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v957_ccc_crossover_kinematics.py
verification/v957_ccc_crossover_kinematics.py
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"""v957 -- CCC.SEAM.KINEMATICS.01 (rounds DL-DLV, 2026-08-24): THE CCC
CROSSOVER KINEMATICS DICTIONARY -- the complete kinematic 2D -> 3D -> 4D
uplift of the seam and its thermal/holonomy dictionary, machine-checked
from corpus objects only. ONE module from six probes (28/28 + 29/29 +
18/18 + 21/21 + 14/14 + 7/7 first-pass gates, zero fails; discovery
probes ccc_crossover_gates_probe.py, ccc_uplift_4d_probe.py,
ccc_fork_kms_holonomy_probe.py, ccc_kernel_gamma_kms_probe.py,
ccc_transfer_selector_probe.py, ccc_kappa_freeze_probe.py, notes DL-DLV,
2026-08-24, re-run identically at promotion, embedded BYTE-EXACT and
executed verbatim).
WHAT IS CERTIFIED [E] (exact identities on corpus objects; the CCC
reading itself stays [C], the cyclic interpretation of origin_theory is
NOT upgraded by this module):
(1) RECIPROCAL STRUCTURE: the proven mu_4 Moebius stabiliser
D_4 = <rho: z->iz, sigma: z->1/z> contains tau = rho^2 sigma:
z -> -1/z with Omega * tau(Omega) = -1 identically (Penrose's
reciprocal gauge as a stabiliser element), tau rho tau^{-1} =
rho^{-1} (clock reversal = the Z_2 sheet parity), tau swaps the
clock poles {0, inf} and fixes the marks +-i; the crossover locus
|Omega| = |tau(Omega)| is the unit circle carrying all four marks.
(2) LOCAL-SCALE RIGIDITY: on the 2D seam the clock alone leaves
exactly ONE constant-curvature conformal modulus (the dilation
family, Liouville-exact) and the reciprocal Z_2 kills it (unique
zero at lambda = 1); Lie-algebra form: so(3,1) clock-commutant =
2 with 1 boost, {clock, tau}-commutant = 0.
(3) STINESPRING UNITARITY: the v221 transport (spectrum {1, (2/3)^6,
(1/3)^6}) is CPTP, NOT unitary; its 9 Kraus operators give an
exact isometry V^dag V = 1_3; ||T^n - P_star||_2 = (2/3)^{6n}
exactly; iterated with fresh environments the global overlap is
exactly invariant while the reduced trace distance decays at the
seam rate and the environment retains the full distinguishability
-- "entropy reset" = reduced channel, no information destroyed.
(4) THE KINEMATIC UPLIFT: stereographic(Hopf(v)) = z1/z2 exactly (the
seam sphere IS the Hopf base of S^3); the v492 deck diag(i, i^-1)
acts freely (X = S^3/Z_4 = the lens boundary of the corpus A_3
ALE XY = Z^4 is a smooth closed 3-manifold); clock diag(i, 1) and
the reciprocal lift J = [[0,-1],[1,0]] normalise the deck and
descend (J to tau, inverting the clock); the four marks lift to
four Hopf fibre circles paired by the deck into the two Z_2 sheet
orbits, Gauss linking number 1; conformal rigidity: the
so(4,1)-commutant of the deck is 4-dimensional su(2)+u(1), ALL
Killing, boost content 0 (infinitesimal Yamabe uniqueness with NO
extra condition; control S^3 keeps 10 with 4 boosts); the full
structure {deck, clock, J} leaves EXACTLY the Hopf fibre rotation;
global uniqueness on the Obata solution manifold (boost family
realised explicitly; deck-invariance defect bounded below over a
200-axis scan, zero only at beta = 0); the 4D bridge: the
inversion x -> x/|x|^2 is a conformal involution of the cone
C^2/Z_4 (pullback |x|^{-4} flat, symbolic), fixes the unit lens
pointwise, swaps the aeon ends, commutes with the deck; cone =
cylinder e^{2s}(ds^2 + g), Omega_- Omega_+ = 1; Lorentzian
shadow: (1/cos^2 T)(-dT^2 + g_{S^3}) is Einstein R_ab = 3 g_ab
(symbolic Ricci), crossover T = pi/2 a spacelike round S^3,
Omega_hat Omega_check = -1 with Omega_check = -cos T vanishing to
first order.
(5) KMS/MODULAR CLOSURE (discrete -> dynamics at the free level): the
transport obeys detailed balance exactly; H_mod = -ln T exists
with spectrum {0, 6 ln(3/2), 6 ln 3} exactly (the seam gap IS a
modular energy); T = e^{-H_mod} (one clock tick = one unit of
KMS time at beta = 1); discrete KMS/OS reflection symmetry and OS
Gram positivity hold; the v526 normalisation beta_angle = 2 pi =
1/(4 c3) is replicated exactly and equals the Gibbons-Hawking
normalisation on a crossover fibre of radius 1/H.
(6) HOLONOMY DICTIONARY: pi_1(X) = Z_4 with the v492 flat glue
character chi(D) = i; the two clock-pole circles descend to the
lens core geodesics with holonomy i (full mu_4); every seam
string carries EXACTLY the 2-torsion chi(D^2) = -1, and D^2 = -1
is trivial on the base while flipping the spinor coordinates --
the v522 statement "gaugeable datum = the 2-torsion {1, (-1)^F};
full wrap = fermion parity" in geometric form (typed [C]
structural correspondence); mark strings link the pole strings
once.
(7) SELECTOR NECESSITY (the gamma-facing theorem): for the v512 mark
family M(delta) = {+-1, +-e^{i delta}} the admissible reflection
axes form the pair-exchanging V_4 for every delta != pi/2 and
close to D_4 EXACTLY at delta = pi/2 -- the order-4 clock
factorises into admissible reflections (rho = tau o s_mid) iff
delta = pi/2; at operator level on the N = 16 Majorana seam ring
(fermionic reordering signs kept, index-level, no Fock space) the
free parent is Theta_a-real on ALL 16 axes (v521 side-blindness
replicated) while the interacting mark family is Theta-real on
exactly {k != 4: 2, k = 4: 4} axes and the clock-axis cuts are
Theta-real ONLY at k = 4; asymmetric couplings break reality.
Theta-reality being NECESSARY for reflection positivity, the
UNIQUENESS half of the v534 selector (delta = pi/2, symmetric
couplings) is DERIVED; the survival half (PD at positive
coupling) and the sign bit stay v534's nonperturbative toy facts,
cited -- reality is sign-blind (checked).
(8) THE FROZEN GATE-6 KERNEL (feeds the preregistered search
experiments/ccc-crossover-disc/, NO data touched): exactly two
decay modes with rates ln(729/64) and ln 729 (ratio ln3/ln(3/2) =
2.709511), frozen defect-class overlap table (+-1/sqrt2, +1/sqrt6)
and (0, -2/sqrt6); disc radius theta_max = eta_rec/(eta_0 -
eta_rec) ~= 1.16 deg (typed externals); kappa derived from the
v526 KMS normalisation up to a declared convention band -- u_rec
<= 8.3e-5 ticks across the WHOLE band, kernel contrast <= 3.3e-4:
the template degenerates robustly to the sharp causal top-hat
disc; freeze hashes v1/v2/v3 pinned and recomputed in-run.
HONEST FENCES. The CCC/cyclic reading stays [C]; WOIT.OS.TWISTOR.01
stays [O] with all kill tests live (this module is kinematics + free
level + necessity direction ONLY -- no interacting A_hol, no OS field
reconstruction, no Einstein dynamics derived); the deck fork
(spacetime lens vs internal bundle) is decided only CONDITIONALLY on
v522's wrap = (-1)^F surviving the interacting level; the cosmology
fork (Starobinsky vs crossover epoch) is priced decidable (r in
[0.0033, 0.0048] vs future sigma(r) ~ 0.001) but NOT decided; the
preregistered disc search is a search target of the [C] reading -- its
null would NOT damage the compiler core. External cosmography numbers
are typed external throughout.
"""
import base64
import hashlib
import io
import contextlib
from tfpt_constants import check, summary, reset
PROBES = {
'ccc_crossover_gates_probe.py': (
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),
'ccc_transfer_selector_probe.py': (
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}
def _run_probe(fn):
"""decode byte-exact, verify hash, execute verbatim, return
(rc, n_pass, n_total)."""
npass_expect, sha_expect, *chunks = PROBES[fn]
raw = base64.b64decode("".join(chunks))
sha = hashlib.sha256(raw).hexdigest()
if sha != sha_expect:
return (99, 0, 0)
g = {"__name__": f"v957_embed_{fn[:-3]}"}
buf = io.StringIO()
with contextlib.redirect_stdout(buf):
exec(compile(raw, fn, "exec"), g)
rc = g["main"]()
passes = g.get("PASS", [])
return (rc, sum(bool(x) for x in passes), len(passes))
def run():
reset()
import numpy as np
import sympy as sp
print("v957 -- CCC crossover kinematics dictionary (six probes"
" embedded byte-exact + in-run recomputation)")
# ------------------------------------------------------------------
# S0/S1: the six probes, byte-exact and green as typed
# ------------------------------------------------------------------
for fn in PROBES:
rc, np_, nt = _run_probe(fn)
exp = PROBES[fn][0]
check(f"probe {fn}: hash pinned, re-run green as typed "
f"({np_}/{nt}, expected {exp}/{exp}, rc={rc})",
int(rc == 0 and np_ == nt == exp), 1, exact=True)
# ------------------------------------------------------------------
# in-run recomputation (independent of the embedded sources)
# ------------------------------------------------------------------
RNG = np.random.default_rng(20260824)
# (1) reciprocal structure
zs = RNG.standard_normal(32) + 1j * RNG.standard_normal(32)
check("tau(z) = -1/z satisfies Omega*tau(Omega) = -1 identically",
int(all(np.isclose(z * (-1.0 / z), -1.0) for z in zs)), 1,
exact=True)
# (3) Stinespring: transport, isometry, exact contraction
LAM2, LAM3 = (2.0 / 3.0) ** 6, (1.0 / 3.0) ** 6
u2 = np.array([1.0, -1.0, 0.0]) / np.sqrt(2)
u3 = np.array([1.0, 1.0, -2.0]) / np.sqrt(6)
J = np.full((3, 3), 1.0 / 3.0)
T = J + LAM2 * np.outer(u2, u2) + LAM3 * np.outer(u3, u3)
kraus = [np.sqrt(T[i, j]) * np.outer(np.eye(3)[i], np.eye(3)[j])
for i in range(3) for j in range(3)]
V = np.zeros((27, 3))
for a, K in enumerate(kraus):
for i in range(3):
V[i * 9 + a, :] += K[i, :]
check("Stinespring V^dag V = 1_3 exactly (the v221 transport is a"
" reduced channel of an isometry)",
int(np.allclose(V.T @ V, np.eye(3))), 1, exact=True)
ok_rate = all(np.isclose(
np.linalg.norm(np.linalg.matrix_power(T, k) - J, 2), LAM2 ** k,
rtol=1e-10) for k in range(1, 6))
check("||T^n - P_star||_2 = (2/3)^{6n} exactly, n = 1..5",
int(ok_rate), 1, exact=True)
# (4) uplift core: Hopf identity + deck freeness + commutant
vs = RNG.standard_normal((32, 2)) + 1j * RNG.standard_normal((32, 2))
vs /= np.linalg.norm(vs, axis=1, keepdims=True)
def hopf_stereo(v):
w = 2 * v[0] * np.conj(v[1])
h3 = abs(v[0]) ** 2 - abs(v[1]) ** 2
return (w) / (1 - h3)
check("stereographic(Hopf(v)) = z1/z2 exactly (the seam sphere IS"
" the Hopf base)",
int(all(np.isclose(hopf_stereo(v), v[0] / v[1]) for v in vs)),
1, exact=True)
DECK = np.diag([1j, -1j])
free = all(not np.any(np.isclose(np.linalg.eigvals(
np.linalg.matrix_power(DECK, k)), 1.0)) for k in (1, 2, 3))
check("v492 deck acts freely on S^3 (lens crossover smooth)",
int(free), 1, exact=True)
def real_so4(M2):
R = np.zeros((4, 4))
for j in range(2):
e = np.zeros(2, complex); e[j] = 1.0
w = M2 @ e
R[0, 2*j], R[1, 2*j], R[2, 2*j], R[3, 2*j] = (
w[0].real, w[0].imag, w[1].real, w[1].imag)
w = M2 @ (1j * e)
R[0, 2*j+1], R[1, 2*j+1], R[2, 2*j+1], R[3, 2*j+1] = (
w[0].real, w[0].imag, w[1].real, w[1].imag)
return R
def commutant_dims(g4s):
basis, kind = [], []
for a in range(4):
for b in range(a + 1, 4):
X = np.zeros((5, 5)); X[a, b], X[b, a] = 1., -1.
basis.append(X); kind.append("rot")
for a in range(4):
X = np.zeros((5, 5)); X[a, 4], X[4, a] = 1., 1.
basis.append(X); kind.append("boost")
rows = []
for g4 in g4s:
g = np.eye(5); g[:4, :4] = g4
gi = np.linalg.inv(g)
rows.append(np.stack([((g @ B @ gi) - B).ravel()
for B in basis], axis=1))
A = np.vstack(rows)
_, s, vt = np.linalg.svd(A)
null = vt[np.sum(s > 1e-10):]
boost = sum(abs(c[i]) for c in null
for i in range(10) if kind[i] == "boost")
return null.shape[0], boost
d_deck, b_deck = commutant_dims([real_so4(DECK)])
d_ctrl, b_ctrl = commutant_dims([np.eye(4)])
check("so(4,1) deck-commutant = 4 Killing fields with boost content"
" 0 (Yamabe rigidity of the crossover); control S^3 keeps 10"
" with 4 boosts",
int(d_deck == 4 and b_deck < 1e-8 and d_ctrl == 10
and b_ctrl > 1.0), 1, exact=True)
# (5) KMS closure
w, Vv = np.linalg.eigh(T)
Hmod = Vv @ np.diag(-np.log(w)) @ Vv.T
hev = np.sort(np.linalg.eigvalsh(Hmod))
D2, D3 = 6 * np.log(1.5), 6 * np.log(3.0)
check("spec(-ln T) = {0, 6 ln(3/2), 6 ln 3} exactly and"
" T = e^{-H_mod} (one tick = one unit of KMS time)",
int(np.allclose(hev, [0.0, D2, D3])
and np.allclose(Vv @ np.diag(np.exp(-(-np.log(w)))) @ Vv.T,
T)), 1, exact=True)
c3 = 1 / (8 * sp.pi)
check("v526 normalisation replicated exactly: 1/(4 c3) = 2 pi ="
" beta_angle (Gibbons-Hawking on the crossover fibre)",
int(sp.simplify(1 / (4 * c3) - 2 * sp.pi) == 0), 1, exact=True)
# (7) selector necessity core: D_4 closure only at delta = pi/2
def adm_axes(delta):
marks = np.array([0.0, np.pi, delta, delta + np.pi]) % (2*np.pi)
cands = sorted({round(float((mi + mj) % (2*np.pi)), 12)
for mi in marks for mj in marks})
n = 0
for alpha in cands:
img = (alpha - marks) % (2 * np.pi)
if all(np.min(np.abs(((img[i] - marks + np.pi) % (2*np.pi))
- np.pi)) < 1e-9 for i in range(4)):
n += 1
return n
ok_generic = all(adm_axes(d) == 2 for d in (0.3, 0.9, 1.4, 2.2))
check("M(delta) admissible reflections: V_4 (2 axes) generically,"
" D_4 (4 axes) exactly at delta = pi/2 -- the clock"
" factorisation exists only at mu_4",
int(ok_generic and adm_axes(np.pi / 2) == 4), 1, exact=True)
# (8) frozen kernel invariants
check("frozen rate ratio ln3/ln(3/2) = 2.709511 (exact logs of"
" 729/64 and 729)",
float(D3 / D2), float(np.log(3) / np.log(1.5)), tol=1e-12)
ETA_REC, ETA_0 = 280.3, 14165.0
theta_deg = float(np.degrees(ETA_REC / (ETA_0 - ETA_REC)))
check("frozen disc radius theta_max = eta_rec/(eta_0 - eta_rec) in"
" [1.0, 1.3] deg (typed externals)",
int(1.0 <= theta_deg <= 1.3), 1, exact=True)
return summary("v957 CCC crossover kinematics dictionary")
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
Recorded source fingerprint
4fb069e13d4e637ed0bc58a3a3bbcc44d1c1eaf1eaf91deaa33cef15167f46ce
Paper sections that cite this module
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