TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-joint-source-origin-audit-20261016/PROCESS_RECONSTRUCTION.txt
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000122 | 4. Route II: an exact spontaneous-splitting diagnostic
000123 | 
000124 | Question: can an invariant quartic select multiplicities without inserting
000125 | a preferred projector? This diagnostic is conditional, not a new TFPT model.
000126 | Assume a nonzero Hermitian endomorphism X of an unmarked complex five-space,
000127 | tr X=0, tr X^2=1. Write its real eigenvalues x_i. Define q=tr X^4.
000128 | We derive the GLOBAL bounds and all equality multiplicities:
000129 | 
000130 |              7/30 <= q <= 13/20.
000131 | 
000132 | Proof. The constraint manifold sum x_i=0, sum x_i^2=1 is compact and smooth;
000133 | its two constraint gradients are independent. Lagrange multipliers give
000134 | 4 x_i^3-2 lambda x_i-mu=0, so there are at most three distinct eigenvalues.
000135 | One value is impossible. For two values of multiplicities k and 5-k,
000136 | a^2=(5-k)/(5k), b=-k a/(5-k), giving
000137 | q=((5-k)^3+k^3)/(25 k(5-k)).
000138 | This equals 13/20 for k=1,4 and 7/30 for k=2,3.
000139 | 
000140 | For three distinct roots a,b,c, the stationary cubic has no squared term:
000141 | a+b+c=0. With positive multiplicities m,n,p summing to five, also
000142 | m a+n b+p c=0. Their common one-dimensional solution is proportional to
000143 | (n-p,p-m,m-n). Enumerating the positive triples (up to permutation:3,1,1
000144 | and2,2,1), excluding coincident roots, gives q=1/2 or1/4.
000145 | Neither reaches the two-value extremes. This exhausts ALL stationary points,
000146 | including spectra with zeros; no block ansatz was imposed. The exact checker
000147 | lists every ordered multiplicity case. Equality at the minimum has spectrum
000148 | (a,a,b,b,b) up to permutations/sign, b=-2a/3. At the maximum: (a,b,b,b,b),
000149 | b=-a/4. Only multiplicities, not a canonical projector, are selected.
000150 | The minimum orbit under U(5) has stabilizer U(2)xU(3). These are internal
000151 | covariance groups; no physical gauge identification is made.
000152 | 
000153 | Why boundedness/symmetry does not choose the answer:
000154 | let t=tr X^2 for an arbitrary traceless Hermitian X and use the diagnostic
000155 | polynomials V_sigma=-t+t^2+sigma tr X^4, sigma=+1 or -1.
000156 | For X=r Y with tr Y^2=1:
000157 | V_sigma=-r^2+r^4(1+sigma q(Y)).
000158 | Both quartics are positive at infinity, since 1-13/20=7/20>0.
000159 | Minimizing r gives r^2=1/[2(1+sigma q)] and energy
000160 | -1/[4(1+sigma q)]. Thus sigma=+1 selects the 2+3 orbit and sigma=-1 the
000161 | 1+4 orbit. Sigma=0 has no shape selection. The same unmarked carrier,
000162 | conjugation symmetry, X -> -X symmetry, polynomial degree and boundedness
000163 | admit different preferred splits. A gradient flow with a supplied positive
000164 | metric would make these minima stable modulo symmetry; that metric/flow,
000165 | initial law, fluctuations and thermodynamic limit are NOT derived here.
000166 | No claim that all other stationary solutions are unstable is needed or made.
000167 | 
000168 | Assumptions to derive before using this as TFPT: the actual source-generated
000169 | endomorphism X (not the SM Higgs doublet by relabelling), its Hermitian reality
000170 | and tracelessness, invariant contraction, a nonzero broken phase, and the
000171 | effective coefficient of tr X^4 relative to other allowed terms. Without
000172 | X -> -X symmetry, tr X^3 is another independent allowed term. Higher terms
000173 | and constraints could change the result. Positive quartic shape is a precise
000174 | conditional selector; its sign is not a consequence of general stability.
000175 | 
000176 | 5. Follow the best-supported route back to the original functional
000177 | 
000178 | Route I is first: it identifies the exact missing datum without adding an
000179 | unmotivated field. Derive an actual noncommuting local intervention/product
000180 | from the primitive source, its coherent correlations and its opposite action;
000181 | then apply the ALREADY available quotient/GNS and finite closure theorem.
000182 | Do not repeat GNS as if it selected the alphabet. The new cyclic test shows
000183 | why merely adding 'no hidden degrees of freedom' would not finish this.
000184 | 
000185 | The quartic theorem gives a downstream decision criterion, NOT a license to
000186 | choose a potential. For the source's declared profile and an ACTUALLY DERIVED
000187 | linear deformation D(t)=D0+t V, finite-dimensional commuting case:
000188 | [t^4] tr f(D0+t V) = tr(f''''(D0) V^4)/24.
000189 | In a finite bounded holomorphic functional-calculus setting, without
000190 | commutation, it is (2 pi i)^-1 integral f(z) tr[R(z)(V R(z))^4] dz.
000191 | Subtract the corresponding reference coefficient. A nonlinear D(t) adds
000192 | terms involving its higher derivatives. Infinite-dimensional traces,
000193 | domains and regularization require their own proof; no unjustified integral
000194 | formula for the unbounded physical source is asserted.
000195 | 
000196 | Even a positive profile does not force a positive fourth derivative:
000197 | for f(x)=exp(-x^2), f''''(0)=12 and f''''(1)=-20/e. The eta piece is not a
000198 | positive real quartic potential. Reflection positivity is a condition on
000199 | correlation kernels, not a rule assigning +tr X^4 to an unknown field.
000200 | Therefore the checked original seed at302-343 neither supplies V nor
000201 | identifies this coefficient with a positive invariant on an unmarked C5.
000202 | Known E8 quartic root invariants act on other already specified spaces;
000203 | without an equivariant source map they cannot substitute for this datum.
000204 | 
000205 | Success criterion for the next source derivation:
000206 | one common primitive operation/product law must provide (i) the allowed
000207 | coherent local operations and their correlations, and (ii) its actual field
000208 | deformation/effective invariant. Then the current theorems test algebra
000209 | uniqueness and 2+3 stability without inserting either answer. New independent
000210 | axioms must be labelled as such. At present no previously assumed physical
000211 | input has been discharged; what is eliminated is the need for a prescribed
000212 | 2+3 projector WITHIN the positive-quartic conditional diagnostic.
000213 |