Reading

If f is Lipschitz with constant L, it fits cell-to-cell at lookahead c once L·δ_t+c < ℓ_t — per-level fitting, which is readability outright where cells nest. Necessity is coarse: a readable map need only be Lipschitz at cell scale — pointwise Lipschitz follows where the cells are metric balls, and the leading window has maps readable at lookahead 0 with unbounded pointwise ratio. Three cover shapes give three regimes — balls of an ultrametric (ℓ_t = δ_t, fitting automatic, margin 0); a partition of a connected space (ℓ_t = 0, the lemma silent — readability a per-map alignment question); an overlapping cover with ℓ_t = ρδ_t (fitting bought at margin log_b(1/ρ)). At overlapping covers a redundant cell's children keep the overlap only in its interior, so the committed stream departs from the per-level bound by at most one digit — both directions realized at scanned witnesses; the bound is exact where cells nest. And readability is metric, not topological: even-position digit extraction is uniformly continuous — readable only at a modulus that grows with depth — and readable at no bounded lookahead.

Scope. Rule at scanned scopes; the two-pole criterion's regimes are its positional instances.

verifier: explore_reading_geometry.py

At partition covers the lemma is silent, and the permanent-wall half of the alignment question is answered by the cover's vertex set ∂ — the cell-boundary points, which stay cell endpoints at every deeper level. For f continuous and strictly monotone at an irrational input y, the emitted digit count grows without bound iff f(y) ∉ ∂. Off ∂ the image sits strictly inside a cell at every depth, and the shrinking image intervals fit those cells one after another; an image on a vertex instead straddles it forever — a vertex lies interior to every image interval, and its two sides agree only to the vertex's own depth — so emission freezes at a computable constant, where a rate stall only slows it. A map's permanent walls are exactly f⁻¹(∂) ∖ ∂, its rationalizing set. Redundant covers dissolve every wall: with a positive Lebesgue number no uniformly continuous map walls, an image interval smaller than that number fitting inside some cell wherever it lands. Wall exposure is therefore the size of the vertex set, and the continued fraction's — all of Q — is the largest a rational-vertex grid can have (base b: Z[1/b] ⊂ Q). The cross-window witness: x² walls permanently at √(1/3) in the continued-fraction grid and reads on in base 10.

Scope. Rule at scanned scopes — the maps x² (walls at every scanned √r) and √x (wall-free: a rational square root forces a rational input); the cure realized at every scanned wall of x².

verifiers: explore_cf_nonlinear.py, explore_cf_redundant.py

An integer Möbius map with det = ±1 reads continued-fraction streams at bounded digit-delay: every nonnegative-entry unimodular map is a positive word in three exact transducers — the Stern–Brocot factorization — and delays add over a fixed word. A map with det ≠ ±1 is not boundedly readable: every one owns a mismatched witness. The sharp witness is 2φ = [3; 4, 4, 4, …] with φ³ = 2 + √5 exactly, so emitting t quotients of 2φ consumes 3t quotients of φ — delay linear. Non-unimodularity does not stall every orbit — (2x, √2) reads at delay zero, since 2√2 keeps √2's rate — it guarantees witnesses exist. Necessity is proved at quadratic streams: readability transports through unimodular factors (Smith normal form), reducing the gate to the scalings x → nx, where φ is a universal witness — a matched rate would force the multiplier automorph of nφ to be a power of φ, which floor rigidity (the period trace is strictly increasing in every digit) reserves for the all-1s word alone, so nφ's stream would be φ's own — but the multiplier ring sits inside the lattice, and φ ∉ Z + Znφ. Serret's theorem — equal CF tails ⇔ GL(2, Z)-equivalence — is the classical shadow: readable multiplication is, again, the window's native units.

Scope. Positive direction proved for maps factoring as nonnegative words, and at quadratic streams outright, with no word route needed; sign-crossing words at general streams unscanned. Necessity proved at quadratic streams from proved pieces (floor rigidity, rate forcing, Serret transport).

verifiers: explore_cf_window.py, explore_cf_conductor.py, explore_cf_flow.py

At a quadratic stream the rate is arithmetic. The tail's period-matrix eigenvalue η equals ε^i(y) — ε the field's fundamental unit, i(y) the unit index of the multiplier order of the lattice Z + Zy — so the cell-scale rate is r(y) = 2 i(y) R_K / ℓ(y) (R_K the regulator, ℓ the period length), and a measured stall ratio is (i_out/i_in)·(ℓ_in/ℓ_out). The delay is an instrument: the stall slope of x → nx measures the unit index of a conductor order. Where the classical index ceiling is not attained, the law follows the realized index and the deficit is the order's ring class number against the field's. And the law is map-blind: any monotone C¹ map with nonzero derivative is conductor-priced at quadratic orbits — the map enters only through which output orbit it selects.

Scope. Rule at 17 measured pairs including four predicted before measurement; the automorph identity η = εⁱ exact at all 53 scanned pairs (the eigenvalue identity itself is classical); map-blindness proved at the flow layer's scope, measured within 1% at all six scanned nonlinear pairs.

verifiers: explore_cf_conductor.py, explore_cf_nonlinear.py, explore_cf_flow.py

Every scanned window's digit process is the shift of an expanding map whose per-digit scale ln |T′| is the roof of a suspension flow, and the flow's entropy is exactly 1 (Rokhlin's and Abramov's formulas): entropy per digit equals scale per digit, a positional window being the constant-roof case. The continued fraction's suspension is the modular geodesic flow (Artin, Series), where a quadratic tail rides a closed geodesic of flow period 2 ln η = 2 i R_K per tail period — the conductor rate is the per-orbit return-time mean (the roof identity: over one tail period the product of complete quotients equals η, verified exactly at 25 tails). On this flow the commitment deficit D(n) = scale(image interval) − scale(committed cell) obeys 0 ≤ D(n) ≤ 3 ln(A + 2) + O(1) for monotone C¹ maps at badly approximable non-vertex outputs (quotients ≤ A): every reader is scale-synced regardless of the map — the scale clock always syncs, only the digit clock forks. Rate forcing: at an eventually periodic tail with quadratic output, emission is (r_in/r_out)·n + O(1) with the ratio rational — digit delay is bounded iff the rates match, else linear with the mismatch sign, both signs realized. At a wall the deficit grows linearly instead: measured slope at (x², √2) is 1.7627 = 2 ln(1 + √2) to four decimals — the cusp excursion that never returns.

Scope. Both proved at stated scope — the bound by scale accounting plus bad approximability, the forcing verified at ten pairs including two witnesses constructed to run ahead. The ergodic vocabulary (suspension, roof, Abramov's formula) is classical import; the reading layer on top is the content.

verifier: explore_cf_flow.py

The window's native redundant cover adds to the Stern–Brocot cells the straddle chains: at a vertex v with Farey parents (l, r), the chain cell S_k(v) runs between the k-th mediants and contains v in its interior at every k. The alphabet is the exact-real-arithmetic classic — the root straddle is the image (1/2, 2) of the modified Stern–Brocot digit M (Niqui's {L, R, M} digit set), and productivity for Möbius maps is classical (Gosper, Vuillemin, Edalat–Potts); what the overlap does and cannot do is the content. The ladder law: the cover cells containing a point z are z's own cell ladder plus a chain segment of exactly K(v, z) = ⌊1/(q_v²|z − v|) − q_far/q_v⌋ rungs at each approached vertex — at a convergent, K equals the next partial quotient verbatim; at a rational the chain is infinite. The cover is graded: rank steps by +1 along every child relation, so overlap adds routes, never length. The two gates then split. The wall gate falls: x² reads √2, √3, √(3/2) through the chain with never-freezing emission, ln k slope equal to the input's scale rate to four decimals — the emitted stream is an infinite redundant representation of the rational output, the continued-fraction mirror of 0.1999… The rate gate stands: at the scanned pair (x/2, 2φ) — the staller of the tree clock, delay counted in Stern–Brocot tree steps rather than in quotients, its emission slope 3/4 — the maximal rank of a cover cell containing the image equals the plain tree position at every sampled depth — ladder spacing is orbit data no overlap changes, so rate forcing survives redundancy verbatim. And a delay's sign belongs to the clock: the inverse reading (2x, φ) of the same two streams stalls at 1/3 in the quotient clock and runs ahead at 4/3 in the tree clock, and (x/2, 2φ) does the reverse (φ³ = 2 + √5 makes all four slopes exact) — only the mismatch belongs to the orbit, and the scale clock is the invariant ruler.

Scope. The ladder law's closed form exact at every scanned node (510 nodes × 12 depths); the rank grading a property on every child relation; the wall cure and the rate gate rules at scanned scopes.

verifier: explore_cf_redundant.py

Every window here gates its multiplication-like maps by the same shape: readable at bounded delay iff the map extends to a Lipschitz self-map of the window's completion. A window's completion is the limit object its nested cells define — the infinite digit strings, with the finite ones dense inside — and it is not the same object at every window (Completions). Every row of the table states that gate instead as unit-ness, and rightly: a window whose digit string carries a shift action has a native unit group, and there the two gates coincide. What the rows cannot show is which of the two is doing the work — the family below answers that, and the answer is not unit-ness.

Nonlinear maps sit under a second gate, orthogonal to units — the wall criterion: x² reads φ at bounded delay (its input and output orbits share a rate) while walling permanently at √2 (its image is a vertex). Redundancy splits the gates by shape, and the split is verified at both redundant covers. The redundant leading window dissolves its gate wholesale, because positional roofs are constant — scale rates always match there, so the only gate was boundary alignment, which any positive Lebesgue number kills. The redundant continued-fraction cover dissolves only the wall gate — its roofs are orbit-pinned, so overlap extends terminated ladders (rationals: walls cured) while never raising the reachable rank at an irrational (rate forcing survives). One purchase — a positive Lebesgue number — two fates, decided by whether the window's roof is constant or carries orbit data.

Scope. The gate's iff is a theorem: delay c is exactly Lipschitz constant 2^c in the agreement ultrametric on digit strings, and a Lipschitz map on a dense subset of a complete space extends uniquely — what it leaves between Lipschitz and discontinuous is the class the reading lemma's even-position extraction inhabits (Completions). Each row a rule at its own scopes; the gated half is a theorem at Zeckendorf (×2), and the last row's own theorem families are stated where they are proved (Quadratic windows). The redundancy split is a rule at the two covers scanned.

verifiers: explore_dual_lipschitz.py, explore_cf_window.py, explore_cf_redundant.py