TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/old_source_embedding/THERMAL-RIGIDITY.txt
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000001 | EXACT RIGIDITY OF THE OLD QUARTER SOURCE INSIDE THE GIVEN COUPLED SOURCE
000002 | 29 October 2026. New continuation; no pinned source modified.
000003 | 
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000009 | 
000010 | 1. Question and exact data
000011 | 
000012 | Reuse FINITE-COUPLED.txt and GRADED-HS-REFLECTION-AND-TIME.txt in the parent
000013 | directory. Let g=g*=g^-1 on the actual finite complex internal space, let
000014 | Phi=Phi* be CONSTANT with {g,Phi}=0, and set A_alpha=-i partial_theta+alpha.
000015 | For the actual Fourier momentum p=k+alpha, put
000016 |   h(p)=p g-i g Phi,  C_beta(p)=(1+exp(beta h(p)))^-1, beta>0.
000017 | The original quarter has alpha=1/4 or3/4, hence no p=0. The original elliptic
000018 | normalization is beta=4pi after its actual length/coordinate identification.
000019 | Let e be a FIXED unit vector with g e=e. The local scalar CAR map is
000020 |   V_e f=f tensor e.
000021 | The question is whether this SAME field, time and state reproduce the old
000022 | free occupation n_beta(p)=(1+exp(beta p))^-1, without fitting any temperature,
000023 | momentum, metric, charge or scalar source.
000024 | 
000025 | 2. The whole coupled covariance, with arbitrary finite degeneracy
000026 | 
000027 | Oddness gives h(p)^2=p^2+Phi^2, and [g,Phi^2]=0. Functional calculus yields
000028 |   C_beta(p)=1/2[ I-h(p) f_beta(sqrt(p^2+Phi^2)) ],
000029 |   f_beta(x)=tanh(beta x/2)/x.
000030 | The term -i g Phi f_beta(sqrt(p^2+Phi^2)) maps the positive g space to the
000031 | negative one, so its expectation on e vanishes. Consequently
000032 |   n_e(p):=<e,C_beta(p)e>
000033 |     =1/2[1-p <e,f_beta(sqrt(p^2+Phi^2))e>].                 (1)
000034 | All masses and signed/degenerate finite blocks are retained in (1).
000035 | 
000036 | 3. Strict monotonicity gives a necessary AND sufficient condition
000037 | 
000038 | For c=beta/2>0 and x>0,
000039 |   -x^2 f_beta'(x)=tanh(cx)-cx sech^2(cx)>0.
000040 | Indeed q(y)=tanh(y)-y sech^2(y) has q(0)=0 and
000041 |   q'(y)=2y sech^2(y)tanh(y)>0 for y>0.
000042 | Thus f_beta is strictly decreasing. The spectral measure of Phi^2 in e is
000043 | a probability measure on [0,infinity), and Phi^2>=0. For p>0 this gives
000044 |   n_e(p)>=n_beta(p),
000045 | with equality if and only if that measure is supported at0, equivalently
000046 |   Phi e=0.                                                   (2)
000047 | For p<0 the inequality reverses, with the same equality condition. In
000048 | particular equality at ONE original nonzero momentum already forces (2).
000049 | This is not a fitted finite window or an inference from two matching traces.
000050 | 
000051 | Conversely (2) implies h(p)e=p e for EVERY momentum. Therefore V_e intertwines
000052 | the full H1-domain operators, all bounded Borel functions, covariance, and
000053 | the full circle group with the SAME parameter. Its selfdual conjugate slots
000054 | and the two oriented boundary faces are transported by the existing chart.
000055 | The whole restricted CAR state, field products, product reflection and time
000056 | are then the old quarter graph source, by the already proved diagram.
000057 | The scalar channel need not be the whole internal source.
000058 | 
000059 | Hence in this class the following are equivalent:
000060 |   (a) one nonzero-p occupation equals the old free value;
000061 |   (b) Phi e=0;
000062 |   (c) the fixed local V_e carries the whole original free scalar source and
000063 |       its time (with the same already fixed beta and boundary orientation).
000064 | 
000065 | 4. Positive classical averaging cannot hide the coupling in this test
000066 | 
000067 | Suppose a positive probability measure is ALREADY GIVEN on such constant
000068 | odd selfadjoint Phi, with g,e,p,beta held fixed. Conditional CAR sources can
000069 | be integrated; their averaged state need not be quasifree. Because each
000070 | occupation difference in section3 has the same sign and is bounded,
000071 |   integral n_e^Phi(p) dmu(Phi)=n_beta(p)
000072 | holds if and only if Phi e=0 for mu-almost every Phi. No Gaussian or Wick
000073 | assumption is made on that averaged state, and no moment approximation is
000074 | needed. If the almost-sure kernel condition holds, ALL conditional restricted
000075 | scalar word functionals coincide, so their positive average coincides too.
000076 | 
000077 | This is NOT a statement about complex determinant/Pfaffian/eta weights,
000078 | time- or theta-dependent fields, changing connections, moving e, a new
000079 | renormalized field map, general interacting quantum bosons, or all P1 worlds.
000080 | Nor is preservation of the old free source imposed as a requirement on a
000081 | full interacting TFPT source. A valid interacting successor can change it.
000082 | It rules out only the claimed exact OLD free embedding through this route.
000083 | 
000084 | 5. Consequence for the actual finite constructor
000085 | 
000086 | v252 has supplied Yukawa and Majorana inputs. Its full finite matrix need
000087 | not have any kernel, and is invertible whenever all four Dirac Yukawa
000088 | matrices are invertible (the left/right block equations prove this even
000089 | with Majorana input). Therefore an exact fixed local scalar old-source
000090 | embedding cannot be obtained from a generic such coefficient merely by
000091 | choosing a different positive-grade vector e or by positive averaging.
000092 | The original source could instead have a genuinely dynamical or geometric
000093 | image; this result does not exclude that possibility.
000094 | 
000095 | The separate actual P2 representation dictionary and old-source review
000096 | identify which additional symmetry assumptions select candidate lines.
000097 | No symmetry, massless neutrino or family invariant line is assumed here.
000098 | The existing classification of all constant reducing projectors remains
000099 | valid. This new scalar occupation test is a sharper direct source check
000100 | and reuses that classification rather than launching another random search.