diff --git a/experiments/universality/two_parameter_cusp_population/ANALYTIC_GUARDS.md b/experiments/universality/two_parameter_cusp_population/ANALYTIC_GUARDS.md new file mode 100644 index 000000000..348a2dabc --- /dev/null +++ b/experiments/universality/two_parameter_cusp_population/ANALYTIC_GUARDS.md @@ -0,0 +1,287 @@ +# 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 `00`, + +``` +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|=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)|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.