TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-source-reconstruction-20261024/ANALYTIC-DOMAIN.txt
Original file SHA-256: 6e25cece95f34c2711a142fa482aa901d6a894551bc4b63b0d71d99e3f065e11
Inclusive original lines: 1-125
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000001 | MATH.SEAM.ESERIES.SOURCE-RECONSTRUCTION.06 / analytic-domain audit and repair
000002 | 24 October 2026
000003 | 
000004 | Verdict
000005 | 
000006 | The universal infinitesimal-boundedness assertion in the original
000007 | thm:well-posed-primitive-dynamics is FALSE. A valid same-domain theorem
000008 | requires the actual relative bound to be smaller than1, or a verified
000009 | bounded zeroth-order perturbation. No conclusion is made here that the
000010 | intended special TFPT operator has no valid realization.
000011 | 
000012 | 1. The original assertion and the invalid step
000013 | 
000014 | Boundary-source section001, lines353-376, chooses a compact self-adjoint
000015 | Dirac anchor A with a self-adjoint elliptic modified APS boundary condition.
000016 | It asserts that every symmetric bounded-coefficient differential B of
000017 | order<=1 is infinitesimally A-bounded. The proof continues at section002,
000018 | lines1-30. Its valid estimates are
000019 | 
000020 |   ||psi||_H1 <= C(||A psi||+||psi||),
000021 |   ||B psi|| <= C1 ||psi||_H1+C0||psi||.
000022 | 
000023 | They imply only
000024 | 
000025 |   ||B psi|| <= C1*C ||A psi||+(C1*C+C0)||psi||.
000026 | 
000027 | The coefficient C1*C is fixed. It cannot be replaced by an arbitrary
000028 | epsilon>0. Same differential order gives graph boundedness, not relative
000029 | bound zero.
000030 | 
000031 | 2. Counterexample within the stated modified APS boundary class
000032 | 
000033 | On H=L2([0,2*pi]), let
000034 | 
000035 |   A=-i*d/dx,
000036 |   Dom(A)={f in H1([0,2*pi]): f(0)=f(2*pi)}.
000037 | 
000038 | Integration by parts shows that the adjoint has exactly the same periodic
000039 | trace condition, so A=A*. Its orthonormal eigenbasis is
000040 | 
000041 |   e_n=(2*pi)^(-1/2) exp(i*n*x),  A e_n=n e_n, n in Z.
000042 | 
000043 | The tangential boundary operator in this one-dimensional example is0 on
000044 | the two endpoints. Its kernel is the complete C2 boundary space. The
000045 | periodic trace line L={(z,z)} is Lagrangian for the Green boundary form,
000046 | so this is precisely a self-adjoint modified APS condition. It is also
000047 | the elliptic transmission condition obtained by cutting the circle.
000048 | 
000049 | Take B=-A on the same domain. It is symmetric, first order, and has
000050 | constant bounded coefficients. Its exact A-relative bound is1: any bound
000051 | with coefficient a gives |n|<=a|n|+b, so a>=1, while a=1,b=0 works.
000052 | In particular the estimate with epsilon=1/2 is impossible.
000053 | 
000054 | Moreover A+B is zero on the proper dense domain Dom(A), whereas its
000055 | adjoint is zero on all of H. It is not self-adjoint on the original
000056 | domain and is not closed there. Thus the theorem's general same-domain
000057 | conclusion also fails, not just the proof of relative bound zero.
000058 | 
000059 | This is a counterexample to the generic mathematical theorem. It is not
000060 | presented as a full P1/P2 physical model.
000061 | 
000062 | 3. Reflection oddness does not lower the differential order
000063 | 
000064 | The original split is D_rel=(D-U_tau D U_tau^-1)/2. Once a unitary bundle
000065 | lift of the geometric reflection is actually specified, its principal
000066 | symbol satisfies
000067 | 
000068 |   sigma1(D_rel)=1/2(sigma1(D)-sigma1(U_tau D U_tau^-1)).
000069 | 
000070 | Thus D_rel has order zero iff the two first-order symbols agree. Projecting
000071 | onto the reflection-odd part alone proves no such equality.
000072 | 
000073 | For example, on the doubled circle let D=-i*d/dx and U_tau f(x)=f(-x).
000074 | Then U_tau D U_tau^-1=-D, so D_ref=0 and D_rel=D is first order. The fixed
000075 | set is{0,pi} and the quotient is an interval. A different specified bundle
000076 | or Pin lift may change the symbol parity. The source does not specify
000077 | that lift or a principal-symbol equivariance identity; the actual intended
000078 | special case therefore needs this check.
000079 | 
000080 | Formal symmetry also does not fix the operator domain. If a reflection is
000081 | used to compare two closed realizations, it must preserve or explicitly
000082 | transport the APS boundary subspace, including the kernel Lagrangian.
000083 | 
000084 | 4. Correct conditional replacement
000085 | 
000086 | Let A be a self-adjoint compact elliptic anchor with the chosen self-adjoint
000087 | D-elliptic boundary condition. Suppose Dom(A) is contained in Dom(B), B is
000088 | symmetric on Dom(A), and
000089 | 
000090 |   ||B psi|| <= a ||A psi||+b||psi||,  a<1.
000091 | 
000092 | Then A+B is self-adjoint on Dom(A) by Kato-Rellich. The two graph norms are
000093 | equivalent, hence the compact inclusion of Dom(A) into H persists and the
000094 | sum has compact resolvent as well.
000095 | 
000096 | A reflection-specific sufficient condition is:
000097 | 
000098 |   (i) a specified unitary reflection lift preserves the chosen domain;
000099 |   (ii) sigma1(D)=sigma1(U_tau D U_tau^-1);
000100 |   (iii) the resulting order-zero D_rel has bounded self-adjoint coefficients.
000101 | 
000102 | Then B=D_rel is bounded symmetric on H, so a=0 and the same-domain result
000103 | follows. If the first-order symbols do not cancel, an actual a<1 estimate
000104 | must instead be derived. The graph estimate by itself is insufficient.
000105 | 
000106 | There is a second notation check before applying any replacement: the
000107 | carrier-source section006, lines271-273, later defines D_geo=D_ref+D_rel=D.
000108 | Thus D_geo+D_rel is not simply the reconstructed D. One must name whether
000109 | the anchor is D_ref, D_geo, or a separately declared operator, and apply
000110 | the estimate to that actual pair. Renaming the operator cannot remove the
000111 | domain and relative-bound requirement.
000112 | 
000113 | References checked at the originals
000114 | 
000115 | Gerald Teschl, Mathematical Methods in Quantum Mechanics, Theorem6.4:
000116 |   https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf
000117 | Bär and Ballmann, Boundary Value Problems for Elliptic Differential
000118 | Operators of First Order (elliptic/transmission conditions, graph norms):
000119 |   https://arxiv.org/abs/1101.1196
000120 | Bär and Ballmann, Guide to Boundary Value Problems for Dirac-Type
000121 | Operators, Theorems4.11-4.12 (self-adjoint domains and kernel normal form):
000122 |   https://arxiv.org/html/1307.3021
000123 | 
000124 | The infinite eigenvalue argument above is the proof. The finite
000125 | discriminant/current checks in checker.py do not test this analytic claim.