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docs(p28): notation key for invented and borrowed terminology

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Several terms across this programme are borrowed from molecular biology and
genetics (hachimoji, scar/scar-map, strand/braid, carrier, chirality, mutation)
and could be misread as claims about biological systems. They are not. This
document states the intended mathematical meaning of each, names the source
field it collides with, and says explicitly what is not being claimed.

Part 1 covers terms actually used in the 2.8 submission (seed, jet, carrier,
mutation, deformation, compact form, certificate, red case, authority tag),
grounded in solution.tex and the certificate scripts. Part 2 covers the
biology-adjacent vocabulary of adjacent repositories and is RECONSTRUCTED FROM
WORKING NOTES -- the author should confirm each entry before circulation.

Also records the load-bearing EXACT / NUMERICAL WITNESS / LITERATURE
distinction: the main claim rests only on EXACT and LITERATURE items, and
numerical enclosures are never used as premises.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01WY6SfRYvm8zFKMX9GcjS8u
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# Notation, invented terminology, and borrowed words
## Why this document exists
This work crosses several fields, and its vocabulary was assembled accordingly.
Some names are invented; others are borrowed from unrelated disciplines because
they were short and evocative, and then given a purely mathematical meaning.
That borrowing creates a specific hazard. A reader who knows the source field
may read a term as a claim about that field. **Several terms used across this
programme are borrowed from molecular biology and genetics. No claim about
biology, genetics, or any physical or biochemical system is made anywhere in
this work.** Every object named below is defined entirely by its equations, and
those equations are the only content.
If a term looks like it is asserting something about the natural world, it is
not. Read the definition, not the word.
---
## Part 1 — Terms used in this submission
These are grounded in `solution.tex` and the certificate scripts, and are the
ones an evaluator of Problem 2.8 will actually meet.
| Term | Meaning here | Ordinary meaning it may collide with |
|---|---|---|
| **seed**, **seed row** | The two official integer initial-condition rows `A_0`, `A_1` of the recurrence. Nothing grows; "seed" only marks where the iteration starts. | Botany; also "seed" in cryptography/RNG, which is closer but still not this. |
| **jet** | Truncated local expansion data of a function at a point — the standard differential-geometry sense (an equivalence class of functions agreeing to finite order). | Aviation; fluid jets; the physics "jet" of particles. None apply. |
| **carrier** | The vector space or module on which an operator or representation acts — the object that *carries* the structure. Standard representation-theory usage. | **Genetics: a carrier of an allele.** Not intended, not implied. |
| **mutation**, **mutant** | Software *mutation testing*: deliberately corrupting a constant to confirm a passing check is capable of failing. E.g. changing `64R-44` to `64R-43` must produce an `AssertionError`. | **Genetic mutation.** Entirely unrelated. The "mutant" is a modified copy of a Python file. |
| **deformation**, **deformed family** | Replacing the constant `R` by a free variable `r = 1/x`, giving a one-parameter analytic family that specialises to the official matrix at `x_0 = 1/R`. Standard deformation-theory usage. | Materials science. |
| **compact form** | A short generating expression (Pascal rows `b_0..b_3`, the row `C(x)`, constants `A, B, S`) that reproduces the large official integer data exactly. A re-encoding, not an approximation. | "Compact" in topology — unrelated here. |
| **certificate** | An executable script whose assertions, if they all pass, witness a stated mathematical fact. | Cryptographic certificates; academic certificates. |
| **red case**, **negative control** | A deliberately false input included to prove the check can fail. Borrowed from experimental science, and used in the same spirit. | — |
| **authority tag** | A per-claim label recording *how* a statement is known: `EXACT` (integer/rational arithmetic), `NUMERICAL WITNESS` (high-precision identification, not proof), `LITERATURE` (cited, not re-proved). | — |
### On `EXACT` versus `NUMERICAL WITNESS`
This distinction is load-bearing and is not merely stylistic. `EXACT` means the
statement is closed under exact arithmetic with no floating point anywhere.
`NUMERICAL WITNESS` means a quantity was *identified* to some number of digits —
strong evidence, **not** a proof. The submission's main claim rests only on
`EXACT` and `LITERATURE` items; numerical enclosures appear as corroboration and
are never used as premises.
---
## Part 2 — Borrowed words from biology, used in the wider programme
These do **not** appear in the Problem 2.8 submission, but they appear in
adjacent repositories that an evaluator may encounter. They are the terms most
likely to be misread as biological claims.
> **Reconstructed from working notes — the author should confirm each definition
> before this document is circulated.**
| Term | Meaning in this work | Source field | Explicitly NOT a claim about |
|---|---|---|---|
| **hachimoji** | A base-8 encoding/decoding scheme (a codec) used to render algebraic invariants as short strings. The name was taken purely because the source system has eight symbols. | Benner et al.'s *hachimoji DNA*, a synthetic 8-letter genetic alphabet (Japanese 八文字, "eight letters"). | DNA, nucleotides, synthetic biology, base pairing, or any biochemical system. It is an integer-encoding convention and nothing else. |
| **scar** (as in `SCAR-013`) | A registry entry recording a route that has been *proved impossible*, together with its witness and a redirect to a live route. A permanent mark left by a closed-off attempt. | **CRISPR / molecular cloning:** a "scar" is residual sequence left after an editing or ligation step. | Gene editing, CRISPR, recombination, or any laboratory procedure. |
| **scar-map** | A tabulated collection of such exclusion records (e.g. the `Δ₇` scar-map). | as above | as above |
| **strand**, **braid** | Strands of a mathematical *braid group* `B_n` — topological curves in a cylinder, and their linear representations (Burau, Lawrence–Krammer–Bigelow). | DNA strands, double helix. | DNA, chromosomes, or any molecular structure. The braid group is a purely topological/algebraic object. |
| **chirality**, **chiral** | Handedness — the orientation or sign convention of a basis or lookup table. | Chemistry (stereochemistry), particle physics. | Molecular handedness or any chemical claim. |
| **codebook** | An information-theoretic dictionary mapping objects to short canonical labels. | Coding theory (correct source); also espionage. | — |
### The general rule
Where a borrowed word could be read as a claim about a natural system, it is
being used **metaphorically for structure only**. The mathematics is defined by
its equations; the name is a label chosen for brevity, and carries none of the
source field's content, mechanisms, or empirical commitments.
---
## Part 3 — Why the vocabulary is like this, stated plainly
The programme spans modular forms, hypergeometric series, braid representations,
formal verification, and computational infrastructure. Working across that range
produces a lot of objects that need names faster than the literature supplies
them. Borrowing a short, memorable word from an unrelated field is efficient for
a solo author and costly for a reader — the cost only becomes visible when the
work is read by someone fluent in the source field.
This document is the correction. It is not an apology for the naming; it is a
key, so that no reader has to guess whether a biological word implies a
biological claim. It does not.
## Part 4 — Reading rule for evaluators
1. Every mathematical object is defined by displayed equations. If a name and an
equation disagree, the equation governs.
2. No statement in this work is a claim about genetics, molecular biology,
chemistry, or any physical system.
3. Where a term is borrowed, Part 1 or Part 2 gives the intended meaning and the
collision it may cause.
4. The distinction between `EXACT`, `NUMERICAL WITNESS`, and `LITERATURE` is
substantive; see Part 1.