# OBSERVATORY — what the digits of a ramified field read The object: a field K totally ramified of degree e over Q_p, cut out by an Eisenstein polynomial F = x^e + Σ_{i=1}^{e−1} p·b_i·x^i − p·d with d a unit, with root π, the valuation v of K normalized by v(π) = 1, and the unit u = p/π^e of O_K = Z_p[π], whose residue ū = 1/d̄ enters the third of the conditions f = 1 (residue degree one), e = (p − 1)·p^n and ū = −1 under which a place has a head (CLOCK.md#the-head-criterion). Written in canonical digits, u = Σ u_r·π^r with every u_r in {0, …, p − 1}, the unit is a string, its digit u_r sitting at **level** r, the power of π it multiplies, and a unit z ≠ 1 at level v(z − 1); this page asks what that string is and what reads it. Its verifiers are triangle.py for the string, tame.py for what a class of units reads off it at an odd prime, cofactor.py and quarter.py for what a root of unity reads, wild.py at 2, weld.py, rung.py, staircase.py and relative.py for the module the units form, precision.py for the digits the jump set reads, and storey.py for the next step of the cyclotomic tower above K and a step of degree p below it. The answer so far, in two lines: the string of u is the polynomial's coefficients re-encoded, digit for digit and level by level, so it is a complete invariant of F and runs backwards into a designer. Where (p − 1) divides e, the least landing past the bend s = e/(p − 1) of each class of units (landing and seat class are bound at OBSERVATORY.md#the-tame-readout) is read off one coefficient level, where F leaves one fixed design: x^e + p at an odd prime and at 2 below the top class, a design of at most four terms at the top class over 2. That makes CLOCK.md's width a closed form at every head (Z₂'s is 1), and at a head, of e ≥ 2 when p = 2 (its field holds ζ_p), the same number is the capped excess over p·s of the seat class's highest landing (OBSERVATORY.md#the-jump-set-as-the-class-maxima): Pagano's jump set, the invariant of the units as a filtered module, is the staircase of the class maxima above its first point, and its last point decides whether the next **storey** of the cyclotomic tower, the step above K, ramifies, and where it breaks, a storey being any step of degree p in a tower of fields. triangle.py is a tool in its own right: given p, e ≥ 2 and any prefix u_0, …, u_r with u_0 ≠ 0, its `design` returns the one Eisenstein polynomial with no coefficient digit above level r whose u begins with that prefix, and its `read` prints the digits of u for any Eisenstein polynomial, in pure Python. ## The coefficient triangle Tier: theorem. Verifier: proof; triangle.py::section_diagonal, triangle.py::section_bijection, triangle.py::section_designer, triangle.py::section_control. Write b_i = Σ_k b_(i,k)·p^k and d = Σ_k δ_k·p^k in base p, and give b_(i,k) the level i + k·e and δ_k the level k·e. Each level r ≥ 1 then carries exactly one coefficient digit, the one at (i, k) = (r mod e, r div e), δ_k when i = 0. The map from (δ₀; the digits at levels 1..r) to (u₀; u₁, …, u_r) is a bijection for every r, triangular, each diagonal entry a unit of F_p: u₀ = 1/δ₀ mod p, and u_r = s_r·c_r + g_r(c_1, …, c_(r−1)), s_r = ±ū^(k+2), with c_r the level-r digit, g_r a function of the lower digits and δ₀, and the sign − exactly at the δ levels. In the limit it is a bijection from all Eisenstein polynomials over Z_p onto the digit strings with u₀ ≠ 0: the string of u is a complete invariant of F, and a prefix u₀..u_r is printed by exactly one choice of the digits at levels 0..r, whatever the digits above; that choice is the prefix's cell. The proof is one fixed point. F(π) = 0 reads π^e = p·D with D = d − Σ b_i·π^i, so u = 1/D, and since p = u·π^e, each p^k in D is u^k·π^(ke): every coefficient digit enters D as one monomial at its own level times a power of u. Two polynomials agreeing below level r cut out rings that agree modulo π^(r+1), since O/π^A is Z_p[π]/(π^A, p − u·π^e) and so fixed by u modulo π^(A − e), which by induction the lower digits fix; read there, raising the level-r digit by one moves D by ∓u^k·π^r, and 1/(D + Δ) = u − u²Δ + u³Δ² − …, whose later terms sit at level 2r or above; so u moves by a multiple of π^r with residue ±ū^(k+2), a unit move, no digit below r moves, and a carry out of level r lands at r + e since p·π^r = u·π^(r+e). Nothing uses the value of p or the parity of e. The levels are not new: that the monomials b_(i,k)·p^k·π^i of an Eisenstein polynomial sit at distinct levels i + k·e is the bookkeeping of Krasner's reduced polynomials as Monge states it (arXiv:1109.4617, Section 2), and the ramification polygon reads coefficient valuations at the same levels (Pauli–Sinclair, arXiv:1504.06671; Doris, arXiv:1803.08023). That literature reads the levels into the extension — its uniformizer changes, its polygon, a reduced set of generators (Guàrdia i Rúbies–Jones–Keating–Pauli–Roberts–Roe, arXiv:2606.01453) — and none of the four prints the string of u or the landing of a unit. The nearest kin is Monge's Lemma 4 (Section 4), which reads the norm of a unit 1 − θπ^m off one coefficient digit at first order: the same one-level shape, aimed at the norm group rather than the p-power map. The norm group does not determine the readouts. By local class field theory the norm group of K over Q_p is its largest abelian subfield's, and a quartic over Q₂ whose three letters are all ramified, none of −1, 2 and −2 being a square or 5 times a square in it (OBSERVATORY.md#the-letters-of-kζ₈k-at-e--4), has no quadratic subfield, so its norm group is all of Q₂^×; such quartics sit at widths 1, 2 and 3. The kinship is the level map: Monge's Lemma 6 runs on min(px, x + e), CLOCK.md's ψ. Its instances: the bijection exhaustive over seven cell sets at six (p, e), 8192 cells in all, every string unmoved by 16,384 lifts of the digits from the first multiple of e past the cell up; the diagonal move s_r exact at 4024 (polynomial, level, step) changes at twelve (p, e), e = 2, 4, 6, 8, 12, 16 at 2, 6, 12, 18 at 3, 4, 20 at 5 and 6 at 7, every residue d̄, levels to 3e + 4; and the designer printing 14 random targets up to level 80. ## The anchor design of the 2e-th roots of unity Tier: theorem; observation (the cell at (p, e) = (3, 6)). Verifier: proof; triangle.py::section_anchor, wild.py::section_control, triangle.py::section_cell, wild.py::section_letters. At e a power of 2, the field Q₂(ζ_(2e)) is cut out by Φ_(2e)(x + 1) = (x + 1)^e + 1, whose root is ζ − 1, and every one of its e + 1 terms is nonzero. Its digits u₁ to u_(3e/2 − 1) are nonetheless printed by the design A_e = x^e + 2x^(e/2) + 4x^(e/4) − 6, their only nonzero digits past u₀ = 1 sitting at r = e/2 and 5e/4, at every e = 2^n ≥ 4: ζ_(2e) makes z^e + 1 vanish, so its string begins with the one prefix at which the element S₀ of the wild readout's proof (OBSERVATORY.md#the-wild-readout) reaches 3e/2, and A_e prints that prefix. With 2 = u·π^e its equation reads 1 − D = π^(e/2) − u·π^e + u·π^(5e/4), so u = 1/D = 1 + (1 − D) + (1 − D)² + …; modulo π^(3e/2), (1 − D)² ≡ π^e and (1 − D)³ ≡ 0, and as v(u − 1) = e/2, u ≡ 1 + π^(e/2) + π^e·(1 − u) + u·π^(5e/4) ≡ 1 + π^(e/2) + π^(5e/4). The design is read against Φ_(2e)(x + 1) at e = 4, 8, 16 and 32, and its own u at e = 64. It prints the digits and not the field. A₄'s field holds ζ₄, but in it the letters of 2 and −2 are unramified, so it is not Q₂(ζ₈), which fills 2 of the 8 cells its prefix leaves open (OBSERVATORY.md#the-letters-of-kζ₈k-at-e--4). At (p, e) = (3, 6) the matching agreement reaches the field, an observation: the cell of Q₃(ζ₉)'s string through level e = 6, reduced to x⁶ + 6x⁵ + 6x⁴ + 3x³ − 24, holds a root of Φ₉ certified by Hensel's lemma at the design and at each of 20 random lifts of its higher digits, and so does x⁶ − 3x⁴ + 3 in another cell (its u₃ is 0, the cell's 1), while x⁶ + 3, whose field holds a cube root of −3 and so is not abelian, holds none. Which agreement forces the field is the continuity of roots (Krasner's lemma); at (3, 6) the design and each of its 20 sampled lifts cut it out, and at 2 agreement through 3e/2 − 1 does not force it at e = 4. ## The tame readout Tier: theorem. Verifier: proof; tame.py::section_readout, tame.py::section_width, tame.py::section_control. Let p be odd and the bend s = e/(p − 1) an integer. A class (c, m), c·p^m = s, is the units of level exactly c, which reach the bend after m p-th powers; a unit z of the class (c, m) has **landing** v(z^(p^(m+1)) − 1). The **departure** of F is l = v(u + 1) = min(e·v_p(d + 1), min_i (e·v_p(b_i) + i)), the least level of a nonzero coefficient digit of (F − x^e − p)/p: l is 0 iff ū ≠ −1 and infinite at x^e + p. Then the least landing of the class (c, m) is p·s + min(l, p^m), and when l < p^m every unit of the class lands there. A class whose units all land at one depth is **rigid**. The class (c₀, M) with m largest, s = c₀·p^M and p ∤ c₀, is the **seat class**. Proof. Write z = 1 + π^c·t with t a unit of residue a. Below the bend each p-th power multiplies the level by p, and the correction it makes to z^(p^j) − 1 sits at relative level (p − 1)(s − c·p^j), at least c(p − 1)²·p^(m−1), which exceeds p^m at every odd p. So y = z^(p^m) has y − 1 = π^s·t^(p^m) up to a factor 1 + η with v(η) > p^m, and y^p − 1 = (p·(y − 1)/u)·B, B = (u + 1) + (W − 1) + u·Σ C(p, k)/p·(y − 1)^(k−1), with v(p·(y − 1)/u) = p·s, W = t^((p−1)p^m)(1 + η)^(p−1), and the sum over 2 ≤ k ≤ p − 1 at level (k − 1)s ≥ p^m. Since a^(p−1) ≡ 1 mod p, t^(p−1) = 1 − (t₁/a)·π + …, with −t₁/a running over all of F_p as the next digit t₁ of z does, and its p^m-th power is 1 − (t₁/a)·π^(p^m) + …, a leading digit keeping its value under each p-th power below the bend. So B is u + 1 plus terms at level p^m or deeper, among them −(t₁/a)·π^(p^m). If l < p^m, v(B) = l for every unit; otherwise v(B) ≥ p^m, with equality at all but one t₁. At a place with a head (f = 1, s = p^n, ū = −1), the class (1, n) is the level-1 units, its least landing is CLOCK.md's arrival A, and so the width A − s − e of CLOCK.md#the-width, as (p − 1)·s = e, is w = min(l, s) at every odd p: one coefficient level of any Eisenstein polynomial of the field, capped at the bend. l itself is not a field invariant (x² + 3 has l infinite, and its translate x² − 6x + 12, cutting out the same field, has l = 1), but min(l, s) is. The row l = 0 is known. Fesenko and Vostokov (Local Fields and Their Extensions, ch. I, (5.7) case (2)) print the graded p-th power at the bend as the map of the residue field adding θ₀ = ū times an element to its p-th power, which at f = 1 is multiplication by 1 + ū, injective iff ū ≠ −1. Beyond that first-order step ch. I gives the kernel's order and when it vanishes, and that U_r, the units ≡ 1 mod π^r, lies in U_(r−e)^p above p·s, but not where a unit of the kernel lands; so every row with l ≥ 1, and the whole wild readout, where θ₀ = 1 and the map, squaring plus the identity, always has the kernel F₂, sit past what it reads. N. Nakagoshi (Nagoya Math. J. 73 (1979) 41–60) describes the unit group of O_K modulo every power of π through Hasse's generators. Where μ_p lies in K, as it does at every l ≥ 1, since there ū = −1 and (p − 1) divides e, the order of the generator at the bend enters his Theorem 3 as an unknown that his Remark (ii) only bounds; beyond the fields his section 9 works out, Q_p(ζ_p) and Q₂(√δ) at δ ≡ 2 mod 4, those rows are not his either. Its instances: 154 (polynomial, class) readings over ten (p, e) at p = 3, 5, 7, from s = 1 to s = 9 and at every departure from 0 to s + 2 and infinity, deeper digits random, all exact, 58 of them rigid on every unit read to two digits; and the width read off CLOCK.md's tick ladder, through the group exponent and not the landing, equal to min(l, s) at the five odd-p places with a head there and ten designed ones, six at s = 5 reading w = 1 through 5 and four at (p, e) = (3, 6), s = 3, reading w = 1 through 3. ## The cofactor readout Tier: theorem. Verifier: proof; cofactor.py::section_control, cofactor.py::section_designs, quarter.py::section_fields, quarter.py::section_admission. Let K hold a primitive p^N-th root of unity ζ, at p = 2 with N ≥ 2 and e a power of 2 at least 4, and write ζ − 1 = π^c·τ, τ a unit, the root's cofactor, i_τ the level of τ's principal part (τ over its Teichmüller digit). With dep the departure, l at an odd prime and at 2 the wild departure λ, the least level where a coefficient digit of F differs from the anchor design's (OBSERVATORY.md#the-wild-readout), and s the bend, min(dep, s) = min(p^(N−1)·i_τ, s): below the bend the departure is p^(N−1) times the level of the root's cofactor. So dep ≥ p^(N−1), and dep < s only at a multiple of p^(N−1). Proof at odd p. With ζ_p − 1 = π^s·τ_p, Φ_p(ζ_p) = 0 gives −p/(ζ_p − 1)^(p−1) = ε with v(ε − 1) = s exactly, and p = u·π^((p−1)s) turns this into u + 1 = 1 − τ_p^(p−1)·ε, so min(l, s) = min(j, s), j the level of τ_p's principal part. For v(Ξ) ≤ s/p, (1 + Ξ)^p − 1 = Ξ^p(1 + δ) with v(δ) ≥ (p − 1)²s/p > s at odd p, so τ_p = τ^(p^(N−1))·(1 + δ′), v(δ′) > s, and a p-th power below the bend multiplies a principal level by p. At 2, e ≥ 4, the root that reads is ζ₄ = √−1. Write ζ₄ − 1 = π^(e/2)·τ₄ and τ₄ = 1 + y, so that i_(τ₄) = v(y), τ₄ being its own principal part. Then ζ₄² = −1 reads u(1 + π^(e/2)·τ₄) = −τ₄², and putting it into the element S₀ of the wild readout's proof (OBSERVATORY.md#the-wild-readout) gives one with the valuation of y(y + π^(e/2)) + π^(5e/4), read below 3e/2. With ξ = v(y(y + π^(e/2))), the **quarter readout**: λ = ξ at ξ < 5e/4, λ = 5e/4 at ξ > 5e/4, λ > 5e/4 at ξ = 5e/4, and ξ is 2i_(τ₄) at i_(τ₄) < e/2 and above e otherwise. Squaring down from ζ to ζ₄, each squaring's correction sits at relative level e − h ≥ 3e/4, h ≤ e/4 the level of the unit squared, so i_(τ₄) = 2^(N−2)·i_τ below e/2, which is the readout. So the values λ ≤ e at which some field holds ζ₄ are exactly the even λ below e, each realized by the design below: an odd λ < e and λ = e never hold ζ₄, while λ > e always does (OBSERVATORY.md#the-jump-set-as-the-class-maxima). It designs rigid seats. Over Q_p(ζ), x^k + a(ζ − 1)x^g − (ζ − 1), 1 ≤ g < k, p ∤ a, is Eisenstein with τ = 1/(1 − aπ^g), and its norm, the homogenized Φ_(p^N) of A = 1 − ax^g + x^k over B = 1 − ax^g, is an Eisenstein polynomial over Q_p with dep = p^(N−1)·g (at 2 for k a power of 2). At an odd prime its seat is rigid iff g < p^(v_p(k)): at p = 3, k = 3, e = 18, l = 3 and 6 sit below p^M = 9, and the jump sets read (1, 3), (4, 2) and (1, 3), (5, 2) (OBSERVATORY.md#the-jump-set-as-the-class-maxima). At 2 it prints every multiple of 2^(N−1) below e as λ. Its instances: at odd p, 26 fields at p = 3 and 5 with N = 2 and 3, every primitive root, and three controls at N = 1 and 2; fourteen designs, l = p^(N−1)·g at each; the four rigid seats at e = 18, their staircase, Pagano's reading and the set rebuilt from the G(m) of OBSERVATORY.md#the-jump-set-as-the-class-maxima agreeing. At 2, 34 fields with N = 2 to 4 and e = 4 to 16 at every root and both of ±ζ₄, every branch of the quarter readout met, and sixteen designs; among 180 fields designed to each λ up to 3e/2 + 1, ζ₄ sits at no odd λ < e and at no λ = e. The family Φ_(p^N)(1 + x^c + p·Ω(x)), Ω zero or drawn at random of degree below c with coefficients below p², whose root π gives ζ = 1 + π^c + p·Ω(π), has τ's principal part past the bend, and at odd p printed l = s at all twelve fields read, which is why it never met a rigid seat. ## The wild readout Tier: theorem. Verifier: proof; wild.py::section_top, wild.py::section_other, wild.py::section_width, wild.py::section_control, wild.py::section_letters. At p = 2 the bend is s = e, and u is always 1 = −1 mod π. Every class (c, m) with c ≥ 2 obeys the tame formula, with l = v(u + 1) the departure from x^e + 2: its least landing is 2e + min(l, 2^m), and the whole class lands there when l < 2^m. The top class (1, n), e = 2^n, does not. Its reading is the **wild departure** λ, the least level where a coefficient digit of F differs from the anchor design's A_e = x^e + 2x^(e/2) + 4x^(e/4) − 6 (x² + 2x − 6 at e = 2), that is, λ = min(e·v₂(d − 3), e·v₂(b_(e/2) − 1) + e/2, e·v₂(b_(e/4) − 2) + e/4 at e ≥ 4, and e·v₂(b_i) + i over the other i). The least landing of the top class is 2e + min(λ, 3e/2), and the whole class lands there when λ < 3e/2. So the width of every head over 2 with e ≥ 2 is w = min(λ, 3e/2): one coefficient level again. The departure is measured from A_e where the odd prime's is measured from x^e + p, and the cap sits at 3e/2 where the odd prime's sits at s. Both designs are the cyclotomic field's string up to the cap. In Q_p(ζ_(p^(n+1))), with π = ζ − 1, the level-1 unit ζ never lands, so its class is not rigid below the cap and its departure reaches the cap. So at every prime the width of a head is how far u's string agrees with that of the cyclotomic field of the same e, capped. Three corollaries follow. At e ≥ 4 neither A_e nor x^e + 2 has a digit at level 1, so u's digit u₁ is 1 iff λ = l = 1, and then every class with m ≥ 1 lands at 2e + 1 alone: the **first-digit lock**. The anchor's own levels are no blind spot: λ = e/2 and λ = 5e/4 each sit one digit off A_e, and their landings are attained like every other. And the prefix under which no level-1 unit lands below 2e + 3e/2 is unique, the anchor's, which ζ_(2e) realizes. The landing is the field's, so min(λ, 3e/2) is the same at every Eisenstein polynomial of the field, and below it u's digits are the anchor's in every uniformizer: the capped departure is an invariant of the field. By the coefficient triangle the width reads the coefficient digits below 3e/2 and no deeper one, each word u₁..u_(3e/2 − 1) printed by exactly one cell of them: the Eisenstein polynomials of degree e over Q₂ fall into 2^(3e/2 − 1) cells, and every class's least landing is a function of the cell. Proof. For z of level c put h_j = (z^(2^j) − 1)/π^(c·2^j). Then h_j = h_(j−1)² + u·π^(e − c·2^(j−1))·h_(j−1), and z^(2^m) + 1 = π^e(u + h_m), so the landing is 2e + v(u + h_m). At c ≥ 2 every correction sits at level 2^m or deeper, and the one at 2^m (c = 2 only) is a constant. With t = (z − 1)/π^c = 1 + t₁·π + …, this gives u + h_m ≡ (u + 1) + (t₁ + [c = 2])·π^(2^m) mod π^(2^m + 1), where the digit t₁ is free. At c = 1 the chain gives h_n ≡ t^e + u·π^(e/2)·t^(e/2) + u²·π^(5e/4)·t^(e/4) mod π^(3e/2), and with t^(e/4) = 1 + π^(e/4)·q, u + h_n = S₀ + π^e·q²(q² + u) there, where S₀ = 1 + u + u·π^(e/2) + u²·π^(5e/4). (At e = 2, h_1 = t² + u·π·t, S₀ = 1 + u + u·π, and with t = 1 + π·y the varying part π²·y(y + u) sits at level 3.) Either v(q) ≥ e/4 or q ≡ 1 mod π^(e/4), so the varying term sits at e + min(e/2, l) or deeper: above v(S₀) when l ≠ e/2, v(S₀) being l when l < e/2 and e/2 when l > e/2, and at 3e/2 or deeper when l = e/2, so it can reach v(S₀) only when v(S₀) ≥ 3e/2. So a level-1 unit lands rigidly at 2e + v(S₀) below that cap. At the cap, g = 1 + π³ has v(g^e − 1) = 5e/2 exactly, so one of z and z·g attains it. The level-r digit of S₀ is u_r plus a function of lower digits, so at each level R ≤ 3e/2, v(S₀) ≥ R holds for exactly one prefix u₁..u_(R−1) of u. At R = 3e/2 that prefix is 1 + π^(e/2) + π^(5e/4), at which S₀ ≡ 2 + π^e = π^e(u + 1) modulo π^(3e/2), so that v(S₀) reaches 3e/2. A_e prints exactly that prefix, so by the coefficient triangle v(S₀) is λ, capped at 3e/2. Since ζ_(2e) satisfies z^e + 1 = 0, the digit string of Q₂(ζ_(2e)) begins with the anchor's. Its instances: 81 polynomials over e = 2, 4, 8, 16 and 32, with every capped departure from 1 to the cap realized at e ≤ 16, all exact, 56 of them rigid on every unit read to four digits; and the top class at all 256 cells of the Eisenstein quartics (coefficient digits at levels 1 to 8), landing at 8 + w, with 128, 64, 32, 16, 8 and 8 cells at widths 1 to 6. 244 readings of the classes with c ≥ 2 over e = 4, 8, 16, 6, 10 and 12. The width read off CLOCK.md's tick ladder equals min(λ, 3e/2) at the ten Eisenstein places with a head over 2 there and at twenty designed ones. At every ramified Q₂(√δ), δ squarefree with |δ| ≤ 400, it agrees with CLOCK.md#a-ramified-quadratic-place-over-2-reads-its-width-off-δ-mod-8. ## The letters of K(ζ₈)/K at e = 4 Tier: rule (the letters are functions of the cell by the mod π⁹ argument below, checked at each of the 256 cells of the Eisenstein quartics against two random lifts of its coefficient digits at levels 9 to 16; 5 is not a square at each cell's polynomial with no coefficient digit past level 8; the letters by w are read off the table printed over those cells). Verifier: wild.py::section_letters. Over a totally ramified quartic K of Q₂, K(ζ₈) = K(ζ₄, √2) is read by three letters: each of −1, 2 and −2 is split (a square in K), unramified (5 times it a square) or ramified. They are functions of the coefficient digits through level 8, since a unit is a square iff it is a square mod π⁹. The width w = min(λ, 6) of the wild readout and the letters are two axes, the filtration and the decomposition, and they meet in one row: above w = 2 the letter of −1 is w's, ramified at 3, unramified at 4 and split at 5 and 6. At w = 1, 3 and 5 all three letters are fixed, 2 and −2 ramified, and −1 ramified at w = 1 as at 3; at w = 2 at most one letter is not ramified, and −1 is split at 16 of its 64 cells and every letter ramified at another 16; at w = 4 and 6, 2 and −2 are either both ramified or both not ramified, and K is Q₂(ζ₈) at 2 of the 8 cells with w = 6. ## Pagano's jump set Tier: known; observation (the coefficient-column readings past e, at 131 fields). Source: C. Pagano, Jump sets in local fields, arXiv:1810.09975 (2018), Theorems 1.4, 1.5, 1.7, 1.8, 1.11 and 10.1, Propositions 3.35, 3.39 and 5.5, and property (P.1) of the reduction maps in the proof of Theorem 9.1. Verifier: relative.py::section_fields, relative.py::section_control. When K holds ζ_p, its principal units U₁ are, as a filtered Z_p-module, a free module modulo one relation, and the relation's orbit under the filtered automorphisms is an extended jump set (I, β): points (i, β(i)) with β strictly decreasing and weight ρ^β(i)(i) strictly increasing, ρ(i) = min(p·i, i + e), CLOCK.md's ψ. Proposition 3.39 reads it inside U₁: for ζ a generator of the p-power roots of unity, of order p^N, the weight of ζ modulo p^n-th powers, the largest v(ζ·h − 1) over h in U₁^(p^n), changes value exactly at n = β(i) − N, and just past there it is ρ^(β(i) − N)(i). His Theorem 10.1 reads the near points, those with p^(β − N)·i < e, off the Eisenstein polynomial g of π over Q_p(ζ_(p^N)), with no separability hypothesis. The **coefficient column** ι is the coefficient a_ι of x^ι in g, weighted by its coefficient-column level l_ι = c·v_ζ(a_ι) + ι, with v_ζ the valuation of Q_p(ζ), v_ζ(ζ − 1) = 1, and c the level of ζ, which is [K : Q_p(ζ)]; over the coefficient columns with l_ι < e and v_p(ι) ≤ v_p(c) the minimal candidates (l_ι/p^(v_p(ι)), v_p(ι) + N) are the near points. His Theorem 1.11 is the case N = 1 under strong separability, where every point is near. The far points, e ≤ p^(β − N)·i, are read by the reduction map of his Section 9: normalize g to a unit 1 + Σ ã_ι·x^ι and, while its weight ω stays below e* = p·e/(p − 1), write ω = ρ^ν(i) with p ∤ i and multiply by the one (1 + θ·x^i)^(p^ν), θ Teichmüller, that raises the weight, reducing mod g; the points are the (i, ν + N) where ν reaches a new minimum. That reads every point with p^(β − N)·i < e*, which is every point but e* itself, since a unit past e* is a p-th power. Read here, not in the source (relative.py). The levels need no polynomial over Q_p(ζ): ζ − 1 = π^c·τ gives 1/τ = −Σ (a_ι/(ζ − 1))·π^ι, so l_ι sits c past the level where the π^ι terms of 1/τ's expansion in the monomials (ζ − 1)^k·π^ι first carry a digit. And, as observations: Theorem 10.1 is that reduction truncated at e, where p-th powering acts as in characteristic p; past e its carries enter, and the coefficient-column readings tried miss them: carried past e, the near formula is off at 56 of 131 fields and the weight e + p^(N−1)·l_ι, the rigid seat's, at 73, and a field whose 1/τ has no digit off its π^0 terms still holds far points (Q₂(ζ₄)((ζ₄ − 1)^(1/8)) reads the far points (5, 4), (13, 3), (29, 2) past its first point (1, 5)). The reading below takes any Eisenstein polynomial over Q_p and reads one digit level. Its instances (relative.py): Theorem 10.1 reproduced at 131 fields with N from 1 to 5 and 504 (field, root) readings, 72 of the fields built over Q_p(ζ) to order. ## The jump set as the class maxima Tier: theorem. Verifier: proof, given Pagano's Theorem 1.5 and the proof of his Proposition 3.35; staircase.py::section_control, staircase.py::section_fields, rung.py::section_classes, weld.py::section_jump_weld, weld.py::section_i, weld.py::section_head, weld.py::section_control, cofactor.py::section_designs. Let K hold ζ_p, so s = e/(p − 1) = c₀·p^M with p ∤ c₀, and e* = p·s, and let p^N be the order of the p-power roots of unity in K. For 0 ≤ m ≤ M the class Γ_m = (s/p^m, m) has least landing e* + min(dep_m, X_m) by the readouts, dep and X being l and p^m, or λ and 3e/2 at the top class over 2. Its highest landing is H(m) = the weight of the jump set's point with the largest β ≤ m, infinite at m < N, where Γ_m holds a primitive p^(m+1)-th root of unity. So every point's weight but the first's is a class maximum, H drops exactly at the β of those points, and the jump set is the staircase of the class maxima above its first point (c₀, M + 1) at e*, its last step at β = N. The landings of Γ_m lie in [min(H(m), e* + X_m), H(m)], both ends attained, a rigid class (dep_m < X_m) being the one value e* + dep_m. No class below N is rigid, so a field holding ζ_(p^N) has l ≥ p^(N−1) at an odd prime (the cofactor readout says more, and l < p^M does occur), and over 2 a field holding ζ_(2e) has λ ≥ 3e/2. At the seat m = M this reads min(H(M) − e*, X) = min(dep, X), CLOCK.md's width at a place with a head, of e ≥ 2 when p = 2, at every N. At a rigid seat with M ≥ N the second point is one coefficient level: with dep = q·e + r, 0 ≤ r < e, (i₂, β₂) = ((s + r)/p^(v_p(s + r)), q + 1 + v_p(s + r)), or (e*, q) when r = 0. At N = 1 a rigid seat has dep ≤ e, so q = 0 or dep = e and (e*, 1); over 2 a field without ζ₄ has λ ≤ e, its top class rigid with H(M) − e* = λ, and λ > e forces ζ₄ into K. At N ≥ 2 every β is at least 2, so at a rigid seat dep = e never occurs, and a rigid top class over 2 with λ in (e, 3e/2) reads q = 1. Proof. Let G(m) be the largest v(ζ_p·h − 1) over h in U₁^(p^m), infinite when ζ_p lies there, that is at m < N. The units of level s among the p^m-th powers are exactly the p^m-th powers of Γ_m. Let y have level s. Above the bend the p-th power carries each level onto the level e higher, and ζ_p ∈ K forces ū = −1, so y^p − 1 sits past e* at every leading digit; if it sits at e* + j, then y^p = y′^p with y′ in U_(s+j), so y = ζ_p^a·y′ and the landing's excess over e* is v(ζ_p^(−a)·y − 1) − s. A power b with ab ≡ −1 mod p turns ζ_p^(−a)·y into ζ_p·y^b and keeps its level, so no landing of Γ_m exceeds e* + G(m) − s, and an h attaining G(m) > s gives y = h^(−1) landing there: H(m) = G(m) + e, with no outside theorem. In Pagano's presentation U₁ is Λ modulo the line through V = Σ p^β(i)·φ_i, Λ the free Z_p-module on one vector φ_i of weight i for each i < e* prime to p and for i = e*, the points' i among them, and V summed over the points; the torsion generator is Σ p^(β(i) − N)·φ_i and ζ_p is ζ̂ = Σ p^(β(i) − 1)·φ_i. His proof of Proposition 3.35 runs on ζ̂ unchanged: modulo p^m·Λ, ζ̂ is the sum over β(i) ≤ m, of weight the least ρ^(β(i) − 1)(i); a multiple of V is p·ζ̂ times a scalar, so adding one multiplies ζ̂ by a unit; and an added element of p^m·Λ raising the weight would put a point's i in the image of ρ, where no i < e* prime to p lies, e* being excluded as in his proof. So G(m) is that least weight, attained at the largest β ≤ m since ρ(i) = min(p·i, i + e) is increasing, and it exceeds s at N ≤ m ≤ M, which gives H. A rigid class has its highest landing at its least, and otherwise H(m) ≥ e* + X_m, which is the bracket. The weight e* + dep pulls back by −e for q + 1 steps to s + r, then by p while p divides. Finally, at N = 1, H(M) − e* > e would put ζ_p·h in U_(e*+1) for an h in U₁^(p^M), M ≥ 1, where every unit is a p-th power, making ζ_p a p-th power. At odd p and at c₀ > 1 the cap p^M = s/c₀ is below e, so the bound is silent there. Its instances: at N = 1, the class maxima found by brute force, G setting only the enumeration's depth, at 238 classes over 101 fields, their staircase the jump set at the 100 fields with every class enumerated (one class exceeded the enumeration's cap), the closed form at 69 rigid seats. At N ≥ 2, 59 fields with N from 2 to 5: over 2 the fields with λ > e at e = 4, 8, 16, and the family of OBSERVATORY.md#the-cofactor-readout at p = 2, 3, 5 up to e = 20. There the staircase of the brute maxima equals Pagano's Proposition 3.39 read verbatim off a lifted root of order p^N at every field, up to four points; 168 classes obey the bracket, none rigid below N; the closed form holds at 20 rigid seats, all over 2 with dep > e (e = 8, λ = 10 reads (5, 3)), and at the four designed rigid seats at p = 3, N = 2 of OBSERVATORY.md#the-cofactor-readout. The seat's reading: the capped excess exact at 106 fields, CLOCK.md's ladder width at 45 places with a head, six weights matching a module-basis computation, and −1 a square at 38 fields with λ > e and over Q₂(√δ) exactly at δ ≡ 7 mod 8, where the capped departure min(λ, 3e/2) is 3 > e = 2. ## The jump set reads M·e digits of u Tier: theorem. Verifier: proof, given the class maxima, Pagano's Theorem 1.8 and Serre's mass formula; precision.py::section_control, precision.py::section_precision, precision.py::section_anchor, precision.py::section_cells. Let K hold ζ_p, s = c₀·p^M. The jump set of K is a function of u modulo π^(M·e + 1), that is of the digits u₁, …, u_(M·e), u₀ being p − 1; at M = 0 it is (c₀, 1) and reads no digit. So the p^(M·e) cells of those digits partition the Eisenstein polynomials with ū = −1, each cell carrying one jump set, and they fall on the jump sets in counts p^(M·e) times the Haar mass of Pagano's Theorem 1.7. Proof. Every point but the first has β ≤ M and i ≤ e*, and ρ^β(e*) = e* + β·e, so its weight is at most e* + M·e: at N ≤ m ≤ M, G(m) ≤ e* + (M − 1)·e < T := e* + (M − 1)·e + 1, and G is infinite below N. The jump set is thus a function of min(G(m), T), which U₁/U_T and ζ_p modulo U_T decide; a power of ζ_p prime to p changes no G(m). O_K/π^A is Z_p[π]/(π^A, p − u·π^e), fixed by u modulo π^(A − e). Take A = T + (p − 2)·s: two fields whose u agree there share O/π^A, and since v(Φ_p′(ζ_p)) = e − s = (p − 2)·s, Hensel puts the one's ζ_p within π^T of a power of the other's. A − e = M·e + 1, as e* + (p − 2)·s = 2e. The count follows: under Haar measure on the coefficients with d̄ = −1 the coefficient digits are uniform, so by the coefficient triangle the cells are equally likely. With (p − 1) dividing e, ū = −1 holds exactly when K contains E = Q_p(ζ_p), since −u is then a (p − 1)-th power. Pagano's Theorem 1.8 gives the mass as a volume of Eisenstein polynomials of degree s over E, and both volumes are Serre's measure on the fields, a field weighing p^(−γ)/n, n its degree over the base, with γ one more than the valuation of its discriminant over its base, in the base's normalized valuation, less its degree over the base (Serre's formula as Pagano's Section 1 states it, the volume form from Serre's first proof, as Pagano notes before Theorem 1.8). For K ⊇ E, with 𝔡 the discriminant and v_E normalized on E, v_p(𝔡_(K/Q_p)) = v_E(𝔡_(K/E)) + (p − 2)·s, so the two γ, v_p(𝔡_(K/Q_p)) − e + 1 and v_E(𝔡_(K/E)) − s + 1, agree, and the factors 1/n, 1/e against 1/s, differ by the constant ratio e/s = p − 1, which normalizing each to a probability absorbs. The bound is attained at the anchors read. The anchor field Q_p(ζ_(p^(M+1))), or its tame twist π^(c₀) = ζ − 1, has the jump set (c₀, M + 1) alone. A field agreeing with it modulo π^(M·e) shares min(G(m), T − 1), so G is infinite below M and G(M) is infinite or e* + (M − 1)·e: its jump set is the anchor's, or the anchor's with (e*, M) at weight e* + M·e, the second exactly when ζ_(p^(M+1)) is not in K. At every anchor read, each of the other p − 1 cells takes the second, so there the digits u₁, …, u_(M·e − 1) do not suffice; that every anchor's siblings do is not proved. Its instances: at 65 random fields over nine (p, e), e = 2, 4, 8, 12, 16 at 2, 6, 12, 18 at 3 and 20 at 5, four designs sharing the first M·e + 1 digits read one jump set; at ζ₄, ζ₈, ζ₁₆, ζ₉, ζ₂₅ and Φ₈(x³ + 1) (c₀ = 3) the anchor's own cell holds the root of unity and every one of its p − 1 siblings adds (e*, M), at weights 6, 16, 40, 15, 45, 48. The cells fall 2, 1, 1 at (2, 2), 128, 32, 32, 16, 16, 16, 8, 4, 2, 2 at (2, 4) and 486, 162, 54, 18, 6, 3 at (3, 6), p^(M·e) times Theorem 1.7's mass exactly (that mass computed by hand, checked against Serre's masses of the quadratics at e = 2), no cell split by its tail. ## The last point gives the break of the next storey Tier: theorem. Verifier: proof, given Pagano's Propositions 3.39 and 5.5 and Fesenko and Vostokov's ch. 3 (2.5); storey.py::section_control, storey.py::section_quadratics, storey.py::section_next, storey.py::section_rigid. Let K hold ζ_p, let ζ, of order p^N, generate its p-power roots of unity, and let (i_N, N) be the last point of the jump set. Then K(ζ_(p^(N+1)))/K is unramified iff i_N = e*, and otherwise totally ramified of degree p with its one break t = e* − i_N. At N = 1 with p | s, when the class (s/p, 1) is rigid this is one coefficient level: t = e − dep, dep being l at an odd prime (rigid iff l < p) and l over 2 at e ≥ 4 (rigid iff l = 1), and λ at e = 2 (rigid iff λ < 3); t = 0 means the storey is unramified. Proof. By Proposition 3.39 at n = 1, the weight of ζ modulo p-th powers, the largest v(ζ·h − 1) over h in U₁^p, is ρ⁰(i_N) = i_N. So K(ζ_(p^(N+1))) = K(a^(1/p)) with a = ζ·h of level i_N, the best such level. Put α = 1 + (ζ_p − 1)·Y: since (ζ_p − 1)^(p−1) is −p times a principal unit, α^p = a becomes Y^p − Y = (a − 1)/(ζ_p − 1)^p up to terms of higher valuation, a right side of valuation i_N − e*. A unit right side with residue outside the image of x^p − x on F_p, which is {0}, gives the unramified extension; a right side of valuation −t, p ∤ t, gives a totally ramified one with break t (Fesenko and Vostokov, Local Fields and Their Extensions, 2nd ed., ch. 3 (2.5)). Below e* a point's i is prime to p. The unramified half is Pagano's Proposition 5.5. At N = 1 the weight i_1 is G(1) of the class maxima, above s when p | s, so H(1) = ρ(i_1) = i_1 + e, and a rigid class has H(1) = e* + dep, which gives i_1 = s + dep. Its instances: the break read with no weight, in a top field Q_p(ζ_top) generated by a root of unity ζ_top of p-power order, with uniformizer ζ_top − 1, as v(ζ_top^(k−1) − 1) − 1 for σ: ζ_top ↦ ζ_top^k generating the Galois group of the top field over the base, equals e* − i_N at Q₂(ζ₈) over Q₂(√2), Q₂(√−2) and Q₂(ζ₄) (1, 1, 3), Q₂(ζ₁₆) over Q₂(ζ₈) (7) and Q₃(ζ₉) over Q₃(ζ₃) (2). At the six ramified quadratics of Q₂, K(ζ₄)/K sorted by whether −1 or −5 is a square matches the last point: (3, 1) at Q₂(√±2) and Q₂(√±10), (4, 1) = (e*, 1) at Q₂(√−5), N = 2 at Q₂(ζ₄), which is CLOCK.md#the-width's reading of the same extension. Over Q₃, x⁶ + 3 reads t = 2 and x¹² + 3 reads t = 4, and x⁶ + 6x⁵ + 6x⁴ + 3x³ + 12 reads i₁ = 9 = e*, an unramified next storey at an odd prime with f = 1, through a seat class that is not rigid. The rigid reading holds at 8 designed fields at p = 2, 3, e from 2 to 18. ## A radical storey adds e to the first defect and passes the rest down Tier: theorem. Verifier: proof, the first defect given the break of the next storey, the theorem above, and its sources; storey.py::section_descent, storey.py::section_boundary. Let K hold ζ_p. Its p^m-power defect κ_m is the largest v(y^(p^m) − ζ_p) over principal units y, infinite iff ζ_p is a p^m-th power: the G(m) of the class maxima, so H(m) = κ_m + e, and κ₁ = i₁ when N = 1. Let K/C be totally ramified of degree p, ζ_p in C, and let L ≤ e_K be the level, capped at e_K, to which π_K^p agrees with an element of C; a radical storey K = C(π_C^(1/p)) has L = e_K, and F(x^p) cuts it out when F cuts out C. Then for m ≥ 2 κ_m(K) = p·κ_(m−1)(C) when p·κ_(m−1)(C) < (m − 1)e_K + L, and κ_m(K) ≥ (m − 1)e_K + L otherwise, given (p − 1)L ≥ e_K. At a radical storey the condition reads κ_(m−1)(C) < m·e_C. A finite κ_j is at most e* + (j − 1)e in any field, so at a finite κ_(m−1)(C) it always holds at an odd prime, and over 2 it fails only at κ_(m−1)(C) = m·e_C, the largest value allowed. With N = 1 in C and in K, when C's next storey is unramified or K/C is cyclic and breaks at some t_(K/C) above t_C = e*_C − κ₁(C), as a radical storey does at e*_C, the first defect is κ₁(K) = κ₁(C) + e_K, so the next storey over K is unramified exactly when C's is, and otherwise breaks where C's does. Up a radical tower C₀ = Q_p(ζ_p) ⊂ C₁ ⊂ ⋯ ⊂ C_n with N = 1 at every storey, at an odd prime each defect descends to a first defect m − 1 storeys down, κ_m(C_n) = p^(m−1)·κ₁(C_(n−m+1)) for m ≤ n + 1, each first defect set by the formula above; over 2 the same holds off the boundary, where only the lower bound is proved. Proof. Write y in O_K as Σ_(ι t_C. Upper numbering passes to quotients, so the first lower break of the compositum over C is t_C, and past it the ramification group is Gal over C(ζ_(p²)), not over K; lower numbering passes to subgroups, so the compositum over K has break t_C, and κ₁(K) = e*_K − t_C by the break of the next storey. A cyclic storey whose own break t_(K/C) lies below t_C, which needs a unit Kummer radicand, has κ₁(K) = p·κ₁(C) + (p − 1)·t_(K/C) by the same count at the second lower break, the two formulas meeting at t_(K/C) = t_C; no such storey is among the instances. Its instances, κ read exactly off the echelon of U₁^(p^m), no search: 81 defects with m ≥ 2 and 24 first defects over 24 storeys at p = 2, 3, e_K from 4 to 54, the towers x^(2·3^n) + 3 over Q₃(√−3) (κ₁ 1, 7, 25, 79) and x^(2^n) − 2 (3, 7, 15, 31, 63; x³² − 2 reads κ₁ to κ₇ as 63, 62, 60, 56, 48, 32, 32), and F(x^p) over random F. Over 2 the boundary was met 4 times, three in the descent and once at the specimen below, each at m = 2 over a C whose next storey is unramified, and each time κ₂(K) = 2·κ₁(C) after all: an observation. The unramified storey stays so, x⁴ − 2x² + 6 over Q₂(√−5) reading κ₁ = 8 = e*. ## Open fronts The digit string is a complete invariant of the polynomial; it depends on the uniformizer π, and the jump set is one function of the string that every uniformizer of K returns alike; which other field invariants the string carries is open, and so are a base field with f > 1 and a bend s = e/(p − 1) that is not an integer.