# NEXAH Science Lab — Family Connection Map Date: `2026-09-30` Status: `CANONICAL_SYNTHESIS_MAP / NO_CLAIM_PROMOTION` Disposition: `V1_BASELINE_SUPERSEDED_FOR_NAVIGATION_BY_V2` Navigation note: this file is retained as the V1 baseline. The current map is [`FAMILY_CONNECTION_MAP_V2_2026-10-05.md`](FAMILY_CONNECTION_MAP_V2_2026-10-05.md), with an [interactive view](NEXAH_FAMILY_CONNECTION_MAP_V2.html) that adds ERITH, Inside/Outside, Tessarec, the F40/Hopf result, The Rest and the later Transversum/Euler fixture. Purpose: connect the currently converging research families without merging their evidence, domains or claim authority. ## 1. Why this map exists The Lab is increasingly reusing the same typed operations across arithmetic, representation, transition analysis, dynamical systems and control: ```text select -> encode -> observe transitions -> compare structure -> derive coordinates -> test events -> retain residuals ``` This is genuine methodological compression. It does not imply that all source systems share one mechanism, one geometry or one physical ontology. The map therefore records both: - where families are joined by an exact map or a reproducible result; - where they share only an analysis method; - where the proposed bridge remains untested; - where two visually similar objects must remain non-equivalent. ## 2. Connection types | Code | Meaning | Scientific consequence | |---|---|---| | `EXACT_MAP` | Declared deterministic map between typed objects | The output follows from the input and map; no empirical novelty by itself | | `EMPIRICAL_LINK` | Reproduced relation under a bound test contract | Valid only for the registered data, operator and scope | | `METHOD_BRIDGE` | Same comparison method can be used in both families | Shared language, not shared mechanism | | `OPEN_BRIDGE` | Testable connection has been formulated but not admitted | No result or architecture promotion | | `NON_EQUIVALENCE` | Similar representations have been explicitly separated | Prevents silent collapse of distinct objects | ## 3. The seven families ### F1 — Observation, representation and residual Core objects: ```text carrier, cut, frame, projection, record, fiber, reconstruction, residual ``` This family supplies the general comparison grammar. It includes Representation Fidelity, Records/Return/Residual, Observer/Frame/Aperture/Cut and RTTG-style provenance. Current status: strongest as a method and in bounded representation audits; external incremental utility remains separately gated. GEN01 adds one bounded exact implementation of the T5/T6/T7 relation path: typed objects are lifted to stored relation objects, adjacent relation objects are compared through one independently recomputed meta-level, and lossless base representations return exactly. Endpoint-only projection is formally non-injective in the registered expanded history space. The `mathcal G_0/mathcal G_1/mathcal G_2` notation remains case-local and stops at one meta-level. GEN02 extends that evidence from one recurrence family to four authored relation families under one exact field contract: directed successor, symmetric equivalence, role-ordered incidence and a ternary-intersection holdout. All stored records, meta-relations, destructive controls and three lossy baselines passed with byte-identical replay. This is an `EMPIRICAL_LINK` between heterogeneous typed-relation fixtures and the shared record grammar, not proof of shared semantics: the semantic registry remains family-specific, and the `n+1` Binder remains a separate unvalidated candidate. ### F2 — Finite arithmetic, address and boundary Core objects: ```text prime value, prime index, residue class, CRT address, finite state grammar, cube boundary pair, Shadow-Cube lift ``` This family supplies exact discrete carriers and feature maps. Root7, Mod6, Zero/Glyph/Iota, Vendessimal grids, palindrome half-edges and cubic boundary pairs remain separate constructions inside the family. Current status: much of the local arithmetic is exact or finitely auditable; the wider geometric and physical readings remain bounded. PIR01 adds one exact finite post-hoc census: in the first one million prime indices, the complete frozen `5–6–7 / 42–41–61` conjunction has the sole hit `n=6`. This remains an F2 arithmetic/address result. It is `NON_EQUIVALENCE` with a PMR mechanism and does not explain PMR01B/PMR01C admissions. ### F3 — Transition operator, current and recurrence Core objects: ```text transition counts, stochastic operator, stationary traffic, directional current, spectrum, invariant subspace, cycle space, Cycle Core ``` This family begins when an ordered source sequence is converted into a transition object. PMR01 and PMR01A belong here. Current status: PMR01 did not establish stable hard sheets. PMR01A found strong split-half stability in global spectra and normalized current, but not a panel-wide stable thresholded event alphabet. ### F4 — Phase, drift and flow coordinates Core objects: ```text raw circular phase, projected phase, spectral phase, phase increment, unwrapping, drift, winding diagnostic, embedded field ``` This family supplies continuous or circular coordinates for ordered motion. Lorenz, Roessler, Halvorsen and modular prime sequences can all be represented with phase coordinates, but those coordinates are not thereby identical. Current status: phase is an available comparative coordinate. A projection-independent universal phase has not been established. The Curve–Janus–ORION binding adds one exact, case-local coordinate fixture. For the historical curve `gamma(p)=(p sin(p)cos(p),p sin^2(p),p cos(p))`, the full Cartesian norm recovers `p` on `p>=0`, while `w=floor(p/2pi)` and `alpha=p mod 2pi` split the same state into winding/shell and residual phase. This is not a prime-specific effect: ORION already records the historical exceptional-prime reading as failed/demoted. CURVE00 verified the exact and numerical fixture with 15/15 checks and byte-identical replay. It also separates the centered K°-Grid from both the historical `kappa=56` layer channel and the later ordinal Kappa path. CURVE01 completed its fully frozen V0.10/V7 200,000-draw production path as a `VALID_NEGATIVE`. Representation, controls, replay and independent finalization passed, while the registered directional effect failed its effect-size, permutation, block and leave-one-out support conditions. The case is closed without a prime law, capability or automatic successor. The finalized scientific-result SHA-256 is `5a02d638d31036e0d4ec442623de0c04887b2221e563f08cb1bc41d8eb15985b`. ### F5 — Connectivity, embedding and topology Core objects: ```text graph connectivity, layout, embedding, quotient, homology, persistent topology, torus or Moebius candidate ``` This family asks whether a global topological object is recovered rather than drawn into the representation. Current status: connectivity and embeddings exist; layout-independent PMR topology has not been established. ### F6 — Synchronization, activation and control Core objects: ```text phase mismatch, gate score, IOTA activation, event alphabet, Kuramoto order parameter, control policy, target and cost ``` This family concerns collective timing or interventions after the underlying coordinate and event rule have been stabilized. Current status: demonstrators and exploratory analyses exist. PMR cross-modulus synchronization has not been executed because its admission gates failed closed. ### F7 — Validation, replay and claim governance Core objects: ```text baseline, constrained null, split-half stability, calibration, blind holdout, replay, result hash, stop rule, claim ceiling ``` This family is transversal. It does not provide another mechanism; it governs every bridge between F1–F6. Current status: operational and repeatedly used. It is the reason negative, mixed and withheld results remain part of Lab knowledge. ## 4. Current family spine ```text F1 OBSERVATION / REPRESENTATION | | EXACT_MAP or METHOD_BRIDGE v F2 FINITE ADDRESS / ARITHMETIC BOUNDARY | | exact construction of counts; | empirical stability of learned operators v F3 TRANSITION OPERATOR / CURRENT / SUBSPACE | | OPEN_BRIDGE under PMR01B: | gauge-aligned spectral phase v F4 PHASE / DRIFT / FLOW COORDINATE | \ | OPEN_BRIDGE \ OPEN_BRIDGE v v F5 CONNECTIVITY / TOPOLOGY F6 SYNCHRONIZATION / CONTROL F7 VALIDATION / REPLAY / CLAIM GOVERNANCE applies across every arrow. ``` This is a dependency map, not a proof chain. A downstream family is not admitted because an upstream picture looks compatible with it. ## 5. Connection register | From | To | Connection | Status | Present evidence boundary | |---|---|---|---|---| | F1 | F2 | A carrier is mapped to typed finite addresses or residues | `EXACT_MAP` when the map is declared | Address does not recover lost source history | | F1/F2 | F7 | Transversum/Two-Cut compares two directed readings of one carrier after declared transport into a common frame; antipode and persistent address are typed separately; the [Euler Antipode Two-Cut instrument](RESEARCH_AREAS/FIELD_GLYPH_NUMBER_ARCHITECTURE_STAGE_0_2026-09-14/NEXAH_EULER_ANTIPODE_TWO_CUT_INSTRUMENT.html) executes the bounded address fixture | `METHOD_BRIDGE`; `EXACT_MAP` for a declared geographic or even-clock involution | `A^2=id` does not imply invariance of every observable; the binder is not a third signal; Euler `13 <-> 3 mod 20` is address geometry only | | F1 | F1 | Typed relations are stored as objects and adjacent relation objects are joined by one typed meta-relation | `EXACT_MAP` in GEN01 and bounded `EMPIRICAL_LINK` across GEN02's four authored families | Shared field contract through one ternary holdout; family semantics remain explicit; not the full DEFG family, n+1 Binder or unlimited recursion | | F2 | F3 | An ordered residue sequence determines transition counts and an estimated operator | `EXACT_MAP` for counts; `EMPIRICAL_LINK` for stability | PMR01A supports spectrum/current stability on development data | | F2 | F1 -> F4 | Prime or integer parameters sample the declared historical curve; norm and phase-shell coordinates are then derived | `EXACT_MAP` for the declared curve | Prime sampling was not statistically exceptional in the historical board; no JANUS mechanism or aperture is inferred | | F2 | F3 | Shadow-Cube features alter the representation of residue states | `METHOD_BRIDGE` | Deterministic lift adds no within-modulus information over the full residue state | | F3 | F4 | A two-dimensional invariant subspace defines an aligned spectral phase | `EMPIRICAL_LINK` for a subset; panel gate failed | PMR01B found stable phase maps for 6/10 moduli, but only 2/10 also passed all constrained nulls | | F4 | F5 | Phase ordering and winding recover global topology | `OPEN_BRIDGE` | Phase embedding or a torus rendering is not topology evidence | | F4 | F6 | Stable phase mismatch or increments define events and synchronization statistics | `OPEN_BRIDGE` | Existing percentile events are exploratory; frozen rules are required | | F5 | F6 | Topology constrains synchronization or control | `OPEN_BRIDGE` | No NEXAH confirmatory result presently binds this arrow | | F1 | F3–F6 | The same cut/projection/residual grammar audits each derived representation | `METHOD_BRIDGE` | Shared audit method does not imply shared dynamics | | F1 | F3 | Declared mirror normals and a seed define reflection generators whose words act on a finite state orbit | `EXACT_MAP` in the Coxeter Orbit Lab | Exact for the selected ADE/H3 construction; it does not establish that an unrelated grid uses the same reflection group | | F1/F3 | F5/F7 | The Word & Orbit Sequencer executes generator letters, words and powers while retaining carrier, orbit phase, return test and representation boundary | `EXACT_MAP` for the registered E8/H3 actions; `METHOD_BRIDGE` for the Fibonacci/Euler shared-index lanes | Animation exposes computed transitions but adds no evidence; concentric lanes do not identify their generators | | F3 | F5 | A finite generator orbit is rendered as a labelled transition graph; for H3, nearest-neighbour hull edges recover the registered Dodecahedron, Icosidodecahedron or Icosahedron | `EXACT_MAP` in the bounded demonstrator | The orbit and hull are mathematical constructions; their 3D→2D rendering is a projection, not extra evidence | | F2/F3 | F5 | LOKI's four sign-coordinate flips generate the Q4 hypercube graph, decomposed into Q3− and Q3+ with eight fourth-coordinate Green Bridges | `EXACT_MAP` as Coxeter type `A1^4` | Exact graph/group presentation; the screen drawing remains a projection and no physical 4D claim follows | | F1 | F5 | MIWA applies an observer-local Iota cut and Pearl aperture to the declared LOKI/Q4 carrier | `EXACT_MAP` for the coded selection; `METHOD_BRIDGE` across other carriers | Selection, projection and residual belong to the observer record and do not alter or identify the source group | | F1 | F5 | The 12 H3 dodecahedron faces define 12 directed aperture normals, identified by duality with the 12 icosahedron vertices; azimuth, elevation and cone opening select active channels | `EXACT_MAP` for polyhedral duality and the declared selector; `METHOD_BRIDGE` to an astrolabe | Astrolabe grid is a coordinate/projection interface, not evidence for a historical dodecahedral instrument or physical coupling | | F1 | F5 | Roman layer stack keeps the 20-node H3 carrier between a declared PP (Projection Plane) and RP (Return Plane), with 12 apertures grouped into 6 exact opposite face pairs; Ghost-Link and iridic layer expose projection/return choices | `EXACT_MAP` for H3 counts and opposite pairs; `METHOD_BRIDGE / OPEN_BRIDGE` for PP/RP, Ghost and iridic expression | Aperture-size profile is a demonstrator parameter; no historical instrument function or physical identity is claimed | | F1 | F5 | Groma–Astrolabium–Dodekaeder research question separates astronomical direction frame, terrestrial right-angle basis, exact H3 axis operator and artifact-specific two-aperture optics | `EXACT_MAP` for ideal geometry; `EXPERIMENTAL` for calibrated replicas; `OPEN` for historical coupling | A working reconstruction proves possibility only; historical use requires independent archaeological evidence and specialist review | | F2 | F3/F5 | Prime values may label nodes or select a finite carrier before an operator is applied | `METHOD_BRIDGE` / `OPEN_BRIDGE` | Prime labels do not cause Coxeter symmetry; a claimed identity requires an explicit bijection preserving generators or adjacency | | F7 | F1–F6 | Nulls, replay, holdout and stop rules govern admission | `EMPIRICAL_LINK` / governance | Reproducibility does not by itself establish truth or universality | ## 6. What has actually compressed The following concepts can now be treated as one reusable methodological spine: ```text typed source -> declared observation or encoding -> transition or relation operator -> invariant and residual comparison -> independent null and replay -> bounded result or fail-closed stop. ``` This compression is supported across multiple Lab families. The following concepts must not yet be compressed into identities: ```text residue phase != physical phase operator subspace != sheet alphabet cycle drawing != unique cycle decomposition phase embedding != emergent torus shared phase notation != common mechanism event coincidence != synchronization synchronization != causal coupling Shadow-Cube lift != new within-modulus information prime-labelled grid != Coxeter root system similar polygon != shared reflection group projected hedron != physical object ``` ### Coxeter Orbit Lab bridge — 5 October 2026 The [Coxeter Orbit Lab](CASE_STUDIES/E8_DEMOS_INTERACTIVE_ORIENTATION_2026-09-21/COXETER_ORBIT_TOPOLOGY_LAB_2026-10-05/README.md) is the maintained teaching and inspection surface for this bridge. It makes one cross-family chain executable without merging the families: ```text F1 declared source/frame -> F3 involutive generators and words -> finite orbit / return structure -> F5 graph, hull, duality and projection -> F7 explicit status and non-equivalence boundary ``` This is stronger than a visual analogy where the generator action and orbit are computed exactly. It remains only a `METHOD_BRIDGE` where an existing NEXAH grid or prime family is compared by appearance, count or labels without an explicit structure-preserving map. The [Word & Orbit Sequencer](CASE_STUDIES/E8_DEMOS_INTERACTIVE_ORIENTATION_2026-09-21/COXETER_ORBIT_TOPOLOGY_LAB_2026-10-05/NEXAH_WORD_ORBIT_SEQUENCER.html) adds an execution layer without changing that status. Its inner clock records letters within one word; its outer clock records completed word powers and orbit phase. The registered E8 word returns after 30 powers and partitions the 240 roots into eight 30-cycles. The H3 `r1-r2` word returns after five powers on the selected dodecahedron orbit. A separate Fibonacci/Euler display shares only an index and remains a `METHOD_BRIDGE`. ## 7. Relation to closed Labreports and active pilots The Labreport Index records only closed Lab knowledge. Its existing reports populate primarily F1, F2 and F7, with bounded contributions to recurrence, embedding and representation. PMR01, PMR01A, PMR01B and PMR01C are development pilots and are not silently converted into closed Labreports by this synthesis. Their identical replay hashes and fail-closed decisions are registered evidence. PMR01B tested the F3-to-F4 bridge: six moduli passed phase-coordinate stability, four passed all nulls, and only `17,41` passed both. The required panel count was five. PMR01C then tested incremental prediction. All ten phase-context arms improved development log loss under the frozen baseline, but only `13,17` beat the phase-label null; five were required. The nominal gain is therefore not panel-wide specific to the learned spectral geometry. ## 8. PMR01B/PMR01C result and next experiment admission PMR01B tested the arrow: ```text F3 transition operator / stable subspace -> F4 gauge-aligned spectral phase ``` PMR01C tested the incremental-utility question and also failed its panel gate. Therefore no F5 topology, F6 synchronization or blind test is admitted. The next responsible action is a non-confirmatory baseline-adequacy review because all moduli selected the largest allowed backoff concentration. No further PMR experiment is activated by this map. ## 9. Claim boundary Permitted: > Several previously separate NEXAH research lines now share a typed > observation–operator–comparison–validation spine. Their family relations > can therefore be registered without merging their evidence or mechanisms. Not permitted: - all NEXAH families are one discovered natural law; - the family spine proves a universal phase or topology; - visual similarity closes an open bridge; - PMR01A already validates synchronization; - family compression promotes a demonstrator into an empirical result.