Observatory

K/Q₂ totally ramified of degree e, f = 1 (so i* = e), any arrival class (c, m) with c·2^m = e. Every arrival spectrum is rigid below its freedom rung, and its minimum is one general staircase on the digits of w = 2/π^e. An intermediate class (c > 1, any even e) reads the digits w₁ … w_2^m−1: L = 2e + min(j₁, 2^m), j₁ the position of the first nonzero digit. The window's own top class (c = 1, forcing e a 2-power) reads w₁ … w_3e/2−1 — half again the window's depth: the walk stops at the first rel r with nonzero effective digit, which is w_r everywhere except the two skeleton positions r = e/2 and r = 5e/4 (one position at e = 2), where it is w_r + 1 — that added 1 a constant, contributed by the binomial game itself and by no digit; freedom at 3e/2. The skeleton positions are the blind rungs: a field whose digits vanish there stops anyway — the pure fields x^e − d, v(d) = 1, collapse at rel e/2. Corollaries: w₁ = 1 locks every class with m ≥ 1 rigid at 2e + 1 (the starters m = 0 grade at the same floor); the unique stopless digit vector is nonzero exactly at {e/2, 5e/4}, and ζ_2e carries it — the vector of 2/(ζ_2e − 1)^e is exactly that skeleton at every 2-power depth. The top readout grows linearly with the window — 2, 5, 11, 23, 47 digits at e = 2, 4, 8, 16, 32.

Scope. Proved for every e and every class by pricing each monomial of the game with its exact multinomial 2-content: six monomials enter below 3e/2, and the staircase, the lock, the blind rungs, and the no-stop vector are one enumeration. f = 1 is essential (digit idempotence; f ≥ 2 Frobenius-twists the digits and the pair-cancellation dies). Engine: the enumeration brute-checked to e = 64; the e = 32 face — five classes past the starter, digit windows 1, 3, 7, 15, 47 — confirmed sight-unseen, ζ₆₄'s vector {16, 40} called by the corollary before measurement.

verifier: explore_readout_proof.py

The landing of an arrival class is the avoidance value of its torsion constellation. The primitive p^m+1-th roots of unity all sit at v(ζ − 1) = c, the class's own level; the landing is i* plus the approach sum Σ v(u − ζ) over the constellation, and a class-c unit approaches each root exactly as deep as the window forces — integer levels, F_p-rational residues. The forcing ends at a fractional level (K(ζ)/K ramified), at a non-rational residue (unramified), or at the constellation's own branching when the torsion is in K — a tree game on the pairwise root distances, the higher-unit filtration of the torsion. Measured at the μ₄ floor over all six ramified quadratics of Q₂, exhaustively: the class-1 landing minimum is 5 / 6 / 7 — the floor p·i* + 1 shared by units starting on the seat itself, plus 0 / 1 / 2 — iff K(i)/K is ramified / unramified / split, the letter computed independently of any orbit; √−5 is the unique unramified field, and two fields never before censused, √10 and √−10, were predicted on √2's chain and measured exactly so. In-field torsion plays the tree: the achieving class-1 unit at ζ₈ approaches the four primitive eighth roots with multiset {2, 2, 3, 3} — collide one antipodal pair, avoid the other. The odd-p face is doubly thin: arrival classes exist iff p(p−1) | e, and the μ_p floor never shows the ramified letter — K(ζ_p) = K((−p/π^e)^1/(p−1)) is a tame unit-root extension, always unramified (Wilson's theorem) — so the third letter is a p = 2 wildness privilege. Beneath it the seat splice is a residue pigeonhole: the orbit chain deviates from the pure two-gear law iff ζ_p ∈ K — the p − 1 torsion directions exhaust F_p^×, and F₂'s single residue is why every p = 2, f = 1 seat splices.

Scope. Rule in range, two censuses. The constellation census: exhaustive units at Q₂, the six ramified quadratics, x³ − 2, x⁶ − 2, and ζ₈; sampled at ζ₁₆ — the four class landings 17, 18, 20, 28 predicted then measured — and ζ₉. The odd-p census: exhaustive at the two ramified quadratics of Q₃, sampled at five e = 6 fields (p = 3) and two e = 20 fields (p = 5); ramification labels certified independently of every orbit (residue tests, constructed torsion elements).

verifiers: explore_arrival_defect.py, explore_tame_face.py

Where the constellation forces a collision — ζ_p ∈ K, by the pigeonhole above — the depth of the forced approach is a Kummer defect. The collided cluster — the p^m preimages of ζ_p among the primitive roots — has cluster product u^pm − ζ_p exactly, so its approach sum never leaves the window: it is bounded by D_m = max_y v(y^pm − ζ_p), the p^m-power Kummer defect of ζ_p in K — window-internal, measured by its own climb — and the per-root approach ends at the level D_m/p^m. At m = 1, writing δ = D₁: the arrival ceiling is (p−1)·i* + δ, and the value menu is δ ∈ {ℓ ∈ (i*, p·i*) : p ∤ ℓ} ∪ {p·i*} ∪ {∞} — exactly one fractional value exists at e = 2, which is why the four quadratics of Q₂ whose letter is ramified share the cut 3/2; at p = 3 the censused sextics x⁶ + 3, x⁶ − 6, and x⁶ + 3x⁴ + 3 all measure δ = 7, cut 7/3. The trichotomy is the defect read at its three ranges: ramified δ < p·i* (fractional cut), unramified δ = p·i* (residue cut at the integer i*; the boundary Artin–Schreier image is {0}, so odd-p, f = 1 unramified specimens exist and are word-designable — the designed word [1, 0, 0, 2, 1, 1, 2] measures δ = 9 with spectrum {12, 15}, ζ₉ certified in its unramified cubic lift), split δ = ∞ (the torsion in-field). Classically the defect is the Kummer conductor: the ramification break of the degree-p extension cut by a unit of defect depth δ — K(ζ_p²)/K at the μ_p² storey — is t = p·i* − δ, verified by direct σ-action at the ζ₉ and ζ₈ models; the textbook 2^k − 1 break ladder of Q₂(ζ_2ⁿ) is the δ = 1 rigid case. At m = 2 the ceiling reads the p²-power defect: x¹⁸ + 3 has D₂ = 21, per-root cut 21/9 = 7/3 again.

Scope. The cluster reduction and the ceiling identity are derivations general in (p, m); the value menu is derived at m = 1 (the two-gear image argument); field verification is a rule in range — seventeen fields at p = 2, 3, 5 (e ≤ 20, m ≤ 2), defects and ceilings measured by independent targeted climbs, torsion certified at the Hensel and Newton margins.

verifier: explore_cutoff_ladder.py

Odd p, K/Q_p totally ramified of degree e, f = 1, integral seat i* = e/(p−1), any class (c, m). One formula: L = p·i* + min(δ_w, p^m), where δ_w = v(w − 1) is the window defect of w = −p/π^e (−p, not p — with this sign, by Wilson's theorem, the leading digit w₀ = 1 iff ζ_p ∈ K) — a different depth from the cutoff law's Kummer defect δ: the designed word [1, 0, 0, 2, 1, 1, 2] above has δ = 9 and δ_w = 3, its spectrum minimum 9 + min(3, 3) = 12. Gate closed — w₀ ≠ 1, δ_w = 0 — every class lands rigid at p·i*. Gate open, the ladder digits stop the walk rigidly and freedom arrives at the truncation p^m. δ_w itself depends on the choice of π; its p^m-truncation is uniformizer-invariant — exactly the invariant content of w − 1. Odd p has no in-window constants and no blind rungs: the wildness criterion c(p−1)² < p — the first deeper-squaring constant undercuts freedom iff it holds — fails at every odd p and every p = 2 intermediate class, and holds exactly at the p = 2 top class, where the always-open gate, the 3e/2 bonus, and the skeleton constants all live. The long p = 2 windows are a wildness privilege with an exact criterion, and the p = 2 intermediate staircase L = 2e + min(j₁, 2^m) is this formula verbatim.

Scope. The enumeration proof is general in (p, c, m, e); field verification is a rule in range at p = 3, 5 (e ≤ 20, m ≤ 2): designed defect ladders δ_w = 1, 2, 3, 5 land on the formula, including floors that beat the previously censused law.

verifier: explore_tame_readout.py

Order the sub-leading coefficient bits of F by level: b_i mod 2 at level i (i = 1 … e−1), d mod 4 at level e, ⌊b_i/2⌋ mod 2 at level e + i (i = 1 … e/2−1) — 3e/2 − 1 bits, the word length exactly. The map bits → window word is unitriangular over F₂: digit r equals the level-r bit plus a function of strictly earlier bits. Hence a bijection — every word in {0,1}^3e/2−1 is printable, each by exactly one reduced design (an F whose coefficients carry only the listed bits), and a greedy level-by-level solve places any chosen word without disturbing its prefix. Designed worlds print their words in live arrival spectra: at every class the sampled spectrum minimum equals the staircase on the placed word. The readout is a lossless programmable channel — the window word is the sub-leading Eisenstein data, re-encoded bijectively.

Scope. Unitriangularity is general in even e, nowhere using a 2-power; the bijection is exhaustive at e = 4, 6, 8 (32 / 256 / 2048 designs, every word realized), placement at e = 16 (27 target words, prefix invariance asserted at every step); the live-spectrum tie is a rule in range at e = 8, 16.

verifier: explore_readout_triangle.py

Every deeper coefficient bit feeds levels ≥ 3e/2, so word(F) = word(reduce(F)), where reduce(F) is the reduced design carrying F's triangle bits. With the triangle's bijection: the word is a complete invariant of the reduced bits — Eisenstein windows sort into exactly 2^3e/2−1 word classes, one per reduced design, and word(F) = word(G) iff reduce(F) = reduce(G). The cyclotomic anchor reduces in closed form: by Kummer's binomial parities, reduce(Φ_2e(x+1)) = x^e + 2x^e/2 + 4x^e/4 − 6 at every 2-power e ≥ 4 — the unique reduced design carrying ζ_2e's skeleton word at every 2-power depth. At depth e = 32 the three-term world x³² + 2x¹⁶ + 4x⁸ − 6 walks its top class stopless to the freedom rung and lands at 112, ζ₆₄'s landing: a designed cyclotomic mimic, exact mod the invisible bits. The mimic option is itself a p = 2 wildness privilege: at odd p a stopless designed sail does not imitate the cyclotomic anchor — it generates the field (x⁶ − 3x⁴ + 3 generates Q₃(ζ₉)).

Scope. The level pricing is the triangle's own derivation, general in even e; the closed form is verified by direct binomial bits and by the Kummer count independently at e = 4 … 64. Engine: 628 non-grid fields — deep-bit perturbations plus a random Eisenstein zoo, e ≤ 16 — each print their reduction's word.

verifier: explore_reduction_law.py

At odd p the channel is thinner and closes at the window edge: for F = x^e + Σ p·b_ixⁱ − p·d Eisenstein over Z_p, the in-window word (w₀, …, w_e) is a function of exactly (b₁ … b_e−1 mod p; d mod p²) — there is no second diagonal, and the deeper coefficient digits first act at level e + 1. The grid → word map is unitriangular with diagonal −w₀², hence a bijection, and the greedy solve designs any word — the landing-15 world x⁶ + 6x⁴ − 15 realizes a landing absent from every censused (3, 1, 1) field. One difference from p = 2: the odd-p word is a π-chart, not a field invariant — it moves under uniformizer twists up to an exact depth (twist depth p^m·k, invariance from k > i† = c(p−1)) — and the invariant is the arrival spectrum together with the field's chart-word set, the solution variety of its forced-digit laws.

Scope. The diagonal pricing and the unitriangularity are general in (p, e), seat-free; censuses exhaustive at (p, e) = (3, 4), (5, 3), (3, 6) — 162 / 500 / 1458 words, all distinct; designed landings and the chart action are rules in range at p = 3, 5.

verifiers: explore_slot_algebra.py, explore_deep_spectrum.py

The two gears are Herbrand's two slopes: ψ(i) = min(p·i, i+e) = p·φ_seat(i) identically, φ_seat the two-slope function breaking at the seat, slopes 1 and 1/p — Herbrand's shape. At m ≥ 2 the game's monomials organize into storeys by j = v_p(A), A the monomial's part total, and the scaling bijection — sizes ×p, offsets fixed — carries the (c, m−1) game's band onto storey j ≥ 1 verbatim: multinomial valuations and digit values preserved (Wilson's theorem in blocks), onsets rel_j = e(m−j) − c(p^m − p^j), and rel_j(c, m) = p^j · rel₀(c, m−j), rel₀ taken in that class's own window, of degree e/p^j. Every storey opens as a copy of the game one window down; the only m-dependence left is a sign, so gate-open the shifted digits are pinned at every storey — w_relj = (−1)^m+1−j and w_relj+p^j = (−1)^m−j·k₁, the fork digit k₁ = w_p^m returning at every storey with alternating sign. Measured in one digit vector: Q₃(ζ₈₁) has k₁ = 2 and pins (2, 2, 1, 1) = (−1, +k₁, +1, −k₁) at rels 84, 87, 136, 137; six single-coefficient perturbations of Φ₈₁(x+1) stick at 81 + the violated rel, exactly. And the filtration is the classical one: with ζ = ζ₈₁ and σ_a: ζ ↦ ζ^a the Galois action, the spectrum i_G(σ_a) = v(ζ^a − ζ) on Q₃(ζ₈₁) is the seat ladder {c·p^j}, lower breaks one below ({0, 2, 8, 26}), upper numbering equally spaced — the textbook cyclotomic ladder measured in the game's own model. The observatory reads the cyclotomic tower's ramification filtration through the unit filtration; the words, pins, and designed sticks above are digit-level fine structure the classical objects do not carry.

Scope. The Herbrand identity is exact, swept p ≤ 7; the scaling bijection is a general derivation with its census face verified exhaustively in range; the pin ladder is a rule in range — the anchor plus six designed perturbations at (p, c, m) = (3, 1, 3), gate-open regime; the m = 3 sign flip was fixed as the falsification point before the run, and held. The Q₃(ζ₈₁) filtration is a measurement, not a general theorem. The p-power map's two slopes and the cyclotomic break ladder are classical (Serre); the contact names the game.

verifier: explore_telescope.py