# Affine Dynamics, Sampling, and Quadrature

## Definitions, proofs, and sharp bounds for a symmetric six-node family

Revised 20 September 2026

This note expresses the constructions in standard mathematical terminology. Its principal additions are a complete classification of positive quadrature for the stated six-node family, explicit signed weights, a cross-ratio criterion for positivity, and the sharp amplification of bounded sample errors. The affine and sampling results establish the maps through which these conclusions can be reused.

All exactness degrees below mean exactness for **every polynomial of degree at most the stated degree**. An integration rule refers to a specified mathematical linear functional. No physical interpretation is required. The distinction between established general methods and their application to this particular family is retained throughout; no literature-wide originality claim is made.

## 1. Affine recurrences and their coordinate changes

Let $c,r\in\mathbb R$ and define

$$
T_{c,r}(x)=c+r(x-c),\qquad H_c(x)=x-c,\qquad L_r(y)=ry.
$$

**Proposition 1.** The coordinate change $H_c$ conjugates the affine map to scalar multiplication:

$$
H_c\circ T_{c,r}\circ H_c^{-1}=L_r,
\qquad T_{c,r}^{\,n}(x)=c+r^n(x-c),\quad n\geq0.
$$

**Proof.** Direct substitution gives $H_c(T_{c,r}(x))=r(x-c)$. Applying this identity repeatedly and then applying $H_c^{-1}(y)=y+c$ proves the iterate formula. $\square$

For $|r|<1$, every iterate converges to $c$. For $0<r<1$, an initial value above $c$ decreases toward $c$, while one below $c$ increases toward $c$. The initial value $c$ is fixed. These sign conditions matter when importing statements about geometric sequences.

For example, $c=1$ and $r=\varphi/(\varphi+\pi)$, where $\varphi=(1+\sqrt5)/2$, select one member of this general family. The generic identities do not establish a special optimality property for that choice.

## 2. Quantization, exact residuals, and the grid origin

For a declared step $h>0$, define

$$
q_h(x)=\left\lfloor\frac{x}{h}\right\rfloor,
\qquad \rho_h(x)=x-hq_h(x).
$$

The defining floor inequalities give

$$
0\leq\rho_h(x)<h,\qquad x=hq_h(x)+\rho_h(x).
$$

Thus $x\mapsto(q_h(x),\rho_h(x))$ is a bijection from $\mathbb R$ to $\mathbb Z\times[0,h)$. The quotient alone loses information. An exact residual restores it. For arbitrary real inputs, this is an identity involving exact real data; a finite digital representation requires an additional representation assumption.

**Proposition 2.** Suppose $0<r<1$, $z_0=x$, and

$$
z_{k+1}=h\left\lfloor\frac{rz_k}{h}\right\rfloor,
\qquad \eta_k=rz_k-z_{k+1}.
$$

Then for $n\geq1$,

$$
r^n x=z_n+\sum_{k=0}^{n-1}r^{n-1-k}\eta_k,
\qquad
0\leq r^n x-z_n<h\frac{1-r^n}{1-r}<\frac{h}{1-r}.
$$

**Proof.** Each $\eta_k$ lies in $[0,h)$. The identity for $n=1$ is its definition. Multiplying the identity for $n$ by $r$ and substituting $rz_n=z_{n+1}+\eta_n$ gives the identity for $n+1$. The finite geometric sum gives the bounds. $\square$

For positive integers $Q,S$ and $d\in\mathbb Z$, the integer version is

$$
d'=\left\lfloor\frac{Qd}{S}\right\rfloor,
\qquad Qd=Sd'+\rho,\qquad 0\leq\rho<S.
$$

Reverse reconstruction uses $d=(Sd'+\rho)/Q$ and must check integer divisibility. An ordered sequence of valid residuals and a terminal state determine the preceding states. Approximating $r$ by $Q/S$ does not by itself prove equality of the corresponding floor maps on any stated domain.

**Proposition 3.** Define the zero-aligned and center-aligned quantizers by

$$
\Pi_{0,h}(x)=h\lfloor x/h\rfloor,
\qquad
\Pi_{c,h}(x)=c+h\lfloor(x-c)/h\rfloor.
$$

Then

$$
H_c\circ\Pi_{c,h}\circ T_{c,r}
=\Pi_{0,h}\circ L_r\circ H_c,
$$

and $\Pi_{c,h}=\Pi_{0,h}$ as functions on $\mathbb R$ if and only if $c/h\in\mathbb Z$.

**Proof.** Substitution proves the first identity. If $c/h$ is an integer, the integer-translation law for the floor gives equality of the quantizers. Conversely, equality at $x=c$ gives $c=h\lfloor c/h\rfloor$, so $c/h$ is an integer. $\square$

The condition cannot be discarded. For $c=1/2$, $h=1$, $r=1/2$, $x=3/4$, one has $T_{c,r}(x)=5/8$ but

$$
H_c(\Pi_{0,1}(T_{c,r}(x)))=-\tfrac12,
\qquad
\Pi_{0,1}(L_r(H_c(x)))=0.
$$

An affine change of coordinates therefore carries the quantization grid's origin as part of its data.

## 3. Polynomial evaluation connects recovery and quadrature

Let $z_0,\ldots,z_5$ be six distinct real numbers, and put

$$
(V_d)_{ij}=z_i^{\,j},\qquad 0\leq i\leq5,\quad0\leq j\leq d.
$$

If $f(t)=\sum_{j=0}^{d}a_jt^j$, its six sampled values are $y=V_da$.

**Proposition 4.** For $0\leq d\leq5$, the map $a\mapsto V_da$ is injective. Two distinct sampled degree-at-most-$d$ polynomials disagree at at least $6-d$ positions.

**Proof.** A nonzero polynomial of degree at most $d$ has at most $d$ distinct roots. Apply this to the difference polynomial. $\square$

Consequently, exact unique decoding with $e$ arbitrarily corrupted records and $r_0$ erased records is guaranteed when

$$
\boxed{2e+r_0<6-d.}
$$

Indeed, two candidate records consistent with the received data could differ at no more than $2e+r_0$ positions, contradicting the distance bound. This is a uniqueness guarantee, not a specified efficient decoding algorithm.

For affine vector-valued records $p_i=m+z_iq$, apply the root argument to a nonzero coordinate of the difference. Here $d=1$, so the guarantee becomes $2e+r_0\leq4$. It assumes trusted labels, known nodes, and exact surviving data. It does not by itself establish tolerance to small errors in every record.

Now let $I$ be a specified linear functional on polynomials and set

$$
\mu_d=(I(1),I(t),\ldots,I(t^d))^{\mathsf T}.
$$

**Proposition 5.** The rule $Q(f)=\sum_iw_i f(z_i)$ is exact through degree $d$ if and only if

$$
\boxed{V_d^{\mathsf T}w=\mu_d.}
$$

**Proof.** For every coefficient vector $a$,

$$
Q(f)=w^{\mathsf T}V_da=(V_d^{\mathsf T}w)^{\mathsf T}a,
\qquad I(f)=\mu_d^{\mathsf T}a.
$$

These expressions agree for all $a$ exactly under the displayed condition. $\square$

At $d=5$, $V_5$ is square and invertible. Hence a unique real, possibly signed quadrature rule exists for any specified $I$ on this polynomial space. Recovery and quadrature are thus connected through evaluation and its transpose. Positivity remains an additional condition.

## 4. A complete positivity criterion

**Theorem 6.** Nonnegative weights satisfying $V_d^{\mathsf T}w=\mu_d$ exist if and only if

$$
P(z_i)\geq0\text{ for every }i
\quad\Longrightarrow\quad I(P)\geq0
$$

for every polynomial $P$ of degree at most $d$.

**Proof.** Necessity follows from $I(P)=\sum_iw_iP(z_i)$.

For sufficiency, let $C=\{V_d^{\mathsf T}w:w\geq0\}$. This is a convex cone. It is closed: if vectors in $C$ converge, the first coordinate of each is the sum of its nonnegative weights. These sums are bounded, so the weight vectors have a convergent subsequence with a nonnegative limit representing the limiting vector.

If $\mu_d\notin C$, choose a nearest point $p\in C$ and let $b=p-\mu_d\ne0$. Minimizing squared distance along the segment from $p$ to any $v\in C$ gives $b^{\mathsf T}(v-p)\geq0$. Taking $v=0$ and $v=2p$ shows $b^{\mathsf T}p=0$. Therefore

$$
b^{\mathsf T}v\geq0\quad(v\in C),
\qquad b^{\mathsf T}\mu_d=-\|b\|^2<0.
$$

Taking $v$ to be each generating column of $V_d^{\mathsf T}$ gives $V_db\geq0$. The polynomial with coefficient vector $b$ is consequently nonnegative at every node but has negative $I$-value, contrary to the hypothesis. $\square$

This is the finite moment-cone alternative written directly for the sample nodes. It makes an obstruction polynomial a complete form of infeasibility certificate, rather than an isolated trick.

## 5. The symmetric six-node family

Define

$$
\varphi=\frac{1+\sqrt5}{2},\quad
0<s<1,\quad A=s+\varphi^2,\quad B=s+1.
$$

The unnormalized nodes are $-A,-B,-s,s,B,A$. Normalize by $A$:

$$
x=(-1,-v,-u,u,v,1),\qquad
u=\frac{s}{A},\quad v=\frac{B}{A},\quad
\alpha=u^2,\quad\beta=v^2.
$$

Here $0<\alpha<\beta<1$. Use the uniform probability average

$$
I(f)=\frac12\int_{-1}^{1}f(t)\,dt.
$$

Its moments are $I(t^{2j})=1/(2j+1)$ and $I(t^{2j+1})=0$. The same weights integrate the uniform average on $[-A,A]$ after rescaling. Weights for the unnormalized integral itself are multiplied by $2A$.

Both $s/(s+\varphi^2)$ and $(s+1)/(s+\varphi^2)$ increase with $s$. Since $(1+\varphi^2)^2=5\varphi^2$, their endpoint bounds give

$$
\alpha<\frac1{5\varphi^2}<\frac15,
\qquad
\alpha+\beta<\frac1{\varphi^2}<\frac23.
\tag{1}
$$

We shall also use

$$
3-\frac5{\varphi^2}>0,\qquad
3-\frac7{\varphi^2}=7\varphi-11>0.
\tag{2}
$$

These inequalities follow directly from $\varphi=(1+\sqrt5)/2$; for the last one, $\sqrt5>15/7$ follows by squaring positive sides.

## 6. Explicit weights and the complete degree classification

Write $\gamma_\alpha,\gamma_\beta,\gamma_1$ for the **total masses on each opposite pair**. Each individual node in a pair receives half that mass.

**Theorem 7.** The unique rule exact through degree five has pair masses

$$
\boxed{
\begin{aligned}
\gamma_\alpha
 &=\frac{2(5\beta-1)}{15(\beta-\alpha)(1-\alpha)},\\[2pt]
\gamma_\beta
 &=\frac{2(1-5\alpha)}{15(\beta-\alpha)(1-\beta)},\\[2pt]
\gamma_1
 &=\frac{3-5(\alpha+\beta)+15\alpha\beta}
 {15(1-\alpha)(1-\beta)}.
\end{aligned}}
\tag{3}
$$

**Proof.** Uniqueness follows from Proposition 5. Reflection produces another degree-five-exact rule, so uniqueness forces equal weights at opposite nodes. The even moments therefore require

$$
\begin{pmatrix}
1&1&1\\
\alpha&\beta&1\\
\alpha^2&\beta^2&1
\end{pmatrix}
\begin{pmatrix}\gamma_\alpha\\\gamma_\beta\\\gamma_1\end{pmatrix}
=\begin{pmatrix}1\\1/3\\1/5\end{pmatrix}.
\tag{4}
$$

For example, apply the rule to $(t^2-\beta)(t^2-1)$. Its average is $2(5\beta-1)/15$, and it vanishes on the other two pairs. This yields the first formula. Applying $(t^2-\alpha)(t^2-1)$ and $(t^2-\alpha)(t^2-\beta)$ yields the other two. The resulting masses satisfy (4), and symmetry gives the odd moments. $\square$

The denominator of every expression in (3) is positive. Bounds (1) and (2) show that $\gamma_\beta$ and $\gamma_1$ are positive throughout $0<s<1$. Thus the only possible negative mass is $\gamma_\alpha$, whose sign is the sign of $5\beta-1$.

Define

$$
s_* =\frac{\sqrt5-1}{4}=\sin18^\circ.
$$

Since $v$ increases, solving $v=1/\sqrt5$ gives

$$
\beta\geq\frac15\quad\Longleftrightarrow\quad s\geq s_*.
\tag{5}
$$

**Theorem 8.** The largest exactness degree achievable with nonnegative weights is

$$
\boxed{
d_+(s)=
\begin{cases}
3,&0<s<s_*,\\
5,&s_*\leq s<1.
\end{cases}}
\tag{6}
$$

The largest degree achievable with unrestricted real weights is exactly five throughout $0<s<1$.

**Proof: degree three is always attainable with strictly positive weights.** Set

$$
t=\frac{2}{3(2-\alpha-\beta)},\qquad
(\gamma_\alpha,\gamma_\beta,\gamma_1)=(t,t,1-2t).
\tag{7}
$$

By (1), all three masses are positive. They sum to one and satisfy

$$
t\alpha+t\beta+1-2t=\frac13.
$$

Assigning half of each mass to each opposite node makes all odd moments vanish, proving degree-three exactness.

**Proof: below the threshold degree four is impossible with nonnegative weights.** Let

$$
P(t)=(t^2-\beta)(t^2-1).
$$

This polynomial is zero at $\pm v,\pm1$ and positive at $\pm u$, but

$$
I(P)=\frac{2}{15}(5\beta-1)<0
$$

when $s<s_*$. Theorem 6 rules out every nonnegative degree-four-exact rule, including asymmetric rules.

**Proof: at and above the threshold degree five is attainable.** Equations (3) and (5) give nonnegative masses. They are all strictly positive for $s>s_*$. At $s=s_*$ they are

$$
(\gamma_\alpha,\gamma_\beta,\gamma_1)
=\left(0,\frac56,\frac16\right).
$$

Thus the four active nodes and their individual weights are

$$
\left(-1,-\frac1{\sqrt5},\frac1{\sqrt5},1\right),
\qquad
\left(\frac1{12},\frac5{12},\frac5{12},\frac1{12}\right).
$$

The inner two nodes carry zero weight. No degree-four-exact rule can have strictly positive weights on all six nodes at this boundary: the same polynomial $P$ has zero average and is positive at exactly those inner nodes.

**Proof: degree six is impossible even with signed weights.** The polynomial

$$
W(t)=(t^2-\alpha)(t^2-\beta)(t^2-1)
$$

vanishes at all six nodes, so every sample rule gives $Q(W)=0$. Direct integration gives

$$
I(W)=-\frac{2}{105}
\left[3-7(\alpha+\beta)+35\alpha\beta\right]<0.
\tag{8}
$$

Indeed, (1) and (2) imply that the bracket is larger than $3-7/\varphi^2>0$. Thus no choice of six real weights is exact on all degree-six polynomials. Together with Theorem 7, this proves the signed maximum and completes (6). $\square$

When $s=\sin(\varepsilon/2)$ with $0<\varepsilon<180^\circ$, the threshold is exactly $\varepsilon=36^\circ$.

| Parameter range | Largest degree with nonnegative weights | Degree-five weight signs | Largest degree with strictly positive weights at all six nodes |
|---|---:|---|---:|
| $0<s<s_*$ | 3 | Inner pair negative; other pairs positive | 3 |
| $s=s_*$ | 5 | Inner pair zero; other pairs positive | 3 |
| $s_*<s<1$ | 5 | All positive | 5 |

The one-arcsecond seed belongs to the first row. This extension varies a declared construction parameter; it does not change that seed by implication.

## 7. The cross-ratio determines the positivity regime within this family

For the ordered outer points $(y_1,y_2,y_3,y_4)=(-A,-B,B,A)$, use the convention

$$
R=\frac{(y_3-y_1)(y_4-y_2)}{(y_3-y_2)(y_4-y_1)}
=\frac{(A+B)^2}{4AB}
=\frac{(1+v)^2}{4v}.
\tag{9}
$$

This cross-ratio is unchanged by a nondegenerate projectivity of the line, with the usual extended-real interpretation if an image point is at infinity. For finite images under $M(t)=(at+b)/(ct+d)$, this follows from

$$
M(x)-M(y)=\frac{(ad-bc)(x-y)}{(cx+d)(cy+d)};
$$

all added factors cancel in (9).

**Corollary 9.** Within the family of Section 5, degree-five quadrature with nonnegative weights exists if and only if

$$
\boxed{R\leq R_*:=\frac{5+3\sqrt5}{10}.}
\tag{10}
$$

Strict inequality gives strictly positive weights on all six nodes; equality gives the four active boundary nodes.

**Proof.** For $0<v<1$,

$$
\frac{d}{dv}\frac{(1+v)^2}{4v}=\frac{v^2-1}{4v^2}<0.
$$

The threshold $v=1/\sqrt5$ corresponds to $R=(5+3\sqrt5)/10$. Apply (5) and Theorem 8. $\square$

In particular, four outer points determine which positive-quadrature regime the specified original family member occupies. The inverse parameter formula is

$$
s=\frac{\varphi}{2}
\left(\sqrt{\frac{R}{R-1}}-\sqrt5\right).
\tag{11}
$$

To verify it, observe that $A-B=\varphi$, $A+B=2s+\varphi+2$, and $\varphi+2=\varphi\sqrt5$. Thus $\sqrt{R/(R-1)}=(A+B)/(A-B)$.

This conclusion depends on membership in the stated family and the original uniform baseline measure. A general projective coordinate change does not preserve uniform measure or polynomial degree. Equation (10) classifies the original family member inferred from the invariant; it is not a claim that arbitrary projective images, equipped with a new uniform measure, have the same quadrature rule.

## 8. Sharp amplification of sample errors

**Theorem 10.** For fixed real weights $w_i$ and additive sample errors $|e_i|\leq\delta$, where $\delta\geq0$,

$$
\sup_{|e_i|\leq\delta}
\left|\sum_iw_ie_i\right|
=\delta\sum_i|w_i|.
\tag{12}
$$

**Proof.** The triangle inequality gives the upper bound. Choose $e_i=\delta\,\operatorname{sgn}(w_i)$, with $e_i=0$ when $w_i=0$, to attain it. $\square$

For the unique degree-five rule, the two outer pair masses are positive and all masses sum to one. Hence its exact amplification factor is

$$
\boxed{
\Lambda(s)=\sum_i|w_i|
=1+2\max(0,-\gamma_\alpha)
=1+\frac{4\max(0,1-5\beta)}
{15(\beta-\alpha)(1-\alpha)}.}
\tag{13}
$$

It is strictly larger than one below the threshold, and equal to one at and above the threshold. Any rule exact on constants satisfies $\sum_i|w_i|\geq|\sum_iw_i|=1$, so the latter value is the smallest possible amplification.

This is a worst-case bound for bounded, independently selectable sample perturbations. It is not a statistical variance statement and does not include floating-point errors in computing nodes or weights. For polynomials of degree at most five, the ideal rule is exact, so (12) bounds the entire error caused by those sample perturbations. For other functions, quadrature approximation error is an additional term.

At the one-arcsecond seed, a floating-point illustration gives pair masses approximately

$$
(-0.2472108572,\ 1.0699869896,\ 0.1772238677)
$$

and $\Lambda\approx1.4944217145$. These displayed decimal values illustrate the exact formulas; they are not used to establish their signs.

The exact one-sided small-seed limit is

$$
\lim_{s\downarrow0}\gamma_\alpha=-\frac2{5\varphi},
\qquad
\lim_{s\downarrow0}\Lambda(s)=1+\frac4{5\varphi}.
\tag{14}
$$

Indeed, $\alpha\to0$, $\beta\to\varphi^{-4}$, and substituting into (3) gives $\gamma_\alpha\to(2/15)(5-\varphi^4)=-2/(5\varphi)$. This is a limit from distinct six-node configurations; $s=0$ itself has coincident inner nodes and is excluded from the family.

The tradeoff below the threshold is therefore explicit: degree-three exactness can retain nonnegative weights and amplification one, while degree-five exactness forces a negative inner pair and the larger factor (13).

## 9. Conditions for composing the connections

| Connection | Required data | What the map preserves or establishes |
|---|---|---|
| Affine recurrence to geometric scaling | Fixed point $c$ and ratio $r$ | Invertible coordinate change; exact iterates |
| Floor quantization to quotient and residual | Step $h$, grid origin, exact residual | Exact reconstruction at that scale |
| Local residuals to global deviation | Ordered recurrence and residuals | Exact weighted error identity |
| Affine recentering through quantization | Transported grid origin | Commuting identity in Proposition 3 |
| Polynomial coefficients to samples | Distinct labeled nodes and degree bound | Injectivity and exact-record distance bound |
| Integral to sample weights | Declared measure and polynomial space | Transposed moment equations |
| Positive weights to obstruction polynomials | Same nodes and functional | Necessary and sufficient criterion in Theorem 6 |
| Four-point cross-ratio to quadrature regime | Membership in Section 5's family; original uniform measure | Exact parameter and positivity classification |
| Signed weights to sample-error bound | Additive bounded-error model | Sharp operator norm in Theorem 10 |

These hypotheses specify when conclusions can pass through several maps. Merely reusing a scalar, name, or diagram does not establish the needed identities.

## 10. Verification and source status

The written arguments above establish the all-parameter claims. The supplementary standard-library Python checker below verifies formal polynomial identities after denominators are cleared, exact calculations in $\mathbb Q(\sqrt5)$ at seven seeds, the boundary weights, the small-seed limit, and the grid-origin counterexample. It uses explicit exceptions, including when Python optimization is enabled.

Both executions completed successfully on 20 September 2026:

```text
python3 verify_sampling_extension.py
12 exact check groups passed.

python3 -O verify_sampling_extension.py
12 exact check groups passed.
```

The symbolic polynomial identities hold for formal $\alpha,\beta$; the seven-seed checks supplement the written inequalities and do not replace them. The checker is not a proof-assistant formalization.

The existing construction and earlier exploration were supplied in the conversation and linked at the [construction archive](https://omega-naught.grok.me/) and [six-point exploration](https://omega-naught.grok.me/geometry/exploration-02). On this check, the live pages could not be retrieved; a direct request to the root returned HTTP 403. Consequently, the additional results are distinguished from the supplied earlier material, not certified absent from the current website. No website publication was performed.

Background definitions previously inspected are available at [Geometric Sequence](https://algebrica.org/geometric-sequence/), [Geometric Series](https://algebrica.org/geometric-series/), [Floor and Ceiling Functions](https://algebrica.org/floor-and-ceiling-functions/), [Linear Maps](https://algebrica.org/linear-maps/), and [Systems of Linear Equations](https://algebrica.org/systems-of-linear-equations/). These references supply standard vocabulary. The family-specific classifications and proofs are presented here, without attributing them to those source pages.

## Appendix: complete supplementary checker

Save the following block as `verify_sampling_extension.py`. Python 3 and its standard library suffice.

```python
"""Exact supplementary checks; Python 3 standard library only.

The accompanying written proofs establish the parameter-wide claims.
Acceptance uses explicit exceptions and is unchanged under python -O.
"""
from fractions import Fraction as F


def require(condition, label):
    if not condition:
        raise ValueError(label)


# Sparse polynomials in two formal variables, with rational coefficients.
def add(*polys):
    out = {}
    for poly in polys:
        for monomial, coefficient in poly.items():
            out[monomial] = out.get(monomial, F(0)) + coefficient
    return {m: c for m, c in out.items() if c}


def scale(poly, scalar):
    return {m: c * scalar for m, c in poly.items() if c * scalar}


def mul(*polys):
    out = {(0, 0): F(1)}
    for poly in polys:
        nxt = {}
        for (i, j), c in out.items():
            for (k, l), d in poly.items():
                m = (i + k, j + l)
                nxt[m] = nxt.get(m, F(0)) + c * d
        out = {m: c for m, c in nxt.items() if c}
    return out


class K:
    """Exact a + b*sqrt(5), with rational a and b."""
    def __init__(self, a=0, b=0):
        if isinstance(a, K):
            self.a, self.b = a.a, a.b
        else:
            self.a, self.b = F(a), F(b)

    def __add__(self, other):
        other = K(other)
        return K(self.a + other.a, self.b + other.b)

    __radd__ = __add__

    def __neg__(self):
        return K(-self.a, -self.b)

    def __sub__(self, other):
        return self + (-K(other))

    def __rsub__(self, other):
        return K(other) + (-self)

    def __mul__(self, other):
        other = K(other)
        return K(self.a * other.a + 5 * self.b * other.b,
                 self.a * other.b + self.b * other.a)

    __rmul__ = __mul__

    def __truediv__(self, other):
        other = K(other)
        den = other.a**2 - 5 * other.b**2
        if not den:
            raise ZeroDivisionError
        return self * K(other.a / den, -other.b / den)

    def __rtruediv__(self, other):
        return K(other) / self

    def __pow__(self, n):
        require(isinstance(n, int) and n >= 0, 'exponent')
        out = K(1)
        for _ in range(n):
            out *= self
        return out

    def __eq__(self, other):
        other = K(other)
        return self.a == other.a and self.b == other.b

    def sign(self):
        if not self.b:
            return (self.a > 0) - (self.a < 0)
        if not self.a:
            return (self.b > 0) - (self.b < 0)
        if self.a * self.b > 0:
            return 1 if self.a > 0 else -1
        difference = self.a**2 - 5 * self.b**2
        if difference > 0:
            return 1 if self.a > 0 else -1
        return 1 if self.b > 0 else -1


def masses(alpha, beta):
    return (
        2 * (5 * beta - 1) / (15 * (beta - alpha) * (1 - alpha)),
        2 * (1 - 5 * alpha) / (15 * (beta - alpha) * (1 - beta)),
        (3 - 5 * (alpha + beta) + 15 * alpha * beta)
        / (15 * (1 - alpha) * (1 - beta)),
    )


def run():
    labels = []
    one = {(0, 0): F(1)}
    a, b = {(1, 0): F(1)}, {(0, 1): F(1)}
    da, db = add(one, scale(a, -1)), add(one, scale(b, -1))
    gap = add(b, scale(a, -1))
    den = scale(mul(gap, da, db), 15)
    na = scale(mul(add(scale(b, 5), scale(one, -1)), db), 2)
    nb = scale(mul(add(one, scale(a, -5)), da), 2)
    nc = mul(add(scale(one, 3), scale(add(a, b), -5),
                 scale(mul(a, b), 15)), gap)
    for j in range(3):
        lhs = add(mul(na, {(j, 0): F(1)}),
                  mul(nb, {(0, j): F(1)}), nc)
        require(lhs == scale(den, F(1, 2*j + 1)), 'formal moment identity')
    labels.append('formal degree-five moment identities')

    sixth = add(scale(one, F(1, 7)),
                scale(add(one, a, b), -F(1, 5)),
                scale(add(a, b, mul(a, b)), F(1, 3)),
                scale(mul(a, b), -1))
    require(sixth == scale(add(scale(one, 3), scale(add(a, b), -7),
                              scale(mul(a, b), 35)), -F(2, 105)),
            'formal sixth-degree obstruction')
    labels.append('formal sixth-degree obstruction')
    fourth = add(scale(one, F(1, 5)),
                 scale(add(one, b), -F(1, 3)), b)
    require(fourth == scale(add(scale(b, 5), scale(one, -1)), F(2, 15)),
            'formal fourth-degree obstruction')
    labels.append('formal fourth-degree obstruction')

    phi, root = K(F(1, 2), F(1, 2)), K(0, 1)
    star = (root - 1) / 4
    rstar = (5 + 3 * root) / 10
    require(phi**2 == phi + 1, 'golden identity')
    require((1 + phi**2)**2 == 5 * phi**2, 'endpoint bound')
    require((3 - 7 / phi**2).sign() > 0, 'uniform obstruction bound')
    require((F(2, 3) - 1 / phi**2).sign() > 0, 'positive cubic rule bound')
    labels.append('exact field identities and endpoint bounds')

    cases = [K(F(1, 1000000)), K(F(1, 10)), K(F(1, 4)), star,
             K(F(1, 2)), K(F(3, 4)), K(F(999999, 1000000))]
    for s in cases:
        A, B = s + phi**2, s + 1
        u, v = s / A, B / A
        alpha, beta = u**2, v**2
        g = masses(alpha, beta)
        require(g[0].sign() == (s - star).sign(), 'inner mass sign')
        require(g[1].sign() > 0 and g[2].sign() > 0, 'other mass signs')
        for j in range(3):
            require(g[0]*alpha**j + g[1]*beta**j + g[2] == F(1, 2*j+1),
                    'exact field moment')
        sixth_value = -F(2, 105)*(3 - 7*(alpha+beta) + 35*alpha*beta)
        require(sixth_value.sign() < 0, 'sixth moment obstruction')
        t = 2 / (3 * (2 - alpha - beta))
        require(t.sign() > 0 and (1 - 2*t).sign() > 0, 'cubic positivity')
        require(t*alpha + t*beta + 1 - 2*t == F(1, 3), 'cubic moment')
        ratio = (1+v)**2 / (4*v)
        require((rstar - ratio).sign() == (s-star).sign(), 'cross-ratio threshold')
        total_variation = sum((x * x.sign() for x in g), K())
        require(total_variation == 1 + (0 if g[0].sign() >= 0 else -2*g[0]),
                'amplification formula and attaining error signs')
        if s == star:
            require(beta == F(1, 5), 'boundary location')
            require(g == (K(0), K(F(5, 6)), K(F(1, 6))), 'boundary masses')
            require(ratio == rstar, 'boundary cross-ratio')
    labels.extend(['field moments and mass signs on seven seeds',
                   'strictly positive cubic rules on seven seeds',
                   'degree-six failure on seven seeds',
                   'cross-ratio threshold on seven seeds',
                   'sharp amplification on seven seeds',
                   'exact boundary weights and cross-ratio'])
    limiting = masses(K(0), 1 / phi**4)
    require(limiting[0] == -2 / (5 * phi), 'limiting negative mass')
    require(1 - 2*limiting[0] == 1 + 4 / (5 * phi), 'limiting amplification')
    labels.append('one-sided small-seed amplification limit')

    c, h, r, x = F(1, 2), F(1), F(1, 2), F(3, 4)
    floor = lambda z: z.numerator // z.denominator
    T = c + r*(x-c)
    require(h*floor(T/h)-c == -F(1, 2), 'unaligned grid counterexample')
    require(h*floor(r*(x-c)/h) == 0, 'recentered comparison')
    require(c + h*floor((T-c)/h)-c == h*floor(r*(x-c)/h), 'aligned grid')
    labels.append('grid-origin compatibility counterexample')
    for label in labels:
        print('PASS:', label)
    print(f'{len(labels)} exact check groups passed.')


if __name__ == '__main__':
    run()
```