Recursive Ray Echoes in a Bounded Lattice

Recursive Ray Echoes in a Bounded Lattice By Dean Kulik Qu Harmonics. [email protected] From Origin Seed to Harmonic Echo Field (Byte 1 → Byte 8) When a ray is launched inside an 8×8 bounded box, each wall reflection “encodes” the ray’s direction into a growing interference pattern. We can think of Byte 1 as the origin seed (the initial directional vector), and Byte 8 as the resulting echo field after multiple recursive bounces. Each bounce mixes the ray’s trajectory like a round of a hash function – the ray reflects (folds) at the boundaries, preserving information about its direction but scrambling its path across the lattice. Over successive reflections, the path’s imprint evolves into a stable pattern that effectively “remembers” the initial vector. The final Byte 8 output is a complex lattice pattern encoding the input direction, much like a cryptographic hash encodes its input. This visual echo field acts as a form of symbolic memory of the ray’s origin. Reflection as a Hashing Operation: Each wall bounce flips the ray’s direction in that axis (like a bit flip) and constrains the coordinates modulo the box size. In code terms, one can simulate this by taking the position modulo 16 (since an 8-unit box reflected is like a 16-unit period) and folding values above 8 back down (if x > 8: x = 16 - x; dx = -dx). This is analogous to how a hashing round folds data within fixed bounds. The ray’s direction vector is effectively “hashed” through recursive reflections – small changes in the input angle produce entirely different path patterns (an avalanche effect), yet the final pattern deterministically depends on the initial vector, preserving its “memory” in distributed form. Lattice Resonances: When Paths Fill the Box vs. Collapse As the ray ricochets, it may either (a) cover the lattice nearly uniformly, or (b) fall into a repeating orbit that traces a limited set of lines. The outcome depends on the rationality of the direction vector’s slope (the ratio of its x and y components relative to the box). This is a classic result in billiard dynamics: “a billiard trajectory with a rational slope is periodic, while one with an irrational slope is dense in the square.” In our context: Harmonic Full-Field Coverage (Ergodic Trajectory): If the ray’s direction is incommensurate with the box dimensions (effectively an irrational slope), the path will not repeat and eventually fill the 8×8 field densely. Each bounce lands at a new location, and over time the trajectory “remembers” every corner of the lattice. The echo field becomes a nearly continuous web of lines – a full memory field where the ray visits almost every region. This is the ideal “truth field”: the input’s influence is spread evenly, leaving a complete imprint. In practice, a direction vector with a high-order rational ratio (very large coprime integers) approximates this dense coverage. For example, an initial direction of [3.0, 0.35] (slope ≈0.1167) produces a path that zig-zags through nearly the entire grid without obvious repetition. The plot below shows how this ray sweeps across almost all lattice lines, creating a dense mesh of echoes: Ray with input direction [3.0, 0.35] yielding near-complete coverage. The blue trajectory densely fills the 8×8 box with 576 segments, producing an echo field that “remembers” the input angle. Red dots mark bounce points on the boundaries. Why 0.35 gives nearly perfect coverage: The angle corresponding to dy/dx = 0.35/3.0 ≈ 0.1167 (about 6.7°) is essentially a high-order rational – in fact 0.35 is 7/20, so the slope is 7/20 ÷ 3 = 7/60 (a rational with a large denominator 60). This means the ray must complete a long sequence of bounces (60 horizontal crossings for every 7 vertical crossings) before it even begins to repeat. In 576 steps it still hasn’t closed a cycle, so it visits a great variety of points. The result is a nearly uniform lattice coverage with finely spaced lines. Each reflection adds a new “layer” to the pattern until the field is saturated with the ray’s presence. In essence, 0.35 hits a sweet spot: it’s rational enough to eventually form a resonance, but with such large period that the interim trajectory appears ergodic (covering the field uniformly). The angle 0.35 acts like a phase attractor for full coverage – at or near this value, the system “converges” to a complete memory field before any repetition interferes. Periodic Lattice Resonance (Partial Coverage): If the ray’s direction slope is a simple rational (small integer ratio), the path will eventually close on itself and start repeating a fixed cycle. In these cases, the trajectory resonates with the box dimensions, forming a stable periodic orbit – a harmonic pattern – but it only covers a subset of the lattice. The memory field is partial: the ray keeps revisiting the same lines, leaving portions of the box untouched. For example, an initial direction [1.0, 0.5] (slope = 0.5, i.e. 1/2) yields a periodic orbit. This slope equals 1/2, meaning the ray’s path aligns such that after a few bounces it returns to its starting point. The result is a striped echo pattern: the ray oscillates between a small number of track lines. The figure below illustrates this partial collapse – the path forms repeating diagonal bands and neglects other regions: Ray with input direction [1.0, 0.50] (slope = 1/2). The trajectory is periodic (1088 segments shown over many cycles) and confined to a set of parallel bands. Red markers indicate the repeating bounce positions. This is a stable lattice resonance but covers only part of the box. In this case, the slope 0.5 = 1/2 leads to a short 4-bounce cycle that closes on itself (hitting a corner and retracing). The pattern is a harmonic standing wave in the box – analogous to a resonant mode where only certain “nodes” are hit. The ray’s memory field here is stable but incomplete: it remembers the input direction as a simple repeating pattern. Mathematically, such a trajectory corresponds to a rational slope (p/q with small integers). It will hit the same boundary points in a fixed sequence, forever cycling. The path has collapsed to a limited set of lattice lines (a partial memory field). Degenerate Interference (Self-Collapse): In extreme cases, the path can retrace itself exactly, yielding almost no coverage. For instance, a 45° angle (slope = 1) will bounce corner-to-corner and follow the same diagonal line back and forth. This destructive interference is a trivial resonance: the ray keeps hitting its own path. The echo field reduces to a single line (or a small set of lines) repeatedly traced – the system has minimal memory of the initial input (basically just one line’s worth). Similarly, purely horizontal or vertical inputs (0° or 90°) simply oscillate back and forth along one axis. These are edge cases where the trajectory’s “echo” contains little information – the input wasn’t trust-aligned to excite the field. In terms of memory, these inputs produce a nearly empty or redundant field (all bounces overwrite the same track). Why do some angles fill the box while others don’t? It comes down to frequency matching with the 8×8 geometry. A ray direction can be seen as having “notes” or frequencies in the x and y directions. If those frequencies are rationally related (forming a simple ratio), the motion in X and Y will sync up periodically, creating a repeating orbit (like a repeating melody). If they are irrationally related, the motion never quite syncs, and the ray will sample new positions indefinitely – akin to a quasi-random sequence. In wave terms, the box supports certain standing wave harmonics; a ray whose angle corresponds to those harmonics will resonate in a fixed pattern. Non-harmonic angles cause a superposition of many modes, effectively spreading the ray everywhere. This is why a slight change in angle can cause a bifurcation from ordered to disordered behavior. Phase Thresholds and Symbolic Convergence Not all “full coverage” trajectories are equal – some achieve a more uniform distribution of bounces than others. We observe special phase thresholds (critical angles) at which the echo field qualitatively changes. The angle 0.35 (as a fraction of the horizontal) appears to be one such threshold that produces a symbolic convergence: the ray’s path converges to a stable, all-encompassing pattern (the “truth field”). We call 0.35 a collapse attractor because at this angle the system collapses any randomness into a coherent coverage of the lattice. Nearby angles may exhibit two distinct regimes (a bifurcation): for example, an angle just off 0.35 might start to wander in a more complex way or split the coverage into two interleaved patterns (a superposition-like outcome). At 0.35 exactly, the pattern “locks in” to a clean resonance that still covers the field completely. In effect, the ray finds a harmonic equilibrium between repeating and exploring. This can be interpreted as the input being trust-aligned – tuned precisely to the box’s dimensions so that the output field is an orderly reflection of the input. Delta-Phase Map: One way to visualize this is to consider a map of coverage vs. angle. As the ray’s angle (phase) varies from 0° to 45°, the coverage goes from zero (just sliding along a wall) up to maximal (some irrational angle). Rational slopes act like “spikes” or resonances on this map – at those angles the pattern repeats (dropping coverage). For extremely high-order rationals (angles very close to irrational), the coverage is nearly full because the repetition period is huge. Thus the coverage graph oscillates with finer structure as angle changes. Angles like 0.35 (7/20 of horizontal component in our normalized units) stand out because they combine a moderately high denominator with alignment to the grid: the trajectory hits every possible horizontal strip (20 distinct heights in 8 units, each 0.4 apart) before repeating. This granularity yields a