TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE39-COMMON-CLUTCH-CLASS-ADDENDUM.txt
Original file SHA-256: a4c940f1116bb8b3dcef63c3732719be8d44620a75e6fea5ed853738fb533f93
Inclusive original lines: 1-131
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 | ALGEBRAIC CLOSURE OF THE COMMON LEADING-DISTANCE CLUTCH CLASS
000002 | ==============================================================
000003 | 
000004 | [OMITTED: non-scientific conversation or execution metadata]
000005 | [OMITTED: non-scientific conversation or execution metadata]
000006 | 
000007 | VERDICT
000008 | -------
000009 | 
000010 | The original three diagonal length spectra exclude EVERY common three-channel
000011 | leading-distance clutch in each of the following two classes.
000012 | 
000013 | Single leg:
000014 | 
000015 |   L_f=a_f P_f x+b_f 1,      a_f>0.                       (1)
000016 | 
000017 | Equal-weight two legs with the same q/p distance list:
000018 | 
000019 |   L_f=a_f(P_f x+Q_f x)+b_f 1,   a_f>0.                  (2)
000020 | 
000021 | Here x is any common real three-component list and P_f,Q_f are arbitrary
000022 | permutations.  Sector-dependent positive scale and offset are allowed.  The
000023 | result is therefore stronger than the earlier test of
000024 | x={1,3,5}/12.
000025 | 
000026 | It is scoped exactly to a COMMON list x and equal left/right magnitude in
000027 | (2).  Different q and p distance spectra, unequal left/right scales, extra
000028 | C6 windings, non-leading mode sums or operator-history dependent readouts
000029 | leave this class and are not excluded.
000030 | 
000031 | 
000032 | 1. SINGLE-LEG CLASS
000033 | -------------------
000034 | 
000035 | For a sorted nondegenerate list x_0<x_1<x_2 define its gap quotient
000036 | 
000037 |   r(x)=(x_1-x_0)/(x_2-x_1).                              (3)
000038 | 
000039 | Permutation does not change the sorted spectrum.  Positive scaling and a
000040 | common offset do not change r.  Thus (1) forces the same r in every sector.
000041 | 
000042 | The original sorted rows and their quotients are
000043 | 
000044 |   L_u={0,3,7}: r_u=3/4,
000045 |   L_d={2,5,7}: r_d=3/2,
000046 |   L_e={3,5,8}: r_e=2/3.                                  (4)
000047 | 
000048 | They are not equal.  Therefore (1) is impossible for every common x.
000049 | 
000050 | 
000051 | 2. COMPLETE SIX-PERMUTATION TWO-LEG CLASSIFICATION
000052 | --------------------------------------------------
000053 | 
000054 | If x has repeated entries, every spectrum in (2) is degenerate for the only
000055 | relative permutations that could otherwise give three values, so it cannot
000056 | produce the three nondegenerate rows (4).  Assume x has three distinct
000057 | entries.  An affine normalization, absorbed into a_f and b_f, puts
000058 | 
000059 |   x=(0,1,t),   t>1.                                      (5)
000060 | 
000061 | Apply the inverse of P_f to all three output coordinates.  The spectrum then
000062 | depends only on the relative permutation
000063 | 
000064 |   sigma=P_f^(-1)Q_f in S3
000065 | 
000066 | through x+sigma x.  The six possibilities are exhaustive:
000067 | 
000068 |   sigma                 spectrum multiset      result
000069 |   ---------------------------------------------------------------
000070 |   identity              {0,2,2t}              r=1/(t-1)
000071 |   (012)                 {1,t,t+1}             r=t-1
000072 |   (021)                 {1,t,t+1}             r=t-1
000073 |   (01)                  {1,1,2t}              repeated
000074 |   (02)                  {2,t,t}               repeated
000075 |   (12)                  {0,t+1,t+1}           repeated.          (6)
000076 | 
000077 | At t=2 the middle transposition becomes fully degenerate; it never supplies
000078 | a third nondegenerate form.  Thus every nondegenerate spectrum in (2) has
000079 | gap quotient
000080 | 
000081 |   r in {t-1,1/(t-1)}.                                   (7)
000082 | 
000083 | All sectors must consequently have the same quotient or its reciprocal.
000084 | The original d and e rows satisfy this relation:
000085 | 
000086 |   r_d=3/2,   r_e=2/3.
000087 | 
000088 | But the u row has
000089 | 
000090 |   r_u=3/4,   1/r_u=4/3,
000091 | 
000092 | which is neither 3/2 nor 2/3.  No t>1 can therefore produce all three
000093 | original rows.  This proves the exclusion of (2) without a scan or a choice
000094 | of particular clutch phases.
000095 | 
000096 | 
000097 | 3. PHYSICAL AND SOURCE SCOPE
000098 | ----------------------------
000099 | 
000100 | For a reflected q/p collar whose two legs carry the same leading magnitude
000101 | dist(alpha+kappa_a,Z), equation (2) is precisely the most generous class
000102 | obtained by allowing arbitrary family relabelings, sector scales and sector
000103 | offsets while retaining equal left/right decay weights.  The obstruction
000104 | above is therefore a complete algebraic exclusion of that proposed common
000105 | leading-distance mechanism.
000106 | 
000107 | The original source is not limited to this class.  It has sector-dependent
000108 | residue packets and actual C6 winding histories:
000109 | 
000110 |   R_u={0,1,3}, R_d={1,2,5}, R_e={2,3,5},
000111 | 
000112 | followed by one length-six winding in each first-generation direction.
000113 | Those data can create the three different gap patterns in (4), but they are
000114 | not consequences of a common leading distance x.  Likewise a full Schwinger
000115 | sum over all modes, unequal two-leg weights, or a further source operator may
000116 | produce another class.  None is ruled out here, and none may be called a
000117 | derivation of the original lengths until its actual operator map reproduces
000118 | the rows (4).
000119 | 
000120 | The established cover pi:C->X_f, sign-line local system and whole B_mu
000121 | endpoint response remain unchanged.
000122 | 
000123 | 
000124 | REUSED SOURCES
000125 | --------------
000126 | 
000127 | [OMITTED: non-scientific conversation or execution metadata]
000128 | [OMITTED: non-scientific conversation or execution metadata]
000129 | [OMITTED: non-scientific conversation or execution metadata]
000130 | [OMITTED: non-scientific conversation or execution metadata]
000131 |