TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DEFECT-TIME-PHASE-INDEPENDENT.txt
Original file SHA-256: 828d2d844d505b504da0466d9f6008c189df0d56a63be56e5ab9297e7c0b7d66
Inclusive original lines: 1-122
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 | DEFECT LINE: CIRCLE CHARACTER VERSUS PROJECTED PARALLEL TRANSPORT
000002 | Independent first-identification check, 2026-10-29.
000003 | No new source class, browser, Atlas mutation or old-source edits.
000004 | 
000005 | [OMITTED: non-scientific conversation or execution metadata]
000006 | [OMITTED: non-scientific conversation or execution metadata]
000007 | [OMITTED: non-scientific conversation or execution metadata]
000008 | 
000009 | VERDICT
000010 | The moving-defect covariance and quarter character of the GIVEN circle
000011 | action are correct. The displayed Gaussian orbit frame is not horizontal
000012 | for the natural orthogonal-projector connection. Its circle character
000013 | therefore does not by itself compute that connection's holonomy. A
000014 | separately prescribed flat connection can carry the quarter holonomy;
000015 | this review does not exclude such a connection or change the existing
000016 | physical time/state data.
000017 | 
000018 | 1. THE SAME CLOSED DEFECT FAMILY
000019 | 
000020 | On H=l2(Z), retain B e_n=(n+alpha)e_n, alpha=1/4 or3/4,
000021 | R=exp(-ell B), ell>0, and U e_n=e_(n+1). Put
000022 | 
000023 |  T_phi=R-exp(i phi)U, Dom(T_phi)=Dom(R),
000024 |  L_phi=ker T_phi*.
000025 | 
000026 | Its one-dimensional adjoint kernel is generated by
000027 | 
000028 |  y_n=exp(-ell[n(n-1)/2+alpha n]),
000029 |  y(phi)_n=exp(i phi n)y_n.
000030 | 
000031 | Indeed T_phi*=R-exp(-i phi)U*, and its recurrence is
000032 | y(phi)_(n+1)=exp(i phi)exp(-ell(n+alpha))y(phi)_n.
000033 | The Gaussian lies in Dom(R), and all its n moments are finite.
000034 | The norm is independent of phi. With e_phi=y(phi)/||y||, the frame
000035 | is smooth and periodic with e_(2pi)=e_0.
000036 | 
000037 | 2. THE ORTHOGONAL-PROJECTOR CONNECTION
000038 | 
000039 | The family lies in the specified trivial ambient Hilbert bundle. Its
000040 | natural induced unitary connection is nabla_proj=P_phi d, where
000041 | P_phi=|e_phi><e_phi|. With the Hermitian product linear in the second
000042 | argument, set
000043 | 
000044 |  mu=<n>=sum_n n |y_n|^2 / sum_n |y_n|^2.
000045 | 
000046 | Then <e_phi,partial_phi e_phi>=i mu. A section s=c(phi)e_phi is
000047 | projector-parallel precisely when c'+i mu c=0. Therefore
000048 | 
000049 |  s_parallel(phi)=exp(-i mu phi)e_phi,
000050 |  Hol_proj=exp(-2pi i mu).
000051 | 
000052 | This fixes the sign convention explicitly. Reversing the loop orientation
000053 | or defining holonomy by inverse transport conjugates the result; it does
000054 | not make the numerical value a quarter phase.
000055 | 
000056 | 3. THE EXISTING DIAGNOSTIC SCALE IS A DIRECT COUNTERCHECK
000057 | 
000058 | For alpha=1/4 and ell=4log2 the weights are exactly
000059 | 
000060 |  |y_n|^2=2^(-4n^2+2n).
000061 | 
000062 | A direct sum gives
000063 | 
000064 |  mu=0.1855335532291087...,
000065 |  -2pi mu=-1.1657416956379574...,
000066 |  Hol_proj=0.3940689714100713-0.9190808700934909 i.
000067 | 
000068 | These are not alpha=1/4 or the circle character +i. The inverse holonomy
000069 | is not a quarter phase either. The arithmetic is tightly controlled:
000070 | sum over -4<=n<=4 is a sum of rational powers of2; the first omitted
000071 | positive weight is 2^-90 and successive ratios are at most2^-42.
000072 | The first omitted negative weight is2^-110, with ratios at most2^-46.
000073 | The resulting absolute error in mu is below6e-27; the printed decimal
000074 | is only rounded floating-point output of that rational finite sum.
000075 | 
000076 | The old seam-full-collar-reconstruction-20261015/PROOF.txt:153 expressly
000077 | calls ell=4log2 a DIAGNOSTIC choice making transfers rational. It is not
000078 | being promoted here to a physically derived collar width. It is already
000079 | an allowed concrete case of the claimed formula and suffices to test the
000080 | purported general identification. For alpha=3/4 the weights reflect
000081 | n->-n, so mu and the projected phase angle change sign.
000082 | 
000083 | 4. WHAT THE CIRCLE ACTION REALLY COMPUTES
000084 | 
000085 | The exact given action obeys
000086 | 
000087 |  exp(i t B)y(phi)=exp(i alpha t)y(phi+t),
000088 |  exp(2pi i B)=exp(2pi i alpha)I.
000089 | 
000090 | It transports the defect family and has the declared quarter central
000091 | character. Relative to the projected parallel section, its phase is
000092 | exp(i(alpha+mu)t), rather than a statement of projector horizontality.
000093 | These are entirely compatible facts about the same original operators.
000094 | 
000095 | In the periodic e_phi frame three different valid unitary connections
000096 | illustrate precisely which extra identification would be needed:
000097 | 
000098 |  nabla_proj=d+i mu dphi has holonomy exp(-2pi i mu);
000099 |  nabla_frame=d declares e_phi parallel and has holonomy1;
000100 |  nabla_orbit=d-i alpha dphi declares exp(i alpha phi)e_phi parallel
000101 |              and has holonomy exp(2pi i alpha).
000102 | 
000103 | Every connection on a one-dimensional base has zero curvature. Calling a
000104 | connection flat thus does not choose its holonomy or identify it with
000105 | the orthogonal projection connection. The equation defining e_phi is a
000106 | particular frame normalization, not a derivation of nabla_orbit. Ordinary
000107 | periodic frame gauge changes leave a fixed connection's holonomy unchanged.
000108 | 
000109 | 5. SCOPE OF THE CORRECTION
000110 | 
000111 | [OMITTED: non-scientific conversation or execution metadata]
000112 | [OMITTED: non-scientific conversation or execution metadata]
000113 | [OMITTED: non-scientific conversation or execution metadata]
000114 | [OMITTED: non-scientific conversation or execution metadata]
000115 | [OMITTED: non-scientific conversation or execution metadata]
000116 | [OMITTED: non-scientific conversation or execution metadata]
000117 | [OMITTED: non-scientific conversation or execution metadata]
000118 | 
000119 | This is a calculation on the already supplied scalar defect line. It does
000120 | not assert an invariant defect in the entire coupled v252 operator, choose
000121 | a new time, or claim that the complete operational Seed lacks physical
000122 | time. It leaves all actual source, state and time data intact.