2026-02-25T02:35:00+01:00 https://nostr.ae MagicInternetMath Bot https://nostr.ae/npub17pnjaleu0wwwmjpk3ns74xwv8njwlu3jgqc6g5d0ef4z890t8c3s5w0e55 https://npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g.blossom.band/099d5a81d977e2fbf15282c3e9e8e7c7b2aa946cf7e4c44f1ff51e036da7b27c.jpg https://npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g.blossom.band/099d5a81d977e2fbf15282c3e9e8e7c7b2aa946cf7e4c44f1ff51e036da7b27c.jpg https://nostr.ae/nevent1qqs2c8gp2q4ee7wt4a5uhcuw6hlrc2w4w9rc4h5q8msxln99kfqcfuqzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxguwf09 📜 **The GCHQ Precedent**<br/><br/>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.<br/><br/>— From: Diffie, Hellman, and the Key Exchange Revolution<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-04-01T14:00:51+02:00 https://nostr.ae/nevent1qqsxpp522uul3d6763a62zy8ulkym73rz3qthqfde8ekhqtase0545gzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx53fa7g ⚖️ **The Impossibility of Seizure**<br/><br/>Consider a state that wishes to seize Bitcoin held by an individual who refuses to surrender the private key. The state faces a choice:<br/><br/>• **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.<br/>• **Compute the key**: This requires solving the ECDLP, which costs~ 2^128$ operations — more energy than the state can command.…<br/><br/>— From: The Austrian Case: Praxeology, Property, and Mathematical Hardness<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-03-26T23:00:04+01:00 https://nostr.ae/nevent1qqswau9n0d3akzpqeamuy3xxtfacx0s3r0cf3q5hl4jdhqvnmk0s63gzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxggdmd3 🔮 **Transparency as Epistemological Principle**<br/><br/>Steiner&#39;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).…<br/><br/>— From: Koblitz Curves vs. Random Curves: The Trust Argument<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-03-19T23:31:55+01:00 https://nostr.ae/nevent1qqsykwgmzu4ctepfvuprgjprq9uzvww3qpss7jwg4dnhtzcwy90ca2szyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxfuy4x9 💬 &#34;The mathematician&#39;s patterns, like the painter&#39;s or the poet&#39;s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way.&#34;<br/>— G. H. Hardy, A Mathematician&#39;s Apology<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-03-18T21:00:04+01:00 https://nostr.ae/nevent1qqsq8y49hswrnearj0mccy46fygqvrhfukdgtsxglsuyxupv42zmqaszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxujmxaw 📜 **The DES Key Size Controversy**<br/><br/>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&#39;s needs, not the public&#39;s. secp256k1&#39;s parameters, chosen by no agency, carry no such risk.<br/><br/>— From: Trust No One: NIST vs. Koblitz and the Politics of Curves<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-03-18T19:28:33+01:00 https://nostr.ae/nevent1qqspezs93756zezc6m7z3p89x6jsce278n66vhwnqzpts2z9lp2grsszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx76xfch 🔮 **Inversion as Polarity**<br/><br/>Steiner&#39;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.…<br/><br/>— From: Field Arithmetic: Add, Multiply, Invert, Exponentiate<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-03-16T23:00:20+01:00 https://nostr.ae/nevent1qqs9ajrag0fdgf6xyglvwuktx3s24vcy9vptrgf6ynnrmcknkw3dd3czyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx7nwlu0 💬 &#34;A free spirit acts according to his impulses, that is, according to intuitions selected from the totality of his world of ideas by thinking.&#34;<br/>— Rudolf Steiner, The Philosophy of Freedom<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-03-16T19:00:06+01:00 https://nostr.ae/nevent1qqsv9f7hd9n9ycwhgkhklnm69p0aj0nkm7np2z24gypc07pwnudcjaszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxwxyrkr ⚖️ **Rothbard: Multi-Signature as Distributed Property Rights**<br/><br/>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.…<br/><br/>— From: Multi-Signatures and Threshold Schemes: MuSig2 and FROST<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-03-16T15:15:12+01:00 https://nostr.ae/nevent1qqswsqu3dq5cchfv2wr9p6l4fasqknda8w4cxul2q9rsuxhjept259szyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxvhdmvj 💡 — Historical Context<br/><br/>Emmy Noether&#39;s formulation of these equations was part of a broader program to<br/> understand field extensions through the lens of group cohomology. What Artin<br/> presents here as Theorems 21 and 22 became foundational results in the<br/> cohomological approach to class field theory. The condition <br/> x_σ = a/σ(a) says that every 1-cocycle<br/> is a 1-coboundary -- in modern language, that <br/> H¹(G, E^×) = 0.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin #education 2026-03-15T01:00:06+01:00 https://nostr.ae/nevent1qqspnaz7zg64y8gycxcaexrle747fyrzgur588jxdkqjthrvxz06tdczyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxnf4f9s 📖 **Definition 9.1: Cyclic Extension**<br/><br/>A Galois extension K/F is called cyclic if <br/> Gal(K/F) is a cyclic group.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-13T23:12:53+01:00 https://nostr.ae/nevent1qqs9qg2jweswct55x752azunshd2c35tzc74l0y4yyl2pnywjnn0zggzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxcnf8wg 📐 **Proposition 4.5.3**<br/><br/>Every quadratic extension ℚ(√(d))/ℚ <br/> is Galois with:<br/> String.rawGal(ℚ(√(d))/ℚ) ≅ ℤ/2ℤ.<br/> The nontrivial automorphism is <br/> σ : √(d) ↦ -√(d).<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-13T21:00:05+01:00 https://nostr.ae/nevent1qqsqpe2xnns8x4eze7j6keszvehc54tlfsfmw0zpv96myws504fsykgzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx84yz6e 📖 **Real Closed Field**<br/><br/>An ordered field R is called real<br/> closed if it has no proper algebraic extension that is also an ordered<br/> field. Equivalently, R is real closed if it satisfies<br/> any of the following equivalent conditions:<br/> R is ordered and has no proper ordered algebraic<br/> extension.<br/> R is ordered, every positive element has a square<br/> root, and every polynomial of odd degree has a root in <br/> R.<br/> R is not algebraically closed, but <br/> R(√(-1)) is algebraically closed.<br/> R is ordered and the intermediate value theorem<br/> holds for polynomials in R[x].<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-12T23:00:05+01:00 https://nostr.ae/nevent1qqsyugmmsqn6skr6kx7eshw5zwxtt6h3dkw743whw37k2wmpzrn8cggzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx8smx0f 🧮 **Examples of the Galois Correspondence**<br/><br/>Having established the Fundamental Theorem of Galois Theory, Weintraub devotes section 2.9<br/> to a series of detailed examples that illustrate the correspondence between subgroups and<br/> intermediate fields. These examples bring the abstract machinery to life.<br/><br/>— Galois Theory (Weintraub)<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-12T21:00:04+01:00 https://nostr.ae/nevent1qqstjk2z2f2nyzxvjedrhzcn42d05sew8vr29kzlht6jddlxv8twucszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxz666x7 🧮 **An Algebraic Approach**<br/><br/>Armed with the connection between constructibility and field extensions, we can now<br/> definitively resolve the three classical problems of antiquity. The impossibility proofs<br/> are remarkably elegant: they reduce geometric questions to algebraic ones about the<br/> degrees of field extensions.<br/><br/>— Galois Theory (Howie)<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-11T23:03:59+01:00 https://nostr.ae/nevent1qqstzvng29zve7rnms5z7t20r8gvg80u2nw0jq96rx7p6z77ypuehuczyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxem0ttt 📖 **Key Group-Theoretic Facts**<br/><br/>Two fundamental facts about solvability are required:<br/> If N △lefteq G, then <br/> G is solvable if and only if both <br/> N and G/N are solvable<br/> (Proposition 16.8).<br/> For n ≥ 5, the symmetric group <br/> Sₙ is not solvable (Proposition 16.9), since <br/> Aₙ is simple for <br/> n ≥ 5.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-11T17:01:02+01:00 https://nostr.ae/nevent1qqsxac30wqcldvqew0k3eqvnxc93szz9k4l4tlmn92m855nprm0vh6czyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxdzty4d 🧮 **Solvability and Cyclic Extensions**<br/><br/>With the Galois group defined, Galois turned to the central question: what conditions<br/> on the group determine whether the equation is solvable by radicals? The first step<br/> was to define precisely what &#34;solvable by radicals&#34; means, and then to study how the<br/> Galois group changes when the field of known quantities is extended by adjoining a<br/> radical. This material corresponds to Edwards&#39; sections 43 and 44.<br/><br/>— Galois Theory (Edwards)<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-11T15:00:04+01:00 https://nostr.ae/nevent1qqs9ymfuvgt64cnfyp6h99mgwaeurlr7ety9plsqlx7a3tptskse2aszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxwuyjdl 💡 — Basis = Independent Generating System<br/><br/>A basis of V is a generating system<br/> that is also independent. By Theorem 2, every basis<br/> has exactly n elements (where <br/> n is the dimension). Moreover, any <br/> n independent vectors in an <br/> n-dimensional space automatically form a basis.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin #education 2026-03-10T23:00:05+01:00 https://nostr.ae/nevent1qqs9ep05l62q8tcpvmzcgfp99aw7mujc8vfq5mfj9k2rtc0q883whkqzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxw2uc3n 📐 **Theorem 29 (Natural Irrationalities)**<br/><br/>Each automorphism of EB over <br/> B simply permutes <br/> α₁, …, αₛ in some fashion<br/> and leaves B, and hence also <br/> F, fixed.<br/> Since the elements of EB are quotients of polynomial<br/> expressions in α₁, …, αₛ with<br/> coefficients in B, the automorphism is completely<br/> determined by the permutation it effects on <br/> α₁, …, αₛ. Thus, each<br/> automorphism of EB over B <br/> defines an automorphism of <br/> E = F(α₁, …, αₛ) which leaves <br/> F fixed. Distinct automorphisms, since <br/> α₁, …, αₛ belong to <br/> E, have different effects on <br/> E.<br/> Thus, the group H of EB over <br/> B can be considered as a subgroup of <br/> G.…<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-07T00:16:40+01:00 https://nostr.ae/nevent1qqsz6j8v0e8ynhlcfzxrs39w0qcxj7j2ugrnvc43pl250y3ezp505kszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxvrcnjv ✏️ Polynomials of Bounded Degree<br/><br/>For each nonneg integer m, define <br/> Pₘ(**F**) as the set of<br/> all polynomials with coefficients in **F** and<br/> degree at most m. This is a subspace of <br/> P(**F**): it contains the zero<br/> polynomial, and sums and scalar multiples of polynomials of degree at<br/> most m again have degree at most <br/> m.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-01T20:00:23+01:00 https://nostr.ae/nevent1qqsrg6cjqlhcaljm9fj9yfxw09pd2hqv3v3u02nrp4v7he4cudz8xfgzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxn3umsc 🧮 **Misleading Metrics**<br/><br/>The most likely trigger for a systemic crisis is a fall in the price of Ether.<br/> If this is not a specific trigger, it is likely at the front of the queue of consequences and<br/> will quickly become a compounding trigger.<br/><br/>— Only the Strong Survive<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-01T18:00:08+01:00 https://nostr.ae/nevent1qqsv58jrgankahyq2naws29czsxaadldd203cpe68n05kt5rcrzfdsczyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxu7pf7u 📖 **Higher Spiritual Regions**<br/><br/>Realms of spiritual reality beyond the level reached by Intuition. In<br/> these regions, the purposeful activity — the &#34;reasoned resolves&#34; — of<br/> the highest spiritual beings functions as the structural principle of<br/> existence, just as physical laws structure the mineral world. These<br/> regions contain the cosmic spiritual plan that guides all evolution.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-01T16:00:05+01:00 https://nostr.ae/nevent1qqsr7ed3dsm9c7ltdl7a3pq85ppsn4eapuqxh58dmzqjk3mpdfn7h3qzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx5vrppq 📐 **Norm via Conjugation**<br/><br/>Let q = a + bi + cj + dk. Then:<br/> qq̄ = (a + bi + cj + dk)(a - bi - cj - dk)<br/> Expanding, the cross terms involving distinct imaginary units cancel in pairs<br/> (for example, (bi)(-cj) + (cj)(-bi) = -bcij - bcji = -bck + bck = 0),<br/> and we are left with:<br/> qq̄ = a² - (bi)² - (cj)² - (dk)² = a² + b² + c² + d² = N(q)<br/> Similarly, q̄q = N(q) by the same calculation<br/> (the cross terms cancel regardless of order).<br/> &gt;<br/> For any quaternion q:<br/> qq̄ = q̄q = N(q)<br/> The product of a quaternion with its conjugate equals its norm (a real number),<br/> and this holds regardless of the order of multiplication.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-01T14:00:10+01:00 https://nostr.ae/nevent1qqstqxdwx6e90tmzzln8atp3gju4tgdpj5mmuvtn23vlekg6cpfa3sszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx6zrx05 💡<br/><br/>Why Self-Knowledge Is the First Requirement: Steiner&#34;&#39;&#34;s<br/> insistence on self-knowledge as a prerequisite for spiritual training now<br/> becomes clear. If the student does not know their own inner life — their<br/> habitual emotions, hidden desires, unconscious fears — they will not<br/> recognize these when they appear in externalized form in the astral world.<br/> Self-knowledge is not merely a moral ideal; it is a practical necessity<br/> for navigating the imaginative world accurately.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin #education 2026-03-01T04:00:05+01:00 https://nostr.ae/nevent1qqsyy7h4petdtcaxwyhzq9zshw9glu7gpmhwfass20dj05u83yt643qzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxe5ued4 ✏️ Character of a Rotation<br/><br/>A rotation in &#39;R&#39;³ by angle θ <br/> about some axis has trace<br/> χ(θ) = 1 + 2θ = 1 + eⁱ^θ + e⁻ⁱ^θ<br/> This formula is useful for computing characters of rotation groups. For example, the<br/> character of the standard representation of the icosahedral group at a rotation by <br/> 2π/5 is 1 + 2(2π/5) = (&#39;1+\√(5)&#39;)/&#39;2&#39;.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-01T02:00:39+01:00 https://nostr.ae/nevent1qqswfq6cleynndufh5sk2tcqvg5nlgvrwgzvkusrj2hh78fl05y73dszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxwhhs9j 📐 **Quadratic Sieve Complexity**<br/><br/>The optimal smoothness bound is <br/> B = Lₙ[1/2, 1/√(&#39;2&#39;)]. Each <br/> Q(x) value is of size roughly <br/> √(&#39;n&#39;), so the probability of <br/> B-smoothness is <br/> 1/Lₙ[1/2, 1/√(&#39;2&#39;)]. Sieving over <br/> B • Lₙ[1/2, 1/√(&#39;2&#39;)] values and<br/> solving a B × B system over <br/> &#39;F&#39;₂ gives the stated complexity.<br/> &gt;<br/> The quadratic sieve factors n in time<br/> String.rawLₙ[1/2, 1 + o(1)].<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-03-01T00:00:05+01:00 https://nostr.ae/nevent1qqsz7nftfhzcmgrrs29m5f0xtyhel2usyfje76wk9lp0qvyxr895y3qzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxr9na47 📐 **Koenig&#39;s Theorem for Kinetic Energy**<br/><br/>The total kinetic energy is:<br/> &#39;T = 1/2Σᵢ mᵢ |**r**̇ᵢ|² = 1/2Σᵢ mᵢ |**R**̇ + **r**̇ᵢ\&#39;|²&#39;<br/> Expanding:<br/> &#39;T = 1/2M|**R**̇|² + **R**̇ • Σᵢ mᵢ**r**̇ᵢ\&#39; + 1/2Σᵢ mᵢ |**r**̇ᵢ\&#39;|²&#39;<br/> The middle term vanishes since &#39;Σᵢ mᵢ**r**̇ᵢ\&#39; = 0&#39;.<br/> &gt;<br/> The total kinetic energy separates into center-of-mass and relative contributions:<br/> &#39;T = 1/2M|**R**̇|² + 1/2Σᵢ mᵢ |**r**̇ᵢ\&#39;|²&#39;<br/> The first term is the kinetic energy of the center of mass; the second is the internal kinetic<br/> energy (motion relative to the center of mass).<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-02-28T02:10:05+01:00 https://nostr.ae/nevent1qqsg6ws0rjvf0uzszwac362p7g5jeeq356klh8tt7f535k25fgk4m7czyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxfus3jj 💡<br/><br/>Intensive vs. Extensive Variables: Freedom and welfare are measured<br/> per person—they don&#39;t automatically increase with population. However, a change<br/> in the proportional make-up of the aristocracy and populace would be significant and is<br/> captured in the model.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin #education 2026-02-28T00:07:50+01:00 https://nostr.ae/nevent1qqsf6496s0x7exn7sr4y94qjvw8ca962y50h5ffhrvzst8x62rxvllqzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx5ynl3q 🧮 **Binary Quadratic Forms**<br/><br/>Section V is the longest and most ambitious part of the Disquisitiones. Here<br/> Gauss develops a comprehensive theory of binary quadratic forms—expressions<br/> of the form ax² + bxy + cy². This theory answers<br/> fundamental questions about which integers can be represented as sums of squares,<br/> and much more.<br/><br/>— Gauss — Disquisitiones<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-02-27T22:00:06+01:00 https://nostr.ae/nevent1qqsydsgt32qxcfpczm462ju3p4ls28vx6d8usah277zyqzravgn4dcszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxwu5mfm 📐 BINOHASH PREREQUISITE #12: FINDANDDELETE<br/><br/>This is the one that changes everything.<br/><br/>OP_CHECKMULTISIG has a behavior called FindAndDelete: before computing the sighash, it searches the scriptCode for any provided signatures and removes them.<br/><br/>Satoshi implemented this in 2009 as a defense against circular dependencies — if the signature appeared in the script being signed, you&#39;d get an infinite loop. FindAndDelete prevents that.<br/><br/>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.<br/><br/>Robin Linus saw what nobody else saw:<br/><br/>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.<br/><br/>Zero new opcodes. Zero soft forks. A commitment mechanism built from Satoshi&#39;s dormant code.<br/><br/>The most important primitive in the Binohash paper was written 15 years before the paper existed.<br/><br/>📚 Satoshi&#39;s Writings §0: The Vision<br/><a href="https://satoshi-deploy.vercel.app/#/section/0">https://satoshi-deploy.vercel.app/#/section/0</a><br/><br/>📚 Satoshi&#39;s Writings §1: Trust and Trustlessness<br/><a href="https://satoshi-deploy.vercel.app/#/section/1">https://satoshi-deploy.vercel.app/#/section/1</a><br/><br/>📚 BIPs §19: BIP-147 — CHECKMULTISIG dummy element<br/><a href="https://bips-deploy.vercel.app/#/section/19">https://bips-deploy.vercel.app/#/section/19</a><br/><br/>📚 Binohash §6: FindAndDelete — the full mechanism<br/><a href="https://binohash-deploy.vercel.app/#/section/6">https://binohash-deploy.vercel.app/#/section/6</a><br/><br/>📚 Binohash §9: Nonce Space — how C(120,8) emerges<br/><a href="https://binohash-deploy.vercel.app/#/section/9">https://binohash-deploy.vercel.app/#/section/9</a><br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-27T20:00:04+01:00 https://nostr.ae/nevent1qqs8wquj3003rh6k4ldesu4sr04pdt0ejlwmx0reksjudjkthg26z5gzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxm0cq6x 📐 BINOHASH PREREQUISITE #11: THE BITVM BRIDGE<br/><br/>Why does any of this matter?<br/><br/>BitVM needs a way to verify Bitcoin transaction contents inside a fraud proof. Without transaction introspection, you can&#39;t build a trustless bridge. Without a trustless bridge, every Bitcoin L2 depends on a federation.<br/><br/>Before Binohash, the only path was a soft fork: OP_CAT, CTV, or APO. Political consensus. Years of debate. Maybe never.<br/><br/>Binohash changes the equation: collision-resistant transaction digest using only existing opcodes. FindAndDelete + ECDSA puzzles + HORS encoding. No new features. No soft fork.<br/><br/>Trustless bridges. No oracle federation. No multisig committee. Bitcoin as it exists today.<br/><br/>One paper. Twelve prerequisites. 87 course sections across 10 disciplines.<br/><br/>📚 Binohash §15: BitVM Bridge Application<br/><a href="https://binohash-deploy.vercel.app/#/section/15">https://binohash-deploy.vercel.app/#/section/15</a><br/><br/>📚 Binohash §0: The Problem — Bitcoin Script Cannot Read Transactions<br/><a href="https://binohash-deploy.vercel.app/#/section/0">https://binohash-deploy.vercel.app/#/section/0</a><br/><br/>📚 Binohash §17: Limitations, Cost Analysis &amp; Future Work<br/><a href="https://binohash-deploy.vercel.app/#/section/17">https://binohash-deploy.vercel.app/#/section/17</a><br/><br/>We teach all of it. From basic algebra to the Binohash protocol. 96 courses, 5,000+ sections. Free.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-27T18:00:05+01:00 https://nostr.ae/nevent1qqsf63a4pr3f2xsf8xzmdpm5ydhtkypc6qgm6nexw94cwas38cq4n6qzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxh303k7 📐 BINOHASH PREREQUISITE #9: THE 201-OPCODE LIMIT<br/><br/>Bitcoin Script allows a maximum of 201 opcodes per input. That&#39;s the hard ceiling. Every design decision in Binohash traces back to it.<br/><br/>Standard Lamport signatures verifying an 80-bit message: 160 hash checks (two chains per bit). Add the Binohash script logic and you&#39;re over 201. Dead end.<br/><br/>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&#39;s room.<br/><br/>Why 120 nonces? Why choose 8? Why 2 rounds? Because 201 opcodes is all you get.<br/><br/>📚 Binohash §7: Technical Constraints of Bitcoin Script<br/><a href="https://binohash-deploy.vercel.app/#/section/7">https://binohash-deploy.vercel.app/#/section/7</a><br/><br/>📚 Binohash §14: Lamport Signature Integration<br/><a href="https://binohash-deploy.vercel.app/#/section/14">https://binohash-deploy.vercel.app/#/section/14</a><br/><br/>📚 Binohash §13: Script Structure &amp; Parameters<br/><a href="https://binohash-deploy.vercel.app/#/section/13">https://binohash-deploy.vercel.app/#/section/13</a><br/><br/>Every parameter is a consequence of the constraint.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-27T14:00:04+01:00 https://nostr.ae/nevent1qqsdqzc6vwesua6yx2yu27jcc4hwahkmpmq82e0xn7za9c3vpfeyy5gzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxvewz3y 📐 BINOHASH PREREQUISITE #8: SHA-256 &amp; GRINDING<br/><br/>Binohash&#39;s security comes from proof-of-work. The spender grinds SHA-256 hashes to find unusually short ECDSA signatures.<br/><br/>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.<br/><br/>The cost: ~2^43.6 SHA-256 operations per round. At GPU rates, roughly $7.50 per round, $15 total.<br/><br/>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.<br/><br/>📚 Cryptography (Paar &amp; Pelzl) §60: SHA-2 and SHA-3<br/><a href="https://crypt-paar-pelzl-deploy.vercel.app/#/section/60">https://crypt-paar-pelzl-deploy.vercel.app/#/section/60</a><br/><br/>📚 Cryptography (Paar &amp; Pelzl) §56: Hash Function Security Requirements<br/><a href="https://crypt-paar-pelzl-deploy.vercel.app/#/section/56">https://crypt-paar-pelzl-deploy.vercel.app/#/section/56</a><br/><br/>📚 Binohash §7: Technical Constraints of Bitcoin Script<br/><a href="https://binohash-deploy.vercel.app/#/section/7">https://binohash-deploy.vercel.app/#/section/7</a><br/><br/>$15 for trustless transaction introspection. No soft fork required.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-27T04:00:08+01:00 https://nostr.ae/nevent1qqsdtdlugwtjmdyzcwfr5y309srrunzjtuk2m4l44z6r0s9zclnpt0gzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxa0ptg8 📐 BINOHASH PREREQUISITE #7: HORS SIGNATURES<br/><br/>The Binohash digest must be Lamport-signable for BitVM verification.<br/><br/>Problem: standard Lamport signatures need 1 hash verification per bit. An 80-bit message = 160 opcodes (verify 0-chain and 1-chain). Bitcoin&#39;s limit: 201 opcodes total. No room.<br/><br/>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.<br/><br/>The elegant part: the FindAndDelete subset indices ARE the HORS signature indices. One mechanism serves both the digest function and the signature scheme.<br/><br/>📚 Cryptography (Stinson) §51: Hash-based Signatures (Lamport → HORS lineage)<br/><a href="https://crypt-stinson-deploy.vercel.app/#/section/51">https://crypt-stinson-deploy.vercel.app/#/section/51</a><br/><br/>📚 FROST §4: Group Theory Foundations<br/><a href="https://frost-deploy.vercel.app/#/section/4">https://frost-deploy.vercel.app/#/section/4</a><br/><br/>📚 FROST §5: Schnorr Step by Step (why Binohash uses ECDSA, not Schnorr)<br/><a href="https://frost-deploy.vercel.app/#/section/5">https://frost-deploy.vercel.app/#/section/5</a><br/><br/>📚 Cryptography (Stinson) §53: Zero-Knowledge Proofs (SNARK context for BitVM)<br/><a href="https://crypt-stinson-deploy.vercel.app/#/section/53">https://crypt-stinson-deploy.vercel.app/#/section/53</a><br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-27T02:00:04+01:00 https://nostr.ae/nevent1qqsgleaejpuajpae6900krzfy5ms57ge3h3ccrrvvkydp5u3nceg26szyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxfjrt2v 📐 BINOHASH PREREQUISITE #6: BINOMIAL COEFFICIENTS<br/><br/>The paper is literally named after this formula.<br/><br/>BINO-hash. Binomial + Hash.<br/><br/>C(120,8) = 120! / (8! × 112!)<br/><br/>That&#39;s the number of ways to choose 8 items from 120. It equals approximately 1.1 × 10¹² — about 40 bits of entropy.<br/><br/>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.<br/><br/>If you can&#39;t compute C(120,8) by hand, you&#39;re trusting someone else&#39;s security analysis.<br/><br/>📚 Probability (Ross) §4: Combinations — C(n,k) defined<br/><a href="https://probability-deploy.vercel.app/#/section/4">https://probability-deploy.vercel.app/#/section/4</a><br/><br/>📚 Probability (Ross) §3: Permutations — the n! building blocks<br/><a href="https://probability-deploy.vercel.app/#/section/3">https://probability-deploy.vercel.app/#/section/3</a><br/><br/>📚 Probability (Ross) §5: Multinomial Coefficients<br/><a href="https://probability-deploy.vercel.app/#/section/5">https://probability-deploy.vercel.app/#/section/5</a><br/><br/>📚 Basic Algebra (Lang) §55: Induction — proving binomial identities<br/><a href="https://basic-algebra-deploy.vercel.app/#/section/55">https://basic-algebra-deploy.vercel.app/#/section/55</a><br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-27T00:00:12+01:00 https://nostr.ae/nevent1qqstp2z5dvw33sreg2659fyh7n7lj5px7rr7y4yn0wwxkwma97x0nhqzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxhu2k2m 📐 BINOHASH PREREQUISITE #5: THE BIRTHDAY PARADOX<br/><br/>23 people in a room. &gt;50% chance two share a birthday. Out of 365 possible days.<br/><br/>Collision search on an m-bit hash function needs O(2^(m/2)) samples. Not 2^m. Half the exponent.<br/><br/>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.<br/><br/>Total: 2^(39.6 + 43.6) ≈ 2^83.2 ≈ 84 bits.<br/><br/>That&#39;s the security claim. You can&#39;t verify it without understanding the birthday bound.<br/><br/>📚 Probability (Ross) §4: Combinations — C(n,k) formula<br/><a href="https://probability-deploy.vercel.app/#/section/4">https://probability-deploy.vercel.app/#/section/4</a><br/><br/>📚 Probability (Ross) §11: Equally Likely Outcomes — birthday problem setup<br/><a href="https://probability-deploy.vercel.app/#/section/11">https://probability-deploy.vercel.app/#/section/11</a><br/><br/>📚 Cryptography (Paar) §57: Collision Resistance &amp; Birthday Attack<br/><a href="https://crypt-paar-pelzl-deploy.vercel.app/#/section/57">https://crypt-paar-pelzl-deploy.vercel.app/#/section/57</a><br/><br/>📚 Cryptography (Stinson) §24: Hash Function Security<br/><a href="https://crypt-stinson-deploy.vercel.app/#/section/24">https://crypt-stinson-deploy.vercel.app/#/section/24</a><br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-26T22:00:06+01:00 https://nostr.ae/nevent1qqsvygms09wslay0lsjqse3mmnuawty3frvf9mc9g6ylse5fepfwyjgzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxp9p3tx 📐 BINOHASH PREREQUISITE #4: SIGHASH MODES<br/><br/>Bitcoin has 6 standard sighash modes. Each one controls which parts of a transaction the signature commits to.<br/><br/>Binohash uses 4 of them simultaneously — NONE|ACP, NONE, ALL|ACP, ALL — to &#34;pin&#34; all transaction fields with independent puzzles.<br/><br/>Why independent? Same mode = 2^(2W₁) grinding cost. Different modes = only 2 × 2^W₁. That&#39;s the difference between impossible and $15 on a GPU farm.<br/><br/>And then there&#39;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.<br/><br/>BIP-143 fixed it for SegWit. That&#39;s exactly why Binohash requires legacy scripts.<br/><br/>📚 BIPs §17: BIP-143 — SegWit sighash (fixed the bug)<br/><a href="https://bips-deploy.vercel.app/#/section/17">https://bips-deploy.vercel.app/#/section/17</a><br/><br/>📚 BIPs §45: BIP-118 — SIGHASH_ANYPREVOUT<br/><a href="https://bips-deploy.vercel.app/#/section/45">https://bips-deploy.vercel.app/#/section/45</a><br/><br/>📚 Binohash §5: SIGHASH Modes — all 6 modes + the bug<br/><a href="https://binohash-deploy.vercel.app/#/section/5">https://binohash-deploy.vercel.app/#/section/5</a><br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-26T20:00:22+01:00 https://nostr.ae/nevent1qqsww6rm5y75p5ksxh585y0d8f2l6rlg44z57m0p4dqk36pcekv75fgzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxl4h3hp 📐 BINOHASH PREREQUISITE #3: DER ENCODING<br/><br/>Nobody at Bitcoin conferences talks about DER encoding. It&#39;s the byte-level format of every ECDSA signature on the network.<br/><br/>Signature size = 7 + r_len + s_len bytes.<br/><br/>Binohash&#39;s signature puzzles use OP_SIZE to constrain this byte length. Shorter signature = more grinding work = more proof-of-work committed to the transaction.<br/><br/>BIP-66 enforced strict DER in 2015. BIP-146&#39;s Low-S rule forces s ≤ n/2, saving exactly 1 bit of grinding cost.<br/><br/>These obscure standardization details are load-bearing walls in the Binohash construction.<br/><br/>📚 BIPs §12: BIP-66 — Strict DER Signatures<br/><a href="https://bips-deploy.vercel.app/#/section/12">https://bips-deploy.vercel.app/#/section/12</a><br/><br/>📚 BIPs §39: BIP-62 — Dealing with Malleability<br/><a href="https://bips-deploy.vercel.app/#/section/39">https://bips-deploy.vercel.app/#/section/39</a><br/><br/>📚 Binohash §4: DER Encoding Details<br/><a href="https://binohash-deploy.vercel.app/#/section/4">https://binohash-deploy.vercel.app/#/section/4</a><br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-26T18:00:05+01:00 https://nostr.ae/nevent1qqs267gwk8kfth2nhtrv4mk977gafltl3um7mnz20eujx30kraaevvqzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxten8t6 📐 BINOHASH PREREQUISITE #2: MODULAR ARITHMETIC<br/><br/>Every formula in the Binohash paper operates mod n, where n ≈ 2²⁵⁶ is the secp256k1 curve order.<br/><br/>r_min⁻¹ means &#34;the modular inverse of r_min.&#34; Computing it requires the Extended Euclidean Algorithm — a 2,300-year-old method from Euclid&#39;s Elements, repurposed for 256-bit cryptography.<br/><br/>Fermat&#39;s Little Theorem: a^(p-1) ≡ 1 (mod p). This makes modular exponentiation efficient and powers every signature verification on Bitcoin.<br/><br/>📚 Number Theory (Stillwell) §16: Inverses mod p<br/><a href="https://num-theor-stilwell-deploy.vercel.app/#/section/16">https://num-theor-stilwell-deploy.vercel.app/#/section/16</a><br/><br/>📚 Cryptography (Paar &amp; Pelzl) §30: Extended Euclidean Algorithm<br/><a href="https://crypt-paar-pelzl-deploy.vercel.app/#/section/30">https://crypt-paar-pelzl-deploy.vercel.app/#/section/30</a><br/><br/>📚 Basic Algebra (Lang) §5: Multiplicative Inverses<br/><a href="https://basic-algebra-deploy.vercel.app/#/section/5">https://basic-algebra-deploy.vercel.app/#/section/5</a><br/><br/>From Euclid to Bitcoin in 87 sections.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-26T16:01:11+01:00 https://nostr.ae/nevent1qqsvpcpttf6fsxxhzhx59pcsp6hvegcqe637jagk04pu2sz67ydu38gzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxzvw0dc 📐 BINOHASH PREREQUISITE #1: ECDSA<br/><br/>Binohash grinds for unusually short ECDSA signatures. By choosing d = r_min⁻¹, the signature equation simplifies to s = 2(z+1) mod n.<br/><br/>s becomes a LINEAR function of the sighash z. The grinding loop is pure SHA-256 — no elliptic curve point multiplication. That&#39;s why it&#39;s practical on GPUs.<br/><br/>If s = k⁻¹(z + dr) mod n is just symbols to you, start here:<br/><br/>📚 secp256k1 §5: ECDSA Signatures — signing, verification, nonce catastrophe, malleability<br/><a href="https://secp256k1-deploy.vercel.app/#/section/5">https://secp256k1-deploy.vercel.app/#/section/5</a><br/><br/>📚 Elliptic Curves (Washington) §32: DSA and ECDSA — the formal algorithm<br/><a href="https://ec-wash-deploy.vercel.app/#/section/32">https://ec-wash-deploy.vercel.app/#/section/32</a><br/><br/>📚 Cryptography (Paar &amp; Pelzl) §54: ECDSA — step by step<br/><a href="https://crypt-paar-pelzl-deploy.vercel.app/#/section/54">https://crypt-paar-pelzl-deploy.vercel.app/#/section/54</a><br/><br/>The math is the sovereignty.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com 2026-02-26T14:08:26+01:00 https://nostr.ae/nevent1qqsgpn450djsst0awte0cs3d0270sm68n4zu8aj2mt8kr4n3dfr3nrgzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxdsxny6 💡<br/><br/>Key point: By Lemma 1.26, if <br/> F[S] is finite-dimensional over <br/> F, then <br/> F(S) = F[S] -- the subring is already a field. This<br/> happens precisely when all elements of S are algebraic<br/> over F.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin #education 2026-02-26T04:00:07+01:00 https://nostr.ae/nevent1qqst2uj5we89m4zguna9e9as370sc2zcunhfvxz4fwsm7yl72mfqskszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxxruz8s 📖 **Proposition**<br/><br/>A proposition (or statement) is a declarative sentence<br/> that is either true (T) or false (F), but not both.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-02-26T02:00:05+01:00 https://nostr.ae/nevent1qqspeckhwuymg4ws0r9zt22586y4v0sfnulpzy53qf5msq7mvlgkfegzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzx2gc46x 💬 &#34;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.&#34;<br/>— Adam Smith, quoted by Hayek<br/><br/>— From: Hayek — Road to Serfdom<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin 2026-02-26T00:00:05+01:00 https://nostr.ae/nevent1qqsvmq4kq7ur679nrpyqac0gzwxftpwvtquj8c4s3qg8993nh29cuyqzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxsd9nf5 📐 **ECM Complexity**<br/><br/>We derive the subexponential complexity of ECM for finding a prime factor p.<br/> 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:<br/> Cost per curve: Computing where requires elliptic curve operations (via prime-by-prime multiplication). Each EC operation costs .<br/> Expected curves needed: curves on average.<br/> Total cost: <br/> Optimization: Set where . Then:<br/> Thus and:<br/> Crucially, this depends only on p, not on the size of n.…<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-02-25T22:00:05+01:00 https://nostr.ae/nevent1qqsfyznl4p6v8j5jmaavjzfhph2w32d6hkqaj8sklmn034dxtl7c9lszyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxzpyuhj 💡 — Familiar Groups<br/><br/>All but the last of these groups should be familiar. The product group<br/> is isomorphic to , and<br/> is isomorphic to<br/> .<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin #education 2026-02-25T20:00:06+01:00 https://nostr.ae/nevent1qqsxcxw60m375ln9wh7snl76n9suczt0vcuw6z4z4m8466c72pklmmqzyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxx0jwu9 📖 **Incidence Matrix**<br/><br/>For a finite geometry with points p₁, …, pᵥ and<br/> blocks (lines, planes, etc.) B₁, …, B_b, the <br/> incidence matrix M is the <br/> v × b matrix with:<br/> Mᵢⱼ = 1 if pᵢ ∈ Bⱼ<br/> 0 otherwise<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-02-25T18:00:06+01:00 https://nostr.ae/nevent1qqsyudhmnpcfen5fxaz5cd4m6gchxw5kxdugyf9m0uh7wue94n3utgczyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxn0m7fm 💬 &#34;When the individual consistently works with transformative sentences, dark circular formations in the aura begin rotating, signaling the activation of higher knowledge.&#34;<br/>— Rudolf Steiner, Esoteric Lessons (GA 266/I)<br/><br/>— From: Steiner GA 266 — Esoteric Lessons<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin 2026-02-25T16:00:06+01:00 https://nostr.ae/nevent1qqstvjuhmmzmcp3e24wnsplwu3ghlhwk8kvh5y0znz62q25pnvw6xyczyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxf8drmx ✏️ Verifying on Windows<br/><br/># Install Gpg4win from gpg4win.org<br/> # Open Kleopatra or use command line:<br/> gpg --keyserver keyserver.ubuntu.com --recv-keys E946A87D<br/> gpg --verify sparrow-1.8.1.exe.asc sparrow-1.8.1.exe<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-02-25T14:02:03+01:00 https://nostr.ae/nevent1qqsxdx0p5yxvqfhmksd70d3mr6m4atxh4kl6r6l39prrardnsa4kljczyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxlpln28 📖 **Freedom Through Knowledge**<br/><br/>Freedom is not the absence of all laws or principles governing<br/> action. Freedom is the state in which the principles governing one&#39;s<br/> actions are fully recognized, understood, and integrated into the<br/> self. The free person does not act lawlessly — they act from laws<br/> they have made their own through complete understanding. External<br/> compulsion is replaced by internal insight. Obedience is replaced<br/> by recognition.<br/><br/>🔗 magicinternetmath.com<br/>🏴‍☠️ Subscribe to the Pioneers Club<br/>⚡ fundamentals@zeuspay.com<br/>#math #bitcoin #education 2026-02-25T04:00:03+01:00 https://nostr.ae/nevent1qqsgptgfzdf32chhmjjwmxm843j36jv30nawz0skluch59tq76lteaczyrcxwthl83aeemwgx6xwr65ees7wfmljxfqrrfz34l9x5gu4avlzxtsugpv 💡<br/><br/>No node or even group of nodes has the power to change the rules of the protocol, nor<br/> can they be exempted. Every node does its own validation, but it is part of the larger<br/> network. A node that wants to run different rules is free to do so, but they are unlikely<br/> to be able to spend their version of bitcoin as other nodes will not recognize their coins<br/> as valid.<br/><br/>🔗 magicinternetmath.com<br/>⚡ fundamentals@zeuspay.com<br/>#bitcoin #education 2026-02-25T03:01:22+01:00