Prove the exact power-tariff transition for direct-flow leaves (no new κ)

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

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+# The power-tariff transition of the fixed direct-flow leaves
+
+## Statement and inherited inputs
+
+This is a scoped proof about the historical raw-p7 leaf measure. It does not
+establish a complete multiplication theorem, a full-flow invoice or a new κ.
+The leaf model and inventory are Ryanson Jonathan / pinterbanget's
+[PR #4467](https://github.com/CrocSwap/integer-mult-bounds/pull/4467).
+DLamdata's [PR #6118](https://github.com/CrocSwap/integer-mult-bounds/pull/6118),
+at `6310550e932de70243fc2aaab6f4885688b6fe21`, proves saturation of its
+conditional log-deficit. We use its tilted nonlattice local-limit framework as
+a cited mathematical input, without replaying its certificate or claiming its
+source-law derivation as new work. The new result is the complete real-power
+moment asymptotic, including its critical and divergent regimes.
+
+Let
+
+```
+P = {1:37305, 2:10695, 3:3360, 4:1530, 5:300,
+     8:1464, 16:448, 32:448, 48:448},       m = 121,
+Z(s) = sum_r P[r] r^s,                    mu(s) = (log Z)'(s),
+a = log 5,                               mu(alpha) = a,
+lambda = t-alpha > 0,                    sigma^2 = (log Z)''(alpha).
+```
+
+The theorem holds for **every fixed real t > alpha**. Its intended application
+uses the raw-p7 crossing exponent `t` of #4467/#6118. We inherit, rather than
+recertify, #6118's numerical identification of that exponent. No numerical root
+or onset estimate is needed for the result below. Since `P` has positive mass
+at 1 and 48, `mu` is strictly increasing from 0 to log48; hence alpha exists
+uniquely and sigma is positive. The support contains 1, 2 and 3, so the log-rank
+law is nonlattice, as in #6118.
+
+For each fixed integer `G >= 0` and real `delta in [0,a]`, put
+`N=m exp(Ga+delta)` and define the fixed-depth leaf measure below. At the closed
+endpoint `delta=a`, **keep this G**: this is a fixed-depth extension, not a
+recalculation of the floor. For actual integer ranks `N >= m`, instead select
+`G=floor(log_5(N/m))` and `delta=log(N/m)-Ga in [0,a)`; that selects the original
+leaf model from this family. Let `c_G(rho)` be the weighted count
+of ordered G-tuples of paid ranks with product rho, retaining `m*rho <= N`.
+Each tuple has weight the product of its P multiplicities. The leaf multiset
+has `c_G(rho)*P[r]` leaves at rank `ell=r*rho`, using **every** terminal r,
+without truncating the terminal rank again. This is the inherited #4467 leaf
+model, separated from the other exit components by #6118.
+
+For any fixed real shift u, define its normalized power tariff
+
+```
+M_G(u,delta) = sum_ell L_N(ell) ell^(t+u)
+              / (N^u sum_ell L_N(ell) ell^t),
+A(u) = m^(-u) Z(t+u)/Z(t),       v = lambda+u.
+```
+
+**Theorem.** Uniformly over `delta in [0,a]` (and therefore along every integer
+rank sequence as `G -> infinity`), the following three regimes hold:
+
+1. If `v > 0`, then
+
+   `M_G(u,delta) -> A(u) lambda/v`.
+
+2. If `v = 0`, then
+
+   `M_G(u,delta) ~ A(u) lambda sigma sqrt(pi/2) sqrt(G)`.
+
+3. If `v < 0`, put
+
+   `R(u) = [Z(t+u)/Z(alpha)] exp(-v a)`.
+
+   Then `R(u)>1` and
+
+   `M_G(u,delta) ~ A(u) lambda sigma sqrt(2*pi*G) exp(-v delta) R(u)^G`.
+
+Here `~` means the ratio tends to one, uniformly in delta. The exponent
+threshold is exactly `t+u=alpha`. The boundary offset disappears from the first
+two leading terms but survives in the third.
+
+## 1. Exact finite identity
+
+Give each prefix step the probability `p_s(r)=P[r]r^s/Z(s)`. Set
+`S_G=sum_i log R_i`, `B=Ga+delta`, and `Y=B-S_G`. Expectations `E_s` refer to
+independent prefix steps with law p_s. The independent, untruncated terminal
+step has law p_t. Weighting the leaf multiset by ell^t therefore gives exactly
+
+`M_G(u,delta) = A(u) E_t[exp(-uY) | S_G <= B]`.                 (1)
+
+For any real v define the finite quantity
+
+`J_G(v,delta)=E_alpha[exp(-vY) 1_{Y>=0}]`.
+
+Changing the prefix measure from t to alpha in numerator and denominator of
+(1) cancels `(Z(alpha)/Z(t))^G exp(lambda B)`. Thus, for every finite G,
+
+`M_G(u,delta) = A(u) J_G(lambda+u,delta)/J_G(lambda,delta)`.    (2)
+
+The denominator is positive because the all-ones prefix is admissible. In
+particular `M_G(0,delta)=1` exactly. No asymptotic interchange or extra terminal
+barrier occurs in (1) or (2).
+
+## 2. Positive tilt
+
+We use the following standard consequence of the nonlattice interval local
+limit theorem, in the precise form used in #6118 §§3–4:
+
+`J_G(w,delta) ~ 1/[w sigma sqrt(2*pi*G)]`, for each fixed w>0. (3)
+
+For clarity, the inputs to (3) are explicit. Under p_alpha, the centered
+variable `log R-a` has zero mean, positive finite variance sigma^2 and a
+nonlattice law. The uniform interval local limit theorem gives convergence of
+the scaled measure near any bounded delta to Lebesgue measure divided by
+`sigma sqrt(2*pi)`. It also bounds the scaled probability of every unit interval
+by a constant. Integrating exp(-w y) on y>=0 follows by compact step-function
+approximation and the summable tail `sum_{j>=J} exp(-w j)`. Atoms at the barrier
+are negligible by the same interval theorem with arbitrarily small fixed
+interval width. These facts give uniformity in bounded delta and the constant
+`integral_0^infinity exp(-wy)dy=1/w`.
+
+This use requires no strong nonlattice condition and claims no convergence
+rate. The classical input is C. Stone, *A local limit theorem for nonlattice
+multi-dimensional distribution functions*, Ann. Math. Statist. 36 (1965),
+546–551, Corollary 1, equation (7):
+[DOI](https://doi.org/10.1214/aoms/1177700165). We cite the source-relative
+specialization in #6118; we do not assert an independent audit of that paper.
+
+Apply (3) at w=lambda and w=v>0 in (2). Their ratio tends to lambda/v,
+proving regime 1. Weak convergence to the exponential law alone would not
+justify its negative-u part: (3) supplies the necessary tail control.
+
+## 3. The critical tilt
+
+When v=0, `J_G(0,delta)=P_alpha(S_G<=Ga+delta)`. The ordinary central limit
+theorem gives
+
+`J_G(0,delta) -> 1/2`                                       (4)
+
+uniformly in bounded delta. Indeed the standardized endpoint is
+`delta/(sigma sqrt(G)) -> 0` uniformly, and the normal distribution is
+continuous at zero. Divide (4) by (3) at w=lambda in (2). The resulting
+constant is `A(u) lambda sigma sqrt(2*pi)/2`, giving regime 2.
+
+## 4. Below the critical exponent
+
+For v<0 put beta=alpha+v=t+u. A second exact change of measure gives
+
+`J_G(v,delta) = exp(-v delta) R(u)^G P_beta(S_G<=Ga+delta)`.   (5)
+
+The signs and offset in (5) follow directly from
+`p_beta(tuple)/p_alpha(tuple)=(Z(alpha)/Z(beta))^G exp(v S_G)`.
+Since beta<alpha and mu is strictly increasing, `mu(beta)<a`. The weak law of
+large numbers gives `P_beta(S_G<=Ga+delta)->1`, uniformly for delta in [0,a].
+For example, the complementary event is contained in `S_G/G>a`, whose
+probability is at most `Var_beta(log R)/(G(a-mu(beta))^2)` by Chebyshev.
+Strict convexity of log Z gives
+
+`log R(u)=log Z(beta)-log Z(alpha)-(beta-alpha)a > 0`.
+
+Divide (5) by (3) at w=lambda in (2) to obtain regime 3.
+
+## 5. Consequence and scope
+
+The normalized leaf tariff has a finite limit precisely when `t+u>alpha`;
+it grows as sqrt(G) at equality and exponentially in G below it. Thus a finite
+mean log-depth does not authorize substituting an arbitrary power tariff or
+exchanging that tariff with the limiting exponential law. The finite limit
+above the threshold is explicit, including the independent terminal-rank
+factor `Z(t+u)/Z(t)`.
+
+This settles the leaf component only. Relative source, boundary and leaf
+masses, their joint normalization, a total refined-induction ceiling, explicit
+finite-depth error bounds, physical delivery and every supplier-admission
+obligation remain open. The fixed historical raw-p7 law is not the selected
+#6030 invoice. Neither the three regimes nor their finite checks change κ.
+
+New proof produced by Nick Petalas. Credit DLamdata (#6118) for the saturation
+framework; Ryanson Jonathan / pinterbanget (#4467/#4465/#4459) for the leaf law
+and diagnostics; zxv (#4436) for the direct flow; Cinna Davis / 1337isnot1337
+(#4405) for refined induction; Jacob Alan Sussman for the lockstep compiler;
+and Stone for the classical local limit theorem. Apache-2.0.

research/direct-flow-leaf-tariff/README.md

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+# Exact power-tariff asymptotics for direct-flow leaves
+
+Scoped proof produced by Nick Petalas. No new κ.
+
+For the fixed historical raw-p7 leaf measure of Ryanson Jonathan's
+[PR #4467](https://github.com/CrocSwap/integer-mult-bounds/pull/4467), this
+package derives the full power-tariff asymptotic from the tilted local-limit
+framework in DLamdata's
+[PR #6118](https://github.com/CrocSwap/integer-mult-bounds/pull/6118)
+(`6310550e932de70243fc2aaab6f4885688b6fe21`). This extends that saturation
+result; its certificate is not rerun or audited here.
+
+Write `Z(s)=sum_r P[r]r^s`, `a=log5`, `mu(alpha)=a`, `lambda=t-alpha>0`,
+`sigma^2=(log Z)''(alpha)`, and `N=121 exp(Ga+delta)`. Define
+
+```
+M_G(u,delta) = sum_leaves ell^(t+u) / (N^u sum_leaves ell^t),
+A(u) = 121^(-u) Z(t+u)/Z(t),       v=lambda+u.
+```
+
+Multiplicities are included in both sums. The new theorem is
+
+```
+v > 0:  M_G -> A(u) lambda/v
+v = 0:  M_G ~ A(u) lambda sigma sqrt(pi G/2)
+v < 0:  M_G ~ A(u) lambda sigma sqrt(2 pi G) exp(-v delta) R(u)^G,
+        R(u)=Z(t+u)/Z(alpha) exp(-v a) > 1.
+```
+
+Each statement is for a fixed real u, uniformly over `delta in [0,log5]` in
+the fixed-depth family: keep G at the upper endpoint. Actual integer ranks use
+`G=floor(log_5(N/121))`, hence `delta in [0,log5)`.
+See [PROOF.md](PROOF.md) for the exact finite quotient, the nonlattice
+local-limit dependency, critical CLT argument and second change of measure.
+In particular, convergence of the mean log-depth does not imply that all
+power tariffs converge. The exact threshold is `t+u=alpha`.
+
+## Evidence and reproduction
+
+The infinite-parameter result is a written scoped proof. The standard-library
+verifier checks its new finite factorization and tilted quotient using exact
+rational arithmetic. It imports no predecessor verifier and computes no
+predecessor root, saturation value or full-flow result. It authenticates two
+main-branch input files and reads their literal P/M inventory without executing
+them.
+
+From the repository root:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/direct-flow-leaf-tariff/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/direct-flow-leaf-tariff/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/direct-flow-leaf-tariff
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/direct-flow-leaf-tariff --check-source-history
+```
+
+Other hosts can use their CPython 3.12 executable. Direct replay prints
+`PASS_DIRECT_FLOW_LEAF_TARIFF_FINITE_IDENTITIES`: 15 finite fixtures,
+225 moment identities, 450 tilted-quotient identities, 210 second-measure-change
+identities, 10,191 accepted ordered
+prefixes counted across fixtures, and eight rejecting controls. The five
+semantic controls omit the terminal factor, truncate the terminal rank, reverse
+the power sign, drop the lockstep offset or reverse the tilt shift. Three
+boundary/source controls reject excessive depth, an empty barrier domain and
+corrupted source bytes. Finite exponent choices are integral so these checks
+are exact; they do not certify the asymptotic theorem by enumeration.
+
+All loops are bounded by depth four, nine paid ranks and fixed fixture lists.
+The largest ordered-word enumeration has 9^4 tuples. Python's managed allocation
+is a Power-of-Ten deviation, with live collections bounded by these fixed
+inputs. If a later version accepts variable depth or inventories, it must
+replace these fixed bounds with enforced resource limits before reuse.
+Functions are below 60 lines, no recursion, callback dispatch or dynamic code
+execution is used, and normal/optimized warning-as-error runs exercise the same
+explicit checks. Ruff, mypy, Pyright and Pylint are unavailable in the local PATH
+and environment; stronger static-analysis coverage is unclaimed. A future
+language/toolchain change should add its available diagnostics to VERIFY.json.
+No generated receipt is committed or hand-written.
+
+## Limitations
+
+The local-limit theorem is a cited classical mathematical dependency, following
+#6118's specialization; independent human and formal proof-kernel review are
+unclaimed. The theorem uses any fixed t>alpha and inherits #6118's identification
+of the raw-p7 crossing exponent for that application. It does not reverify that
+identification. No finite-depth rate or practical rank threshold is supplied.
+Relative source/boundary/leaf masses, a complete direct-flow invoice, native
+operators, compiler realization, physical costs and selected supplier admission
+are unproved here. The historical raw-p7 law is not the selected #6030 invoice;
+all five selected framework assumptions remain.
+
+## Open next questions
+
+- Determine the full source/boundary/leaf-normalized exit coefficient, using
+  compatible tilted measures and retaining the offset. The missing non-leaf
+  asymptotics and relative masses block a complete refined-induction ceiling.
+- Give explicit finite-depth errors for these three regimes. Start with a
+  quantitative interval local-limit bound for the fixed log{2,3,5} support;
+  recurrent finite-support phases prevent assuming strong nonlattice decay.
+
+The relevance, first steps and blockers are also in [TASK.json](TASK.json).
+
+## Credit
+
+New theorem and verifier produced by Nick Petalas. DLamdata (#6118) supplies the
+saturation framework; Ryanson Jonathan / pinterbanget (#4467/#4465/#4459)
+supplies the leaf law and diagnostics; zxv (#4436) supplies the direct flow;
+Cinna Davis / 1337isnot1337 (#4405) supplies refined induction; Jacob Alan
+Sussman supplies the lockstep compiler; Stone supplies the classical local
+limit theorem. All predecessor files and credit remain unchanged. Apache-2.0.

research/direct-flow-leaf-tariff/SOURCES.json

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