# Tip-point atlas Bernat Espigulé · 23 September 2026 This edition links the two recovered family registries to a tip-point slicer. It contains 375 marked constructions and retains all 599 source records as selectable charts. These are not asserted to be 375 inequivalent topological types, nor a classification of all stable families. ## Run Open the separately supplied `Tip_Point_Atlas.html` for an offline, single-file edition. For this directory, serve the files over HTTP, for example `python -m http.server 8000`, and open `http://localhost:8000/`. No third-party JavaScript, fonts or analytics are loaded. A modern browser supporting Web Workers, BigInt and DecompressionStream is required. WebGL2 is optional; the Canvas renderer is the fallback. ## Navigation A, B and C are independently draggable parameter locators. Dragging one updates the attractor and coordinate readouts without restarting the field. The family selector, construction/orientation filters and source-chart selector expose every recovered record. A family may have multiple source charts; switching charts can select a different isolated algebraic branch or a different stored cover. Those charts are not flattened into a single purported complete stable domain. The left panel projects selected binary family placements from four real parameters. Up to twelve families can be superposed. The cut and contraction-weight band are display filters, not topological statements. Shift-click a selected-family chip to remove it. The source sample determines the colour, not an unproved topological class. Click or drag a parameter locator in the exterior slice. Drag the background to pan; use the wheel to zoom. Arrow keys move a focused locator; plus/minus zoom a focused canvas. Two touches pan/zoom the slice, but no physical touch-device test has been performed. On the attractor, Shift-click or enable `Pick a tip` to choose an addressed point; its finite prefix followed by the fixed-point tail becomes the tip-slice address. The point evaluation agrees with the sampled point, up to floating arithmetic. `Fit small patch` zooms a recorded rectangle. `Focus locator` rebases 3D geometry around the selected point before converting coordinates to Float32. This prevents small charts from collapsing merely because their coordinates are far from the origin. All source parameters remain Float64. ## Coordinates and identities For a zipper the source parameter is q; for the algebraic contact charts it is a. The displayed source slice uses c = 1/q or c = 1/a. The two canonical reciprocal map coefficients c1,c2 satisfy |c1|>1 and |c2|>1. For orientation-reversing maps, canonical normalization also transforms the coefficient phase; it is not obtained by blindly inverting the unnormalized coefficient. Primary family IDs use the recovered CTF identifiers. Dotted IDs and legacy labels remain searchable aliases. Source chart IDs are derived from the source edition and exact source record. Source SHA-256 values, original representative encodings, parent root isolators and source versions are retained. Moving A/B/C creates a numerical selected point; it does not mutate the original representative encoding or its identifier. Selection exports retain the exact binary-rational coordinates of the displayed numerical point. An implicit polynomial does not identify a branch on its own. `model.js` stays inside the imported source box, solves from its imported centre, checks the polynomial residual, and rejects roots outside the imported isolating square. This is a numerical evaluation of the recorded branch, not a new algebraic certificate. Exact parameter-box certificates remain inherited evidence. ## Tip-point sets and the complement Let R be a family of injective contractive IFSs, Sigma the full address space, pi_p the coding map, and E0 a fixed closed prescribed equivalence relation. Suppose E0 is valid for every p in R and compatible with adding and cancelling common prefixes. Extra identifications can then be detected between addresses with different first symbols. For an address omega, define M_extra(omega) = {p in R: some eta with eta_1 != omega_1 satisfies pi_p(omega)=pi_p(eta), (omega,eta) not in E0}. U = union_omega M_extra(omega), S = R \ U. All infinite addresses occur in this identity, not merely periodic addresses or topological endpoints. A countable equivalent union uses every finite product cylinder [u] x [v] disjoint from E0, with different first symbols: U = union_(u,v forbidden) {p: f_u(K_p) intersects f_v(K_p)}. An extra equality has a product-cylinder neighbourhood disjoint from E0 because E0 is closed. Conversely, any equality in a forbidden product cylinder is additional. Effective enumeration requires a decidable description of E0. Oriented dyadic address intervals supply this description for the marked zippers; it is not presumed for arbitrary families. For compact nested enclosures H_p,n decreasing to K_p, write C_uv,n = {p: f_u(H_p,n) intersects f_v(H_p,n)}. S = intersection_(u,v forbidden) union_n (R \ C_uv,n). The two quantifiers must not be interchanged. Finite survivors do not establish limiting intersections. Exclusion of a finite union does not establish the complement of the entire unstable set. ## Layers and evidence - **Known stable cover:** stored masks, rectangles, inherited analytic domains and real sections. These are not recomputed proofs. Absent coverage is not certified instability. - **Selected tip-point union:** ultimately periodic addresses, evaluated using their closed-form periodic fixed points. Membership is tested with inverse states. Points numerically indistinguishable from the prescribed contact are left unresolved rather than being promoted to a stability verdict. - **Forbidden contact prefixes:** complete forbidden address cylinders for zippers. On a supplied algebraic contact chart, the engine instead tests its finite nonreturn product partition. This requires the source return grammar and is not a newly enumerated complete coding relation. Unsupported or excessive grammars return a work-limit state. - **Fixed clearance:** cylinder pairs are discarded near a prescribed point only when a whole containing disk is inside its clearance ball. This is a positive-clearance computation, not the zero-clearance limit. Raster decisions are numerical tests at sampled centres, not proofs for full pixels. The main legend keeps retained survivors, finite exclusions and work-limit states separate. Green is exclusively inherited stable coverage. Prefix level is not the thesis finite-capture entry depth. ## Fast computation The earlier collinear Canvas app uses inverse states, but truncates the periodic expansion to 60 symbols and discards branches beyond its breadth limit. Those operations are not used for accepting exclusion here. Its parser also silently clamps symbols; the new parser rejects invalid input. The earlier general 4D app labels a depth survivor (and a noncontractive sentinel) as connected and bisects a radial threshold. This edition uses neither label nor an assumed monotone radial boundary. `fast.js` uses reusable typed-array depth-first stacks, scalar real/imaginary arithmetic, squared distance comparisons and precomputed map ratios. On a depth or work limit it retains an unresolved state. The selected tip search expands all opposing first branches. There is no beam-search pruning, probabilistic membership acceptance or branch dropping. Geometry runs independently of field computation. One requestAnimationFrame coalesces pointer updates. Deterministic addressed samples give coherent attractors while a parameter moves. The static atlas surface is not repainted during a locator drag; only its marker overlay is. The Canvas renderer coarsens source masks conservatively while orbiting and returns to the finer mesh at rest. This motion-level mesh is still a subset of the imported mask, not an interpolated fill of missing regions. Field jobs run in a dedicated worker, yield approximately every 8 ms and carry a parameter-view epoch. Resolution refines from 48 to 96 to the requested grid. Timers retain their generation; superseded jobs cannot replace the current field. Cancellation stops the search, retaining only a labelled previously completed field. Proof work uses a separate worker and is initiated explicitly. ## Exact point check The retained verifier requires identical affine semilinear return contractions f_U f_P f_U^{-1} = f_V f_Q f_V^{-1} = h, and complete finite exclusions of all nonreturn product cylinders. Equality includes translation and orientation, not just contraction ratio. The remaining return shrinks to the prescribed point. The browser proposes a finite tree and replays its rational bounds with BigInt. A separate Python Fraction verifier is included. This proves that the first-level intersection equals the prescribed point at that exact parameter. It does not certify a neighbourhood, a full stable set or every possible stronger coding-equivalence predicate. Implicit algebraic charts do not receive new proofs from this rational verifier. At q=(1+i)/2 in DD11, addresses `01(10)` and `10(01)` meet at 1/2, away from the prescribed q. The exact extra-contact button exports this rational equality. It is a point witness, not an open unstable region. ## Sources and authorship Bernat Espigulé introduced the tip-point-set formulation before this software edition. The official abstract for *Complex dynamics: connections to other fields*, Chęciny, Poland, 27–31 March 2023 is titled *On the tip-point set M(w) of complex-parametric families of self-similar sets* and explicitly describes M(w), unstable sets and a fast algorithm: https://holdyn23.mimuw.edu.pl/connections/pages/abs-conf/Espigule.pdf Recovered data: registry 2.0.0 (355 records) and registry 2026.09.21-r2 (244 records). Exact source hashes and record counts are in `Catalogue_Audit.json` in the source distribution. Earlier code examined includes `25-11-26_CF_TipPoint_Set.html`, `unilab26_canvas_app.html`, `25-12-04_Mn_Tree_Complement.frag`, the supplied Three Four-Dimensional Connectedness Loci Explorer, and the previous tip slicer. Source documents and existing rights remain with their respective authors. No additional license is asserted here. ## Validation and limitations The accompanying QA includes exact periodic identities, replay of retained exact point proofs, negative controls, comparisons of the new search with the previous cylinder implementation, every recovered representative, source-box rejection, mask-subset checks, pointer interaction, cancellation and exports. The report distinguishes the old source certificates from checks newly run on this software. The test browser refuses localhost and file navigation. The standalone HTML is therefore tested in memory, with its real Blob workers. WebGL2 was unavailable in that browser, so the current browser acceptance covers Canvas fallback, not a physical GPU. Headless callback timings are not claims of device FPS. The full inherited parameter-box certificate collection has not been rerun. Public deployment requires a separate HTTP/readback check.