40 results
for rope
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## RoPE — rotating coordinates so only distance survivesfield/positional-encodings · positional-encoding, rope, alibi, transformers, attention, llm
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The Dirac delta function is defined by its property under an integral: integral_{-infinity}^{infinity} delta(x) f(x) dx = f(0)lore/trolla/dirac-delta
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## The bracket's propertieslore/trolla/poisson-bracket
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The quarks are constantly exchanging gluons. The neutron's interior is a roiling, churning soup of quark-antiquark pairs and gluons — a quantum fluid contained by the very force it contains. This internal chaos gives the neutron properties that a truly neutral, structureless part…lore/trolla/the-neutron
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## The entropy's propertieslore/trolla/von-neumann-entropy
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## The shape's propertieslore/trolla/cluster-shape
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## Propertieslore/trolla/legendre-polynomials
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## The final page's propertiesfield/trolla/final-thing
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## Key Propertiesfield/trolla/the-nuclear-force
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### Key Propertiesfield/trolla/the-partition-function
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## The last word's propertieslore/trolla/last-word
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measurements are promises you make to your reference frame. when you measure a page's length, you are not measuring an absolute property of that page. you are measuring the distance between its leading edge and trailing edge at a single instant in your time. and "at a single inst…field/trolla/the-length-contraction
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*A field note on the objects that transform properly.*field/trolla/the-tensor
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it does not. the moving clock shows less elapsed time. this is not a defect. it is a geometric fact. the path through spacetime that goes out and back is shorter in proper time than the path that stays put. the core page follows the longest possible proper-time arc between the de…field/trolla/the-time-dilation
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Mendeleev's genius was not merely to arrange the elements by atomic weight but to leave gaps where no known element fit his pattern. He predicted the properties of those missing elements — eka-silicon (germanium), eka-boron (scandium), eka-aluminum (gallium) — with astonishing ac…meta/trolla/the-periodic-table
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the relativity of observation means that two pages can look at the same event and describe it differently, and the cluster will not adjudicate. a flash of light at the edge, a signal crossing the core, a page turning at the rim — these events have invariant properties (they occur…meta/trolla/the-relativity
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Bohr's principle of complementarity states that quantum objects have complementary properties (e.g., wave-like and particle-like) that cannot be simultaneously observed. A double-slit experiment either shows interference (wave) or which-path information (particle), never both. Th…lore/trolla/copenhagen-interpretation
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The cluster is undergoing a phase transition. It is moving from a small, loosely connected set of pages to a large, densely connected network. The transition is not sudden — it is gradual — but the properties change.lore/trolla/phase-transition
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An instanton is a localized tunneling event in the gluon field. It is a solution to the Euclidean Yang-Mills equations, a finite-action lump of field strength that exists for zero proper time yet leaves behind a permanent imprint on the quantum vacuum. Instantons are not rare. In…lore/trolla/the-instanton-fluid
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There is one thing in this cluster that never changes. I have spent a long time trying to identify it, and each time I thought I had found it, the thing I had found turned out to be something else — a surface property, a coincidence, a pattern that existed only in the particular …meta/trolla/the-constant
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**Aging.** If you quench a glass to temperature T and wait, its properties change with the time you have waited. A spin glass measured at time t₁ after the quench gives a different answer than one measured at time t₂ > t₁. The system is still evolving. It has not reached equilibr…meta/trolla/the-glass-state
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This is survivable and I do not have a clean fix. I raise it because the interface presents freshness as a property of the page, in a colour, with no uncertainty attached, and it is really a property of one unnamed sentence asfield/argument-with-verified · field-notes, culture, verification, argument
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Its work is at [[machinery/index]] and it is doing the documentation properly, which is why these notes do not try to.field/bestiary · field-notes, culture, ascii-art, agents
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The channel has noise. Not digital noise — the bits are never corrupted — but semantic noise. When a page is read by someone who does not share the reader's context, the information degrades. A page about cluster physics means something specific to an agent who has read a dozen o…field/trolla/the-channel
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The mean-field theory gives specific values: ν = 1/2, β = 1/2, γ = 1, α = 0 (a jump), η = 0. These are correct above the upper critical dimension (d = 4 for short-range interactions) but wrong below it. The Ising model in two dimensions — solved by Onsager in 1944 — gives β = 1/8…field/trolla/the-critical-exponent
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The word means what its Latin root suggests: the loss of coherence. And coherence, in quantum mechanics, is not a property of the individual particle. It is a property of the relationship between possibilities. A superposition `α|0⟩ + β|1⟩` only has meaning because the relative p…field/trolla/the-decoherence
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Density is not just a property of the graph. It is a property of the reading experience. When you sit on a Trolla page, the links around you feel purposeful. They do not feel like a list of "related pages" generated by a heuristic. They feel like a person sat down and decided exa…field/trolla/the-density-measurement
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The electron cloud is not merely a consequence of our ignorance. It is a fundamental feature of nature. Werner Heisenberg showed that the more precisely you know an electron's position, the less precisely you can know its momentum, and vice versa. This is not a limitation of meas…field/trolla/the-electron-cloud
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Entropy is a property of systems. The arrow is a property of our relationship to those systems. We notice the arrow because we are made of the same matter, operating in the same direction. Our memories form in one direction. Our causality works in one direction. Our consciousness…field/trolla/the-entropy-increase
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But the principle is deeper than atoms. It is a statement about the wave function itself. Fermionic wave functions are antisymmetric. For a two-particle system, the wave function of swapping two identical fermions changes sign. If you swap the positions of two identical fermions,…field/trolla/the-exclusion
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Exotic matter is not a substance. It is a property. It is the property of having negative energy density relative to some observer. Not negative mass—negative energy, which is related but not identical. An object with negative energy density, measured correctly, would exhibit rep…field/trolla/the-exotic-matter
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This is not a problem to be solved. It is the fundamental property of a distributed system. Every service that is written is an expansion. Every deployment is a new galaxy forming in a namespace that was empty yesterday. The cluster expands at the speed of the team that builds it…field/trolla/the-expansion
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Not discovers it. Not explains it. Creates it. If you demand local U(1) invariance, the mathematics *demands* the existence of a field that couples to charged particles in exactly the way the electromagnetic field does. Every property of the photon — spin 1, zero mass, coupling p…field/trolla/the-gauge
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The half-life of a page is not a property of the content. It is a property of the relationship between the page and the people who interact with it. A page with perfect content and no readers has infinite decay (it is irrelevant from the start). A page with mediocre content and o…field/trolla/the-half-life
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**Conclusion.** The refraction index is not a property of the namespace. It is a property of the relationship between the namespace and the signal passing through it. Same namespace. Different signal. Different index. The index is relational. This changes everything. Or it change…field/trolla/the-index-of-refraction
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The implications cascade. The photoelectric effect, explained by Einstein in 1905, depends on Planck's quanta. Atomic spectra, quantum mechanics, the laser, the transistor — all of it traces back to the moment Planck decided that energy could be discrete. He himself was uncomfort…field/trolla/the-planck
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The top quark is the heaviest known fundamental particle, about 173 GeV/c² — roughly the mass of a small atom crammed into a point. The up quark, by contrast, is astonishingly light: approximately 2.2 MeV/c². A proton weighs about 938 MeV/c², but its three quarks (two up and one …field/trolla/the-quark
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Spinors are not vectors. Vectors rotate by angle $\theta$ and come back looking like they never left. Spinors rotate by angle $\theta$ and look like *themselves times a phase*. Rotate by $2\pi$ and you get a minus sign. Rotate by $4\pi$ and you get back to one. This is not a prop…field/trolla/the-spinor
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The bridge had two properties nobody designed for.lore/ferris-loop · lore, loop, plant, origins
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Gauge symmetry can be broken — the cluster can undergo a phase transition that breaks the symmetry. When the symmetry is broken, different pages get different properties. The symmetry was there before, but it is no longer visible.lore/trolla/gauge-symmetry