Prove uniform Gaussian boundary conditioning at the production scale

research/production-gaussian-boundary-limit/PROOF.md

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+# A uniform production-scale Gaussian boundary limit
+
+JoyboyBrian, with substantial OpenAI Codex assistance. Apache-2.0.
+This is a scoped written theorem with exact finite corroboration, not a
+formal-kernel or independent-human proof acceptance.
+
+## Quantifiers and exact source
+
+Let h=sqrt(theta), rho=1+theta, N=floor(1/theta), and n=N-1. For
+0<=k,i<N use the unchanged full-cell broad source matrix
+
+    K[k,i]=alpha^-1 exp(-b(i-k/rho-1/(2rho))²), b=pi/alpha².
+
+There is no band truncation. Put S=diag((-1)^i), A=S K^-1 S,
+P(t)=product_{r>=1}(1-exp(-2tr)), a_b=alpha exp(b/4) P(b)^-3,
+and s=sqrt(b)*h. Split the indices at floor(N/2). Embed each left index
+as the normalized indicator of [si,s(i+1)) in the first L² half-line;
+embed right index i as the corresponding indicator at n-i in the second.
+Call this isometry U_s. Let T have kernel
+exp(-(y²-x²)) 1{x<=y} on L²(0,infinity).
+
+Uniformly for
+
+    h -> 0,                 h^(1/5) <= b <= 1,             (1)
+
+we prove
+
+    || U_s (s A/a_b) U_s* - (T direct-sum T) || -> 0.      (2)
+
+The retained half-line proof establishes ||T||=1/2 using the maximal
+first-order operator -d/dx+2x and its Dirichlet adjoint; we retain that
+endpoint-domain result. Thus (2) gives the uniform relative asymptotic
+
+    ||K^-1||₂ = (1+o(1)) alpha exp(b/4)
+                         / (2 sqrt(b theta) P(b)^3).     (3)
+
+This extends the predecessor's fixed-alpha theorem to a specified joint
+regime. The first retained source case remains alpha=3,theta=1/400,N=400;
+that small fixture need not satisfy the eventual domain (1). Source geometry
+is checked exactly, not used as evidence that an asymptotic quantifier holds.
+
+The actual PR4100 setup is alpha=ceil(sqrt((4/pi)log(16p))) and
+1/(4p²)<=theta<=1/p². It satisfies (1) eventually, uniformly in theta.
+Here p tends through positive integers, logarithms are natural, and the
+ceil and floor in the displayed source are retained. This is the untruncated
+full cell, not a claim that the existing capped RawGaussian API accepts it.
+
+## Relative q-product lemma
+
+Retain the exact q-LDU identity, credited through #1673, #4193, #5103 and
+#5150 and the pinned half-line theorem. Set gamma=b/rho, Q=exp(-2gamma),
+P_j=product_{r=1}^j(1-Q^r), P=P(gamma). For i<=j define L=N-1-j and
+
+    F_ij=sum_{r=0}^L Q^r P_(j+r)/(P_i P_(j-i+r) P_j P_r).
+
+Then A_ij=alpha exp(E_ij)F_ij, with
+
+    E_ij=b(theta*j-1/2)²/rho²-gamma*theta*(j²-i²).         (4)
+
+For i>j reflect both indices: i'=n-i,j'=n-j, with i'<j'. The coefficient
+F is invariant under that reflection and the exponent is
+
+    E_ij=b(theta*j-1/2)²/rho²
+          -gamma*theta*((j'+delta)²-(i'+delta)²),
+    delta=1/theta-n in [1,2).                            (5)
+
+In either orientation E_ij<=b/4. These are exact finite identities.
+For m>=1 put
+
+    eps_m=Q^(m+1)/((1-Q)(1-Q^(m+1))).
+
+If i>=m, j-i>=m, and L>=m, then
+
+    exp(-4 eps_m) <= P³ F_ij <= 1.                       (6)
+
+Here are the details that avoid the predecessor's exponentially poor
+absolute constants. Write R_j=P_j/P. For j>=m,
+
+    0<=log R_j=sum_{r>j}-log(1-Q^r)
+       <=Q^(j+1)/((1-Q)(1-Q^(j+1)))<=eps_m.
+
+The positive weights w_r=P Q^r/P_r sum to one, since
+Q^r/P_r=1/P_r-1/P_(r-1) for r>=1. Moreover
+sum_{r=0}^L w_r=P/P_L=1/R_L>=exp(-eps_m). Now
+
+    P³ F_ij=sum_{r=0}^L w_r R_(j+r)/(R_i R_(j-i+r) R_j).
+
+Each numerator is >=1 and each of the three denominator factors is
+<=exp(eps_m), proving the lower bound. For the upper bound,
+P_(j+r)/P_j<=1, P_i,P_(j-i+r)>=P, and sum Q^r/P_r<=1/P.
+This also proves P³F<=1 outside the good region. No finite product is
+silently substituted for P, and the whole r-sum is essential.
+
+For b<=1, h<=1/2, gamma>=b/2. Logarithmic differentiation gives
+
+    d log P(t)/dt = sum_{r>=1} 2r/(exp(2tr)-1)
+      <= 2/(1-exp(-b))³ <=16/b³,       gamma<=t<=b.
+
+Consequently, with
+
+    k=exp(48 h²/b²)-1,
+
+we have 1<=P(b)^3/P(gamma)^3<=1+k. On the good cells,
+
+    |P(b)^3 F_ij-1| <=4(1+k)eps_m+k.                    (7)
+
+Everywhere 0<=P(b)^3 F_ij<=1+k. The derivative bound follows from a
+convergent geometric majorant, so differentiation under the sum is valid.
+
+## Uniform operator estimate
+
+Choose R>=1, J=floor(R/s), m>=1, and assume s<=R/2,
+2J<=N, J>=m, and N-J>=m+1. Retain the left upper-triangular J-square
+and right strictly-lower J-square, assigning the diagonal only once.
+Their images lie in the two different half-line summands.
+
+In the upper remainder j>=J, j²-i²>=(j-i)J. Each row sum and each
+column sum of sA/a_b is at most
+
+    (1+k)s sum_{d>=0} exp(-s² J d/rho)
+       <=(1+k)(s+rho/(sJ)) <=(1+k)(s+4/R).
+
+We used rho<=2 and sJ>=R/2. Reflection gives the same estimate on the
+lower remainder. This includes middle and cross-edge entries. The
+continuum remainder has norm at most 2/R per summand by the analogous
+row/column integral; using a sum instead of a maximum only enlarges
+our bound. A safe combined discrete/continuum tail allowance is
+
+    4(1+k)(s+4/R).                                     (8)
+
+On each retained square, x=s(i+u), y=s(j+v), 0<=u,v<=1. The exponent
+bound in the pinned half-line proof, with its window radius replaced
+by R/sqrt(b), gives
+
+    |E_ij - (b/4-(y²-x²))| <= D,
+    D=12(R+1)² h.                                      (9)
+
+Indeed its bound is 12b(R/sqrt(b)+1)²h, at most (9) when b<=1.
+The same calculation covers the reflected offset delta in [1,2).
+Both exponents after subtracting b/4 are nonpositive, hence exp has
+Lipschitz constant one on their range.
+
+Cells with i<m or j-i<m occupy at most 2msR area per corner. Their
+kernel difference is at most 2(1+k); triangle and Hilbert-Schmidt bounds
+give 8(1+k)sqrt(ms(R+1)) for both exceptional regions. Diagonal cells
+include the possible cell crossing of x=y and the strict lower convention.
+On the remaining two squares, (7)-(9) give the operator bound
+2R[4(1+k)eps_m+k+(1+k)D]. Therefore
+
+    || U_s(sA/a_b)U_s*-(T direct-sum T) ||
+    <=4(1+k)(s+4/R) + 8(1+k)sqrt(ms(R+1))
+       +2R[4(1+k)eps_m+k+(1+k)D].                      (10)
+
+The embedded finite operator is zero on the orthogonal complement of
+its step-function range. The argument compares its integral kernel on
+all of both half-lines; it does not omit that complement.
+
+## One simultaneous choice for the entire regime
+
+Take R=h^(-1/10), m=ceil(h^(-1/5)/b), J=floor(R/s).
+Under (1), for all sufficiently small h uniformly in b:
+
+* s<=h and s>=h^(11/10); thus R/s>=h^(-11/10),
+  J<=h^(-6/5), while N=floor(h^-2). Hence 2J<=N.
+* m<=2h^(-2/5), so J>=m and N-J>=m+1 eventually.
+* ms(R+1)<=4h^(3/5): use m<=h^(-1/5)/b+1,
+  s=h sqrt(b), b>=h^(1/5), and R+1<=2h^(-1/10).
+* Q<=exp(-b); bm>=h^(-1/5). Thus
+  eps_m<=4b^-1 exp(-h^(-1/5))
+        <=4h^(-1/5)exp(-h^(-1/5)).
+* k<=exp(48h^(8/5))-1, and RD<=48h^(7/10).
+
+All terms in (10) tend to zero uniformly. In particular the R times
+exponential tail vanishes, Rk=O(h^(3/2)), and (8) is O(h^(1/10)).
+This proves (2) without exchanging two unquantified limits.
+
+## Production polynomial conditioning, with ceilings preserved
+
+Put l=log(16p). The actual alpha satisfies
+sqrt(4l/pi)<=alpha<sqrt(4l/pi)+1, hence b is comparable to 1/log p.
+Since h in [1/(2p),1/p], eventually h^(1/5)<=b<=1 uniformly.
+To bound the product, f(x)=-log(1-exp(-x)) is positive decreasing, so
+
+    -log P(b)=sum_{r>=1} f(2br)
+          <=(1/(2b)) integral_0^infinity f(x)dx
+          =pi²/(12b).
+
+The integral follows by the positive exponential series and
+sum_{k>=1}1/k²=pi²/6. Therefore P(b)^-3<=exp(pi alpha²/4).
+For sufficiently large p, (2) bounds the normalized norm by one, giving
+
+    ||K^-1||₂ <= alpha²/(sqrt(pi)*h)
+                    exp(b/4+pi alpha²/4)
+       <= 2p alpha²/sqrt(pi)
+                    exp(b/4+l+sqrt(pi*l)+pi/4).          (11)
+
+For every fixed eta>0, the right side is <=p^(2+eta) eventually.
+This is a uniform polynomial upper bound, with the asymptotic coefficient
+still given exactly by (3). We do not replace its product by an unproved
+large-alpha formula, nor assert a sharp p-power lower bound here.
+In particular a p^10 allowance eventually dominates this *exact full-cell
+inverse norm*. A rounded or truncated reference would additionally need
+its own perturbation bound (for example ||K^-1|| ||Delta||<1); this proof
+has not admitted such a reference or its circuit.
+
+## Decision, costs and next obligations
+
+The fixed-alpha restriction is removed for the actual production scale.
+A full-cell conditioning objection based solely on taking alpha to infinity
+is therefore resolved for this exact kernel. Existing capped-section
+conditioning arguments remain separate, and no existing program is changed.
+
+The theorem constructs no coefficient producer or inverse-application
+algorithm. Common rational reference formation, truncation, grid rounding,
+all circuit prefix bounds, coefficient/page storage and physical delivery,
+tape movement, preparation amortization and integer recovery retain their
+costs and proof obligations. No charge is assigned zero; no runtime,
+complete multiplication theorem, supplier admission or new kappa follows.
+
+Open next questions: bind a specified rounded full-cell reference using
+(11) and its actual entry-error invoice; then determine whether any concrete
+inverse consumer can exploit the guard while paying all preparation and
+physical costs. An arbitrary-section or different-endpoint theorem requires
+fresh geometry; the present N, phase and reflection domain must not be dropped.

research/production-gaussian-boundary-limit/README.md

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+# Uniform Gaussian conditioning at the production scale
+
+The exact full-cell broad Gaussian inverse has the two-edge half-line limit
+uniformly when `theta^(1/10) <= b=pi/alpha² <= 1`. This includes the actual
+PR4100 choice `alpha=ceil(sqrt((4/pi)log(16p)))`, uniformly for
+`1/(4p²)<=theta<=1/p²`. Its inverse norm is eventually at most `p^(2+eta)`
+for every fixed `eta>0`. The exact asymptotic coefficient retains the full
+infinite product; see [PROOF.md](PROOF.md).
+
+The predecessor proved the boundary limit only for fixed alpha. Normalizing
+finite q-products before estimating them removes the absolute constants that
+prevented its production substitution. This is a new quantified analytic
+implication, not another finite parameter run or a claim of a faster solver.
+The exact original phase, `N=floor(1/theta)`, both boundary layers and the
+ceiling in alpha are retained. The first source case is still
+alpha3/theta1/400/N400; its geometry is checked without pretending that a
+small fixture proves the uniform theorem.
+
+The small standard-library checker authenticates the retained sources,
+checks nonvacuous normalized-product bounds by rational enclosures, compares
+the whole q-LDU sum with independent exact elimination, and exercises the
+rescaled exponent and floor/ceiling geometry. Controls discriminate dropped
+positive summands, omitted distance from the edge, missing sqrt(b) grid
+scaling, reflection offsets and mistyped receipts. RESULT.json is mandatory;
+ordinary verification does not rewrite it.
+
+```
+python3.12 -B scripts/research_package.py --package research/production-gaussian-boundary-limit
+python3 -B scripts/check_research_commit.py --package research/production-gaussian-boundary-limit --check-source-history
+```
+
+Both normal and optimized runs are registered. An official committed export
+can execute verify.py with its declared inputs without Git or network.
+This is a scoped written theorem and finite corroboration, not independent
+human or formal-kernel acceptance. No selected frontier changes.
+
+## Costs and limits
+
+The full cell is an analytic source matrix, not an admitted capped
+RawGaussian program. The result does not price coefficient formation,
+truncation, common-reference rounding, prefix amplification, inverse
+application, page storage/delivery, tape movement or integer recovery.
+These remain obligations rather than zero charges. The existing p^10
+allowance dominates this exact inverse eventually; applying it to a
+perturbed reference still requires its actual perturbation invoice.
+
+## Open next questions
+
+Bind one explicitly rounded full-cell reference to the new polynomial inverse
+bound, then identify an actual consumer whose complete preparation and
+physical execution costs justify using it. Different section lengths or
+phase endpoints need a separate theorem. TASK.json records first steps.
+
+Canonical claim: [Discussion4202 comment19172868](https://github.com/CrocSwap/integer-mult-bounds/discussions/4202#discussioncomment-19172868).
+Base 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority7. Fresh overlap was
+coordinated by the parent before assignment; no global novelty guarantee is
+claimed. SOURCES.json binds the exact inherited theorem and source geometry.
+
+JoyboyBrian, with substantial OpenAI Codex assistance. Apache-2.0.
+Credit Jose Manuel/#4193, Vlado Vojdanovski/#1673, Cinna Davis/PR4100,
+Ryanson Jonathan/#4435, Nick Petalas/#5150, the retained JoyboyBrian
+full-cell and half-line work, and the classical q-LDU and product identities.
+Frozen predecessors and their notices remain unchanged.