scaling

which more concretely means that if in a given function any individual term
has a UV scaling of $\mathcal{O}(z^{k})$, it must combine with other terms
such that together they scale as $\mathcal{O}(z^{k-1})$, or the zero cannot be
satisfied. We remark this observation is curiously similar to how permutation
invariance automatically improves any BCFW scaling $z^{\textrm{odd}}$ by one
power to $z^{\textrm{even}}$ McGady and Rodina (2015).

##  VI Beyond scalars

It is natural to ask how constraining the amplitude zero is for YM amplitudes.
In this case, the zero condition needs to set to zero dot products involving
polarization vectors as well. For instance, the simplest zero at 4-point
requires
$p_{1}{\cdot}p_{3}{=}e_{1}{\cdot}p_{3}{=}e_{3}{\cdot}p_{1}{=}e_{1}{\cdot}e_{3}{=}0$.
From the nature of the constraint, it is immediately clear the zero cannot fix
the full amplitude: terms containing different tensor structures, such as
$(e{\cdot}e\,e{\cdot}e\,p{\cdot}p)$ and $(e{\cdot}e\,e{\cdot}p\,e{\cdot}p)$,
cannot be related. One can ask how much of these tensor structures can be
fixed.

Up to 6-point explicit checks, we find that combined with the BCJ color-
kinematic duality requirement (which similarly only acts within these tensor
structures) Bern  _et al._ (2008), the amplitude zero uniquely fixes these
objects! The extra requirement simply demands that amplitudes satisfy the so-
called BCJ relations

$\sum_{i=2}^{n-1}k_{1i}A(2,\ldots,i,1,i+1,\ldots,n){=}0\,,$ (22)

where $k_{1i}{=}\sum_{j=2}^{i}s_{1j}$. This new uniqueness observation is
highly non-trivial and unexpected, as at 6-points the ansatz for each tensor
structure contains around $10^{4}{-}10^{5}$ terms, but remains completely
mysterious.

##  VII Outlook

In this work we explored the newly discovered property of amplitude zeros,
which are part of a radical new picture of amplitudes, where the physical
factorization (a consequence of unitarity) is replaced by splitting, a
property still lacking a physical meaning, but natural from new ”surface
geometry” approach to amplitudes Arkani-Hamed and Figueiredo (2024b). Our
results show the property provides important insight in several directions: it
can used as a new definition of amplitudes (at least in Tr($\phi^{3}$), NLSM,
and partially in YM), reveals the true extent of previous observations (a
secret UV scaling of independent subsets), and also points the way towards a
sharper proof of a long list of uniqueness observations Arkani-Hamed  _et al._
(2018b); Carrasco and Rodina (2019); Rodina (2019a, b, c, 2020), which still
lack a completely satisfactory understanding, that truly explains how
unitarity emerges as a by-product.

The close connection we found between amplitude zeros and BCFW recursion (from
UV scaling to factorization/splitting) strongly suggests a new constructive
approach may exist for general amplitudes, or even some picture in which the
two are completely unified. This relationship can potentially also lead to a
physical reason for the zeros and splitting behavior, or instead may offer an
understanding of the still unexplained improved (and amplitude-defining) UV
scaling of various EFT Carrasco and Rodina (2019).

Finally, the fact that zeros plus color-kinematic duality uniquely fix YM
tensor structures is perhaps the most surprising finding, as it is remarkable
tree level YM can still contain such hidden structures almost in plain sight.
This suggests there may exist a completely new structure centered on these
unique YM building blocks (these objects are also relevant for 4D helicity
amplitudes and were recently considered in Pavao (2023)). Combining the
composition rule Carrasco  _et al._ (2020, 2021) with the amplitude zero
requirement may provide further insight into such objects, using polarization
vectors as building blocks in the composition.

Obtaining similar uniqueness results at loop level, as well as without
assuming locality, starting with Tr$(\phi^{3})$, are other fascinating
possibilities we leave for future work.

Acknowledgments: The author would like to thank Jeffrey Backus, Carolina
Figueiredo, Nima-Arkani Hamed, and Song He for valuable discussions.

##  Appendix A Appendix A: $N$-point proof

In this Appendix we generalize the proof that each amplitude zero uniquely
fixes the different corresponding subsets. Most of the steps are identical and
do not require any further discussion, but we repeat them here nonetheless for
the sake of clarity. First, we write the $n$-point amplitude as a sum of
subsets (as defined in the main text) and impose the zero condition
$s_{1i}{=}0$, $i\neq 2,n$, which gives

$\left.A_{n}\right|_{\textrm{zero}}=\left.\sum
S_{i}\right|_{\textrm{zero}}=\left.\sum\mathcal{D}_{i}\frac{1}{\mathcal{P}_{i}}\right|_{\textrm{zero}}=0\,.$
(23)

Then, we choose a subset $(m)$, which can extract from the above relation by
cutting all (non-boundary) corresponding propagators in $\mathcal{D}_{m}$,
starting with diagrams that have only non-boundary propagators (so they can be
uniquely selected by cuts), and working down from there. It is easy to show
that the zero does not induce extra degeneracies for such propagators. The
fact that the zero only involves non-planar invariants $s_{1i}$, ensures the
propagators remain one-to-one. We obtain

$\left.\left.\mathcal{P}_{m}\right|_{\textrm{zero}}\right|_{\mathcal{D}_{m}-\textrm{cut}}=0\,.$
(24)

We must show the above holds even away from the cut. First, the fact that the
above vanishes implies it must be expressible in a form

$\mathcal{P}_{m}=\sum L_{i}C_{i}\,,$ (25)

where $L_{i}$ are the linear relation induced between the basis elements on
the locus of zero+cut. We must show that the zero alone induces the same
relations as zero+cut. First, all the boundary propagators are given by

$\displaystyle s_{1,2,\ldots,j}$ $\displaystyle=$ $\displaystyle
s_{12}+\sum_{i=2}^{j}s_{1i}+\sum_{i=3}^{j}s_{2i}+s_{3\ldots j},\quad j\leq
n-2\,,$ $\displaystyle s_{2,\ldots,j}$ $\displaystyle=$
$\displaystyle\sum_{i=3}^{j}s_{2i}+s_{3\ldots j},\quad j\leq n-2\,,$
$\displaystyle s_{2,\ldots,n-1}$ $\displaystyle=$ $\displaystyle-
s_{12}-\sum_{i=3}^{n-1}s_{1i}\,.$ (26)

Since the only dependence on the cut non-boundary propagators in $\mathcal{D}$
is through the identical terms in $s_{3\ldots j}$, it is clear one obtains the
same linear relations with or without this cut. These relations are the same
as those given in eq.(20), namely

$\displaystyle L_{j}=s_{12\ldots j}-s_{2\ldots j}+s_{12},\quad
j=3,\ldots,n-1\,.$ (27)

This implies

$\left.\mathcal{P}_{m}\right|_{\textrm{zero}}=0\,,$ (28)

proving that subset $(m)$ vanishes independently on the zero, and this can be
repeated for all subsets.

Next we must prove that subset zero implies any two neighbor diagrams in the
subset have equal coefficient. As before, we consider two diagrams in this
subset which differ by exactly one boundary propagator, as shown in Figure 3.

Figure 3: Two neighbor diagrams differ only by the blue propagator.

These have a form

$\frac{1}{P_{L}P_{R}}\left(\frac{x_{i}}{P_{A}}+\frac{x_{i+1}}{P_{B}}\right)\,.$
(29)

We need to prove $P_{A}=-P_{B}$ on the amplitude zero plus ($P_{L},P_{R}$)-cut
locus. We can write

$\displaystyle P_{A}$ $\displaystyle=$ $\displaystyle
s_{12}+\sum_{k=3}^{i}s_{1k}+P_{L}\,,$ (30) $\displaystyle P_{R}$
$\displaystyle=$ $\displaystyle s_{12}+\sum_{k=3}^{j-1}s_{1k}+P_{B}\,,$ (31)

so, under the zero condition, and on the cut $P_{L}=P_{R}=0$, we find
$P_{A}=-P_{B}$ and conclude the two diagrams cancel if they have equal
coefficients, $x_{i}=x_{i+1}$. This can be repeated for all pairs in the
subset, and then for all subsets, proving that one amplitude zero uniquely
fixes all subsets. The final step is to combine this result with constraints
from distinct zero conditions on other legs, which as discussed in the main
text implies the full amplitude is indeed uniquely fixed after imposing some
sufficient number of zeros.

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