OMEGA object exploration 02: projective recognition, record recovery, and positive integration
20 September 2026
This note derives three further connections from the six labeled baseline marks. It includes a perspective invariant, exact error-correction bounds, and a sharp obstruction to positive integration at the canonical seed angle. Allowing the seed angle to vary as an explicitly declared extension reveals a boundary at exactly 36 degrees.
The derivations use the marked-kernel definitions in Paper VII v1.6.0 and Paper VIII, Section 2. They are additional consequences developed in this exploration, using established mathematical methods. Historical priority for the particular applications has not been assessed.
|Connection                           |Result                                                                                                          |Essential conditions                                                                              |
|-------------------------------------|----------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------|
|Projective geometry                  |A cross-ratio of four outer marks recovers the seed parameter within the declared family.                       |Correct labels; a nondegenerate line projectivity; the prescribed golden extensions.              |
|Exact record recovery                |Six affine landmark records correct two arbitrary corrupted positions or reconstruct four missing positions.    |Known labels and seed; remaining records exact; affine coordinate model.                          |
|Recovery after perspective projection|Six scalar image records correct one arbitrary corruption or reconstruct three missing records.                 |Known seed; finite, nondegenerate line projectivity; exact remaining records.                     |
|Positive baseline integration        |The canonical six nodes admit positive degree-three exactness but no nonnegative rule exact through degree four.|Uniform measure on the full baseline interval; nodes restricted to the six marks.                 |
|A parameter boundary                 |At seed angle 36 degrees, the four outer marks support a positive degree-five rule.                             |This varies the seed angle; it is a related family member, not the canonical one-arcsecond kernel.|
The common definitions are
$$
\varphi=\frac{1+\sqrt5}{2},\qquad
\varepsilon=\frac{\pi}{648000},\qquad
\delta=\frac{\varepsilon}{2},\qquad
s=\sin\delta,\quad c=\cos\delta.
$$
Write
$$
A=s+\varphi^2,\qquad B=s+1,\qquad
(z_0,z_1,z_2,z_3,z_4,z_5)=(-A,-B,-s,s,B,A).
$$
The points, in the order $(d_2,d_1,d,b,b_1,b_2)$, satisfy
$$
p_i=\alpha+a(cu+z_i v),\qquad a>0,\quad
|u|=|v|=1,\quad u\cdot v=0.
\tag{1}
$$
The proofs below hold for the stated parameter ranges with $0<s<1$. The actual object uses the fixed value $s=\sin(\pi/1296000)$. Varying $s$ is an explicit family extension used to analyze recognition and the integration boundary.
1. Four outer marks carry a perspective-invariant seed parameter.
For four distinct finite points with line coordinates $t_0,t_1,t_2,t_3$, define the ordered cross-ratio by
$$
\mathcal C(t_0,t_1;t_2,t_3)
=\frac{(t_2-t_0)(t_3-t_1)}
{(t_2-t_1)(t_3-t_0)}.
\tag{2}
$$
The order matters; a different convention can give one of the other usual cross-ratio values.
For the four outer marks $(d_2,d_1,b_1,b_2)$, their normalized coordinates are $(-A,-B,B,A)$. Therefore
$$
R=\frac{(A+B)^2}{4AB}
=1+\frac{(A-B)^2}{4AB}
=1+\frac{\varphi^2}{4(s+\varphi^2)(s+1)}.
\tag{3}
$$
The last step uses $A-B=\varphi^2-1=\varphi$.
To prove projective invariance directly, let
$$
h(t)=\frac{\beta_1t+\beta_0}{\gamma_1t+\gamma_0},
\qquad
D=\beta_1\gamma_0-\beta_0\gamma_1\ne0,
$$
with no pole at the four selected points. Then
$$
h(x)-h(y)
=\frac{D(x-y)}
{(\gamma_1x+\gamma_0)(\gamma_1y+\gamma_0)}.
$$
Substitution in (2) cancels both copies of $D$ and every denominator factor. Thus the measured cross-ratio is unchanged. Translation, scale, reversal of the coordinate axis, and nondegenerate perspective projection along the line are covered.
The parameter can be recovered explicitly. Since
$$
A+B=2s+\varphi^2+1=2s+\varphi\sqrt5,
$$
equation (3) gives
$$
\frac{R}{R-1}
=\frac{(A+B)^2}{(A-B)^2}
=\left(\frac{2s}{\varphi}+\sqrt5\right)^2.
$$
The expression on the right has a positive square root, so
$$
\boxed{\quad
s=\frac{\varphi}{2}
\left(\sqrt{\frac{R}{R-1}}-\sqrt5\right),
\qquad
\varepsilon=2\arcsin s.
\quad}
\tag{4}
$$
The inverse is unique on $0<s<1$, because
$$
R'(s)
=-\frac{\varphi^2(2s+\varphi^2+1)}
{4(s+\varphi^2)^2(s+1)^2}<0.
$$
For the canonical one-arcsecond seed,
$$
R\approx1.24999916250720811554354836295459.
$$
High-precision substitution in (4) returns one arcsecond. The exact conclusion follows from the algebra and positive-root choice, not from the decimal computation.
This supplies an additional observation of the kernel: the seed parameter is recoverable from the labeled four-point cross-ratio within the prescribed family. The same quantity can be expressed using the outer ray slopes, because central projection to any transverse line is a line projectivity.
This observation does not recover absolute scale, center location, or ambient orientation. It also cannot certify the entire marked outline from four points. Those are separate data and proof obligations. At the canonical angle, $R$ is only about $8.37\times10^{-7}$ below $5/4$, so practical measurement precision matters. The result is an exact perspective identity, not an experimentally established image-recognition accuracy claim.
2. The six marked positions form an exact evaluation code.
For fixed trusted $s$ and labels, put
$$
m=\alpha+ac\,u,\qquad q=a\,v.
$$
Equation (1) becomes
$$
\boxed{\quad p_i=m+z_iq,\qquad i=0,\ldots,5.\quad}
\tag{5}
$$
Each coordinate is an affine polynomial in the known evaluation coordinate $z_i$. Define the distance between two six-position records to be the number of labeled positions at which their entire position vectors differ.
Suppose two candidate baselines $m+zq$ and $\widetilde m+z\widetilde q$ agree at two distinct nodes $z_i,z_j$. Subtracting the two equalities gives
$$
(z_i-z_j)(q-\widetilde q)=0.
$$
Since the nodes are distinct, $q=\widetilde q$, and then $m=\widetilde m$. Thus two different candidate records agree at at most one position and differ at at least five.
This lower bound is attained in the affine record family: choose nonzero $\Delta q$ and set $\Delta m=-z_j\Delta q$. The two records then agree exactly at node $z_j$. Accordingly the minimum record distance is five.
It follows that:
	●	Up to two arbitrary corruptions of position records can be uniquely corrected.
	●	Up to four erasures, whose locations are known, can be reconstructed.
	●	With $e$ corruptions and $r$ erasures, the sufficient unique-recovery condition is $2e+r\le4$.
For the mixed case, two candidates each within $e$ errors of the retained observations must agree at at least $6-r-2e$ retained nodes. Under the stated inequality this is at least two, so they are equal.
A constructive decoder is short. For each retained pair $i\ne j$, compute
$$
q_{ij}=\frac{p_j-p_i}{z_j-z_i},
\qquad
m_{ij}=p_i-z_iq_{ij}.
\tag{6}
$$
Reject $q_{ij}=0$. Predict all six positions and count exact agreements with retained records. With six received records and at most two errors, the true candidate agrees at at least four positions. It occurs among the candidates generated from two correct records. Two different candidates cannot both have four agreements, because their agreement sets would overlap at two positions. Return that unique candidate, or report failure if none is admissible.
Three arbitrary corruptions cannot always be corrected. Take two valid affine records differing at all six positions, and build an observed record using three positions from each. It is three errors away from each candidate. That ambiguity cannot be resolved using these six records alone.
This is the degree-one polynomial-evaluation mechanism underlying Reed-Solomon-type codes. The general polynomial-evaluation construction and its distance calculation are established in coding theory; the present use treats landmark vectors as records over exact real or symbolic coordinates. See Guruswami, Algorithmic Results in List Decoding, Section 2.3, for the standard finite-field construction. Primary author manuscript.
The guarantee concerns sparse exact corruptions. Ordinary measurement noise can alter every position slightly; its treatment requires a specified tolerance and a separate stability analysis. The trusted label-to-$z_i$ association and seed parameter are part of the decoder's inputs. Beyond the stated error bound, a decoder may encounter ambiguity or may favor another valid record.
Recovery of the six baseline positions is also distinct from recovery of the full spatial placement. From $m,q$, one obtains $a=|q|$ and $v=q/a$. If a correct center $\alpha$ is retained, then
$$
u=\frac{m-\alpha}{ac}.
$$
The conditions $|u|=1$ and $u\cdot v=0$ must be checked. Without the center or equivalent frame information, the baseline alone leaves an ambient placement ambiguity.
3. The recovery bound changes under a projective observation.
Suppose the retained image coordinate has the form
$$
y_i=h(z_i),\qquad
h(t)=\frac{at+b}{ct+d},\qquad ad-bc\ne0,
\tag{7}
$$
and all six values are finite. Here $s$ and the node labels remain trusted.
For two such maps, equality at a node is equivalent to
$$
(at+b)(c't+d')-(a't+b')(ct+d)=0.
$$
The numerator has degree at most two. Three distinct zeros therefore force the two rational functions to agree identically. Distinct projective records agree at at most two nodes and differ at at least four.
The bound four is attained. Pick two distinct nodes $r_0,r_1$, and choose a sufficiently small nonzero $\kappa$ such that no relevant denominator vanishes. The map
$$
h_\kappa(t)=
\frac{(1-\kappa(r_0+r_1))t+\kappa r_0r_1}
{1-\kappa t}
$$
has determinant $(1-\kappa r_0)(1-\kappa r_1)\ne0$, and
$$
h_\kappa(t)-t
=\frac{\kappa(t-r_0)(t-r_1)}{1-\kappa t}.
$$
Thus it agrees with the identity at exactly those two nodes and differs at the other four. The minimum record distance is four.
|Coordinate model                    |Data that determine one candidate|Arbitrary corruptions guaranteed correctable|Erasures guaranteed recoverable|Mixed condition|
|------------------------------------|--------------------------------:|-------------------------------------------:|------------------------------:|---------------|
|Affine positions $m+z_iq$           |Two labeled positions            |2                                           |4                              |$2e+r\le4$     |
|Projective line coordinates $h(z_i)$|Three labeled positions          |1                                           |3                              |$2e+r\le3$     |
To decode a single corruption in the projective case, fit the projectivity through each retained triple, rejecting repeated image values, singular maps, or poles at required nodes. Retain a candidate only if it agrees with at least five of the six received values. Existence follows by choosing three correct records; uniqueness follows because two five-position agreement sets overlap at at least four nodes.
Two-error ambiguity is real. Use the identity and $h_\kappa$ above, retain their two common positions, and choose two of the other positions from each candidate. The resulting observation is two errors from each valid projective record.
This identifies a precise change in recoverability when the representation changes. The affine decoder applies to affine coordinates or to correctly rectified data. Raw perspective coordinates require the projective model.
4. The six marks support positive degree-three integration, with a degree-four obstruction.
Choose a new, explicit task: approximate the uniform average of a scalar function along the full baseline interval,
$$
I(f)=\frac1{2A}\int_{-A}^{A}f(t)\,dt.
\tag{8}
$$
The interval includes the small central gap between $-s$ and $s$. It is a declared baseline observable; it is not integration over the circular outline, and it is not the sphere measure used by the icosahedral rules.
Restrict samples to the existing six nodes and require nonnegative weights. Define
$$
D=2A^2-s^2-B^2,\qquad
w=\frac{2A^2}{3D},\qquad w_A=1-2w.
$$
Use total pair masses $w,w,w_A$ at ${\pm s}$, ${\pm B}$, and ${\pm A}$ respectively:
$$
Q(f)=\frac w2[f(-s)+f(s)+f(-B)+f(B)]
+\frac{w_A}{2}[f(-A)+f(A)].
\tag{9}
$$
Consider the range
$$
0<s<s_*=\frac{\sqrt5-1}{4}.
\tag{10}
$$
It is equivalent to $A>\sqrt5\,B$, since
$$
A-\sqrt5 B=(\sqrt5-1)(s_*-s).
$$
In this range $A^2>5B^2$ and $B^2>s^2$. Hence $D>0$ and $w>0$. Also
$$
w_A=\frac{\frac23A^2-s^2-B^2}{D}>0,
$$
because its numerator is greater than $\frac73B^2-s^2>0$.
The weights sum to one, all odd moments vanish by symmetry, and
$$
w(s^2+B^2)+(1-2w)A^2=\frac{A^2}{3}=I(t^2).
$$
Thus (9) integrates every polynomial of degree at most three exactly with strictly positive weights at all six marks.
For the canonical angle, the total pair masses are approximately
$$
(w_s,w_B,w_A)=
(0.359563193388285,\ 0.359563193388285,\ 0.280873613223430).
$$
Each individual node receives half its pair mass.
To show that degree four is impossible for any nonnegative weights on these nodes, take
$$
P(t)=(t^2-B^2)(t^2-A^2).
\tag{11}
$$
It vanishes at the four outer nodes. At the inner pair both factors are negative, so $P(\pm s)>0$. Therefore every nonnegative weighted rule on the six nodes has $Q(P)\ge0$.
However, direct integration gives
$$
\begin{aligned}
P(t)&=(t^2-B^2)(t^2-A^2),\\
I(P)
&=\frac{A^4}{5}-(A^2+B^2)\frac{A^2}{3}+A^2B^2\\
&=\frac{2A^2}{15}(5B^2-A^2)<0,
\qquad 0<s<s_*.
\end{aligned}
\tag{12}
$$
Exactness on all degree-four polynomials would require $Q(P)=I(P)$, a contradiction. This argument does not assume symmetric weights.
The canonical seed lies in (10): $\sin(\pi/1296000)<\pi/1296000<1/4<s_*$. Therefore three is the maximum degree of exactness attainable with nonnegative weights on the six canonical nodes for the measure (8).
If negative weights are permitted, interpolation at the six distinct nodes gives a rule exact through degree five. Its weights are unique and symmetric. The total mass at the inner pair is forced by (11) to equal
$$
\frac{I(P)}{P(s)}
=
\frac{2A^2(5B^2-A^2)}
{15(s^2-B^2)(s^2-A^2)}
<0,
\qquad 0<s<s_*.
$$
Thus the failure of positivity can be located explicitly in the degree-five rule.
Positive integration with prescribed nodes is an established subject: node locations impose constraints on the available positive weights and exactness. The present obstruction and weights follow directly from the displayed OMEGA coordinates. For related general work, see van den Bos and Sanderse, A geometric approach for the addition of nodes to an interpolatory quadrature rule with positive weights. Primary author manuscript.
5. Varying the seed reveals an exact 36-degree boundary.
At $s=s_*$, the ratio $A/B$ is exactly $\sqrt5$. Moreover,
$$
s_*=\frac{\sqrt5-1}{4}=\sin\frac{\pi}{10}.
$$
This trigonometric identity can be verified without a decimal approximation. The positive number $r=s_*$ is less than $1/2$ and satisfies
$3r-4r^3=1-2r^2$. Set $\theta=\arcsin r\in(0,\pi/6)$. The identity becomes
$\sin(3\theta)=\cos(2\theta)=\sin(\pi/2-2\theta)$.
Both arguments lie in $(0,\pi/2)$, where sine is injective. Therefore
$3\theta=\pi/2-2\theta$, so $\theta=\pi/10$.
Since $\varepsilon=2\delta$ and $s=\sin\delta$, the boundary is
$$
\boxed{\varepsilon_*=\frac{\pi}{5}=36^\circ.}
$$
After normalizing the full interval to $[-1,1]$, the four outer nodes are then
$$
-1,\quad-\frac1{\sqrt5},\quad\frac1{\sqrt5},\quad1.
$$
Assign them probability weights
$$
\frac1{12},\quad\frac5{12},\quad\frac5{12},\quad\frac1{12},
$$
and assign zero to the innermost pair. Odd moments vanish. The zeroth, second, and fourth moments are respectively
$$
1,\qquad
\frac16+\frac56\frac15=\frac13,\qquad
\frac16+\frac56\frac1{25}=\frac15.
$$
These identities prove exactness through degree five. This is the four-node Gauss-Lobatto configuration, obtained here directly from the boundary condition on the golden extensions. The weights are strictly positive on its four used nodes and nonnegative on all six available marks.
The 36-degree case changes the seed angle and therefore the kernel instance. It does not replace or reinterpret the canonical one-arcsecond seed. It identifies an exact boundary in an explicitly declared family of related constructions.
The roles and limits of the connections are now explicit:
|Operation                         |Preserved or recovered information                                                          |Information or property requiring additional work                                                 |
|----------------------------------|--------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------|
|Projective observation of the line|Ordered cross-ratio; seed parameter within the declared family                              |Euclidean scale, pose, full outline, and measurement-error accuracy                               |
|Affine record decoding            |Six exact marked positions within the corruption budget                                     |Trusted center/frame for full placement; a noise model for approximate data                       |
|Projective record decoding        |The six image positions within its smaller corruption budget                                |Euclidean rectification and calibration                                                           |
|Positive six-node integration     |Uniform baseline moments through degree three                                               |Degree four is prohibited at the canonical angle under the stated positivity and node restrictions|
|Changing the seed to 36 degrees   |The construction formulas remain defined and the outer nodes support degree-five integration|This is a different member of the parameterized family                                            |
Verification and reproducibility. The companion Python script imports no OMEGA implementation. It computes exact rational functions over $\mathbb Q(\sqrt5)(s)$, with $s$ an indeterminate, and supplies exact affine and projective decoder prototypes. Decimal values use 70-digit working precision only for illustration.
The checker reports its group count from len(checks). It retains the original 19 mathematical groups, including all 15 choices of two corrupted affine record locations, all 15 choices of four affine erasures, all six choices of one corrupted projective location, and all 20 choices of three projective erasures, plus two groups verifying malformed-input rejection. It also constructed the ambiguity examples and checked the quadrature moments, obstruction, and 36-degree boundary identities. The corruption values in those examples are particular exact fixtures; the written proofs establish the general recovery bounds.
Run the supplied script with Python and SymPy 1.14.0:
```bash
python3 OMEGA_Object_Exploration_02.py --output exploration_results.json
```
The program raises explicit errors on failed checks and writes its result only after success. It checks symbolic identities and decoder examples, not a proof-assistant certificate or a practical camera-noise model.
Source context: the coordinates and labels were read from the Paper VII v1.6.0 public source package and checked against Section 2 of Paper VIII. The construction, metric, and Gram sections examined there do not develop the cross-ratio inverse, these landmark decoding budgets, or this baseline positivity obstruction. That comparison is limited to those reviewed sections and is not an exhaustive literature or archive priority search.