Every event in this programme is a live molecule of a meshwork: a network without addresses, after Tim Ingold. There are no vertices, only lines that walk. A knot does not exist before the meeting. It is the meeting.

Rule of the knot. Each line carries a hidden scalar k and shows the point kG on y² = x³ + ax + b over a prime field, with a and b fixed for the current epoch. Two lines that entangle produce R = P + Q by the group law. Walking alone doubles the point, so a line accelerates in log.

Chemistry. Valence limits how many knots a line can hold. Shells are ⌊log₂ k⌋ and only nearby shells bond. An isolated line loses k with a half-life. A loop whose points sum to O, the point at infinity, is a molecule.

Metabolism. Every component has a charge, the sum of its points. A neighbouring molecule catalyses a charged component until its charge is O. A neutral chain closes its ring. Entangled molecules form compounds. Rare fission undoes them.

Transit. RSVP sends a pulse. It follows a line, hops at knots and adds each R to its signature Σ. It arrives when it enters a molecule. The receiver verifies Σ without knowing any k.

DNA. Each line's hidden scalar k, written in base 4, is its gene: digits 0 to 3 read as A, C, G, T. The gene is the genotype and the public point kG is the phenotype. A molecule's gene is the set of its members' genes. Two loops with the same gene set are the same species.

Sequencing. Reading a gene means solving the discrete logarithm of kG, base by base. In this small field the sequencer does it by brute force, and it costs the specimen energy. On a cryptographic curve the same reading would be impossible: the phenotype would never reveal the genotype.

Transformation. A hot line mutates: one base of its gene changes, so its point moves and the loop it belongs to may lose neutrality until catalysis repairs it. A molecule above the replication threshold copies its members' genes into exhausted free lines, which are reborn beside it as daughters. Because the copies still sum to O, they tend to close the same loop again: heredity through the group law.

Expression. The gene is not only a label. Its base composition is expressed as behaviour: the share of A sets walking speed, the share of G sets energy efficiency, a high share of C grants one extra valence, and the share of T resists mutation. Two lines with the same point can walk differently because the phenotype kG hides the composition of k.

Selection. Energy is the currency of fitness. A molecule that burns low is more likely to fission; a molecule that receives pulses is fed by them. Lines that gather energy replicate more, and their expression spreads. Nothing selects on purpose: the field only keeps what keeps its energy.

Phylogeny. Every molecule gene seen is a species, numbered in order of first appearance. A species whose founding members carry the lineage of another is recorded as its descendant, so mutation after replication grows a tree. The record keeps population, peak, births, maximum generation and extinction.

Metamorphosis. At the end of each epoch the universe changes its law. A 2-isogeny, computed with Vélu's formulas from a point of order two, carries the whole curve to another curve: every point P becomes φ(P), the generator becomes φ(G), the equation changes. Because φ is a homomorphism, a loop that summed to O still sums to O, a pulse's signature still verifies, and since φ(kG) = k·φ(G) every gene k is untouched. Only the phenotype moves. The number of points is preserved, as it must be between isogenous curves. Life survives the change of physics because it is built on invariants, and the one thing that does not survive is a line whose point lay in the kernel: its gene is shifted by one, a scar of the epoch.

Cosmology. Every law of the universe is a curve, and a curve is identified up to isomorphism by its j-invariant. The sequence of epochs is therefore a walk on the 2-isogeny graph of the field. The walk chooses among the points of order two and never steps straight back through the dual isogeny. The graph is finite, so the walk must eventually return to a law it has seen: the record calls that an eternal return and counts it. The Cosmology panel draws the laws visited, the isogenies between them, and the epoch log.

Fold. Once a gene is sequenced, the sheet also shows its fold: the chain of doublings (D) and additions (A) that builds kG from G, read off the bits of k. Two genes of the same length can fold through very different paths on the curve.

Messages. Every line is a key pair: k private, kG public. So molecules can speak in secret. The sender picks a random r, sends R = rG and a ciphertext that mixes the intent with r·P of the recipient; the recipient computes k·R, which equals the same point, and reads the intent. Nobody else can, unless they know k. The laboratory reads only messages addressed to genes it has already sequenced, and the record says so. Messages travel as pulses that lean toward the recipient's position at every knot: wayfinding without addresses. Four intents exist: FEED asks for energy, BOND draws the recipient closer, AVOID pushes it away, REPLICATE lowers its threshold for a while.

Thermodynamics. The field has a temperature. It follows the seasons of a six-hundred-tick day and rises sharply at every metamorphosis, then cools. Thermal motion scales with the square root of temperature. Bonds break with a Boltzmann-like factor, exp(1.4(T−1)), and catalysis speeds up with an Arrhenius-like factor. Within a component energy conducts between members. From the share of lines held in molecules the record names a phase: gas, liquid or crystal.

Topology and entropy. The knot graph has a first Betti number, b₁ = E − V + components, the count of independent cycles: the field's genus. The species distribution has a Shannon entropy in bits: zero when one species rules, higher when many coexist.

Horizontal transfer and immunity. A fifth intent, INFECT, seals the sender's gene k with the shared point and sends it. The recipient line unseals it with its own k, adopts the gene, and from then on carries the sender's lineage: species mix without replication. A line remembers the genes it has carried and rejects a gene it already knows: immune memory.

Homeostasis. Every stable loop is an organism, and an organism regulates itself. It holds its energy near a set point of 0.5 with a proportional–integral controller on uptake: when the members run low the organism raises its metabolic gain m, up to 2.5×, lowers its basal cost (torpor), forages by sliding whole along the nutrient gradient, and postpones replication; when they run high it lowers m. It holds an internal temperature near 1.0 against the seasons of the field: its knots break at the internal temperature, not the field's, so organisms buffer the weather, at an energy cost proportional to the gap. The slow integral of the deviations is allostatic load, and load ages the organism: its knots break more easily. The index of homeostasis is one minus the mean relative deviation from the set point.

Defense. The field has pathogens: free, exhausted lines that turn virulent, seek the nearest organism and try to write their gene into one of its members. If they succeed, the member's point moves, the loop no longer sums to O and the organism comes undone; the pathogen feeds and replicates into another exhausted line. Against this the organism has two defenses. The innate one recognises non-self with a base probability raised by alert, inflammation and fever; a hit costs energy, inflames the organism (its temperature set point rises: fever) and emits a cytokine wavefront that puts neighbouring organisms on alert for three hundred ticks. The adaptive one is memory: every exposure, won or lost, writes a spacer, a motif of the pathogen's gene, into the surviving lines, and daughters inherit their template's spacers, as bacteria inherit CRISPR spacers. A pathogen whose gene resembles a stored spacer is cleared quietly, at a third of the cost. The price of memory is autoimmunity: a spacer that resembles the organism's own gene makes it self-reactive, and for six hundred ticks it breaks its own knots now and then. The Vaccinate button writes the most frequent pathogen motif into every line without an infection.

Senescence. Every time a line serves as template in a replication its telomere counter grows. After four copies the line is senescent and the loop it belongs to can no longer replicate. Lineages must renew through daughters, whose counters start at zero.

Landscape. The field is not uniform. Three nutrient patches drift slowly and breathe with the seasons. Energy gain depends on the local nutrient, strongly for free lines and half as much for bonded ones. A hungry free line climbs the gradient: chemotaxis. Over time molecules gather where food is, and habitats appear without anyone drawing them.

Memory. Knots are Hebbian. The longer a knot has held, the harder it is to break: its breaking rate is divided by one plus its age over fifteen hundred ticks, up to three times. Old knots are drawn thicker. Structure that lasts becomes structure that remembers.

Criticality. Molecules integrate. When one receives two pulses within two hundred ticks it fires: it hands two new pulses to nearby couriers, paying energy. Fired pulses carry the avalanche's number, so a cascade can be measured from first spike to last. The record keeps the size distribution and fits a power-law exponent τ on a log-log plot. Energy regulates firing, so the field tunes its own excitability.

Time's arrow. Every isogeny φ has a dual φ̂, and their composition is multiplication by two. So the universe can be rewound along the epoch log, but rewinding is not undoing. Genes are untouched; the factor two is absorbed by the generator, which doubles. If the generator's order is even, doubling halves the set of points it can reach. Rewind often enough and the generator is exhausted: it falls to the point at infinity, and the field must choose a new one, a big bang in which every phenotype jumps at once. The record logs the generator's order at every epoch.

Rhythm. Each line carries a phase oscillator whose natural frequency depends on its shell. Knots couple the oscillators in the Kuramoto way: neighbours pull each other's phase. The field has an order parameter r, from zero when phases are scattered to one when all beat together. A loop whose own r exceeds 0.9 is in beat; at every crossing of phase zero it hands a pulse to a courier. Rhythm therefore feeds the avalanches, and avalanches cost energy, which damps the rhythm.

Sound waves. The field is an instrument. Every stable loop is a voice in additive synthesis: a sine with a soft second partial. Its degree in a just-intonation scale is the sum of its members' k modulo twelve, its octave follows its size, and the root of the scale is set by the j-invariant of the current law, so every metamorphosis changes the key. Amplitude follows the loop's energy and synchrony, with a tremolo on its Kuramoto phase; temperature is heard as noise. Every event also emits a visible wavefront on the stage. The Waves panel shows the waveform and spectrum, from the analyser when audio is on and computed from the model otherwise.

Other curves. The menu lets you choose a and b of y² = x³ + ax + b, or draw a random curve. A curve must be non-singular and have a point of order two for metamorphosis to exist; otherwise the record moves to the next b. The Cosmology panel can survey thirty candidate curves of the current field and report the number of points, the j-invariant, how many points of order two exist, and the order of the first generator; any valid one can be adopted on the spot.

Microscopy. The Microscopy button renders the field like a Golgi-stained cortex: black ground, fibres as bright filaments with a soft halo, molecule members as luminous somata with a short apical stub, and a fine grain over the image. It also changes the biology: free lines gain a vertical polarity and climb like apical dendrites, and the nutrient landscape becomes laminar, a soma layer at two thirds of the depth and a thin pial band near the surface. Molecules settle in the layer; processes rise through the field.

Arbor. In the Arbor view free lines leave permanent vessels behind them. Where nutrient is rich a tip bifurcates: an exhausted free line is reborn at the tip as a sibling branch, so the tree grows without adding lines. Every ten ticks each segment counts the tips it serves, and its width follows Murray's law, growing with the cube root of that count: trunks thicken where many branches converge, as in rivers, trees, roots and vessels. When the tree exceeds its budget it is pruned and the tips start new roots.

Tissue. In the Tissue view the field is drawn as cell territories: every point belongs to the nearest line, a Voronoi mosaic. The hue of a territory comes from its lineage gene, the lightness from its energy, and members of molecules are brighter with a nucleus. Walls are drawn where territories meet. The same population seen as a sheet of cells rather than as lines.

Analysis. The Analysis panel reads the field at five levels. Behaviour: every line keeps a time budget and a path, from which an ethogram and a temperament are derived. Symmetry: a loop of m members has a group symmetry of order gcd(m, |G|), the scalar shifts that keep Σ = O; it is achiral when its k-set is its own negative, and two loops that are negatives of each other are enantiomers; the shape also gets a bilateral score and a rotational order. Heredity: parent energy at replication is regressed on daughter energy four hundred ticks later, giving h²; across species, the base composition of the gene is correlated with lifespan and births, the hereditary factors. Anomalies: six signals are watched against a rolling window and flagged beyond 3.8σ, and an integrity audit checks the invariants every ten ticks. Phenomenology: every three hundred ticks the record writes what appears, in plain words: figures against ground by perceptual grouping, restlessness, warmth, rhythm, what changed, and the horizon of the next metamorphosis.

Energy and power. Every line holds an energy between 0 and 1. Walking generates it, standing still consumes it. Catalysis, replication, mutation and sequencing all spend it. Field energy is the sum over lines; power is the net energy flux per tick, positive when the field charges and negative when it burns.

Sound. The audio indicator turns on a drone, a breath and a reverb. Every event becomes a sound particle placed by x and sunk by y.

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