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Original ledger classification: Float-track certificates (declared tolerances) + exact Clifford combinatorics + [C] toy fence

Recorded statement: The interacting Klein-Landau semigroup -- the clock survives the interaction as a positive transfer exactly on the mu4-commensurate steps (one module, 6 checks, ~2 s; discovery probe interacting_semigroup_probe.py 6/6, verdict…

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Paper context: input 5 tfpt 3 e8 audit bootstrap

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WOIT.GAMMA.SEMI.01

Original ledger classification: Float-track certificates (declared tolerances) + exact Clifford combinatorics + [C] toy fence

Original claim text

The interacting Klein-Landau semigroup -- the clock survives the interaction as a positive transfer exactly on the mu4-commensurate steps (one module, 6 checks, ~2 s; discovery probe interacting_semigroup_probe.py 6/6, verdict CLOCK-SURVIVES-INTERACTING; the gamma net-dynamics slice after v615/v616). FREE BASELINE [E-float]: at g = 0 all seven steps tau_k (dims 29, 22, 16, 11, 7, 4, 2) are Hermitian with normalized vacuum; even steps PD; odd steps k = 1, 3, 5 indefinite with EXACTLY ZERO one-particle diagonal; k = 7 parity-trivial -- the v524 pattern at N = 16. THE INTERACTING LAW [MEASURED, central]: at g > 0 exact Hermiticity survives PRECISELY on {4, 6, 7} and is lost on {1, 2, 3, 5} (dev > 0.1): the semigroup CONTRACTS to the mu4-commensurate window. THE CLOCK ACTS [MEASURED]: k = 4 stays Hermitian PD ((11,0,0)) over g in {1/32..8} -- the mu4 clock is a positive transfer on the interacting net. THE MECHANISM [E, exact]: the survivor interaction is alpha_4-invariant EXACTLY and not alpha_1/alpha_2 invariant -- translation invariance breaks to the quartet stabilizer {0, 4, 8, 12}. SUBSTEP TYPING [E/measured]: k = 6 even quartet-contained (protected); k = 5 quartet-contained but odd (NOT protected -- both conditions needed); k = 7 parity-trivial. THE READING [C]: the reconstructed rotation group of the interacting toy is the CLOCK tower, not the full circle; one toy, one interaction class; gamma proper stays open on A_hol. WOIT.OS.TWISTOR.01 does not move.

Original classification detail

CLOCK-SURVIVES-INTERACTING: free v524 pattern reproduced at N = 16; at g > 0 Hermiticity survives exactly on {4, 6, 7} (mu4-commensurate); clock step PD over the whole grid; mechanism = alpha_4-invariance exact (quartet stabilizer); 6/6 checks

Dependencies as recorded

WOIT.GAMMA.TOY.01; WOIT.BETA2.OS.01; SEAM.STRADDLE.CONE.01; v524; v534; v615; v616

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Registered verification modules

  • v621_interacting_semigroup.pyRegistered verification module · core
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    WOIT.GAMMA.SEMI.01: die wechselwirkende Klein–Landau-Halbgruppe — **DIE CLOCK ÜBERLEBT DIE WECHSELWIRKUNG ALS POSITIVER TRANSFER, GENAU AUF DEN μ₄-KOMMENSURABLEN SCHRITTEN**: bei g = 0 reproduziert sich das v524-Muster bei N = 16 exakt (7 Schritte hermitesch, Dimensionen 29,22,16,11,7,4,2; gerade Schritte PD; ungerade k = 1,3,5 indefinit mit EXAKT NULL Einteilchen-Diagonale; k = 7 paritäts-trivial); **bei g > 0 überlebt exakte Hermitizität GENAU auf {4, 6, 7}** und geht auf {1,2,3,5} verloren (dev > 0.1) — die Halbgruppe KONTRAHIERT aufs μ₄-Fenster; **die Clock (k = 4) bleibt hermitesch PD (11,0,0) über das ganze Gitter g ∈ {1/32..8}**; Mechanismus exakt: die Survivor-Wechselwirkung ist α₄-invariant EXAKT und nicht α₁/α₂-invariant (Quartett-Stabilisator {0,4,8,12}); k = 5 zeigt: Quartett-Einbettung allein schützt nicht (gerade Parität UND Alignment nötig); Lesart: die rekonstruierte Rotationsgruppe des wechselwirkenden Toys ist der CLOCK-TURM, nicht der volle Kreis (6 checks, ~2 s)

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