Classify equal-weight cubic repairs by a Pell recurrence (no κ)

research/cubic-pell-family/PROOF.md

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+# A Pell family for the fixed cubic repair
+
+## Scope and provenance
+
+This answers Open next question 2 of Nick Petalas's
+[PR6653](https://github.com/CrocSwap/integer-mult-bounds/pull/6653): its
+certificates at arities 7 and 97 have defect weights `(-r,r/2,r/2)` on
+directions `(0,d,e)`; does this pattern extend to an infinite family?
+The answer is yes. We also classify **this fixed weight pattern with rational
+directions**, not all three-query repairs.
+
+The fixed cubic reader comes from PR6200. The repair interface and cited
+three-query lower bound are apoursh's PR6212; PR6215 treats another direction
+family. AAAmirinu's PR6246 and CubeJerry's PR6250 supply the earlier
+power-of-two and power-of-two-plus-one branches. PR6653 supplies the explicit
+question, the two seed certificates and their predecessor attribution. Its
+node lemma is consistent with the calculation below, but the proof here uses
+direct coefficient comparison. The source manifests identify the exact bytes
+consulted. Classical Pell descent and power-sum identities are not claimed as
+new mathematics.
+
+This is a polynomial identity and a literal restoration program. Query counts
+are local operation counts, not a physical cost or a complete recursive
+invoice. There is no new supplier admission or κ claim.
+
+## 1. The defect and fixed weight pattern
+
+Let the integer arity satisfy $r>2$. All $r$ children are retained:
+
+```text
+repeat r times:
+    y += a^3
+    a += b
+```
+
+For arbitrary entry values $y=Y,a=A,b=B$, their endpoint is
+
+$$
+ a=A+rB,\qquad b=B,\qquad
+ y=Y+\sum_{i=0}^{r-1}(A+iB)^3.
+$$
+
+The target eager endpoint is $a=A,b=B,y=Y+rA^3$. Thus the defect to remove is
+
+$$
+ D_r(A,B)=3S_1A^2B+3S_2AB^2+S_3B^3,
+$$
+
+where
+
+$$
+ S_1=\frac{r(r-1)}2,\qquad
+ S_2=\frac{r(r-1)(2r-1)}6,\qquad
+ S_3=\frac{r^2(r-1)^2}4.
+$$
+
+For rational directions $d,e$, consider the fixed pattern
+
+$$
+ W_{r,d,e}(A,B)
+ =-rA^3+\frac r2(A+dB)^3+\frac r2(A+eB)^3.                 \tag{1}
+$$
+
+**Theorem 1 (classification of the fixed pattern).** For every integer $r>2$,
+the identity $W_{r,d,e}=D_r$ holds with rational $d,e$ if and only if there
+is a positive integer $n$ such that
+
+$$
+ r^2-3n^2=1,\qquad
+ \{d,e\}=\left\{\frac{r-1-n}{2},\frac{r-1+n}{2}\right\}.       \tag{2}
+$$
+
+In every such case the directions are integers, and after ordering them they
+satisfy $0<d<e<r$. Consequently, allowing noninteger dyadic or arbitrary
+rational directions adds no arities within (1).
+
+*Proof.* The $A^3$ coefficient of (1) vanishes. Its $A^2B$ and $AB^2$
+coefficients agree with those of $D_r$ exactly when
+
+$$
+ d+e=r-1,\qquad d^2+e^2=\frac{(r-1)(2r-1)}3.                \tag{3}
+$$
+
+Hence
+
+$$
+ de=\frac{(r-1)(r-2)}6,\qquad
+ (e-d)^2=\frac{r^2-1}{3}.                                  \tag{4}
+$$
+
+If $n=|e-d|=p/q$ in lowest terms with $q>0$, then
+
+$$
+ 3p^2=(r^2-1)q^2.
+$$
+
+Since $p^2$ and $q^2$ are coprime, $q^2\mid3$, forcing $q=1$.
+Thus $n$ is an integer and (2) follows. It is positive because $r>2$.
+
+Conversely, suppose (2). Equations (3) and (4) hold. The remaining moment is
+also correct:
+
+$$
+ \frac r2(d^3+e^3)
+ =\frac r2\bigl((d+e)^3-3de(d+e)\bigr)
+ =\frac{r^2(r-1)^2}{4}=S_3.
+$$
+
+All four coefficients therefore agree.
+
+Modulo 4, $r^2-3n^2=1$ implies that $r$ odd makes $n$ even, and $r$
+even makes $n$ odd. Thus $r-1\pm n$ is even and both directions are
+integers. Furthermore
+
+$$
+ (r-1)^2-n^2=\frac{2(r-1)(r-2)}3>0,
+$$
+
+so $n<r-1$ and $d>0$. Positivity of $n$ gives $d<e$, while $n<r+1$
+gives $e<r$. This proves all assertions. ∎
+
+There is a useful exact residual before imposing the Pell equation. For
+arbitrary $r,n$, set $d=(r-1-n)/2$, $e=(r-1+n)/2$, and
+
+$$
+ P=r^2-3n^2-1.
+$$
+
+Direct expansion gives
+
+$$
+ W_{r,d,e}-D_r
+ =-\frac{rP}{8}B^2\bigl(2A+(r-1)B\bigr).                  \tag{5}
+$$
+
+Here the displayed formula for $D_r$ is interpreted polynomially if $r$
+is not an integer. Formula (5) identifies the failure off the Pell curve;
+agreement on a special input such as $B=0$ does not certify the identity.
+
+## 2. Every positive solution and an elementary recurrence
+
+Define integer pairs by
+
+$$
+ (r_0,n_0)=(1,0),\qquad
+ r_{k+1}=2r_k+3n_k,\quad n_{k+1}=r_k+2n_k.                \tag{6}
+$$
+
+Equivalently, $r_k+n_k\sqrt3=(2+\sqrt3)^k$. Equation (6), without any
+irrational arithmetic, preserves $r_k^2-3n_k^2=1$. For $k\ge1$, both
+coordinates are positive and strictly increase. The first pairs are
+
+$$
+ (1,0),\ (2,1),\ (7,4),\ (26,15),\ (97,56),\ (362,209),
+ \ (1351,780).
+$$
+
+**Theorem 2 (exhaustiveness).** Every positive integer solution of
+
+$$
+ r^2-3n^2=1
+$$
+
+is exactly one pair $(r_k,n_k)$ with $k\ge1$. The nondegenerate repairs
+of Theorem 1 correspond precisely to $k\ge2$.
+
+*Proof.* For any positive solution, $n<r\le2n$. The right equality occurs
+only at $(r,n)=(2,1)$, because $r^2=3n^2+1\le4n^2$, with equality only
+when $n=1$. Apply the integral inverse of (6):
+
+$$
+ r'=2r-3n,\qquad n'=2n-r.                                \tag{7}
+$$
+
+We have $r'>0$, since $r>\sqrt3n>3n/2$, and $0\le n'<n$.
+The norm remains $r'^2-3n'^2=1$. Thus descent terminates at $n'=0$,
+where the positive first coordinate must be 1. Reversing the descent gives
+(6), and the deterministic inverse makes the index unique. At $k=1$,
+$r=2,d=0,e=1$, so two directions coincide; this excluded boundary does not
+contradict the lower-bound statement for $r>2$. ∎
+
+## 3. Infinitely many arities outside the earlier classes
+
+For comparison with the classes listed by PR6653, use the even-index
+subsequence
+
+$$
+ R_j=r_{2j},\qquad N_j=n_{2j},\qquad j\ge0.
+$$
+
+It has the recurrence
+
+$$
+ (R_0,N_0)=(1,0),\qquad
+ R_{j+1}=7R_j+12N_j,\quad N_{j+1}=4R_j+7N_j.             \tag{8}
+$$
+
+All $R_j$ are odd. They also satisfy
+
+$$
+ R_{j+2}=14R_{j+1}-R_j,
+$$
+
+so modulo 9 their values repeat $1,7,7$. Indeed the first four residues
+are $1,7,7,1$, the next is 7, and the two-coordinate recurrence then repeats
+its initial state. Therefore
+
+$$
+ R_j\equiv7\pmod9\quad\hbox{when }j\equiv1,2\pmod3.    \tag{9}
+$$
+
+These arities lie outside the residue classes
+
+$$
+ r\bmod9\in\{0,1,2,5,8\}
+$$
+
+covered by the cited PR6212/PR6215 constructions. They are at least 7, so
+are not the exceptional arities 3 and 4. Being odd and greater than 1, they
+are not powers of 2.
+
+To exclude the power-of-two-plus-one branch, square the fundamental solution:
+
+$$
+ R_j=r_j^2+3n_j^2=1+6n_j^2.
+$$
+
+For every $j\ge1$, $R_j-1=6n_j^2$ is positive and divisible by 3; it
+cannot be a power of 2. Hence $R_j\ne2^s+1$ for every nonnegative integer
+$s$.
+
+The indices $j\ge1$ with $j\not\equiv0\pmod3$ give infinitely many
+strictly increasing arities outside **all the earlier infinite classes
+listed by PR6653**. This comparison is with those precise classes, not a
+claim to have classified every construction in the literature or every later
+contribution. Its first values are
+
+$$
+ 7,\ 97,\ 18817,\ 262087,\ 50843527,\ 708158977,\ldots.
+$$
+
+The first two are Petalas's already established finite certificates. The
+uniform family and the complete fixed-pattern classification are the added
+result; those seed examples are not claimed anew.
+
+For example, at $r=18817,n=10864$, the directions are $d=3976,e=14840$
+and the defect weights are $(-18817,18817/2,18817/2)$.
+
+## 4. Three queries and three shears restore the dirty inputs
+
+For any $k\ge2$, take the ordered integer directions $0<d<e<r$ from
+Theorem 1. After the retained children, execute:
+
+```text
+a += (e-r)*b       # a = A+eB
+y -= (r/2)*a^3
+a -= n*b           # a = A+dB, since e-d=n
+y -= (r/2)*a^3
+a -= d*b           # a = A
+y += r*a^3
+```
+
+All three shear coefficients are nonzero integers. All three query weights
+are nonzero dyadic rationals. The program uses **three appended cube queries
+and three source shears**. It writes no auxiliary register and never changes
+$b$. Its correction to $y$ is $-W_{r,d,e}=-D_r$, so its exact endpoint is
+
+$$
+ (y,a,b)=(Y+rA^3,A,B).
+$$
+
+This holds as a polynomial identity, hence for arbitrary incoming $Y,A,B$
+over the rationals or reals; neither source nor destination is assumed clean.
+There is no deletion, replacement or reordering of the $r$ retained child
+queries. Visiting directions in the order $e,d,0$ makes the last query
+occur at the restored source, avoiding a separate fourth restoring shear.
+
+The cited PR6212 lower bound excludes one- and two-query dyadic repairs for
+$r>2$, including its stated general dyadic-frame model. Combining that
+predecessor theorem with this construction makes the dyadic repair query
+minimum **exactly three at these arities**, under that same model. The lower
+bound is a dependency, not re-proved by the present finite checker.
+In particular, query coefficients are fixed independently of input data;
+the model has no output feedback into sources, nonlinear source changes,
+intermediate observations or access to child interiors. One signed weighted
+cube accumulation counts as one query, with source changes additional.
+
+## 5. What this does not settle
+
+The classification concerns the weight shape in (1). Failure of the Pell
+equation excludes that shape only; it does not exclude other weights,
+directions or dyadic frames, nor prove a four-query minimum at any other
+arity. A verified finite prefix corroborates the formulas but is not the
+proof of the infinite family; Sections 1–3 supply that proof.
+
+The three-query/three-shear schedule has growing coefficients. Its native
+query and shear costs, routing, precision, storage, full caller and recursive
+invoice remain unpriced. The formal operation counts do not imply a saving
+over eager restoration, a physical realization, or an improvement to the
+selected conditional bound. No independent human or formal proof-kernel
+acceptance is claimed.
+
+Prepared by **apoursh** with substantial **OpenAI Codex and subagent
+assistance**. Retain Nick Petalas, CubeJerry, AAAmirinu and the cited source
+lineage above. Apache-2.0.