📐 β = 5z has order exactly 93 $\\\\beta^{93} = 1$, while $\\\\beta^{3}$ and $\\\\beta^{31}$ are not 1. Therefore $\\\\beta$ has multiplicative order 93. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📐 Symbol count and surplus A seed of $n$ bytes occupies $m = \\\\lceil 8n/5 \\\\rceil$ symbols, with a surplus of $r = 5m - 8n$ bits, and $0 \\\\le r \\\\le 4$. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 Expanded codeword The expanded codeword of a codex32 string is the five expansion symbols 3, 3, 0, 13, 19 followed by every data symbol after the separator — header, payload and checksum alike. Its length is $5 + (\\\\text{number of data symbols})$ and this is what BIP 93 and its implementations mean when they say the length of a codex32 string. From: codex32 Learn more:

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📐 The five constants are one generator scaled by a basis Let $g$ be the generator, packed as $G_0 = \\\\texttt{0x19dc500ce73fde210}$. Then $G_i$ is $g$ with every coefficient multiplied by $2^i$ in GF(32), and for a departing symbol $t$ the XOR of the selected constants equals $t \\\\cdot g$. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 Master seed format Generic codex32, plus the requirement that the payload be one of the six application lengths: 26, 32, 39, 45, 52 or 103 symbols — corresponding to master seeds of 16, 20, 24, 28, 32 or 64 bytes. From: codex32 Learn more:

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📖 The de Bruijn criterion A proof assistant satisfies the de Bruijn criterion if all proofs it accepts are ultimately certified by a small, fixed kernel whose only job is to check terms. All the convenient machinery — automation, tactics, decision procedures — must produce a term the kernel independently accepts, and is therefore not trusted itself. From: codex32 Learn more:

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📖 Polynomial over GF(32) A polynomial over GF(32) is a finite expression $f(X) = a_0 + a_1 X + a_2 X^2 + \\\\cdots + a_d X^d$ in which every coefficient $a_i$ is an element of GF(32). If $a_d \\\\ne 0$, the degree of $f$ is $d$. The zero polynomial has no degree. From: codex32 Learn more:

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📐 Eight-symbol detection Any two equal-length valid codex32 strings within a checksum period that differ in at most eight symbols are in fact the same string. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📐 ℤp is a field when p is prime For $p$ prime, $\\\\mathbb{Z}_p$ is a field with $p$ elements. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📐 A field has no zero divisors If $ab = 0$ in a field and $a \\\\ne 0$, then $b = 0$. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 Derivation Computing the share at an index not currently held, by interpolating the held shares at that index — column by column, exactly as in Part 3. From: codex32 Learn more:

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📐 A one-symbol one-time pad over GF(32) has perfect secrecy With a uniformly random key used once, every ciphertext is equally likely under every message, so observing the ciphertext leaves the attacker\ Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📐 Every order divides 1023 The order of any nonzero element of GF(1024) divides $1023 = 3 \\\\cdot 11 \\\\cdot 31$, so it is one of 1, 3, 11, 31, 33, 93, 341, 1023. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📐 The conjugation map The map $t \\\\mapsto t^{32}$ is a field automorphism of GF(1024). In coordinates it is $(a + bz) \\\\mapsto (a+b) + bz$. It fixes exactly the elements of GF(32), it swaps the two roots of $x^2 + x + 1$, and applying it twice returns the original element. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📐 Any single transcription error is detected If a valid codex32 string is transcribed with one wrong symbol, the result fails verification. The wallet that would have been silently wrong is loudly wrong instead. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 The bech32 alphabet The 32 characters, in field-value order: qpzry9x8gf2tvdw0s3jn54khce6mua7l The character at position $i$ in that string represents the field element $i$, counting from zero. From: codex32 Learn more:

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📐 Creation always produces a string that verifies Any string assembled by appending the checksum that the creation procedure produces passes verification. In the formalization this is Checksum.verify_create; the companion result encoded_seed_checksum_valid carries it through the encoder, proving that every serialized master seed has a valid checksum. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 Weight The weight of a polynomial is the number of its nonzero coefficients. A polynomial of weight at most 8 is one that differs from zero in at most eight places, however spread out. From: codex32 Learn more:

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📐 The long-only verifier is strictly more permissive There are strings that Checksum.verifyLong accepts and that Checksum.verify rejects. The BIP gives one explicitly. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 Parity check A parity check on strings of length $n$ is a linear equation $h_1 u_1 + h_2 u_2 + \\\\cdots + h_n u_n = 0$ with fixed coefficients $h_i$ in GF(32). A code defined as the set of strings satisfying a fixed collection of parity checks is automatically linear, since the solution set of a system of homogeneous linear equations is a subspace. From: codex32 Learn more:

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