Prove a strictly weaker mirror-capacity criterion (no κ)

research/rank-sensitive-mirror/PROOF.md

@@ -0,0 +1,285 @@
+# Rank-sensitive complement capacity for the capped mirror flow
+
+This contribution is produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped written mathematical proof with exact finite corroboration.
+The capped-flow and normalized recurrence contracts are cited premises. No
+native unit realizing the example, physical supplier, new kappa, or complete
+multiplication theorem is supplied.
+
+## 1. Premises and the new source charge
+
+Use zxv's arbitrary-height capped flow from PR4436/4483 and Ryanson Jonathan's
+corrected PR4459 mirror identity. Dillon Amburgey's PR4532 supplies the
+critical-root tail construction and shifted-logarithmic majorant. These results
+are used as premises, without auditing or replaying their examples.
+
+The nonempty integer histograms have target ranks `2 <= k <= m=max(k)` and
+paid ranks `1 <= r < m`, with multiplicities `W_k` and `P_r`. Let
+
+```text
+T(t) = sum_k W_k k^t,       P(t) = sum_r P_r r^t,
+0 < t < 1,                 C = floor(n/m).
+```
+
+Fix a real ambient scale `Lambda>m`. Restrict to integers `n` for which
+`n<=Lambda C`. It suffices that `C >= (m-1)/(Lambda-m)`, since
+`n<=mC+m-1`. This is a finite base-range restriction, not a substitution
+of `n/m` for the integer cutoff.
+
+The cited mirror identity expresses every inner source through a pair `k>r`
+and an integer base product `z`, in the window
+
+```text
+C/k < z <= C/r,          source rank s=rkz.
+```
+
+Consequently `s>rC>=rn/Lambda`. For the pure power, concavity gives
+
+```text
+n^t-(n-s)^t >= t n^(t-1) s >= eta_r(t) s^t,
+eta_r(t) := t (r/Lambda)^(1-t).                         (1)
+```
+
+The inequality includes `s=n`. Every source rank is at most `n`, by the
+inherited flow contract. The usual uniform charge uses only `eta_1`; (1)
+retains the paid letter `r` attached to that particular source window.
+
+The same charge works for the shifted-logarithmic function
+
+```text
+f(0)=0,    f(x)=A x^theta (c+log x)^p for x>=1,
+tau(x)=theta+p/(c+log x),    0<theta<1,    c>=p/(1-theta).
+```
+
+Indeed `f` is increasing and concave on `[1,infinity)`, `tau<=1`, and
+`f'(x)=tau(x)f(x)/x`. For integer `1<=s<n`, concavity gives
+`f(n)-f(n-s)>=tau(n)f(n)s/n`. For `s=n`, use `f(0)=0` and `tau(n)<=1`.
+Thus, at `t=tau(n)`,
+
+```text
+[f(n)-f(n-s)]/f(n) >= eta_r(t) (s/n)^t.                 (2)
+```
+
+This is the new replacement for the uniform source-capacity step. It does
+not alter the flow, its paid positions, or the productive boundary.
+
+## 2. Weighted mirror inequality and complete leaf funding
+
+Partition the `y=z/C` axis at the endpoints `1/k,1/r`, using the inherited
+half-open windows. Write
+
+```text
+E_J(t) = sum_(r>k, J subset (1/r,1/k]) P_r W_k (rk)^t,
+S_(J,r)(t) = sum_(k>r, J subset (1/k,1/r]) P_r W_k (rk)^t,
+D_J(t) = sum_r eta_r(t) S_(J,r)(t).
+```
+
+Suppose a constant `0<=b<1` satisfies
+
+```text
+E_J(t) <= b D_J(t) for every elementary cell J.          (3)
+```
+
+A zero `D_J` must have zero `E_J`. Let `K_t` be the sum of `eta_r(t)s^t`
+over all actual inner sources, with their multiplicities, and let `I_t`
+be their ordinary `t`-potential. Integrating (3) against the exact
+nonnegative capped base measure gives
+
+```text
+B_t <= b K_t,          K_t >= eta I_t,
+eta := eta_1(t) in (0,1).                               (4)
+```
+
+Here `B_t` is boundary-exit potential. There is no assumed density,
+equidistribution, or limit measure. The actual nonlinear complement capacity
+dominates `K_t` after the common normalization in (2).
+
+Let `F_t` be leaf-exit potential and `A_t` the net occupation surplus after
+root sources. The cited exact cancellation is
+
+```text
+I_t = B_t + F_t + A_t.
+```
+
+The new sufficient funding implication is
+
+```text
+A_t >= 0 and (1-b) eta A_t >= (1-eta) F_t
+       imply B_t+F_t <= K_t.                            (5)
+```
+
+To prove it, suppose `K_t<B_t+F_t`. Then cancellation and (4) imply
+`K_t>=eta I_t>eta(K_t+A_t)`, hence
+`(1-eta)K_t>eta A_t`. Also (4) gives
+`F_t>K_t-B_t>=(1-b)K_t`. Multiplying these inequalities contradicts
+the hypothesis of (5). This proof uses no division by `F_t` and covers
+zero leaf or boundary mass. With (2), exit charge is therefore funded by
+the actual inner-source complement capacity. Root complement capacities
+are nonnegative and remain available.
+
+In particular, the inherited bounds
+
+```text
+F_t <= P(t)^(G+1),      A_t >= Q_ell(t) P(t)^(G+1)
+```
+
+are sufficient whenever
+
+```text
+Q_ell(t) >= (1-eta)/((1-b)eta).                          (6)
+```
+
+Leaves have not been dropped. Condition (6) replaces the uniform mirror
+threshold; it does not assert that a favorable mirror test alone funds a flow.
+
+Allowing `Lambda>m` to be chosen makes this a weaker sufficient mirror test.
+Indeed, if the old uniform ratio is `beta<t m^(t-1)`, choose `Lambda`
+sufficiently close to `m` that `beta<eta_1(t)`. Since every `r>=1`,
+`D_J>=eta_1 S_J`, so (3) holds with `b=beta/eta_1<1`. The example below
+proves that the converse fails. No inclusion is asserted for an arbitrary
+fixed, excessively large value of `Lambda`.
+
+## 3. Critical-root corollary
+
+Assume the same finite-flow and normalized recurrence contracts as PR4532.
+Suppose, for some `theta in (0,1)`,
+
+```text
+T(theta)=P(theta),
+mu_T(theta)>mu_P(theta),
+E_J(theta)<=b_* D_J(theta) for every J, with b_*<1.       (7)
+```
+
+Then there exist positive `A,c,p` and an integer base cutoff `n0` such that
+the conditional normalized leaf recurrence has
+
+```text
+L(0)=0,        L(n)<=A n^theta(c+log n)^p for all n>=1.  (8)
+```
+
+The following specifies exactly which predecessor steps are used and which
+capacity step changes. The moment crossing and positive mean gap give positive
+integers `a,b_h`, an integer height base `M>1`, and positive rational tail
+parameters `s_0,lambda` as in PR4532 section 2.1--2.2. The mixed and root
+tail factors are strictly below one, and forward ratio ordering holds.
+This tail argument uses the mean gap, not mirror capacity.
+
+Every nonzero `D_J(t)` is positive and continuous near `theta`; identically
+zero source cells have no exit under (7). Choose `b_bar` strictly between
+`b_*` and one. On a sufficiently short right interval at `theta`, (3)
+holds with `b_bar`, both tails are bounded by constants `U,V<1`, and
+`d(log(T/P))/dt>=d>0`. Also
+
+```text
+eta_1'(t) = Lambda^(t-1)(1+t log Lambda)>0,
+eta_1(t)>=eta_*:=eta_1(theta)>0.
+```
+
+The inherited shifted-logarithmic occupation estimate is
+
+```text
+Q_ell(tau(n)) >=
+(1-U^ell)(exp(K_ell)-1)-Q M^lambda V^ell,
+K_ell = d p(a ell+1)/(c+log m+(ell+1)log M+log 2),
+```
+
+where `Q` is a finite upper bound for `T/P` on that interval. This is the
+same occupation/root-tail estimate, not a new invoice or tail inequality.
+Its right side increases to `exp(dpa/log M)-1` as `ell` increases. Choose
+`p` so that this limit exceeds
+
+```text
+(1-eta_*)/((1-b_bar)eta_*).
+```
+
+Choose `c>=p/(1-theta)`, then a sufficiently large integer `ell0` so that
+the occupation estimate exceeds this threshold, `tau(n)` lies in the
+certified right interval, and `n<=Lambda floor(n/m)` above the base cutoff.
+All these demands are finite. Increasing `ell0` preserves them. Put
+`n0=m M^ell0` and use elementary evaluation through `n0`.
+
+For `n>n0`, set `ell=floor(log_M floor(n/m))`, so `ell>=ell0`.
+The occupation estimate, monotonicity of `eta_1`, and (6) fund all exits
+by the source capacities (2). The inherited exit comparison
+`f(e)/f(n)<=(e/n)^tau(n)` applies unchanged. Choosing
+`A=n0^(1-theta)/(c+log n0)^p` funds the elementary base range since `f/x`
+is decreasing. Every recursive rank is strictly below `n`, so induction
+proves (8). No predecessor native premise has been discharged.
+
+The mean gap is an explicit hypothesis in this corollary. The weighted
+condition is not used to infer it. This avoids transferring PR4532's
+unweighted integral argument to a different mirror inequality.
+
+## 4. Exact integer-histogram separation
+
+Take the abstract histograms
+
+```text
+W_16=1, W_64=2;      P_1=2, P_4=6, P_36=1;
+theta=1/2, m=64,    Lambda=289/4, sqrt(Lambda)=17/2.
+```
+
+Both moments at `theta` are exactly 20. Their tilted-log mean difference is
+
+```text
+mu_T-mu_P = (19 log 2-3 log 3)/5 > 0,
+```
+
+since `2^19>3^3`. The only cell with exits is `(1/36,1/16]`. On it,
+
+```text
+E=24,       S_(J,1)=32,       S_(J,4)=192,
+E/S = 24/224 = 3/28 > 1/16.
+```
+
+Thus PR4532's uniform mirror hypothesis fails, even with its more favorable
+ambient constant `m=64`. For the new charge,
+
+```text
+eta_1=1/17,     eta_4=2/17,
+D_J=32/17+384/17=416/17,
+E/D_J=51/52<1.
+```
+
+All other cells have zero exit coefficient. The floor guard holds uniformly
+for `C>=8`, since `64C+63 <= (289/4)C`. In particular `n>=512` suffices.
+The local mirror calculation uses exactly the cap and costs declared in
+sections 1--2.
+
+There is also an exact two-tail positive control. Choose `a=b_h=1`,
+`M=1024`, `s_0=lambda=1/2`. Then
+
+```text
+T(1)=144, P(1)=62, T(0)=3,
+u=1024^(-1/2)(144/20)(62/20)=279/400<1,
+v=1024^(1/2)(3/20)^2=18/25<1,
+144/20 > 62/20.
+```
+
+The corollary therefore applies to this abstract recurrence at exponent
+`1/2`. This is not a native supplier with saving `1/2`. No native word
+realizing these histograms has been given.
+
+At the exact exponent, (6)'s threshold is 832. The algebraic values
+`B=51,F=1,A=832,I=884,K=52` attain every constraint in (4)--(6) and give
+`K=B+F`. The shifted-log proof uses a strictly larger bound on a right
+interval. The number 832 is not a claimed sufficient compiler height.
+
+## 5. Verification boundary
+
+The standalone verifier computes only this new toy profile and the new
+capacity/funding implications. It uses exact rational and integer arithmetic.
+It enumerates the toy's capped base measures in two ways and checks actual
+nonlinear square-root complement capacities at 49 finite rank/height fixtures.
+Those checks corroborate (1)--(6); (8) rests on the written argument and cited
+flow/tail premises. No p7/p16 or selected native replay is performed.
+
+The contribution only replaces a sufficient complement-capacity test. It
+does not solve the full distribution-dependent mirror question in PR4532,
+prove an actual profile's physical costs, or discharge any of the selected
+record's five inherited assumptions. Arbitrary-input native correctness,
+ownership/restoration, chronological batch widths, adapters, movement,
+precision, stock, tape execution, analytic transfer and exact recovery remain
+separate obligations. Independent human and formal proof-kernel review are
+unclaimed.