Research: exact parity constants and q-series for the projection recurrence

research/projection-parity-constants/PROOF.md

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+# Exact parity constants for the projection recurrence
+
+Status: a proposed scoped mathematical solution with exact finite corroboration.
+Prepared with OpenAI Codex assistance for this task, 2026-10-24 (Asia/Shanghai).
+New text and code: Apache-2.0. No claim of prior literature priority.
+
+## 1. Question, sources, and result
+
+The source is CrocSwap/integer-mult-bounds at
+`ff3cc58e370d0f678e529246f8c73e8094e18fcf`.
+Krzysztof Wojt's PR #5193, `research/projection-geometric-tail`, proves
+the order of convergence and explicitly leaves exact leading parity constants
+and elementary/q-series expressions for the modes open. This note solves
+those two scoped questions. It does not classify monotone seeds.
+
+At block four start with
+
+\[
+a_4=(0,27456,48576,68320,86592).
+\]
+
+For k >= 5, use the unstrengthened recurrence
+
+\[
+a_k(r)=\sum_s N_k(r,s)a_{k-1}(s),
+\]
+
+where the only possibly nonzero entries are
+
+\[
+\begin{aligned}
+N_k(r,r)&=2^{k-r}-1+4^k-2^{k+r},\\
+N_k(r,r-1)&=2(2^k-2^{k-r})+(2^r-1)2^{k-1},\\
+N_k(r,r-2)&=(2^r-1)(2^{r-1}-1)2^{k-r}.
+\end{aligned}
+\]
+
+The last entry is zero when r < 2; omit indices outside the old profile.
+Set T_m=a_m(m)/a_m(1). Let
+
+\[
+\mathcal K=\prod_{i=1}^{\infty}
+\left(\frac{1-2^{-i}}{1+2^{-i}}\right)^2>0.
+\]
+
+**Theorem.** As integer m tends to infinity,
+
+\[
+\boxed{
+T_m=\frac{439277}{149175}
++2^{-m}\left(\frac{815083}{596700}
++(-1)^m\frac{323}{28}\mathcal K\right)
++O(4^{-m}).
+}
+\tag{1}
+\]
+
+Consequently the exact even and odd scaled limits are respectively
+
+\[
+\frac{815083}{596700}+\frac{323}{28}\mathcal K
+=1.535225897075989\ldots,
+\]
+\[
+\frac{815083}{596700}-\frac{323}{28}\mathcal K
+=1.196743266657880\ldots.
+\]
+
+The full scaled sequence has no limit, since K > 0. The known limit
+439277/149175 is recovered, not claimed as new. Exact coefficient extraction
+from (7) below also gives a nonrecursive q-series expression for every mode
+and hence for every finite T_m.
+
+## 2. The credited four-mode reduction
+
+Here is a self-contained verification of the reduction from #5193, which is
+credited to that work. Define
+
+\[
+\begin{aligned}
+h_r&=3/2+2^{-r}-2^r,\\
+B_r&=2^{r-1}+3/2-2^{1-r},\\
+C_r&=2^{r-1}-3/2+2^{-r}.
+\end{aligned}
+\]
+
+Let e_j(r)=0 for r<j, e_j(j)=1, and for r>j put
+
+\[
+(h_j-h_r)e_j(r)=B_re_j(r-1)+C_re_j(r-2).
+\tag{2}
+\]
+
+Because h_r strictly decreases, this determines e_j. With
+
+\[
+d_k=(2^{k-1}-1)(2^{k+1}+1),
+\]
+
+the raw row operator is d_k I + 2^k H, where H is the lower-banded operator
+with coefficients h_r,B_r,C_r. Thus H e_j=h_j e_j. At the new endpoint
+r=k the raw diagonal coefficient is zero, so no undefined old value is needed.
+Solving the unit triangular seed system gives
+
+\[
+(c_1,c_2,c_3,c_4)=
+\left(1,\frac{17}{39},\frac{6181}{14586},\frac{99085}{239811}\right).
+\]
+
+Define
+
+\[
+p_{m,j}=\prod_{k=5}^{m}
+\frac{(2^{k-j}-1)(2^{k+j}+1)}{d_k}.
+\]
+
+The seed equations and induction with H give, for 0 <= r <= m,
+
+\[
+\frac{a_m(r)}{a_m(1)}=\sum_{j=1}^4 c_jp_{m,j}e_j(r).
+\tag{3}
+\]
+
+## 3. A new q-series expression for each eigenmode
+
+Fix j >= 1, and write Q=1/2 and s=2^j. Use the convention
+
+\[
+(z;Q)_n=\prod_{i=0}^{n-1}(1-zQ^i),\qquad (z;Q)_0=1.
+\]
+
+For n >= 0 define v_n by
+
+\[
+e_j(j+n)=\frac{(Q^{n+1};Q)_j}{(Q;Q)_j}v_n.
+\tag{4}
+\]
+
+Thus v_0=1. To check the substitution, put y=2^r. Equation (2) becomes
+
+\[
+2(y-s)(sy+1)e_j(r)
+=s(y-1)(y+4)e_j(r-1)+s(y-1)(y-2)e_j(r-2).
+\]
+
+The ratio of the prefactors in (4) at consecutive r is
+(y-1)/(y-s). Cancellation therefore gives, for n >= 1,
+
+\[
+2(1+s^{-2}Q^n)v_n
+=(1+4s^{-1}Q^n)v_{n-1}+(1-2Q^n)v_{n-2}.
+\tag{5}
+\]
+
+At n=1 the last coefficient is zero, and we set v_{-1}=0.
+
+Let V_j(z)=sum_{n>=0} v_n z^n near zero. Multiplying (5) by z^n
+and summing gives
+
+\[
+(1-z)(2+z)V_j(z)+\frac{(z-2/s)^2}{2}V_j(z/2)
+=2(1+s^{-2}).
+\tag{6}
+\]
+
+Iterating this functional equation yields the explicit expression
+
+\[
+\boxed{
+V_j(z)=(1+s^{-2})\sum_{\ell=0}^{\infty}
+\frac{(-s^{-2})^\ell(sz/2;Q)_\ell^2}
+{(z;Q)_{\ell+1}(-z/2;Q)_{\ell+1}}.
+}
+\tag{7}
+\]
+
+At z=0 the geometric series gives V_j(0)=1. Near zero the series is
+absolutely convergent and satisfies (6); comparing coefficients uniquely
+recovers (5). This also establishes that it is the generating function.
+
+More generally, (7) converges locally uniformly away from
+z=1,2,4,... and z=-2,-4,-8,... . On any such compact set its finite products
+are uniformly bounded and its denominator products are bounded away from zero,
+while s^{-2ell} decays geometrically. Removing the one vanishing factor at
+any listed point makes the series locally uniformly convergent there too.
+Hence V_j is meromorphic with at most simple poles at these points.
+
+## 4. Three explicit poles determine the leading terms
+
+Define the pole coefficients
+
+\[
+A_j=\lim_{z\to1}(1-z)V_j(z),\quad
+D_j=\lim_{z\to2}(1-z/2)V_j(z),\quad
+E_j^-=\lim_{z\to-2}(1+z/2)V_j(z).
+\]
+
+We use the classical q-Gauss identity (NIST DLMF 17.6.1):
+
+\[
+{}_2\phi_1(a,b;c;Q,c/(ab))
+=\frac{(c/a;Q)_\infty(c/b;Q)_\infty}
+{(c;Q)_\infty(c/(ab);Q)_\infty},\qquad |c/(ab)|<1.
+\tag{8}
+\]
+
+Here the basic hypergeometric series includes the denominator (Q;Q)_ell.
+All denominator parameters below avoid its forbidden nonnegative powers
+Q^{-n}, and both arguments have absolute value s^{-2}<1.
+
+At z=1 equation (7) gives
+
+\[
+A_j=\frac{1+s^{-2}}{1+Q}
+{}_2\phi_1(s/2,s/2;-Q^2;Q,-s^{-2})
+=\frac{(-Q;Q)_{2j}}{(-Q;Q)_j^2}.
+\tag{9}
+\]
+
+All terms with ell >= j vanish because s/2=Q^{1-j}; the last potentially
+nonzero term is ell=j-1.
+For example A_1=5/6. The final equality follows by (8) and cancellation
+of the shifted infinite products; it can also use terminating q-Chu-Vandermonde.
+
+Taking the pole at z=2 in (6), where V_j(z/2) contributes A_j, gives
+
+\[
+D_j=\frac{(1-s^{-1})^2}{2}A_j.
+\tag{10}
+\]
+
+At z=-2 equation (7) instead gives
+
+\[
+\begin{aligned}
+E_j^-&=\frac{1+s^{-2}}3
+{}_2\phi_1(-s,-s;-1;Q,-s^{-2})\\
+&=\frac{1+s^{-2}}3
+\frac{(Q^j;Q)_\infty^2}{(-1;Q)_\infty(-Q^{2j};Q)_\infty}\\
+&=\frac{(-Q;Q)_{2j}}{6(Q;Q)_{j-1}^2}\mathcal K.
+\end{aligned}
+\tag{11}
+\]
+
+The last equality uses (-1;Q)_infty=2(-Q;Q)_infty and
+K=(Q;Q)_infty^2/(-Q;Q)_infty^2.
+
+The meromorphic description proves the coefficient asymptotic
+
+\[
+v_n=A_j+D_j2^{-n}+E_j^-(-2)^{-n}+O(4^{-n}).
+\tag{12}
+\]
+
+For completeness this is not an unproved transfer heuristic: subtract the
+simple-pole parts at 1, +/-2, and +/-4. The remainder is analytic on |z|<8,
+so Cauchy's coefficient estimate on |z|=5 is O(5^{-n}). The two subtracted
+parts at +/-4 have coefficients O(4^{-n}). Their constants need not be
+computed to obtain (12). All statements here are for fixed j.
+
+## 5. Returning to the physical block index
+
+Define the rational constants
+
+\[
+F_j=\frac{(-Q;Q)_{2j}}{(Q;Q)_j(-Q;Q)_j^2},\qquad
+G_j=\frac{(-Q;Q)_{2j}}{6(Q;Q)_{j-1}^2(Q;Q)_j}.
+\]
+
+Since n=m-j, (4) and (12), including the first-order change in the
+finite product in (4), yield
+
+\[
+e_j(m)=F_j\left(1-\frac{s-s^{-1}}2\,2^{-m}\right)
++(-1)^m(-1)^j sG_j\mathcal K\,2^{-m}+O(4^{-m}).
+\tag{13}
+\]
+
+For 1 <= j <= 4 the mode product has the rational limit
+
+\[
+P_j=\frac{\prod_{i=5-j}^{3}(1-Q^i)}
+{\prod_{i=6}^{4+j}(1+Q^i)},
+\tag{14}
+\]
+
+with empty products equal to 1. In fact its exact finite formula is
+
+\[
+p_{m,j}=P_j
+\frac{\prod_{i=m+2}^{m+j}(1+Q^i)}
+{\prod_{i=m-j+1}^{m-1}(1-Q^i)}.
+\tag{15}
+\]
+
+Thus, by expansion of these finitely many factors,
+
+\[
+p_{m,j}=P_j(1-h_j2^{-m}+O(4^{-m})).
+\tag{16}
+\]
+
+Multiplication with (13) gives
+
+\[
+p_{m,j}e_j(m)=P_jF_j
++2^{-m}\left[P_jF_j\frac{s-s^{-1}-3}{2}
++(-1)^mP_j(-1)^j sG_j\mathcal K\right]+O(4^{-m}).
+\tag{17}
+\]
+
+The three exact sums from (3) are
+
+\[
+\sum_j c_jP_jF_j=\frac{439277}{149175},
+\]
+\[
+\sum_j c_jP_jF_j\frac{2^j-2^{-j}-3}{2}
+=\frac{815083}{596700},
+\]
+\[
+\sum_j c_jP_j(-1)^j2^jG_j=\frac{323}{28}.
+\]
+
+They prove (1). The accompanying exact checker reconstructs these sums.
+
+## 6. Exact numerical enclosures and verification boundaries
+
+Let K_N be the product defining K truncated after N. For 0 <= x < 1,
+
+\[
+1-\left(\frac{1-x}{1+x}\right)^2=\frac{4x}{(1+x)^2}\le4x.
+\]
+
+Applying product(1-d_i) >= 1-sum d_i to finite tails and then taking
+limits proves, for N >= 3,
+
+\[
+K_N(1-2^{2-N})\le\mathcal K\le K_N.
+\tag{18}
+\]
+
+This both proves positivity and provides rational certified enclosures for
+the two constants. The verifier uses N=80, an original integer recurrence,
+a separately implemented mode recurrence, the rational identities above,
+and rejection controls. Those bounded checks corroborate, rather than replace,
+the universal analytic proof and the classical q-Gauss identity.
+
+## 7. Credit, scope, and remaining questions
+
+- Jacob Alan Sussman #4628: recurrence, seed, and proposed engine interpretation.
+- Maximilian Lutz #4643: independent projection counts and recurrence audit.
+- Nick Petalas #5151: exact rational limiting value and general seed theorem.
+- JoyboyBrian #5184: strict finite-block descent and unattained infimum.
+- Krzysztof Wojt #5193: fixed triangular operator, four-mode representation,
+  and sharp geometric order; these are explicit inputs and credited here.
+- Eumemic #288, Swapnil Jain, and the retained certificate/engine lineage
+  retain their original credit. Q-Gauss summation is classical, not new here.
+
+Sources:
+
+- https://github.com/CrocSwap/integer-mult-bounds/pull/5193
+- https://github.com/CrocSwap/integer-mult-bounds/tree/ff3cc58e370d0f678e529246f8c73e8094e18fcf/research/projection-geometric-tail
+- https://github.com/CrocSwap/integer-mult-bounds/pull/5151
+- https://dlmf.nist.gov/17.6.E1
+- https://dlmf.nist.gov/17.6.E2
+
+This is an arithmetic theorem about the specified recurrence. The inherited
+hook-rank certificates, engine interpretation, physical caller, dirty-state
+restoration, complete costs, and multiplication theorem are not established
+by it. No new kappa or selected-frontier change is claimed. No human review
+or proof-kernel verification is claimed. Public novelty is limited to the
+reviewed source snapshot and targeted live overlap search; unpublished work
+and every historical discussion reply were not exhaustively inspected.
+
+Open next questions: characterize monotone seed profiles; determine whether
+the q-series expression can be reduced further for finite m; resolve the
+independent inherited hook-rank obligations. These are separate tasks.