TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/collar-current-source-bridge-20261021/PROOF.md
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000001 | # Two-face collar, actual face currents, and relative endpoint transport
000002 | 
000003 | Research ID: `UR.COLLAR.FACE_CURRENT.01`. Date: 2026-10-21.
000004 | Verdict: **conditional exact source bridge; physical P1 intervention not selected**.
000005 | This combines existing TFPT data, not new general current-algebra mathematics.
000006 | 
000007 | ## 1. Original data and explicit premises
000008 | 
000009 | Use the original selfadjoint circle operator `B_alpha=-i d_theta+alpha`,
000010 | `alpha=1/4` or `3/4`, its actual periodic/twisted domain, and the original
000011 | positive normal length `ell`. The original product equation gives the closed
000012 | transfer `T=exp(-ell B)` and its orthogonal two-face graph projection
000013 | 
000014 | `G=[[A,C],[C,1-A]]`, `A=(1+exp(-2ell B))^-1`, `C=1/(2 cosh(ell B))`.
000015 | 
000016 | For two-sided unbounded B, T is not declared bounded. G is bounded and is
000017 | the projection onto its closed graph. The circle Fourier blocks `G_r` are
000018 | rank-one 2x2 projections with `tr_faces(G_r)=1`.
000019 | 
000020 | As in the original collar proof, ADD the charged CAR algebra on both faces
000021 | and the pure gauge-invariant quasifree state with covariance G. This field
000022 | and statistical premise is NOT derived from P1. Its second-face occupation
000023 | is `n_r=(1+exp(beta(r+alpha)))^-1`, `beta=2ell`.
000024 | The original one-face negative sea is `P_-=1_(B<0)`.
000025 | Do not identify a geometric graph projection, a mixed occupation, a finite
000026 | native code projector or the physical admissibility projector with each other.
000027 | 
000028 | Original sources reused:
000029 | * `seam-full-collar-reconstruction-20261015/PROOF.txt:63-143`, especially its
000030 |   explicit CAR premise and selected labeled second face at lines96-116.
000031 | * `joint-native-algebra-selection-20261016/ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt:33-161`.
000032 | * `joint-native-algebra-selection-20261016/ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt:36-99`.
000033 | * `joint-native-algebra-selection-20261016/ORIGINAL-P1-VARIATION-GAP-AND-EVENT-SELECTION.txt:37-134`.
000034 | All these paths are relative to `experiments/theory-contracts/`.
000035 | 
000036 | ## 2. The same graph fixes the one-face current cocycle
000037 | 
000038 | Let `P0,P1` be the two labeled FACE projections, not native 3+2 marks.
000039 | For a smooth real circle function f, let `A_j(f)=Pj M_f` on the doubled
000040 | one-particle space. In the G-Fock representation, normal ordered quadratic
000041 | generators exist: `[G,A_j(f)]` is Hilbert-Schmidt. For fixed Fourier shifts
000042 | this follows because `G_r-G_(r+k)` decays at either tail and `C_r` decays;
000043 | smooth Fourier coefficients allow the convergent sum. Constant face terms
000044 | are also allowed because C is Hilbert-Schmidt on this gapped circle.
000045 | 
000046 | Their central commutator is obtained by the Wick contraction
000047 | 
000048 | `c_G(A,B)=Tr[G A(1-G)B G - G B(1-G)A G]`.
000049 | 
000050 | For second-face multiplication, the summand is
000051 | `n_r(1-n_s)(f_(r-s) g_(s-r)-g_(r-s) f_(s-r))`.
000052 | Combining the k and -k terms and using the absolutely convergent telescoping
000053 | sum `sum_r(n_r-n_(r+k))=k`, k>0, gives EXACTLY
000054 | 
000055 | `c_1(f,g)=sum_(k>0) k (conj(f_k)g_k-f_k conj(g_k))`.
000056 | 
000057 | It is independent of ell and alpha. The opposite face has `c_0=-c_1`;
000058 | mixed-face central commutators vanish. The latter follows from the symmetric
000059 | coefficient `C_r C_s`, whose antisymmetric Fourier sum cancels.
000060 | Consequently common multiplication on BOTH faces has `c_common=0`.
000061 | This also follows directly from `tr_faces G_r=1` in the convergent trace.
000062 | For nonzero Fourier modes G-normal-ordering and the original sea normal
000063 | ordering agree. The zero mode in section3 uses the ORIGINAL relative Q;
000064 | its finite offset from G-normal-ordering is `Tr(n_ell-P_-)` and is retained.
000065 | 
000066 | Thus the original second face has precisely the original current relation
000067 | `[J_m,J_-k]=m delta_(mk) I`, not a newly fitted current level. Finite windows
000068 | in the checker are controls of this proof, not finite CCR representations.
000069 | Their boundary commutators are retained; no infinite identity is inferred
000070 | from a finite trace of a commutator.
000071 | 
000072 | For the source profiles `f=cos(theta)`, `g=sin(theta)`, `c_1=-i/2`, while
000073 | the common two-face cocycle is zero. More stringently, choose
000074 | `f=sin(4theta)`, `g=1-cos(4theta)`. BOTH vanish at all four original marks
000075 | `theta=j pi/2` and are quarter-periodic. Their cocycle is `c_1=-2i`.
000076 | With `W(f)=exp(iJ(f))`, the current rectangle is
000077 | `W(sf)W(tg)W(-sf)W(-tg)=exp(2ist)I` on the second face,
000078 | whereas its common two-face counterpart is I. This is a conditional
000079 | coherent-history diagnostic, not an executed detector or probability law.
000080 | 
000081 | ## 3. Complete current state and time, with all charge carry retained
000082 | 
000083 | Wick contraction in the SAME G-state gives, for m>0,
000084 | 
000085 | `<J_m J_-m>=sum_r n_r(1-n_(r+m))=m/(1-exp(-beta m))`,
000086 | `<J_-m J_m>=m/(exp(beta m)-1)`.
000087 | 
000088 | Each term of the second sum is exp(-beta m) times its counterpart in the
000089 | first; their difference telescopes to m. No spectral cutoff is used.
000090 | In this circle source the particle/hole product is convergent, so the
000091 | restricted thermal state is normal in the original negative-sea Fock
000092 | representation. This extra circle-specific fact is not asserted for an
000093 | arbitrary collar or arbitrary Fock representation.
000094 | 
000095 | Reuse the original FULL charge-sector/Hall decomposition. The unscaled
000096 | relative excitation energy is `H0=8K_alpha` with
000097 | `E_alpha(q)=q(q-1)/2+alpha q` and oscillator energies m. Thus
000098 | 
000099 | `Z_rel(beta)=sum_(q in Z) exp(-beta E_alpha(q)) * prod_(m>=1)(1-exp(-beta m))^-1`.
000100 | 
000101 | This equals the original full particle/hole product. No charge is replaced
000102 | by q mod4; all winding carry remains. The formula is relatively normal
000103 | ordered: a given absolute sea/eta contribution is retained separately,
000104 | not reset or declared zero.
000105 | For `f_0=average(f)`, the whole current characteristic function is
000106 | 
000107 | `omega_ell(W(f))=chi_ell(f_0) exp[-1/2 sum_(m>0) m coth(ell m)|f_m|^2]`,
000108 | `chi_ell(t)=sum_q exp(-2ell E_alpha(q)+itq)/sum_q exp(-2ell E_alpha(q))`.
000109 | 
000110 | The nonzero-mode expression follows from the thermal oscillator trace in
000111 | that original, complete boson-fermion/Hall decomposition. Smooth profiles
000112 | retain EVERY Fourier mode. Existing source time acts as
000113 | `U(t)J_-m U(t)^*=exp(-itm/8)J_-m`; geometric ell is not physical time.
000114 | After expressing Heisenberg profiles in this convention, every finite
000115 | ordered Weyl correlation follows by the original BCH cocycle and this
000116 | characteristic function. If a Schrodinger history retains a final U(T),
000117 | it must also be retained: in a thermal trace its relative factor is obtained
000118 | by `beta -> beta+iT/8`, including the ratio `Z_rel(beta+iT/8)/Z_rel(beta)`.
000119 | The prescribed absolute time phase remains separate. Removing it without
000120 | normalization would lose information.
000121 | 
000122 | Moreover `Tr|n_ell-P_-| <= [exp(-2ell alpha)+exp(-2ell(1-alpha))]/[1-exp(-2ell)]`.
000123 | Hence the long-collar limit gives the original negative sea in trace norm,
000124 | `chi_ell ->1` and `coth(ell m)->1`, recovering the existing vacuum Weyl kernel.
000125 | This selects the vacuum only within this GIVEN operator/CAR/long-collar
000126 | class. It neither selects the original source nor makes sign-only data
000127 | sufficient to reconstruct time magnitudes.
000128 | 
000129 | ## 4. Two different full lifts with the same face marginal
000130 | 
000131 | For `g_phi=M_exp(i phi)`, common source conjugation gives
000132 | `G_common=(g_phi direct_sum g_phi) G (g_phi^* direct_sum g_phi^*)`.
000133 | A second-face action gives instead
000134 | `G_face=(I direct_sum g_phi) G (I direct_sum g_phi^*)`.
000135 | Their second-face occupations both equal `g_phi n g_phi^*`, but their
000136 | off-diagonal blocks are respectively `g_phi C g_phi^*` and `C g_phi^*`.
000137 | They therefore differ on the full source. Section2 also distinguishes their
000138 | coherent phase laws. Equality of a reduced covariance cannot identify them.
000139 | This does not say that the two physical contexts must both be available.
000140 | The distinction needs access to the full source or coherent control of the
000141 | lifts. If only one-face conjugation channels are available and the other face
000142 | never recouples, those channels agree; a central phase alone is not an
000143 | ordinary unmonitored-channel discriminator.
000144 | 
000145 | ## 5. A concrete geometric relative endpoint family