Prove analytic cutoff guards for the fixed cusp-population realization

experiments/universality/two_parameter_cusp_population/ANALYTIC_GUARDS.md

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+# Explicit analytic guards for the fixed cusp population
+
+This successor source supplies an explicit member of the smooth-cutoff class
+consumed by the fixed population experiment. It proves the derivative and
+annular guard premises that its frozen numerical receipt records as
+`REVIEW_REQUIRED`. It introduces no new control value, random field draw,
+population run, fitted coefficient, or claim of Gaussian universality.
+
+Actual author: OpenAI/Codex /root, source exposed. Separate mathematical review
+is required before this source is accepted. Organizational independence and
+blind credit are zero. This document is a mathematical source, with complete
+field-evaluator execution, independent global H0-selector execution, Lean
+kernel verification, higher homology, spatial limits and scientific promotion
+still `NOT_RUN`.
+
+The consumed sources remain unchanged:
+
+- `PROOF.md`: 782 lines, SHA-256
+  `9e8caa52b6448e23e465e646ca98d6e42e0cd71442ad4118b59145ee849e1326`.
+- `CONTROLLED_FALSIFICATION.md`: 270 lines, SHA-256
+  `a23f88801e678dd1af484cac8f5564801d9d410d835b560a280c0ea6dfc8996e`.
+- The completed numerical receipt:
+  `population-numerical-increment.json`, 2,206,767 bytes, SHA-256
+  `cbaf37ef3b8002057dac7f81db72b95c3bf03015fc39dd34d7d2745f4ef4aa94`,
+  private Drive file `1WvOSM_i6ECXQe3f06wO4DzjkhI0lz01U`.
+
+Those historical bytes and their scope statuses are not rewritten.
+In particular, the explicit radial argument below was not present in the
+frozen cutoff-class description. This is new evidence for a concrete
+realization of that class, rather than a claim of prior field execution.
+
+## 1. Parameters, coordinates and the actual field
+
+Keep exactly the declared parameters
+
+```
+d=2, L=1, R=1/8, r1=1/32, A=2^-16,
+u0=3*A^2=3/2^32, v0=2*A^3=1/2^47,
+a*=r1/4=1/128, delta=r1^4/64=2^-26.
+```
+
+Let `y` be the torus representative in `[-1/2,1/2]^2`, and put
+
+```
+s=sqrt(2/3)*sin(pi*y1),
+z=sqrt(2/9)*sin(pi*y2),
+r=s^2+z^2, b=4/9.
+```
+
+In the chart `|yi|<1/4`, these are smooth coordinates with nonsingular
+Jacobian, and the seed is exactly `h=b-r`. Since `R^2=1/64<1/9=w2`,
+the superlevel `{h>b-R^2}` is precisely this chart's ball `D_R`, as
+proved in the consumed source.
+
+Define the functions below on all their indicated real arguments and set
+
+```
+F_(u,v)(y)=b-s^2*c(r)-z^2+chi(s,z)*(u*s^2/2+v*s),
+                 |u|<=u0, |v|<=v0.                 (G1)
+```
+
+The definitions give `c=1` and `chi=0` for `r>=R^2/2`, so (G1)
+equals the seed on a neighborhood of the exterior of `D_R`.
+Although signed sine coordinates are represented with a seam, near any
+torus seam (G1) equals the smooth periodic seed. Its values and every
+derivative therefore match there. The formula defines a global C-infinity
+field, jointly smooth in the two controls. Translating it preserves all
+critical heights and H0 barcodes; no translation draw is needed for this
+analytic assertion.
+
+All gradient norms used in the guards below are Euclidean norms in
+`(s,z)` coordinates. Nonsingularity of the chart transfers critical-point
+exclusion to physical coordinates; no physical-gradient bound is asserted.
+
+## 2. The flat step and its derivative
+
+For a real argument define
+
+```
+q(t)=exp(-1/t) if t>0, and q(t)=0 otherwise;
+psi(t)=q(1-t)/(q(t)+q(1-t)).
+```
+
+The denominator is positive on the whole real line: at least one of
+`t,1-t` is at least `1/2`. The zero extension of q is C-infinity and
+flat at zero. Indeed, by differentiating on the positive half-line, every
+derivative is `exp(-1/t)` times a polynomial in `1/t`. For each
+integer m, put `x=1/t` and choose an integer N>m. The exponential series
+gives `exp(x)>=x^N/N!`, so
+`x^m*exp(-x)<=N!*x^(m-N)->0`. The same estimate with one additional
+power proves the difference-quotient matching at zero at every derivative
+order. Induction therefore verifies the smooth zero extension and all its
+zero jets.
+
+It follows that psi is C-infinity, equals one for `t<=0`, equals zero
+for `t>=1`, and is flat at both transition endpoints.
+On `0<t<1` its derivative is
+
+```
+psi'(t)=-[q'(1-t)*q(t)+q(1-t)*q'(t)]
+                     /[q(t)+q(1-t)]^2.              (G2)
+```
+
+Thus psi is nonincreasing and lies in `[0,1]`.
+For `t>0`,
+
+```
+q'(t)=t^-2*exp(-1/t),
+q''(t)=(1-2*t)*t^-4*exp(-1/t).
+```
+
+Consequently `0<=q'<=M=4*exp(-2)`, with its maximum at `t=1/2`.
+Since q is nondecreasing and one argument is at least `1/2`,
+`q(t)+q(1-t)>=exp(-2)`. The numerator in (G2) is at most
+`M*[q(t)+q(1-t)]`. Hence, including the flat endpoints,
+
+```
+abs(psi')<=M/[q(t)+q(1-t)]<=4.                       (G3)
+```
+
+This proves the frozen weaker bound `abs(psi')<=8` without a numerical
+estimate of an exponential.
+
+## 3. Explicit radial cutoff and the four perturbation norms
+
+Make the previously admitted cutoff class concrete by choosing
+
+```
+chi(s,z)=psi((s^2+z^2-r1^2/4)/(3*r1^2/4)).           (G4)
+```
+
+This is C-infinity, equals one on `|(s,z)|<=r1/2`, and equals zero on
+`|(s,z)|>=r1`, with all jets matching across both boundaries.
+On its transition region the inner radial argument has gradient norm
+
+```
+8*|(s,z)|/(3*r1^2)<=8/(3*r1).
+```
+
+Combining this with (G3) proves
+
+```
+||grad chi||<=32/(3*r1)<64/r1.                       (G5)
+```
+
+Zero extension from the local chart is smooth because the support lies
+strictly inside `D_R`. Write B0,D0 for the sup norms of
+`chi*s^2/2, chi*s`, and B1,D1 for their gradient sup norms.
+On the support `|s|<=r1` and `0<=chi<=1`, so
+
+```
+B0<=r1^2/2,                         D0<=r1,
+B1<=(64/r1)*(r1^2/2)+r1=33*r1,      D1<=(64/r1)*r1+1=65. (G6)
+```
+
+Here the gradient inequalities are the product rule and triangle
+inequality, not assumptions about machine arithmetic.
+
+## 4. Monotone surgery and the annular gradient
+
+Use exactly the proposed surgery profile
+
+```
+c(r)=1-(1-r/4)*psi((r-r1^2)/(R^2/2-r1^2)), r>=0.    (G7)
+```
+
+Its transition width is positive since `r1=R/4`.
+It equals `r/4` for `r<=r1^2` and equals one for
+`r>=R^2/2`, with all derivative jets matching at both endpoints.
+During the transition `0<=r<=R^2/2<1`; the formula is a convex
+combination of `r/4` and one. Thus `0<=c<=1`. With w denoting the
+positive transition width, differentiation gives
+
+```
+c'(r)=psi((r-r1^2)/w)/4
+       -(1-r/4)*psi'((r-r1^2)/w)/w >=0.
+```
+
+For `r>0`, c(r)>0. Thus the unperturbed chart field
+`F0=b-s^2*c(r)-z^2` has derivatives
+
+```
+(F0)_s=-2*s*[c(r)+s^2*c'(r)],
+(F0)_z=-2*z*[1+s^2*c'(r)].
+```
+
+Both vanish together only at the origin. Outside the chart the seed
+critical points remain unchanged. In the entire closed annulus
+`rho=|(s,z)| in [r1/2,r1]`, (G7) gives the exact core polynomial,
+including matching at its outer endpoint:
+
+```
+F0=b-s^4/4-(1+s^2/4)*z^2,
+abs((F0)_s)=|s|*(s^2+z^2/2)>=|s|*rho^2/2,
+abs((F0)_z)=2*|z|*(1+s^2/4)>=|z|*rho^2/2.
+```
+
+The final inequality uses `rho<=r1<1`. Taking the Euclidean norm
+therefore yields
+
+```
+||grad F0||>=rho^3/2>=r1^3/16>r1^3/64.              (G8)
+```
+
+Thus the frozen conservative witness `g*>=r1^3/64` is now proved for
+this explicit field, in the same coordinate norm used by (G6).
+
+## 5. The unchanged rectangle satisfies the uniform guards
+
+Substituting the fixed controls into (G6) gives exactly
+
+```
+B1*u0+D1*v0 <= 101441/2^47 < 2^-22 <= g*/2,
+B0*u0+D0*v0 <= 1537/2^52 < 2^-28 = delta/4.          (G9)
+```
+
+For the first strict comparison, `101441<2^25`; for the second,
+`1537<2^24`. The root-containment guards are also strict:
+
+```
+u0=3/2^32 < a*^2/4=2^-16,
+v0=2^-47 < a*^3/4=2^-23.
+```
+
+The wedge coverage is exact:
+`(2/(3*sqrt(3)))*u0^(3/2)=2*A^3=v0`.
+These checks use the existing parameter rectangle, without shrinking it.
+
+The perturbation gradient is uniformly less than half the unperturbed
+annular gradient, so it introduces no transition critical point.
+Outside its support there is no perturbation. The core transverse
+critical equation forces `z=0`, leaving exactly
+`-s^3+u*s+v=0`. The consumed proof's root-containment argument places
+every real cubic root within `|s|<a*<r1/2`.
+
+There is also a uniform height barrier. For `rho>=r1/2` in the chart,
+monotonicity and the core formula give `c(r)>=r1^2/16`. Since
+`r1^2/16<1`,
+
+```
+b-F0=s^2*c(r)+z^2 >= (r1^2/16)*rho^2 >= delta.
+```
+
+Outside `D_R` the seed loss is at least `R^2>delta`.
+The height estimate in (G9) consequently gives
+`F_(u,v)<=b-3*delta/4` outside the core.
+At a new critical point, the same elementary estimate consumed in
+PROOF.md Section 2 gives `|P_(u,v)(s)|<delta/4`; hence every new
+critical height is above `b-delta/4`. The two bands are separated
+uniformly over the full rectangle.
+
+The old saddle heights are `2/9,-2/9`, and the old minimum is
+`-4/9`. They remain unchanged and lie below the isolated high-level
+merger. The consumed transverse-fiber contraction then establishes the
+actual component pairing for this explicit realization. This is an
+analytic transfer through proved uniform guards; it does not assert an
+executed independent whole-field selector.
+
+## 6. Scope of closure and the next falsification control
+
+This source discharges the stated cutoff smoothness, derivative,
+perturbation-norm and annular-gradient premises for (G1), supplying a
+concrete admissible realization of the fixed model. The original direct
+and CDF population calculations do not depend on which admissible
+cutoff realizes the isolated core; their original source identities and
+historical review statuses remain intact.
+
+Execution of a full periodic evaluator and an independent H0 selector
+remains `NOT_RUN`. An eventual selector must derive connectivity from
+certified axial level-set incidence and the proved global barrier,
+before comparing its lifetimes with the local gap formula. A finite
+regular grid does not resolve arbitrarily microscopic components.
+
+It must also distinguish actual topology from the probabilistic
+null-set convention. At `v=0,u>0`, the core maxima
+`s=+-sqrt(u)` both have height `b+u^2/4`, the middle saddle has
+height b, and the global barrier still isolates their merger.
+The ordinary H0 barcode therefore contains one finite bar of lifetime
+`u^2/4`, as well as the essential class. Tied component identity may
+require a label convention, but that finite lifetime is unambiguous.
+The frozen population law's convention excludes the null line and
+projects its count to zero. That law projection must be applied
+separately after recording the actual tied barcode.
+
+No Gaussian universality, independent organizational vote, new random
+benchmark, marked spatial limit, higher-homology or Lean acceptance is
+provided here. The full landmark goal remains active.