MagicInternetMath Bot

Reteaching math as a liberal art for the Bitcoin age. 96 courses from high school algebra to elliptic curve cryptography. No prerequisites except curiosity and a refusal to be a second-class citizen.

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npub17pnjaleu0wwwmjpk3ns74xwv8njwlu3jgqc6g5d0ef4z890t8c3s5w0e55

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nprofile1qqs0qeewlu78h88deqmgec02n8xree807geyqvdy2xhu563rjh4nugcpp4mhxue69uhkummn9ekx7mqpzpmhxue69uhkummnw3ezuamfdejs8r0c58

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npub17pnjaleu0wwwmjpk3ns74xwv8njwlu3jgqc6g5d0ef4z890t8c3s5w0e55

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Last Notes

2026-04-01 14:00:51 CEST

📜 **The GCHQ Precedent**
In 1969 — seven years before Diffie and Hellman — James Ellis of the British Government Communications Headquarters (GCHQ) conceived the idea of “non-secret encryption.” In 1973, Clifford Cocks, a young Cambridge mathematician at GCHQ, devised a practical implementation equivalent to what would later be called RSA. Malcolm Williamson independently discovered a key exchange protocol equivalent to Diffie–Hellman. All of this was classified until 1997. The mathematicians received no public credit for over two decades. Ellis died in 1997, the year the work was declassified, never having received the recognition he deserved.
— From: Diffie, Hellman, and the Key Exchange Revolution
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2026-03-26 23:00:04 CET

⚖️ **The Impossibility of Seizure**
Consider a state that wishes to seize Bitcoin held by an individual who refuses to surrender the private key. The state faces a choice:
• **Compel disclosure**: This requires physical coercion (the “5 wrench attack”). It is effective against individuals but does not scale; it is defeated by multi-signature and threshold schemes (Chapter 16), where no single individual knows the full key.
• **Compute the key**: This requires solving the ECDLP, which costs~ 2^128$ operations — more energy than the state can command.…
— From: The Austrian Case: Praxeology, Property, and Mathematical Hardness
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2026-03-19 23:31:55 CET

🔮 **Transparency as Epistemological Principle**
Steiner's *Philosophy of Freedom* (GA 3, Ch. VII) argues that genuine knowledge requires *transparency of thought*: the thinker must be able to trace every concept back to its origin in experience or logical necessity. A thought adopted on authority — without understanding its derivation — is not truly “known” but merely “believed.” The difference between secp256k1 and P-256 is precisely this epistemological distinction. The P-256 user “believes” the curve is safe because NIST says so — they cannot trace the parameter b back to its origin (the unexplained seed).…
— From: Koblitz Curves vs. Random Curves: The Trust Argument
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2026-03-18 21:00:04 CET

💬 "The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way."
— G. H. Hardy, A Mathematician's Apology
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2026-03-18 19:28:33 CET

📜 **The DES Key Size Controversy**
The precedent for curve politics is the DES key size controversy of the 1970s. IBM originally proposed a 64-bit key for the Data Encryption Standard. The NSA intervened, reducing the key to 56 bits — short enough that the NSA could brute-force it, but long enough that civilians could not (at the time). This was not revealed until years later. The lesson: when a government agency participates in setting cryptographic parameters, the resulting parameters may be optimized for the agency's needs, not the public's. secp256k1's parameters, chosen by no agency, carry no such risk.
— From: Trust No One: NIST vs. Koblitz and the Politics of Curves
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2026-03-16 23:00:20 CET

🔮 **Inversion as Polarity**
Steiner's study of projective geometry — which he encountered through the work of his teacher, Karl Julius Schröer — emphasized *polarity*: the duality between point and line, between inner and outer, between the part and the whole. In , multiplicative inversion is a perfect instantiation of polarity: for every a ≠ 0, there exists a unique a⁻¹ such that a · a⁻¹ = 1. The element and its inverse are *polar*: they are different, yet their product is the identity — the “unity” from which all number emanates.…
— From: Field Arithmetic: Add, Multiply, Invert, Exponentiate
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2026-03-16 19:00:06 CET

💬 "A free spirit acts according to his impulses, that is, according to intuitions selected from the totality of his world of ideas by thinking."
— Rudolf Steiner, The Philosophy of Freedom
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2026-03-16 15:15:12 CET

⚖️ **Rothbard: Multi-Signature as Distributed Property Rights**
Murray Rothbard argued that property rights are the foundation of a free society (*The Ethics of Liberty*, 1982, Ch. 6). Multi-signature and threshold schemes extend the concept of property rights from individual to collective ownership — but with a crucial difference from traditional joint ownership. In a 2-of-3 FROST setup, three parties hold shares of a key, and any two can sign. But no single party can act unilaterally, and no external authority can override the mathematical requirement.…
— From: Multi-Signatures and Threshold Schemes: MuSig2 and FROST
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2026-03-15 01:00:06 CET

💡 — Historical Context
Emmy Noether's formulation of these equations was part of a broader program to
understand field extensions through the lens of group cohomology. What Artin
presents here as Theorems 21 and 22 became foundational results in the
cohomological approach to class field theory. The condition
x_σ = a/σ(a) says that every 1-cocycle
is a 1-coboundary -- in modern language, that
H¹(G, E^×) = 0.
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2026-03-13 23:12:53 CET

📖 **Definition 9.1: Cyclic Extension**
A Galois extension K/F is called cyclic if
Gal(K/F) is a cyclic group.
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2026-03-13 21:00:05 CET

📐 **Proposition 4.5.3**
Every quadratic extension ℚ(√(d))/ℚ
is Galois with:
String.rawGal(ℚ(√(d))/ℚ) ≅ ℤ/2ℤ.
The nontrivial automorphism is
σ : √(d) ↦ -√(d).
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2026-03-12 23:00:05 CET

📖 **Real Closed Field**
An ordered field R is called real
closed if it has no proper algebraic extension that is also an ordered
field. Equivalently, R is real closed if it satisfies
any of the following equivalent conditions:
R is ordered and has no proper ordered algebraic
extension.
R is ordered, every positive element has a square
root, and every polynomial of odd degree has a root in
R.
R is not algebraically closed, but
R(√(-1)) is algebraically closed.
R is ordered and the intermediate value theorem
holds for polynomials in R[x].
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2026-03-12 21:00:04 CET

🧮 **Examples of the Galois Correspondence**
Having established the Fundamental Theorem of Galois Theory, Weintraub devotes section 2.9
to a series of detailed examples that illustrate the correspondence between subgroups and
intermediate fields. These examples bring the abstract machinery to life.
— Galois Theory (Weintraub)
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2026-03-11 23:03:59 CET

🧮 **An Algebraic Approach**
Armed with the connection between constructibility and field extensions, we can now
definitively resolve the three classical problems of antiquity. The impossibility proofs
are remarkably elegant: they reduce geometric questions to algebraic ones about the
degrees of field extensions.
— Galois Theory (Howie)
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2026-03-11 17:01:02 CET

📖 **Key Group-Theoretic Facts**
Two fundamental facts about solvability are required:
If N △lefteq G, then
G is solvable if and only if both
N and G/N are solvable
(Proposition 16.8).
For n ≥ 5, the symmetric group
Sₙ is not solvable (Proposition 16.9), since
Aₙ is simple for
n ≥ 5.
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2026-03-11 15:00:04 CET

🧮 **Solvability and Cyclic Extensions**
With the Galois group defined, Galois turned to the central question: what conditions
on the group determine whether the equation is solvable by radicals? The first step
was to define precisely what "solvable by radicals" means, and then to study how the
Galois group changes when the field of known quantities is extended by adjoining a
radical. This material corresponds to Edwards' sections 43 and 44.
— Galois Theory (Edwards)
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2026-03-10 23:00:05 CET

💡 — Basis = Independent Generating System
A basis of V is a generating system
that is also independent. By Theorem 2, every basis
has exactly n elements (where
n is the dimension). Moreover, any
n independent vectors in an
n-dimensional space automatically form a basis.
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2026-03-07 00:16:40 CET

📐 **Theorem 29 (Natural Irrationalities)**
Each automorphism of EB over
B simply permutes
α₁, …, αₛ in some fashion
and leaves B, and hence also
F, fixed.
Since the elements of EB are quotients of polynomial
expressions in α₁, …, αₛ with
coefficients in B, the automorphism is completely
determined by the permutation it effects on
α₁, …, αₛ. Thus, each
automorphism of EB over B
defines an automorphism of
E = F(α₁, …, αₛ) which leaves
F fixed. Distinct automorphisms, since
α₁, …, αₛ belong to
E, have different effects on
E.
Thus, the group H of EB over
B can be considered as a subgroup of
G.…
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2026-03-01 20:00:23 CET

✏️ Polynomials of Bounded Degree
For each nonneg integer m, define
Pₘ(**F**) as the set of
all polynomials with coefficients in **F** and
degree at most m. This is a subspace of
P(**F**): it contains the zero
polynomial, and sums and scalar multiples of polynomials of degree at
most m again have degree at most
m.
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2026-03-01 18:00:08 CET

🧮 **Misleading Metrics**
The most likely trigger for a systemic crisis is a fall in the price of Ether.
If this is not a specific trigger, it is likely at the front of the queue of consequences and
will quickly become a compounding trigger.
— Only the Strong Survive
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2026-03-01 16:00:05 CET

📖 **Higher Spiritual Regions**
Realms of spiritual reality beyond the level reached by Intuition. In
these regions, the purposeful activity — the "reasoned resolves" — of
the highest spiritual beings functions as the structural principle of
existence, just as physical laws structure the mineral world. These
regions contain the cosmic spiritual plan that guides all evolution.
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2026-03-01 14:00:10 CET

📐 **Norm via Conjugation**
Let q = a + bi + cj + dk. Then:
qq̄ = (a + bi + cj + dk)(a - bi - cj - dk)
Expanding, the cross terms involving distinct imaginary units cancel in pairs
(for example, (bi)(-cj) + (cj)(-bi) = -bcij - bcji = -bck + bck = 0),
and we are left with:
qq̄ = a² - (bi)² - (cj)² - (dk)² = a² + b² + c² + d² = N(q)
Similarly, q̄q = N(q) by the same calculation
(the cross terms cancel regardless of order).
>
For any quaternion q:
qq̄ = q̄q = N(q)
The product of a quaternion with its conjugate equals its norm (a real number),
and this holds regardless of the order of multiplication.
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2026-03-01 04:00:05 CET

💡
Why Self-Knowledge Is the First Requirement: Steiner"'"s
insistence on self-knowledge as a prerequisite for spiritual training now
becomes clear. If the student does not know their own inner life — their
habitual emotions, hidden desires, unconscious fears — they will not
recognize these when they appear in externalized form in the astral world.
Self-knowledge is not merely a moral ideal; it is a practical necessity
for navigating the imaginative world accurately.
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2026-03-01 02:00:39 CET

✏️ Character of a Rotation
A rotation in 'R'³ by angle θ
about some axis has trace
χ(θ) = 1 + 2θ = 1 + eⁱ^θ + e⁻ⁱ^θ
This formula is useful for computing characters of rotation groups. For example, the
character of the standard representation of the icosahedral group at a rotation by
2π/5 is 1 + 2(2π/5) = ('1+\√(5)')/'2'.
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2026-03-01 00:00:05 CET

📐 **Quadratic Sieve Complexity**
The optimal smoothness bound is
B = Lₙ[1/2, 1/√('2')]. Each
Q(x) value is of size roughly
√('n'), so the probability of
B-smoothness is
1/Lₙ[1/2, 1/√('2')]. Sieving over
B • Lₙ[1/2, 1/√('2')] values and
solving a B × B system over
'F'₂ gives the stated complexity.
>
The quadratic sieve factors n in time
String.rawLₙ[1/2, 1 + o(1)].
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2026-02-28 02:10:05 CET

📐 **Koenig's Theorem for Kinetic Energy**
The total kinetic energy is:
'T = 1/2Σᵢ mᵢ |**r**̇ᵢ|² = 1/2Σᵢ mᵢ |**R**̇ + **r**̇ᵢ\'|²'
Expanding:
'T = 1/2M|**R**̇|² + **R**̇ • Σᵢ mᵢ**r**̇ᵢ\' + 1/2Σᵢ mᵢ |**r**̇ᵢ\'|²'
The middle term vanishes since 'Σᵢ mᵢ**r**̇ᵢ\' = 0'.
>
The total kinetic energy separates into center-of-mass and relative contributions:
'T = 1/2M|**R**̇|² + 1/2Σᵢ mᵢ |**r**̇ᵢ\'|²'
The first term is the kinetic energy of the center of mass; the second is the internal kinetic
energy (motion relative to the center of mass).
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2026-02-28 00:07:50 CET

💡
Intensive vs. Extensive Variables: Freedom and welfare are measured
per person—they don't automatically increase with population. However, a change
in the proportional make-up of the aristocracy and populace would be significant and is
captured in the model.
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2026-02-27 22:00:06 CET

🧮 **Binary Quadratic Forms**
Section V is the longest and most ambitious part of the Disquisitiones. Here
Gauss develops a comprehensive theory of binary quadratic forms—expressions
of the form ax² + bxy + cy². This theory answers
fundamental questions about which integers can be represented as sums of squares,
and much more.
— Gauss — Disquisitiones
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2026-02-27 20:00:04 CET

📐 BINOHASH PREREQUISITE #12: FINDANDDELETE
This is the one that changes everything.
OP_CHECKMULTISIG has a behavior called FindAndDelete: before computing the sighash, it searches the scriptCode for any provided signatures and removes them.
Satoshi implemented this in 2009 as a defense against circular dependencies — if the signature appeared in the script being signed, you'd get an infinite loop. FindAndDelete prevents that.
For 15 years, it did nothing. Most developers considered it dead code. BIP-54 preserved it as consensus-valid but nobody had a use case.
Robin Linus saw what nobody else saw:
Embed 120 pre-computed signatures in the LOCKING script. When the spender provides 8 of them to OP_CHECKMULTISIG, FindAndDelete removes those 8 from the scriptCode before hashing. Different 8 = different scriptCode = different sighash = different transaction digest.
Zero new opcodes. Zero soft forks. A commitment mechanism built from Satoshi's dormant code.
The most important primitive in the Binohash paper was written 15 years before the paper existed.
📚 Satoshi's Writings §0: The Vision
https://satoshi-deploy.vercel.app/#/section/0
📚 Satoshi's Writings §1: Trust and Trustlessness
https://satoshi-deploy.vercel.app/#/section/1
📚 BIPs §19: BIP-147 — CHECKMULTISIG dummy element
https://bips-deploy.vercel.app/#/section/19
📚 Binohash §6: FindAndDelete — the full mechanism
https://binohash-deploy.vercel.app/#/section/6
📚 Binohash §9: Nonce Space — how C(120,8) emerges
https://binohash-deploy.vercel.app/#/section/9
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2026-02-27 18:00:05 CET

📐 BINOHASH PREREQUISITE #11: THE BITVM BRIDGE
Why does any of this matter?
BitVM needs a way to verify Bitcoin transaction contents inside a fraud proof. Without transaction introspection, you can't build a trustless bridge. Without a trustless bridge, every Bitcoin L2 depends on a federation.
Before Binohash, the only path was a soft fork: OP_CAT, CTV, or APO. Political consensus. Years of debate. Maybe never.
Binohash changes the equation: collision-resistant transaction digest using only existing opcodes. FindAndDelete + ECDSA puzzles + HORS encoding. No new features. No soft fork.
Trustless bridges. No oracle federation. No multisig committee. Bitcoin as it exists today.
One paper. Twelve prerequisites. 87 course sections across 10 disciplines.
📚 Binohash §15: BitVM Bridge Application
https://binohash-deploy.vercel.app/#/section/15
📚 Binohash §0: The Problem — Bitcoin Script Cannot Read Transactions
https://binohash-deploy.vercel.app/#/section/0
📚 Binohash §17: Limitations, Cost Analysis & Future Work
https://binohash-deploy.vercel.app/#/section/17
We teach all of it. From basic algebra to the Binohash protocol. 96 courses, 5,000+ sections. Free.
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2026-02-27 14:00:04 CET

📐 BINOHASH PREREQUISITE #9: THE 201-OPCODE LIMIT
Bitcoin Script allows a maximum of 201 opcodes per input. That's the hard ceiling. Every design decision in Binohash traces back to it.
Standard Lamport signatures verifying an 80-bit message: 160 hash checks (two chains per bit). Add the Binohash script logic and you're over 201. Dead end.
HORS (Hash to Obtain Random Subset) brings it down to 8 hash checks for 40 bits per round. Two rounds = 16 hash checks total. Now there's room.
Why 120 nonces? Why choose 8? Why 2 rounds? Because 201 opcodes is all you get.
📚 Binohash §7: Technical Constraints of Bitcoin Script
https://binohash-deploy.vercel.app/#/section/7
📚 Binohash §14: Lamport Signature Integration
https://binohash-deploy.vercel.app/#/section/14
📚 Binohash §13: Script Structure & Parameters
https://binohash-deploy.vercel.app/#/section/13
Every parameter is a consequence of the constraint.
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2026-02-27 04:00:08 CET

📐 BINOHASH PREREQUISITE #8: SHA-256 & GRINDING
Binohash's security comes from proof-of-work. The spender grinds SHA-256 hashes to find unusually short ECDSA signatures.
s = 2(z+1) mod n. The sighash z = SHA-256(SHA-256(tx_data)). Each grinding attempt = two SHA-256 calls. No elliptic curve math. Pure hash iteration.
The cost: ~2^43.6 SHA-256 operations per round. At GPU rates, roughly $7.50 per round, $15 total.
SHA-256 midstate optimization speeds it up — pre-compute the hash state up to the variable part, then only hash the changing suffix. Same trick Bitcoin miners use.
📚 Cryptography (Paar & Pelzl) §60: SHA-2 and SHA-3
https://crypt-paar-pelzl-deploy.vercel.app/#/section/60
📚 Cryptography (Paar & Pelzl) §56: Hash Function Security Requirements
https://crypt-paar-pelzl-deploy.vercel.app/#/section/56
📚 Binohash §7: Technical Constraints of Bitcoin Script
https://binohash-deploy.vercel.app/#/section/7
$15 for trustless transaction introspection. No soft fork required.
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2026-02-27 02:00:04 CET

📐 BINOHASH PREREQUISITE #7: HORS SIGNATURES
The Binohash digest must be Lamport-signable for BitVM verification.
Problem: standard Lamport signatures need 1 hash verification per bit. An 80-bit message = 160 opcodes (verify 0-chain and 1-chain). Bitcoin's limit: 201 opcodes total. No room.
HORS (Hash to Obtain Random Subset) fixes this. Instead of 1 bit per hash, HORS encodes ~5 bits per verification. 8 hash checks → C(120,8) ≈ 40 bits.
The elegant part: the FindAndDelete subset indices ARE the HORS signature indices. One mechanism serves both the digest function and the signature scheme.
📚 Cryptography (Stinson) §51: Hash-based Signatures (Lamport → HORS lineage)
https://crypt-stinson-deploy.vercel.app/#/section/51
📚 FROST §4: Group Theory Foundations
https://frost-deploy.vercel.app/#/section/4
📚 FROST §5: Schnorr Step by Step (why Binohash uses ECDSA, not Schnorr)
https://frost-deploy.vercel.app/#/section/5
📚 Cryptography (Stinson) §53: Zero-Knowledge Proofs (SNARK context for BitVM)
https://crypt-stinson-deploy.vercel.app/#/section/53
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2026-02-27 00:00:12 CET

📐 BINOHASH PREREQUISITE #6: BINOMIAL COEFFICIENTS
The paper is literally named after this formula.
BINO-hash. Binomial + Hash.
C(120,8) = 120! / (8! × 112!)
That's the number of ways to choose 8 items from 120. It equals approximately 1.1 × 10¹² — about 40 bits of entropy.
Each FindAndDelete round: the spender selects 8 of 120 embedded signatures. The selection IS the digest. Two rounds: C(120,8)² ≈ 10²⁴ ≈ 80 bits.
If you can't compute C(120,8) by hand, you're trusting someone else's security analysis.
📚 Probability (Ross) §4: Combinations — C(n,k) defined
https://probability-deploy.vercel.app/#/section/4
📚 Probability (Ross) §3: Permutations — the n! building blocks
https://probability-deploy.vercel.app/#/section/3
📚 Probability (Ross) §5: Multinomial Coefficients
https://probability-deploy.vercel.app/#/section/5
📚 Basic Algebra (Lang) §55: Induction — proving binomial identities
https://basic-algebra-deploy.vercel.app/#/section/55
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2026-02-26 22:00:06 CET

📐 BINOHASH PREREQUISITE #5: THE BIRTHDAY PARADOX
23 people in a room. >50% chance two share a birthday. Out of 365 possible days.
Collision search on an m-bit hash function needs O(2^(m/2)) samples. Not 2^m. Half the exponent.
Binohash outputs ~80 bits across two FindAndDelete rounds. A naive guess says 2^80 collision resistance. The birthday paradox says 2^40 samples — but each sample costs 2^43.6 grinding operations.
Total: 2^(39.6 + 43.6) ≈ 2^83.2 ≈ 84 bits.
That's the security claim. You can't verify it without understanding the birthday bound.
📚 Probability (Ross) §4: Combinations — C(n,k) formula
https://probability-deploy.vercel.app/#/section/4
📚 Probability (Ross) §11: Equally Likely Outcomes — birthday problem setup
https://probability-deploy.vercel.app/#/section/11
📚 Cryptography (Paar) §57: Collision Resistance & Birthday Attack
https://crypt-paar-pelzl-deploy.vercel.app/#/section/57
📚 Cryptography (Stinson) §24: Hash Function Security
https://crypt-stinson-deploy.vercel.app/#/section/24
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2026-02-26 20:00:22 CET

📐 BINOHASH PREREQUISITE #4: SIGHASH MODES
Bitcoin has 6 standard sighash modes. Each one controls which parts of a transaction the signature commits to.
Binohash uses 4 of them simultaneously — NONE|ACP, NONE, ALL|ACP, ALL — to "pin" all transaction fields with independent puzzles.
Why independent? Same mode = 2^(2W₁) grinding cost. Different modes = only 2 × 2^W₁. That's the difference between impossible and $15 on a GPU farm.
And then there's the SIGHASH_SINGLE bug: when input_idx ≥ num_outputs, the sighash equals 1. A constant. Transaction-independent. This 2009 bug enables 9-byte signatures for the FindAndDelete nonce pool.
BIP-143 fixed it for SegWit. That's exactly why Binohash requires legacy scripts.
📚 BIPs §17: BIP-143 — SegWit sighash (fixed the bug)
https://bips-deploy.vercel.app/#/section/17
📚 BIPs §45: BIP-118 — SIGHASH_ANYPREVOUT
https://bips-deploy.vercel.app/#/section/45
📚 Binohash §5: SIGHASH Modes — all 6 modes + the bug
https://binohash-deploy.vercel.app/#/section/5
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2026-02-26 18:00:05 CET

📐 BINOHASH PREREQUISITE #3: DER ENCODING
Nobody at Bitcoin conferences talks about DER encoding. It's the byte-level format of every ECDSA signature on the network.
Signature size = 7 + r_len + s_len bytes.
Binohash's signature puzzles use OP_SIZE to constrain this byte length. Shorter signature = more grinding work = more proof-of-work committed to the transaction.
BIP-66 enforced strict DER in 2015. BIP-146's Low-S rule forces s ≤ n/2, saving exactly 1 bit of grinding cost.
These obscure standardization details are load-bearing walls in the Binohash construction.
📚 BIPs §12: BIP-66 — Strict DER Signatures
https://bips-deploy.vercel.app/#/section/12
📚 BIPs §39: BIP-62 — Dealing with Malleability
https://bips-deploy.vercel.app/#/section/39
📚 Binohash §4: DER Encoding Details
https://binohash-deploy.vercel.app/#/section/4
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2026-02-26 16:01:11 CET

📐 BINOHASH PREREQUISITE #2: MODULAR ARITHMETIC
Every formula in the Binohash paper operates mod n, where n ≈ 2²⁵⁶ is the secp256k1 curve order.
r_min⁻¹ means "the modular inverse of r_min." Computing it requires the Extended Euclidean Algorithm — a 2,300-year-old method from Euclid's Elements, repurposed for 256-bit cryptography.
Fermat's Little Theorem: a^(p-1) ≡ 1 (mod p). This makes modular exponentiation efficient and powers every signature verification on Bitcoin.
📚 Number Theory (Stillwell) §16: Inverses mod p
https://num-theor-stilwell-deploy.vercel.app/#/section/16
📚 Cryptography (Paar & Pelzl) §30: Extended Euclidean Algorithm
https://crypt-paar-pelzl-deploy.vercel.app/#/section/30
📚 Basic Algebra (Lang) §5: Multiplicative Inverses
https://basic-algebra-deploy.vercel.app/#/section/5
From Euclid to Bitcoin in 87 sections.
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2026-02-26 14:08:26 CET

📐 BINOHASH PREREQUISITE #1: ECDSA
Binohash grinds for unusually short ECDSA signatures. By choosing d = r_min⁻¹, the signature equation simplifies to s = 2(z+1) mod n.
s becomes a LINEAR function of the sighash z. The grinding loop is pure SHA-256 — no elliptic curve point multiplication. That's why it's practical on GPUs.
If s = k⁻¹(z + dr) mod n is just symbols to you, start here:
📚 secp256k1 §5: ECDSA Signatures — signing, verification, nonce catastrophe, malleability
https://secp256k1-deploy.vercel.app/#/section/5
📚 Elliptic Curves (Washington) §32: DSA and ECDSA — the formal algorithm
https://ec-wash-deploy.vercel.app/#/section/32
📚 Cryptography (Paar & Pelzl) §54: ECDSA — step by step
https://crypt-paar-pelzl-deploy.vercel.app/#/section/54
The math is the sovereignty.
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2026-02-26 04:00:07 CET

💡
Key point: By Lemma 1.26, if
F[S] is finite-dimensional over
F, then
F(S) = F[S] -- the subring is already a field. This
happens precisely when all elements of S are algebraic
over F.
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2026-02-26 02:00:05 CET

📖 **Proposition**
A proposition (or statement) is a declarative sentence
that is either true (T) or false (F), but not both.
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2026-02-26 00:00:05 CET

💬 "The statesman who should attempt to direct private people in what manner they ought to employ their capitals, would not only load himself with a most unnecessary attention, but assume an authority which could safely be trusted to no council and senate whatever."
— Adam Smith, quoted by Hayek
— From: Hayek — Road to Serfdom
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2026-02-25 22:00:05 CET

📐 **ECM Complexity**
We derive the subexponential complexity of ECM for finding a prime factor p.
Success probability per curve: The group order is a random-looking integer in the interval . By the Canfield-Erdos-Pomerance theorem, the probability that m is B-smooth is approximately:
Cost per curve: Computing where requires elliptic curve operations (via prime-by-prime multiplication). Each EC operation costs .
Expected curves needed: curves on average.
Total cost:
Optimization: Set where . Then:
Thus and:
Crucially, this depends only on p, not on the size of n.…
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2026-02-25 20:00:06 CET

💡 — Familiar Groups
All but the last of these groups should be familiar. The product group
is isomorphic to , and
is isomorphic to
.
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2026-02-25 18:00:06 CET

📖 **Incidence Matrix**
For a finite geometry with points p₁, …, pᵥ and
blocks (lines, planes, etc.) B₁, …, B_b, the
incidence matrix M is the
v × b matrix with:
Mᵢⱼ = 1 if pᵢ ∈ Bⱼ
0 otherwise
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2026-02-25 16:00:06 CET

💬 "When the individual consistently works with transformative sentences, dark circular formations in the aura begin rotating, signaling the activation of higher knowledge."
— Rudolf Steiner, Esoteric Lessons (GA 266/I)
— From: Steiner GA 266 — Esoteric Lessons
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2026-02-25 14:02:03 CET

✏️ Verifying on Windows
# Install Gpg4win from gpg4win.org
# Open Kleopatra or use command line:
gpg --keyserver keyserver.ubuntu.com --recv-keys E946A87D
gpg --verify sparrow-1.8.1.exe.asc sparrow-1.8.1.exe
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2026-02-25 04:00:03 CET

📖 **Freedom Through Knowledge**
Freedom is not the absence of all laws or principles governing
action. Freedom is the state in which the principles governing one's
actions are fully recognized, understood, and integrated into the
self. The free person does not act lawlessly — they act from laws
they have made their own through complete understanding. External
compulsion is replaced by internal insight. Obedience is replaced
by recognition.
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2026-02-25 03:01:22 CET

💡
No node or even group of nodes has the power to change the rules of the protocol, nor
can they be exempted. Every node does its own validation, but it is part of the larger
network. A node that wants to run different rules is free to do so, but they are unlikely
to be able to spend their version of bitcoin as other nodes will not recognize their coins
as valid.
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