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Original ledger classification: Identity [E] (exact integral/spectral computations; honest negative on the kinematic selection; RP-slot open)

Recorded statement: The RP form on the 81-twins -- spectrally BLIND, integrally ANCHOR-VISIBLE (one module, 11 checks, ~1 s; discovery probe rp_bit_form_probe.py 11/11, verdict ANCHOR-VISIBLE): with the corpus reflection theta(X) = Sigma X^T Sigma (v10) and…

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Paper context: TFPT — Introduction and reading guide

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DIAMOND.BIT.RPFORM.01

Original ledger classification: Identity [E] (exact integral/spectral computations; honest negative on the kinematic selection; RP-slot open)

Original claim text

The RP form on the 81-twins -- spectrally BLIND, integrally ANCHOR-VISIBLE (one module, 11 checks, ~1 s; discovery probe rp_bit_form_probe.py 11/11, verdict ANCHOR-VISIBLE): with the corpus reflection theta(X) = Sigma X^T Sigma (v10) and the declared states (trace; anchor (1,1,2); ladder (1,2,4); unit), the RP kernel K(x,y) = omega(theta(x) y) on the Hermite basis of each twin order has IDENTICAL inertia on both twins for every state (trace (4,0,3); vector states (3,1,3)) and is never PSD -- no kinematic v534-style positivity selection exists (honest negative, consistent with v567's certificate coincidence and v534's nonperturbative lesson); but the INTEGRAL Smith towers of the anchor-state Gram SEPARATE the twins -- q32 = 2: (2,2,18,18,18,18,0) vs q32 = 0: (2,2,2,18,18,18,0), one slot 18 vs 2 = a factor 9 = N_fam^2 (the compiler twin carries an EXTRA 3^2 of divisibility); the ladder state separates identically while trace/unit are blind: the bit is visible to the RP form ONLY through anchor-coupled states; trace discriminant -6561 = -81^2 on both twins; q32 = 1 collapses to a 5-dim order (v568 reproduced); and the blindness is ROBUST under the compiler's own flow -- the V-semigroup-deformed kernels K_n = omega(theta(x) V^n y), n = 1..3, keep identical inertia on both twins for every state (the deformation moves the inertia, never differentially): a dynamical selection needs genuine interaction, not a semigroup weighting. THE GIBBS PENCIL SEPARATES (the first one-parameter dynamical distinction): along the anchor-Hamiltonian Gibbs family omega_g ~ tr(e^{-gH} .), H = I + Pi_dom (v566), det(K_0 + s K_Pi) factors (6561/4096)(s^2+16s+16)^2(15s^2-16s-16) on the compiler twin vs -(6561/4096)(s^2+16s+16)^3 on the alt (6561 = 81^2) -- the compiler's extra factor has its root s* = (8 - 4 sqrt 19)/15 INSIDE the physical Gibbs range: the compiler twin's kernel degenerates at a finite critical temperature, the alt twin's never does (a rank-drop marker, honest: not a positivity selection) -- and THE CRITICAL MODE IS IDENTIFIED: the null space at s* is one-dimensional, rank 1, supported entirely on the seam row (the radical / A2-Ext-carrier row of v566 S5), fixes the seam line (X e3 = e3), is a fixed point of the sheet flow (V X = X) and is annihilated by the dominant projector: the Gibbs criticality lives exactly in the Ext-carrier slot -- the typed resonance with the alignment bit deepens to a SHARED CARRIER; and the half-sided restrictions stay blind (Levi (3,2) / radical (1,1) inertia identical on both twins across the Gibbs range): the RP-positivity selection resists every kinematic and half-sided design tried -- what remains is the genuinely INTERACTING construction (the v534 FK-quartet analogue), typed open. NAMED LIMITS: no physical selection claimed; the dynamical/interacting RP question stays the open RP-slot; the alignment-bit resonance stays typed; no marker moves. [2026-07-31 V: the interacting follow-up is executed -- v584: rank-one Ext-source coupling separates the twins directionally in the unit box (exact thresholds, pencil-factor numerators, positive-coupling healing of exactly the compiler twin); see DIAMOND.BIT.EXTSOURCE.01.]

Original classification detail

ANCHOR-VISIBLE: spectrally blind (same inertia, never PSD), integrally separated exactly through anchor/ladder states (Smith slot 18 vs 2, factor 9 = N_fam^2); trace discriminant -81^2; 11/11 checks (half-sided restrictions blind too; the interacting construction is the one remaining RP route) (V-flow robustness; Gibbs pencil: compiler Gibbs-critical at s* = (8-4sqrt19)/15, alt never; critical mode = seam-row flow-fixed direction, the Ext-carrier slot); dynamical RP-slot open

Dependencies as recorded

DIAMOND.BIT.SELECT.01; QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; v10; v23; v534; v568

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Registered verification modules

  • v572_rp_bit_form.pyRegistered verification module · flavor
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    DIAMOND.BIT.RPFORM.01: the RP form on the 81-twins — spectrally BLIND, integrally ANCHOR-VISIBLE: with the corpus reflection θ(X) = ΣXᵀΣ and the declared states (trace, anchor (1,1,2), ladder (1,2,4), unit), the RP kernel has identical inertia on both twins for every state and is never PSD (no kinematic v534-style positivity selection exists — honest negative, consistent with v567), but the INTEGRAL Smith towers of the anchor-state Gram separate them: (2,2,18,18,18,18,0) vs (2,2,2,18,18,18,0) — one slot, 18 vs 2, a factor 9 = N²fam: the compiler twin carries an extra 3² of divisibility, the ladder state separates identically, anonymous states are blind — the bit is visible to the RP form ONLY through anchor-coupled states; trace discriminant −6561 = −81² on both; q32 = 1 collapses to dim 5 (v568 reproduced); robust under the compiler V-flow (semigroup-deformed kernels keep identical inertia); the Gibbs pencil of the anchor Hamiltonian separates DYNAMICALLY: the compiler twin's kernel degenerates at a finite critical temperature s* = (8−4√19)/15 inside the Gibbs range (det factors with an extra quadratic; constant 81²), the alt twin's never does — a rank-drop marker, not a positivity selection; the critical mode is IDENTIFIED: seam-row supported (the A₂-Ext-carrier row), seam-line fixing, sheet-flow fixed point — the alignment-bit resonance deepens to a shared carrier; the half-sided (Levi/radikal) Restriktionen bleiben ebenfalls blind — the interacting RP construction is the one remaining route, typed open (11 checks, ~1 s)

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