# From the shared clock to the signed mismatch

23 October 2026. This extends the existing seed construction without choosing
an empirical target. Time splitting and space expanding retain their distinct
physical meanings; the six state labels below are internal clock/closure labels.

## Start with the update already present

Use T(i,b)=(i+1 mod 3,b XOR 1), with the existing minimal three-step closure
and the original inversion on every tick. Consider scalar readouts f on its
six states. Every readout decomposes uniquely into a pole-even part and a
pole-odd contrast. Write these as

    f_even(i,b)=s_i,
    f_odd(i,b)=(-1)^b x_i.

This is the decomposition of a function under the actual binary exchange,
not a new edge sign chosen for the color target. For the contrast,

    f_odd(T(i,b))-f_odd(i,b) = -(-1)^b (x_(i+1)+x_i).

Thus the plus sign in the earlier mismatch follows from comparing opposite
poles at successive positions. It was previously introduced as a signed
amplitude representation. Here it is obtained directly for scalar contrast
readouts of the existing clock.

## A quadratic observable induced by that update

Let D=I-T on readouts, using the uniform counting inner product. Then

    L=D^T D=2I-T-T^T.

Restricted to the even and odd readouts, respectively, it gives

    L_even = [[2,-1,-1],[-1,2,-1],[-1,-1,2]],
    L_odd  = [[2, 1, 1],[ 1,2, 1],[ 1, 1,2]].

The half factor in the restriction E^T L E / 2 compensates for counting both
poles; the readout norm has the same factor. Along the explicit clock orbit
the values are x,-y,z,-x,y,-z. Their consecutive squared differences sum to

    2[(x+y)^2+(y+z)^2+(z+x)^2].

Each paired clock link separately restricts to the corresponding rank-one
edge matrix (e_i+e_j)(e_i+e_j)^T. Their sum is exactly the earlier color
mismatch operator. This connects the clock and edge constructions at the
operator level, including their normalization. It does not yet give an
experimentally identified quantum amplitude or make each link an independently
available physical control.

Exact checks cover all eight triangle flip assignments. Balanced transport
has one odd-sector zero mode; odd inversion has none. For the primitive
flip-every-tick case, the spectra are {0,3,3} and {1,1,4}. These are collective
readout modes, not spacetime dimension counts.

## What selects this quadratic observable?

The clock by itself does not single out every quadratic observable. Exact
rational linear constraints give the following real matrix-space dimensions:

| Conditions on a six-state operator H | Dimension |
|---|---:|
| Commutes with T | 6 |
| Also real symmetric | 4 |
| Also supported on the diagonal and neighboring clock links | 2 |
| Also annihilates the constant readout | 1 |

The last class is precisely H=aL. Analytically, real symmetry, clock stationarity
and neighbor support give H=uI+v(T+T^T); H1=0 requires u=-2v. Nonnegative cost
selects a>=0. The overall scale is not fixed by these structural constraints.
No empirical scale was selected or optimized. The conditions specify the
uniqueness class; deriving their physical role remains part of the program.

## Rotation carried by the contrast clock

On contrast coordinates the clock acts as S(x,y,z)=(-y,-z,-x). Set K=2S-I.
Exact multiplication gives K^2=-3I+4*ones. On x+y+z=0 this reduces to K^2=-3I,
so J=K/sqrt(3) obeys J^2=-I. The clock therefore supplies a complex structure
on this real two-dimensional contrast plane. It supplies one complex plane,
not yet three independently complex color amplitudes.

The symmetric mismatch L also discards the direction distinguishing opposite
clock frequencies. Consequently a scalar function of L alone cannot reproduce
the directed update T. An evolution law must retain the clock information
alongside a cost or binding observable rather than silently substituting one
for the other.

## Reproduction and status

`python clock_transport_bridge.py` checks the full matrices, restrictions,
three edge terms, all signings, rotation identity and constraint dimensions
using integer/Fraction arithmetic. It writes `clock-transport-results.json`.
It uses no third-party package, data fit, random search or rank tolerance.

`ClockTransportBridge.lean` contains four supporting identities. Its Mathlib
check returned 0 errors and 0 warnings, with the standard axiom base only.
This advances the clock-to-mismatch bridge. Quantum amplitudes, physical
gauge-covariant transport and the observer-readout law still need their
identification through seed evolution; none is classified as a fit here.

For methodological context, D'Ariano and Perinotti derive propagation from
explicit update principles rather than a generator census alone:
https://arxiv.org/abs/1306.1934. Their assumptions are not silently adopted here.