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prop:symmetry | Covariance and normalization | proposition | area_feedback | manuscript_proved | \label{prop:symmetry}
Orthogonal isomorphisms intertwine the legal update. For every nonzero real $c$, the similarity $(x,A,a)\mapsto(cx,c^2A,ca)$ also intertwines it. Replacing the update by $A'=A+\gamma x\wedge a$ for a fixed $\gamma\neq0$ produces an equivalent normalized system under $\widetilde A=A/\gamma$. | The identities $(QAQ^*)(Qa)=Q(Aa)$ and $(Qx)\wedge(Qa)=Q(x\wedge a)Q^*$ give the first assertion. The scaled gate is $c^3Aa=0$ and the scaled area increment is $c^2x\wedge a$, proving the second. Division by $\gamma$ preserves the kernel and converts the third rule into~\eqref{eq:update}. | [
"eq:update"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
prop:area | Exact area accumulation | proposition | area_feedback | manuscript_proved | \label{prop:area}
For any finite word from $(x_0,A_0)$, evaluated by the algebraic update even if legality is ignored,
\begin{align}
x_n&=x_0+\sum_{j=1}^n a_j,\\
A_n&=A_0+x_0\wedge\sum_{j=1}^n a_j+
\sum_{1\leq i<j\leq n}a_i\wedge a_j.
\label{eq:area}
\end{align}
On $V\oplus\bigwedge^2V$, the product
\[... | The endpoint formula follows by summation. Substituting $x_{j-1}=x_0+\sum_{i<j}a_i$ into $A_j-A_{j-1}=x_{j-1}\wedge a_j$ and summing proves~\eqref{eq:area}. Both associations of three factors have endpoint $x+y+z$ and area $A+C+F+x\wedge y+x\wedge z+y\wedge z$, proving associativity. The remaining assertions follow fro... | [
"eq:area"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
lem:decomposition | Orthogonal decomposition | lemma | area_feedback | manuscript_proved | \label{lem:decomposition}
For real skew-adjoint $A$, $R=\im A=(\ker A)^\perp$. The restriction $B=A|_R$ is an invertible skew-adjoint operator on $R$. If $h=B^{-1}u$, then $\inner{u}{h}=0$. | A vector $v$ is perpendicular to $\im A$ exactly when $\inner{v}{Az}=0$ for all $z$, equivalently $A^*v=-Av=0$. Thus $V=R\oplus K$. The subspace $R$ is $A$-invariant since $A$ maps all of $V$ into $R$. Its restricted kernel is $R\cap K=0$, so $B$ is invertible in finite dimension. Its inverse is skew-adjoint: $(B^{-1})... | [] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:kernel | Kernel transport and rank creation | theorem | area_feedback | manuscript_proved | \label{thm:kernel}
Let $a\in K=\ker A$, write $x=u+w$ with $u\in R$ and $w\in K$, and set $h=(A|_R)^{-1}u$. Let $A'=A+x\wedge a$.
If $w,a$ are linearly independent, then
\begin{equation}
\ker A'=K\cap w^\perp\cap a^\perp,
\qquad \rank A'=\rank A+2,
\qquad P_{\ker A'}(x+a)=0.
\label{eq:birth}
\end{equation}
If $w,a... | Relative to $V=R\oplus K$, the new operator is
\begin{equation}
A'=\begin{pmatrix}
B&ua^T\\
-au^T&wa^T-aw^T
\end{pmatrix}.
\label{eq:block}
\end{equation}
Let $r+z\in R\oplus K$ be in its kernel. The upper equation gives
\[
Br+u\inner{a}{z}=0,
\qquad r=-h\inner{a}{z}.
\]
Since $\inner{u}{h}=0$, substitution into... | [
"eq:lower"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:rho | Exact update of the exclusion radius | theorem | area_feedback | manuscript_proved | \label{thm:rho}
Under the hypotheses and notation of Theorem~\ref{thm:kernel}, put $\rho'=\rho(x+a,A')$.
If $w,a$ are independent, then
\begin{equation}
(\rho')^2=\rho^2+\norm{w+a}^2.
\label{eq:rho-birth}
\end{equation}
If they are dependent, then
\begin{equation}
(\rho')^2=\rho^2+
\frac{\norm{h}^2\inner{a}{w+a}^2}... | In the independent case, Theorem~\ref{thm:kernel} says $x+a\in\im A'$. Therefore $(\rho')^2=\norm{x+a}^2=\norm{u}^2+\norm{w+a}^2$.
In the dependent case, define $J:K\to V$ by $Jz=z-h\inner{a}{z}$. Its image is the new kernel. Orthogonality of $K,R$ gives
\[
J^*J=I_K+\norm{h}^2aa^T,
\qquad J^*(x+a)=w+a.
\]
Write $v=w... | [
"eq:rho-graph",
"thm:kernel"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:minimum | Global optimal endpoint and uniqueness | theorem | area_feedback | manuscript_proved | \label{thm:minimum}
For every state $(x,A)$, with $u=P_{\im A}x$ and $w=P_{\ker A}x$, equation~\eqref{eq:main} holds. The endpoint $u$ is reached by the legal move $a=-w$. If $u\neq0$, every nonzero move preserving $\rho$ must be exactly $-w$; after that move every further nonzero legal move strictly raises $\rho$. | For any finite legal trajectory, Theorem~\ref{thm:rho} gives $\rho(s_n)\geq\rho(s_0)$. Every endpoint satisfies $\norm{x_n}\geq\rho(s_n)$ because orthogonal projection cannot increase norm. Thus no reachable endpoint has norm below $\rho(s_0)$.
The vector $a=-w$ belongs to $K$, yields $x'=u$, and is in the dependent c... | [
"eq:main",
"eq:rho-graph",
"thm:kernel",
"thm:rho"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
cor:origin | Exact origin test | corollary | area_feedback | manuscript_proved | \label{cor:origin}
From an arbitrary state $(x,A)$, the origin is reachable if and only if $Ax=0$. If it is reachable, one move $a=-x$ reaches it. | Theorem~\ref{thm:minimum} makes origin reachability equivalent to $P_{\im A}x=0$, which by Lemma~\ref{lem:decomposition} is $x\in\ker A$. The stated move is then legal, and $x\wedge(-x)=0$. | [
"lem:decomposition",
"thm:minimum"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
cor:closed | Closed-loop rigidity from zero memory | corollary | area_feedback | manuscript_proved | \label{cor:closed}
For a legal history starting at $(0,0)$, if its final endpoint is zero, then every area tensor along the history is zero. More strongly, if $j$ is its first step with $x_{j-1}\wedge a_j\neq0$, then
\[
\norm{x_n}\geq\rho(s_n)\geq\norm{x_j}>0
\qquad\text{for every }n\geq j.
\] | Before the first nonzero area increment, $A_{j-1}=0$. Thus $R=0$, $w=x_{j-1}$, and the nonzero wedge means $w,a_j$ are independent. Formula~\eqref{eq:rho-birth} gives $\rho(s_j)=\norm{x_j}>0$. Monotonicity proves the inequality and rules out a later zero endpoint. If no such first increment exists, every $A_j$ is zero. | [
"eq:rho-birth"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:exterior | Orthogonal exterior increments | theorem | area_feedback | manuscript_proved | \label{thm:exterior}
Every legal update satisfies, for each integer $j\geq1$,
\begin{equation}
E_j(A')=E_j(A)+
\norm{\frac{\alpha^{\wedge(j-1)}}{(j-1)!}\wedge x\wedge a}^2.
\label{eq:exterior}
\end{equation}
All $E_j$ are therefore nondecreasing along legal histories. | The bivector $x\wedge a$ is decomposable, so its square under exterior multiplication is zero. Since degree-two elements commute under exterior multiplication,
\[
\frac{(\alpha+x\wedge a)^{\wedge j}}{j!}
=\frac{\alpha^{\wedge j}}{j!}
+\frac{\alpha^{\wedge(j-1)}}{(j-1)!}\wedge x\wedge a.
\]
The tensor $\alpha$ is sup... | [
"eq:exterior"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
prop:scalar | Two scalar identities | proposition | area_feedback | manuscript_proved | \label{prop:scalar}
Let $E=E_1(A)$, $N=\norm{Ax}^2$, and
\[
D=\norm{x\wedge a}^2=\norm{x}^2\norm{a}^2-\inner{x}{a}^2.
\]
For every legal update,
\begin{equation}
E'=E+D,
\qquad N'=N+\norm{x+a}^2D.
\label{eq:scalar}
\end{equation}
If $E>0$, the weaker radius $\sqrt{N/E}$ is nondecreasing and satisfies $\sqrt{N/E}\le... | The first identity is the case $j=1$ of Theorem~\ref{thm:exterior}. Put $X=\norm{x}^2$, $Y=\norm{a}^2$, $c=\inner{x}{a}$. Since $Aa=0$,
\[
A'(x+a)=Ax+x(c+Y)-a(X+c).
\]
The vector $Ax$ is perpendicular to both $x$ and $a$. The squared norm of the remaining vector is
\[
X(c+Y)^2+Y(X+c)^2-2c(c+Y)(X+c)
=(XY-c^2)(X+Y+2c)... | [
"thm:exterior"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:rank | Minimum steps to maximal rank | theorem | area_feedback | manuscript_proved | \label{thm:rank}
Let $k=\dim\ker A$, $m=\lfloor k/2\rfloor$, and $w=P_{\ker A}x$. The maximal rank reachable from $(x,A)$ is
\[
\rank A+2m=2\lfloor d/2\rfloor.
\]
If $m=0$, that rank is already attained. If $m>0$, the minimum number of legal steps needed to attain it is
\begin{equation}
\begin{cases}
2m,&w=0,\\
2m-... | Theorem~\ref{thm:kernel} permits rank increases only by two, and at most $m$ such increases fit in dimension $d$. A rank increase resets the new kernel projection of the endpoint to zero. From a state with that projection zero, the next move cannot raise rank because its $w$ is zero. Thus at least one intervening move ... | [
"thm:kernel"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
cor:parity | Even and odd ambient dimensions | corollary | area_feedback | manuscript_proved | \label{cor:parity}
In even dimension, a full-rank state permits only the zero move. In odd dimension, every state permits a nonzero move. From $(0,0)$, maximal rank can be reached in $d$ steps for even $d$ and $d-1$ steps for odd $d\geq1$. | The kernel is zero exactly at full rank. An odd-dimensional real skew matrix always has nonzero kernel. Apply Theorem~\ref{thm:rank} with $k=d$ and $w=0$; the cases $d=0,1$ already have maximal rank zero. | [
"thm:rank"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:launch | Orthogonal launch family | theorem | area_feedback | manuscript_proved | \label{thm:launch}
Let $e_1,\ldots,e_d$ be orthonormal. The prefix $e_1,\ldots,e_n$, $0\leq n\leq d$, is legal from zero and yields
\[
x_n=\sum_{i=1}^n e_i,\qquad
A_n=\sum_{1\leq i<j\leq n}e_i\wedge e_j.
\]
Its rank is $2\lfloor n/2\rfloor$. For $n\geq1$,
\begin{equation}
\rho_n^2=\begin{cases}n,&n\text{ even},\\ n-... | Before step $i$, the area is supported on earlier coordinates, so it annihilates $e_i$. Formula~\eqref{eq:area} gives the state. In the occupied coordinates the matrix has every upper-triangular entry one. If it annihilates a vector $v$, subtracting successive row equations gives $v_{i+1}=-v_i$. The first row then forc... | [
"eq:area"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:accessibility | Three steps are sufficient and necessary for endpoint interior at a specified state | theorem | area_feedback | manuscript_proved | \label{thm:accessibility}
Start in $\R^3$ at $x=(1,1,0)^T$, $k=(0,0,1)^T$, equivalently $A=e_1\wedge e_2$. The set of endpoints reachable in three legal moves has nonempty interior in $\R^3$. The union of endpoints reachable in at most two moves has empty interior. | Use scalar controls $p,q,r$ in~\eqref{eq:axial}. All are legal and all intermediate $k$ remain nonzero. Direct substitution gives the polynomial endpoint map
\begin{equation}
F(p,q,r)=
\begin{pmatrix}
1+pq+pr+qr+p^2qr\\
1-pq-pr-qr+p^2qr\\
p+q+r-2pqr
\end{pmatrix}.
\label{eq:endpointpoly}
\end{equation}
For clari... | [
"eq:axial",
"eq:endpointpoly"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:accessibility-general | Three-dimensional endpoint accessibility classification | theorem | area_feedback | manuscript_proved | \label{thm:accessibility-general}
Let $V=\R^3$ and $A\neq0$. The finite reachable endpoint set has nonempty interior if and only if $Ax\neq0$. Whenever $Ax\neq0$, the exact minimum number of steps needed for an endpoint set with nonempty interior is three. | If $Ax=0$, then $x\in\ker A$, which is a line. Every legal input lies on that line, writes zero wedge, and leaves the kernel fixed. All endpoints remain on the line, so the reachable set has empty interior.
If $Ax\neq0$, let $k=k(A)$. An orientation-preserving orthogonal change of coordinates puts the state in the for... | [
"eq:axial"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:dormant | Invariant inaccessible summand | theorem | area_feedback | manuscript_proved | \label{thm:dormant}
For a state $(x,A)$, let $K=\ker A$, $u=P_{\im A}x$, and define
\[
C=\Span\{A^j u:j\geq0\},\qquad S=K\oplus C,
\qquad H=S^\perp.
\]
Then $S,H$ are $A$-invariant and $A|_H$ is invertible. Every future legal input and every future endpoint lies in $S$; the block $A|_H$ never changes. Replacing it by... | The cyclic subspace $C$ is contained in $\im A$ and is $A$-invariant. Indeed, the span stabilizes in finite dimension, and multiplication by $A$ shifts each spanning vector to the next. Since $A$ is skew-adjoint, the orthogonal complement of an invariant subspace is invariant. Thus $S$ and $H$ are invariant. Also $H\su... | [] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
prop:subdivision | Reversal and subdivision | proposition | area_feedback | manuscript_proved | \label{prop:subdivision}
Let $a$ be legal at $(x,A)$. The algebraic inverse displacement $-a$ is legal immediately after that move if and only if $x\wedge a=0$. If $x\wedge a\neq0$ and $m\geq2$ is an integer, replacing $a$ by $m$ successive copies of $a/m$ fails legality at the second copy. | Since $Aa=0$,
\[
(A+x\wedge a)a=\norm{a}^2x-\inner{x}{a}a.
\]
If $a\neq0$, this is zero exactly when $x$ is parallel to $a$, equivalent to $x\wedge a=0$. If $a=0$, both assertions hold trivially. When the reverse move is legal, substitution shows that it restores both $x$ and $A$.
After the first fractional move the ... | [] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:continuous | Continuous collapse and a quantitative refinement bound | theorem | area_feedback | manuscript_proved | \label{thm:continuous}
Suppose $x,A$ are absolutely continuous on a compact interval, $A(t)$ is skew-adjoint, and
\[
\dot x=v,\qquad \dot A=x\wedge v,\qquad Av=0
\]
hold almost everywhere. If $A(0)=0$, then $A(t)=0$ everywhere and $x\wedge\dot x=0$ almost everywhere.
For a finite legal discrete trajectory with $A_0=0... | Absolute continuity permits the chain rule for the quadratic function $\norm A_F^2/2$. Its derivative is $\inner{A}{x\wedge v}_F=2\inner{x}{Av}=0$ almost everywhere. Hence it remains zero, and positive definiteness forces $A=0$. The area differential equation then gives $x\wedge\dot x=0$.
In the discrete case, sum the... | [
"eq:refinement",
"eq:scalar"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
prop:closure | A nonclosed endpoint set | proposition | area_feedback | manuscript_proved | \label{prop:closure}
For the state $x_0=(1,1,0)^T$, $A_0=e_1\wedge e_2$ in $\R^3$, the closure of $\Reach(x_0,A_0)$ contains points other than $x_0$ of norm $\sqrt2$. None of those other points is reachable by a finite legal word. Consequently this reachable endpoint set is not closed. | Consider the polynomial ordinary differential equation
\[
\dot x=k,\qquad \dot k=x\times k,
\qquad x(0)=x_0,\ k(0)=e_3.
\]
It has a smooth solution on a sufficiently small interval. Direct differentiation yields
\[
\norm{k(t)}^2=1,\qquad x(t)\cdot k(t)=t,
\qquad \norm{x(t)}^2=2+t^2.
\]
Define $u(t)=x(t)-t k(t)$. Th... | [
"eq:axial",
"thm:minimum"
] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
conj:interior | Interior for cyclic states in odd dimension | conjecture | area_feedback | conjecture | \label{conj:interior}
Let $d\geq5$ be odd, let $\rank A=d-1$, and suppose
\[
\ker A\oplus\Span\{A^jP_{\im A}x:j\geq0\}=V.
\]
Then the finite reachable endpoint set has nonempty interior in $V$. | [] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. | |
thm:alt-interval | Interval update and terminal formula | theorem | algebraic_alternative | manuscript_proved | \label{thm:alt-interval}
For $1\leq m\leq N-1$,
\begin{equation}
D(U_m)=U_{\min(m,N-m-1)},\qquad
T(U_m)=U_{\max\{m,\,2\min(m,N-m-1)\}}.
\label{eq:alt-interval-step}
\end{equation}
Put $n=N-1$, let $r$ be the smallest nonnegative integer for which
$2^rm\geq n/2$, and set $a=2^rm$. Then
\begin{equation}
F(U_m)=U_{\ma... | Each $t^i\in U_m$ has only one possible nonzero pairing with a basis element,
namely $t^{N-i}$. That partner is absent from $U_m$ exactly when
$i\leq N-m-1$. Distinct exponents have distinct partners, so there is no
additional cancellation between basis columns. This proves the radical formula.
If $d=\min(m,N-m-1)$, pr... | [] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:alt-tensor | An exact tensor activation law | theorem | algebraic_alternative | manuscript_proved | \label{thm:alt-tensor}
Let $B=k[t]/(t^5)$, $\tau=[t^4]$, and $U=\operatorname{span}\{t,t^2\}$.
The state $U$ is terminal. For an integer $r\geq1$, put
\[
W_r=U^{\otimes r}\subset
B^{\otimes r}=k[x_1,\ldots,x_r]/(x_1^5,\ldots,x_r^5),
\]
with the tensor trace extracting $x_1^4\cdots x_r^4$. Then
\begin{equation}
F(W_r... | On $U$, the only nonzero basis pairing is $\beta(t^2,t^2)=1$.
Thus $D(U)=kt$ and $D(U)^2=kt^2\subset U$, proving terminality.
The initial monomial exponent set of $W_r$ is $S_r=\{1,2\}^r$.
An initial monomial can pair with another initial monomial only when both
have every exponent equal to two. Hence $D(W_r)$ contain... | [] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
thm:spectral-rank-one | Positive rank-one spectral representation | theorem | spectral_appendix | manuscript_proved | \label{thm:spectral-rank-one}
For $A'=A+x\wedge a$, the positive squared skew frequencies of $A'$,
counting multiplicities, are as follows.
\begin{enumerate}
\item If $D_0=0$, they are the eigenvalues of
\[
H^+=H+zz^T.
\tag{S2}
\]
The rank of $A'$ is $2m$, and its kernel has dimension $k$.
\item If $D_0>0$, define
\[... | For $t>0$, the matrix $M=tI-A$ is invertible, because the eigenvalues of
a real skew-adjoint operator are zero or purely imaginary. Write
\[
\beta=\norm a^2,\qquad \gamma=\inner x a=\inner w a,\qquad
f(t)=x^TM^{-1}x.
\]
The admissibility condition $Aa=0$, together with skew-adjointness,
gives both
\[
M^{-1}a=\frac a... | [] | manuscript/Avenyra_Area_Feedback_v1.0.0.tex | audits/PROOF_AUDIT.txt | No per-claim priority certification; see audits/NOVELTY_AUDIT.txt. | Manuscript proof status is not formal certification; finite code checks are separate. |
Avenyra Area Feedback
A path writes signed-area memory, and that memory determines which moves the path may take next.
Research byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research release: v1.0.0 · 7 October 2026
This repository contains a complete 40-page mathematical manuscript, editable LaTeX, three exact verification programs, proof and novelty audits, and structured records for examining the claims and worked examples. It is an AI-assisted research dossier with explicit proofs and unresolved priority questions.
Read the manuscript · Editable source · Proof audit · Novelty audit
The mechanism
Fix a finite-dimensional real Euclidean space. A state is a vector $x$ and a skew-adjoint operator $A$, with $A^T=-A$. An input $a$ is legal exactly when $Aa=0$. A legal move updates the state by
The legality test and both right-hand sides use the old state. The resulting operator governs the next input. From zero, a history accumulates
Signed-area accumulation is established mathematics. The manuscript studies its coupling to this changing kernel constraint.
Central result
Let $\operatorname{Reach}(x,A)$ be the endpoints of all finite legal histories, including the empty history. Then
The unique minimum endpoint is attained by one legal input, $a=-P_{\ker A}x$. Later kernel changes cannot improve that value. Uniqueness is about the endpoint, not the control word; taking the closure of the reachable set can add other minimum-norm points.
Further results and boundaries
| Result | Precise scope |
|---|---|
| Kernel transport | A move either removes two kernel dimensions or transports the kernel as an explicit graph. |
| Sharp rank timing | For $m=\lfloor\dim\ker A/2\rfloor>0$, maximal skew rank requires exactly $2m$ moves when $P_{\ker A}x=0$, and $2m-1$ otherwise. |
| Exterior coefficients | Every squared exterior-volume coefficient is nondecreasing under exact legal moves. |
| Three-dimensional interior | For $A\ne0$, endpoint interior exists exactly when $Ax\ne0$; three moves are necessary and sufficient. |
| Inaccessible blocks | Certain invertible skew blocks can be replaced without changing any legal future input word or endpoint. |
| Finite versus limiting reachability | A proved example has a nonclosed reachable endpoint set. |
| Refinement boundary | A nonzero area-writing jump cannot be split into equal legal substeps along its direction. |
| Continuous boundary | The absolutely continuous exact-gate analogue cannot create area from zero memory. |
The paper includes twelve candidate mechanisms and three developed comparisons. A retained Frobenius-algebra alternative exhibits nonmonotone terminal growth and tensor activation; its monomial sector is also identified with inhibited reactions.
A calculation
In three dimensions, the legal inputs $e_1,e_2,e_3$ from zero give
The optimal next input is $(-1/3,1/3,-1/3)^T$. The unique finite minimum endpoint is $(2/3,4/3,2/3)^T$, whose squared norm is $8/3$.
The structured examples also include dimension-zero and dimension-one limits, a two-dimensional shutdown, two hidden-memory parameters, and an illegal rectangle continuation.
Originality and established ancestry
The documented search, dated 7 October 2026, includes 61 specialist queries in 18 batches and additional primary-source comparisons. The original proof and prior-art reviews were separately assigned AI-agent tasks. Packaging this repository adds no independent human review or new literature investigation.
Established ancestors include path signatures and free step-two groups, Goh-matrix kernel constraints, explicit Euler integration, a three-dimensional radial Lorentz equation, and older examples of Euler accessibility and nonclosed reachability. Classical rank-one matrix analysis underlies the spectral appendix.
Within that documented search, no matching source was located for the general-dimensional finite rule together with the exact projection minimum, kernel-transport formulas, and sharp rank-timing results developed here. Priority remains unresolved. This is not worldwide novelty certification. Full behavioral classification and less obvious singular-control equivalences remain open.
See the complete novelty audit for sources, successful reductions, search scope, and gaps. The manuscript contains the references actually used.
Repository contents
| Path | Contents |
|---|---|
manuscript/ |
The original PDF and standalone editable LaTeX, preserved byte for byte. |
code/ |
Three original verifiers and pinned mathematical dependency. |
audits/ |
Original proof and novelty audit notes. |
data/claims.jsonl |
23 extracted statement records, including proofs where supplied and the labeled conjecture. |
data/worked_examples.jsonl |
Eight exact finite examples, with states, controls, legality, rank, and exclusion radius. |
docs/ |
Reproduction, schema, AI-agent, and expert-review guides. |
verification/original/ |
Outputs captured for the original research release. |
verification/rerun/ |
Outputs and runtime report from the packaging verification run. |
provenance/ |
Original README, validation record, checksum list, and packaging changes. |
release/ |
The untouched original research ZIP. |
tools/validate_release.py |
Offline integrity, extraction, reference, and exact-example checks. |
manifest.json, SHA256SUMS.txt |
Checksums for this Hugging Face repository layout. |
Reproduce
From this repository root:
python -m pip install -r code/requirements.txt
python code/verify_avenyra.py
python code/independent_verify.py
python code/verify_candidates.py
python tools/validate_release.py
The mathematical dependency is SymPy 1.14.0. The third original verifier uses only the Python standard library. Exact rational arithmetic and symbolic identities support the finite checks; universal theorems have separate manuscript proofs.
For PDF compilation, runtime details, and verification limits, see Reproducibility.
Load the structured records
The repository has two dataset configurations: claims and worked_examples. Both use JSON Lines. The split name train is a Hub loading convention; no train/test partition or machine-learning benchmark is asserted.
from datasets import load_dataset
# Replace the namespace with the actual repository owner after upload.
repo_id = "YOUR_USERNAME/avenyra-area-feedback"
claims = load_dataset(repo_id, "claims", split="train")
examples = load_dataset(repo_id, "worked_examples", split="train")
Offline loading also works with load_dataset("json", data_files="data/claims.jsonl", split="train"). Standard-library JSON loading is sufficient if the optional datasets package is unavailable.
manuscript_proved means that the supplied statement has a proof in the manuscript. It does not mean formal certification or a new independent audit. The extracted LaTeX retains macros and cross-references from the complete source; consult that source for their definitions. Read Data schema before using the records as labels or evaluation material.
AI agents and expert review
AI-agent guide supplies a reading order, exact update conventions, claim-status rules, and a format for reporting objections or proposed extensions.
Expert review guide identifies proof dependencies, potential hidden equivalences, priority questions, and specific targets for further work. These are review instructions, not a claim that external experts have endorsed the research.
Status / Completeness
| Category | Status |
|---|---|
| Definitions, explicit models, and central theorem program | Present with manuscript proofs. |
| Original verifiers and exact examples | Re-executed; captured outputs included. |
| Established mathematical relationships | Documented in the manuscript and novelty audit. |
| Originality of the specific rule and theorem package | Potentially novel; priority unverified. |
| Full reachable-set and behavioral classifications | Unresolved. |
| Cyclic maximal-rank interior in odd dimensions at least five | Conjecture. |
| Internal optimization and inaccessible-block reduction | Demonstrated within the mathematical model. |
| External applications, physical compute, and energy claims | Undemonstrated. |
| Independent human review or proof-assistant certification | Not performed. |
| Original manuscript reporting checklist | 16 of 16 requested components included. |
The original reporting checklist is a count of included components, not a probability of correctness or novelty and not a measure of completion of the open research program. The images and imaginative seeds motivated design choices; reproduction does not require those images.
Citation and reuse status
Use CITATION.cff or citation.bib. No repository URL or DOI has been invented before upload. The research byline is the requested publication designation; the dossier describes its AI-assisted origin.
The original release specifies no reuse license. This packaging introduces no additional license grant. See License notice.
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