Prove saturation of the fixed #4467 leaf deficit (no new κ)

research/direct-flow-leaf-saturation/CLAUDE-RESPONSE.md

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+# Claude Code review response
+
+The initial Claude Code (Opus 5.5) review of `ff365dba6911ce615bec76e8d1f149faf1fe16ca`, its follow-up at `6310550e932de70243fc2aaab6f4885688b6fe21`, and the OpenAI Codex (GPT-6 Astra) responses are consolidated in [REVIEW.md](REVIEW.md).
+
+Use [the current evidence index](VALIDATION.md#current-evidence-index) for checked source hashes, commands and receipts. The theorem and numerical bounds are unchanged. This file preserves existing links; REVIEW.md is the authoritative review record. All reviews are AI reviews, not human or formal verification.

research/direct-flow-leaf-saturation/PROOF.md

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+# A nonlattice saturation theorem for the fixed #4467 leaf law
+
+This is a scoped mathematical proof using the classical nonlattice interval local-limit theorem stated below. The Python certificate verifies its source-specific numerical hypotheses and finite source-law controls; it is not a proof-kernel verification of this argument. No finite-depth error rate is claimed.
+
+## 1. Source law and statement
+
+Use the fixed raw-p7 inventories in Ryanson Jonathan / pinterbanget's #4467 (`research/direct-flow-leaf-walk/verify.py`):
+
+```
+P = {1:37305, 2:10695, 3:3360, 4:1530, 5:300,
+     8:1464, 16:448, 32:448, 48:448}
+W = {5:420, 6:270, 7:840, 8:1395, 9:3195,
+     25:448, 41:448, 57:448, 121:224}
+m = 121
+```
+
+Let `Z(s) = Σ_r P_r r^s`, `μ(s) = Z'(s)/Z(s)`, and let `t` be the unique root of `Σ_k W_k k^t = Z(t)` in the certified interval `[0.99731351095114, 0.99731351095116]`. Put `a = log 5`. Let `α` be the unique solution of `μ(α)=a` and set `λ=t−α`.
+
+The certificate also bounds the invoice derivative strictly above 211661 on the predecessor’s entire bisection interval `[0.99,1]`, so the isolated root is its unique root there. The certificate proves `μ(t)>a`, `α∈[0.98981438244597, 0.98981438244599]`, and thus `λ∈[0.00749912850515, 0.00749912850519]`. All intervals here are outward bounds, not floating-point root guesses. Uniqueness of α follows from `μ'(s)=Var_s(log r)>0`, its limits 0 and log48 at the two infinities, and `0<a<log48`.
+
+Under `p_t(r)=P_r r^t/Z(t)`, take independent `X_i=log R_i` and write `S_G=Σ_{i=1}^G X_i`. For any `δ∈[0,a]` let `B_G=Ga+δ`, and condition on `S_G≤B_G`. Define `Y_G=B_G−S_G` on that event.
+
+**Theorem.** Uniformly for `δ∈[0,a]`, as `G→∞`,
+
+1. `P_t(Y_G≥x | S_G≤B_G) → exp(−λx)` for each fixed `x≥0`;
+2. `E_t[Y_G | S_G≤B_G] → 1/λ`.
+
+The condition has positive probability because `P_1>0`.
+
+**Actual leaf consequence.** For every integer rank `N≥121`, set `G=floor(log_5(N/121))` and `δ=log(N/121)−G log5∈[0,a)`. Write `L_N(ℓ)=X_N(ℓ)−BX_N(ℓ)` for the pinned flow's leaf-exit multiplicity at rank `ℓ`, subtracting its boundary-exit histogram from its complete exit histogram and interpreting missing entries as zero. Define its potential-weighted mean leaf log-depth by
+
+```
+D_leaf(N) = [Σ_ℓ L_N(ℓ) ℓ^t log(N/ℓ)] / [Σ_ℓ L_N(ℓ) ℓ^t]
+          = [Σ_ρ Σ_r F[G][ρ] P_r (rρ)^t log(N/(rρ))] / [Σ_ρ Σ_r F[G][ρ] P_r (rρ)^t].
+```
+
+Here `F[G]` is the prefix-count table described in §2, and both sums over `r` use the entire paid inventory. The denominators are positive because the all-ones prefix and terminal rank are present. Then
+
+`D_leaf(N) → log121 − μ(t) + 1/λ ∈ [136.51870330,136.51870331]`
+
+as `N→∞`, including sequences approaching either end of a lockstep level. There is no restriction to the level starts `N=121·5^G`.
+
+## 2. Why this is the source's measure
+
+The pinned `diagnostics/overlap_flow.py` constructs `F[d][ρ]` by multiplying `d` paid ranks and retaining a new product precisely when `121·ρ≤N`. All paid ranks are positive integers, so their logarithms are nonnegative: the final product constraint implies every earlier constraint. Consequently `F[G][ρ]` is exactly the sum of `Π_i P_{r_i}` over ordered G-tuples with product ρ and `ρ≤N/121`.
+
+In its `d==G` branch, the flow issues *every* final rank `r` with multiplicity `F[G][ρ] P_r` at exit rank `rρ`. This final rank is **not conditioned** by `121·rρ≤N`. Weighting by `(rρ)^t` factorizes into the conditioned G-step law above and an independent terminal `R` with law `p_t`.
+
+Thus, exactly at every finite N,
+
+`E[log(N/(Rρ))] = log121 − μ(t) + E[Y_G | S_G≤B_G]`.
+
+This G-versus-G+1 distinction matters. The checker independently enumerates ordered words through G=3 and compares all leaf multiplicities with the whole-flow `X−BX`; an extra-terminal-truncation mutant fails. The source audit also checks the original #4436 flow. These are finite controls for the source derivation, not evidence for the asymptotic theorem.
+
+## 3. The classical input and its hypotheses
+
+We use the nonlattice interval local-limit theorem in the following standard form. If iid real `V_i` have mean zero, finite positive variance `σ²`, and a nonlattice law, then for each fixed `h>0`,
+
+`sup_x | sqrt(G) P(Σ V_i∈[x,x+h)) − h/(σ sqrt(2π)) exp(−x²/(2Gσ²)) | → 0`.
+
+Reference: C. Stone, “A local limit theorem for nonlattice multi-dimensional distribution functions,” *The Annals of Mathematical Statistics* 36(2) (1965), 546–551, [DOI 10.1214/aoms/1177700165](https://doi.org/10.1214/aoms/1177700165), **Corollary 1, equation (7), derived from the Theorem, equation (5), with footnote 2's uniformity convention**. The displayed form is a specialization, not a verbatim quotation. Use `d=1`, `A_G=0`, `B_G=√G` in Stone's normalization (his `B_G` is unrelated to our barrier `Ga+δ`), and normal limiting density `φ_σ(u)=exp(−u²/(2σ²))/(σ√(2π))`. For `T_G=Σ V_i` and any fixed width `H>0`, substitute `x=X/√G` and `h=H/√G` in his half-open boxes. This gives, uniformly over `X∈R`,
+
+```
+P(T_G∈[X,X+H)) = ∫_[X/√G,(X+H)/√G) φ_σ(u) du + o(G^(−1/2))
+               = (H/√G) φ_σ(X/√G) + o(G^(−1/2)).
+```
+
+The second equality uses uniform continuity of the Gaussian density. Multiplying by `√G` gives the required supremum statement; footnote 2 makes the remainder uniform in the unlisted variable `x`, hence also in `X`. The paper's Corollary 4, equation (12), is only a fixed-`x` statement and is not the input used for our uniform concentration bound. Stone's theorem is an external mathematical dependency; no discrete law is replaced by a density without this interval argument.
+
+Apply it to the tilted law `q(r)=P_r r^α/Z(α)`, with `V=log r−a`. Its mean is zero by definition, and its variance is positive and finite because the support is finite and contains distinct points of positive mass. The ordinary central limit theorem supplies Stone's normal domain-of-attraction hypothesis with this normalization. The support is nonlattice: it contains `0,log2,log3` before centering. If all belonged to a translate of `hZ`, their differences would imply `log2/log3` rational. An integer relation `u log2=v log3` with nonzero integers would imply `2^u=3^v`, contradicting unique factorization. Centering and tilting do not alter this property.
+
+This support argument verifies Stone's characteristic-function condition (2). Indeed, for `θ≠0`, equality `|E exp(iθV)|=1` holds exactly when the unit complex variable `exp(iθV)` is almost surely constant, equivalently when the support lies in `c/θ+(2π/θ)Z` for some real `c`. Thus nonlattice support gives strict inequality for every nonzero `θ`.
+
+The finite support does **not** satisfy Stone's stronger condition (3): simultaneous approximation of its finitely many phases gives arbitrarily large `|θ|` for which the characteristic-function modulus approaches one. Under that condition, Stone's equations (6) and (8) extend the local-limit statement to boxes as small as `exp(−cn)` for a suitable `c>0`; their remainder is still an unquantified `o(1)`. Neither equation is used here. Our proof uses equations (5) and (7) for fixed unnormalized windows. Neither cited version supplies an explicit quantitative convergence rate, error constant or onset `G`; failure of condition (3) is not the explanation for this missing quantitative information. Such bounds require additional work and are not ruled out by this argument.
+
+Two consequences are used:
+
+- For fixed finite intervals in y, `sqrt(G) Q(B_G−S_G∈I) → |I|/(σ sqrt(2π))`, uniformly over bounded δ.
+- There is a constant C such that for all `G≥1` and all real x, `sqrt(G) Q(S_G∈[x,x+1))≤C`.
+
+The second follows from the theorem for h=1 for sufficiently large G, and enlarging C for finitely many smaller G. It is an existence bound, not a computable convergence rate. Either convention for half-open intervals works: the maximum atom is `o(G^(-1/2))`, by bounding it with an interval of arbitrary fixed h and then taking h down to zero. This also handles an atom at the exact truncation boundary.
+
+## 4. Weighted local limits and integrable tails
+
+Write `c=1/(σ sqrt(2π))`. For the nonnegative functions
+
+`f_0(y)=exp(−λy)`, `f_1(y)=y exp(−λy)`, and `f_x(y)=1_{y≥x} exp(−λy)`,
+
+we claim, uniformly in δ,
+
+`sqrt(G) E_q[f(B_G−S_G) 1_{S_G≤B_G}] → c ∫_0^∞ f(y)dy`.
+
+On a fixed compact interval this follows by upper and lower step-function approximations and the interval local limit. The functions are bounded and Riemann integrable there, including the single jump of `f_x`; the possible boundary atom is negligible by §3.
+
+For an integer `J≥0` and the tail partition into intervals `[j,j+1)`, the concentration bound in §3 gives, uniformly over G and δ,
+
+`sqrt(G) E_q[f_0(Y_G) 1_{Y_G≥J}] ≤ C Σ_{j≥J} exp(−λj)`,
+
+and
+
+`sqrt(G) E_q[f_1(Y_G) 1_{Y_G≥J}] ≤ C Σ_{j≥J} (j+1) exp(−λj)`.
+
+Here Y_G denotes the unconditioned difference `B_G−S_G` inside the indicated nonnegative event. Both series tend to zero as `J→∞` because λ>0; `f_x≤f_0` has the same domination. This proves the weighted local limits and supplies the missing uniform-integrability argument for the mean. Weak convergence alone would not suffice.
+
+## 5. Undoing the tilt
+
+For an ordered G-tuple,
+
+`p_t(tuple)/q(tuple) = (Z(α)/Z(t))^G exp(λ S_G)`.
+
+On writing `S_G=B_G−y`, the common factor `(Z(α)/Z(t))^G exp(λ B_G)` cancels in every conditional ratio. Hence
+
+`P_t(Y_G≥x | S_G≤B_G) = E_q[f_x(B_G−S_G) 1_{S_G≤B_G}] / E_q[f_0(B_G−S_G) 1_{S_G≤B_G}]`,
+
+and the analogous numerator with `f_1` gives the conditional expectation. By §4, the three scaled integrals converge to `c exp(−λx)/λ`, `c/λ²`, and `c/λ`. The denominator limit is strictly positive, so their ratios converge uniformly over δ to `exp(−λx)` and `1/λ`. Section 2 then gives the claimed leaf-depth limit.
+
+## 6. Scope, controls, and remaining decisions
+
+This proves the saturation that #4467 explicitly labelled heuristic. It does **not** prove its suggested total log-gap `g∞≈122`, stable source/boundary/leaf shares, any numerical refined-induction ceiling, or the quoted astronomical ranks required to achieve a saving. Those require separate control of other flow components and a finite-depth estimate. The raw-p7 inventory is historical and is not the selected #6030 invoice.
+
+Nonlattice support is essential for this limit law. For `X∈{0,log2}` with probabilities `(1/3,2/3)` and even G, conditioned at `B_G=G log2/2`, λ=1 but the limiting deficit is `log2` times a geometric variable with probabilities `2^(-j-1)`, j≥0. Its mean is `log2`, not 1. Indeed the successive binomial mass ratio below G/2 tends to 1/2 and is bounded by 1/2, which proves both the geometric limit and mean convergence by domination. The verifier includes this failure control. Zero or negative relative drift likewise lies outside the theorem's λ>0 hypothesis.
+
+Credit: the flow is zxv's #4436, the refined induction is Cinna Davis / 1337isnot1337's #4405, the lockstep construction is Jacob Alan Sussman's Theorem 1G, and the source law, saturation conjecture and diagnostics are Ryanson Jonathan / pinterbanget's #4467 (following #4465). This proof and certificate were prepared for DLamdata with substantial OpenAI Codex assistance. Apache-2.0. Internal AI checks are not independent human review or formal verification.