Relational Echo Calculus (REC): A Transformation-First Mathematical Communication Prototype

Introduction: Premise and Prior Work Arithmetic-based interstellar primers, including Lincos, precede this proposal; attempts to decode interstellar messages already discuss symbol relations and functions . Research on emergent agent communication and generalization is well established . The transformation group below is likewise standard . Our contribution is an experimental package and test protocol; we do not claim the underlying algebra is new. This Internet-Draft is a faithful conversion of the author's manuscript "Relational Echo Calculus (REC): A Transformation-First Mathematical Communication Prototype", Version 0.1, dated 21 October 2026 (session REC-20261021-01; status: exploratory design; originality: not established; provenance: user-directed and AI-assisted drafting and implementation). Mathematical notation has been rendered in plain text. The figure of the original manuscript is described in prose. It is published for discussion; it is not a protocol specification.

Definitions and Operations Fix n >= 1, and let the underlying vertex space be V_n = {0,1}^n. A "motif" is a finite subset M of V_n; empty motifs are allowed. A "bond" connects x, y in V_n when d_H(x,y) = 1, where d_H(x,y) is the number of coordinates i (1 <= i <= n) with x_i != y_i. An "echo" is an action on motifs, induced by The set of echoes is the automorphism group G_n, isomorphic to (Z_2)^n semidirect S_n, of order 2^n * n!. Define "fusion" A |_| B = A union B and the "resonance signature" where the notation denotes a multiset (multiplicities retained), not an ordinary set. This notation is purely descriptive and does not presuppose a receiver knows numbers. The transmission uses repeated before/after demonstrations to teach such concepts. Proposition 1 (Echo invariance). For all M contained in V_n and all T in G_n, R(T(M)) = R(M); also the multiset of degrees in the subgraph induced by M is invariant. Moreover, T(A |_| B) = T(A) |_| T(B). Proof. A coordinate permutation leaves the count of differing coordinates between any pair unchanged. XOR with a fixed vector cancels when the two resulting points are compared, hence d_H(Tx,Ty) = d_H(x,y). All pairwise distance multiplicities and distance-one edges are preserved under a bijection. The last identity follows because the image of a union under a function is the union of the images. Proposition 2 (A coarse-signature collision). There are motifs with identical resonance signatures that lie in different G_4-orbits. Proof. Interpret integers as four-bit vertices. Choose A = {0,1,2,12} and B = {0,1,6,7}. Direct enumeration gives R(A) = R(B) = {{1,1,2,2,3,3}} (multisets). In A, two distance-one bonds share vertex 0, yielding induced degree multiset {{0,1,1,2}}. In B, the bonds are disjoint, yielding {{1,1,1,1}}. Degrees are invariant by the preceding proposition, so no echo can map A onto B.

Higher-Dimensional Interpretation The vertices V_4 are a tesseract's vertices in a combinatorial sense; V_5 similarly gives a five-dimensional cube. More dimensions increase the transformation family to 2^n * n! without fundamentally changing the primitive relation. The original manuscript includes a figure visualizing a projection of the four-dimensional hypercube and the two four-vertex motifs of Proposition 2; the projection can overlap unrelated points in two dimensions, and dashed-looking projected geometry should not be interpreted as actual intersection in four dimensions.

Glyph Interpretation Experiment The symbols g1, g2, g3 designate three hidden echoes. An observed example consists of a symbol, a visible before-state motif, and a visible after-state motif. A white-box symbolic reference decoder enumerates G_4 and intersects the compatible echo sets across demonstrations. Unique recovery in this constrained setup establishes only internal consistency; it does not demonstrate genuine alien-language acquisition, general agent learning, or message universality. Reproduction commands and generated results are in the README.md and data.json files of the accompanying research package.

Competing Communication Hypotheses Conjecture (H2: Invariance-first learning advantage). With matched information budgets and target tasks, agents given transformation-first instructional demonstrations decode unseen structural relationships more reliably than agents given arithmetic-first instruction. We compare this to H1 (arithmetic-first superiority) and H3 (hybrid superiority). The primary task and its correct answer must be identical across conditions. Primer content and ordering vary, with matched supervision, signal length, noise, and training compute. See PROTOCOL.md in the accompanying research package for a blind comparison design, exact primary metrics, negative controls and held-out dimensions.

Falsifiers, Contradictions and Evidentiary Limits The H2 advantage claim fails under adequately powered, information-matched testing if H2 does not significantly improve decoding across designated recipient architectures. H3 fails if the hybrid does not outperform the best single-method comparator. Null controls with randomly reassigned symbols should yield chance behavior, otherwise leakage or structural shortcuts are suspected. A claimed decoding improvement is not evidence of nonhuman authorship or physical alien intelligence. Equal distance signatures do not certify symmetry equivalence; the exhibited collision is a safety case preventing false promotion. No cross-agent experiment or blinded independent validation has yet occurred.

Provenance, Novelty and Reproducibility Established: hypercube group actions, Hamming distance, invariant multiset calculations. Derived: the stated collision and its proof. Conjectural: the comparative communication advantage. Potential project-specific synthesis: naming, ordering and benchmark formulation, whose global novelty remains unverified. Test statuses are recorded separately for exact toy checks (run) and planned H1/H2/H3 experiments (not run). Attribution is to the research lead as project sponsor, with explicit AI assistance, and does not imply independent invention by a human researcher.

IANA Considerations This document has no IANA actions.

Security Considerations This document defines no protocol and no on-the-wire format. The considerations below concern the integrity of the proposed experiment. Decoding experiments are vulnerable to leakage: if answer keys, symbol assignments or expected outcomes are available to the agents under test (for example through training data or shared context), measured decoding success can reflect memorization rather than inference. Blinded comparisons therefore need held-out answer material, randomized symbol reassignment controls and documented custody of evaluation keys. Equality of a coarse invariant (such as the resonance signature) must not be treated as proof of equivalence; Proposition 2 shows that doing so admits false matches. Results from synthetic, AI-generated messages must not be presented as evidence about real signals.