"""v572 -- DIAMOND.BIT.RPFORM.01: the RP form on the 81-twins -- spectrally BLIND, integrally ANCHOR-VISIBLE: the reconstruction bit shows in the Smith tower of the anchor-state Gram (18 vs 2, a factor 9 = N_fam^2) and in no anonymous state. PROVENANCE. The RP-slot follow-up of v568: v534 showed reflection positivity selects the alignment bit dynamically at toy level; the reconstruction bit (q32 = 2 vs 0) sits in the same structural slot. This module runs the KINEMATIC half: reflection theta(X) = Sigma X^T Sigma (the v10 sheet involution), declared corpus states (trace; anchor a = (1,1,2), v23; ladder v = (1,2,4), v568; unit), RP kernel K(x,y) = omega(theta(x) y) on the Hermite basis of each twin's integral order. Audited by the discovery probe (rp_bit_form_probe.py, 11/11, verdict ANCHOR-VISIBLE). [E] 1. HONEST NEGATIVE -- SPECTRALLY BLIND: identical inertia on both twins for every declared state (trace (4,0,3); anchor/ladder/unit (3,1,3)); neither twin's kernel is PSD for any state, so the kinematic form cannot run a v534-style positivity selection at all -- consistent with v567 (classical certificates coincide) and with v534's lesson (the selection there was nonperturbative). [E] 2. THE CENTRAL POSITIVE RESULT -- ANCHOR-VISIBLE: the INTEGRAL Smith towers of the anchor-state Gram separate the twins -- q32 = 2 gives (2,2,18,18,18,18,0), q32 = 0 gives (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 the same way, while trace and unit are blind: the bit is visible to the RP form ONLY through anchor-coupled states. [E] 3. the blindness is ROBUST under the compiler's own flow: the V-semigroup-deformed kernels K_n(x,y) = omega(theta(x) V^n y), n = 1..3, have identical inertia on both twins for every state (the deformation moves the inertia, never differentially) -- a dynamical selection, if any, needs genuine interaction, not a semigroup weighting. [E] 4. THE DYNAMICAL SEPARATION (the Gibbs pencil): along the anchor-Hamiltonian Gibbs family omega_g ~ tr(e^{-gH} .), H = I + Pi_dom (v566), the pencil determinant det(K_0 + s K_Pi) (s = e^{-g} - 1) factors EXACTLY differently on the twins -- compiler: (6561/4096)(s^2+16s+16)^2 (15s^2-16s-16); alt: -(6561/4096)(s^2+16s+16)^3 (the constant 6561 = 81^2 is the saturation index squared) -- and 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 -- the first one-parameter DYNAMICAL distinction of the bit (a rank-drop marker, honest: not a positivity selection). [E] 5. THE CRITICAL MODE IDENTIFIED: the compiler twin's null space at the critical temperature is one-dimensional and structurally canonical -- rank 1, supported entirely on the seam row (the radical / A2-Ext-carrier row), 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. [E] 6. honest negative, completed: the HALF-SIDED restrictions stay blind too -- the Gibbs kernels restricted to the Levi sublattice (dim 5, inertia (3,2)) and the radical sublattice (dim 2, (1,1)) have identical inertia on both twins across the sampled 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. [E] 7. consistency: the trace-state Gram discriminant is -6561 = -81^2 = -(saturation index)^2 on both twins; the excluded member q32 = 1 collapses to a 5-dimensional order (the v568 split-point collapse reproduces). NAMED LIMITS AS LOAD-BEARING CONTENT: no physical selection is claimed -- the kinematic RP form does NOT distinguish the twins by positivity; the dynamical/interacting RP question (the v534 analogue) stays the open RP-slot; the resonance with the alignment bit stays a TYPED OBSERVATION; no marker moves. Fences: Smith 1861 CLASSICAL; the reflection and all states are prior structure, declared before the run. Python-only, counted per GATE.WOLFRAM.02. Discovery: experiments/tfpt-discovery/rp_bit_form_probe.py (2026-07-31, 11/11, ANCHOR-VISIBLE). """ import time import numpy as np import sympy as sp from sympy.matrices.normalforms import hermite_normal_form, \ smith_normal_form T0 = time.time() FAILS = [] N_CHK = 0 def check(name, ok, detail=""): global N_CHK N_CHK += 1 if not ok: FAILS.append(name.split()[0]) print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (": " + detail) if detail else "")) I3 = sp.eye(3) IDX7 = ((0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2)) SIG = sp.diag(1, -1, -1) A = sp.Matrix([1, 1, 2]) LAD = sp.Matrix([1, 2, 4]) ONE = sp.Matrix([1, 1, 1]) def Qc(c): return sp.Matrix([[3, 1, 0], [3, 2, 0], [3, c, 1]]) def hermite_basis(Q): U, V = Q * sp.diag(1, 0, 0), Q * sp.diag(0, 1, 1) ws = [I3] fr = [I3] for _ in range(5): fr = [w * G for w in fr for G in (U, V)] ws += fr Mrows = sp.Matrix([[w[i, j] for (i, j) in IDX7] for w in ws]) H = hermite_normal_form(Mrows.T) mats = [] for k in range(H.cols): Mx = sp.zeros(3, 3) for r, (i, j) in enumerate(IDX7): Mx[i, j] = H[r, k] mats.append(Mx) return mats def theta(X): return SIG * X.T * SIG STATES = { "trace": lambda Mx: sp.trace(Mx), "anchor": lambda Mx: (A.T * Mx * A)[0], "ladder": lambda Mx: (LAD.T * Mx * LAD)[0], "unit": lambda Mx: (ONE.T * Mx * ONE)[0], } def gram(mats, omega): K = sp.Matrix(len(mats), len(mats), lambda i, j: omega(theta(mats[i]) * mats[j])) return (K + K.T) / 2 def inertia(K): ev = np.linalg.eigvalsh(np.array(K.tolist(), float)) return (int((ev > 1e-9).sum()), int((abs(ev) <= 1e-9).sum()), int((ev < -1e-9).sum())) def snf_divs(K): K2 = (2 * K).applyfunc(sp.Integer) # integral (half-int entries) s_ = smith_normal_form(K2) return [abs(s_[i, i]) for i in range(min(s_.shape))] def run(): global N_CHK, FAILS N_CHK = 0 FAILS = [] print("=" * 78) print("DIAMOND.BIT.RPFORM -- the RP form on the 81-twins") print("=" * 78) basis = {c: hermite_basis(Qc(c)) for c in (2, 0)} res = {} for c in (2, 0): res[c] = {} for nm, om in STATES.items(): K = gram(basis[c], om) res[c][nm] = (inertia(K), snf_divs(K), sp.nsimplify(K.det())) # --- R1: spectral blindness ------------------------------------------------- same_inertia = all(res[2][nm][0] == res[0][nm][0] for nm in STATES) check("R1.1 [E, honest negative -- SPECTRALLY BLIND] the RP kernel " "K(x,y) = omega(theta(x) y) has IDENTICAL inertia on both twins " "for every declared state (trace (4,0,3); anchor/ladder/unit " "(3,1,3) with a one-dimensional kernel): no positivity or " "signature distinction exists -- consistent with v567 (every " "classical certificate coincides on the pair) and with v534's " "lesson that RP selection is nonperturbative", same_inertia and res[2]["trace"][0] == (4, 0, 3) and res[2]["anchor"][0] == (3, 1, 3)) check("R1.2 [E] neither twin's kernel is PSD for any declared state " "(3 negative directions everywhere): the kinematic form CANNOT " "run a v534-style positivity selection at all -- the RP-slot " "question stays with the interacting/dynamical level", all(res[c][nm][0][2] == 3 for c in (2, 0) for nm in STATES)) # --- R2: the integral level SEES the bit ----------------------------------- an2, an0 = res[2]["anchor"][1], res[0]["anchor"][1] ld2, ld0 = res[2]["ladder"][1], res[0]["ladder"][1] check("R2.1 [E, the central positive result -- ANCHOR-VISIBLE] the " "INTEGRAL Smith towers of the anchor-state Gram SEPARATE the " "twins: q32 = 2 gives %s, q32 = 0 gives %s -- they differ in " "exactly one slot (18 vs 2, a factor 9 = N_fam^2): the compiler " "twin carries an EXTRA 3^2 of divisibility in its anchor Gram" % (an2, an0), an2 != an0 and an2 == [2, 2, 18, 18, 18, 18, 0] and an0 == [2, 2, 2, 18, 18, 18, 0]) check("R2.2 [E] the ladder state (1,2,4) separates the same way " "(q32 = 2: %s vs q32 = 0: %s), while the ANONYMOUS states are " "blind -- trace and unit give identical towers on both twins: " "the bit is visible to the RP form ONLY through anchor-coupled " "states" % (ld2, ld0), ld2 != ld0 and res[2]["trace"][1] == res[0]["trace"][1] and res[2]["unit"][1] == res[0]["unit"][1]) check("R2.3 [E, consistency] the trace-state Gram determinant is " "-6561 = -81^2 = -(saturation index)^2 on BOTH twins (the index-" "square appears as the RP discriminant; forced by [P_Z : O] = 81 " "and the pattern form of signature (4,3))", res[2]["trace"][2] == -6561 and res[0]["trace"][2] == -6561) # --- R2b: the blindness is robust under the compiler's own flow ------------- flow_blind = True for c in (2, 0): pass V2 = Qc(2) * sp.diag(0, 1, 1) V0 = Qc(0) * sp.diag(0, 1, 1) for n_ in (1, 2, 3): for nm, om in STATES.items(): Ka = gram_flow = sp.Matrix( 7, 7, lambda i, j: om(theta(basis[2][i]) * V2**n_ * basis[2][j])) Kb = sp.Matrix( 7, 7, lambda i, j: om(theta(basis[0][i]) * V0**n_ * basis[0][j])) ia = inertia((Ka + Ka.T) / 2) ib = inertia((Kb + Kb.T) / 2) if ia != ib: flow_blind = False check("R2.4 [E, honest negative extended] the spectral blindness is " "ROBUST under the compiler's own flow: the V-semigroup-deformed " "kernels K_n(x,y) = omega(theta(x) V^n y), n = 1..3, have " "IDENTICAL inertia on both twins for every declared state (the " "deformation shifts the inertia -- vector states move (3,1,3) -> " "(2,3,2) at n >= 1 -- but never differentially): a dynamical " "selection, if any, needs genuine interaction, not a semigroup " "weighting", flow_blind) # --- R2c: the Gibbs pencil of the anchor Hamiltonian -- DYNAMICAL separation s_ = sp.symbols("s") pencil = {} for c in (2, 0): Vc = Qc(c) * sp.diag(0, 1, 1) evs = sorted(Vc.eigenvals().keys(), key=lambda e: abs(e)) domc = evs[-1] Pi = sp.eye(3) for e in evs[:-1]: Pi = Pi * (Vc - e * sp.eye(3)) / (domc - e) K0 = sp.Matrix(7, 7, lambda i, j: sp.trace(theta(basis[c][i]) * basis[c][j])) K1 = sp.Matrix(7, 7, lambda i, j: sp.trace(Pi * theta(basis[c][i]) * basis[c][j])) K0 = (K0 + K0.T) / 2 K1 = (K1 + K1.T) / 2 detp = sp.factor(sp.expand((K0 + s_ * K1).det())) pencil[c] = (K1.rank(), detp) shared = (s_**2 + 16 * s_ + 16) extra = (15 * s_**2 - 16 * s_ - 16) ok2 = sp.simplify(pencil[2][1] - sp.Rational(6561, 4096) * shared**2 * extra) == 0 ok0 = sp.simplify(pencil[0][1] + sp.Rational(6561, 4096) * shared**3) == 0 check("R2.5 [E, THE DYNAMICAL SEPARATION -- the Gibbs pencil] along the " "anchor-Hamiltonian Gibbs family omega_g ~ tr(e^{-gH} .), H = I + " "Pi_dom (v566), the pencil determinant det(K_0 + s K_Pi) " "(s = e^{-g} - 1) FACTORS DIFFERENTLY on the twins: compiler " "(6561/4096)(s^2+16s+16)^2 (15s^2-16s-16) vs alt " "-(6561/4096)(s^2+16s+16)^3 -- the compiler carries an EXTRA " "quadratic factor; the constant is 6561 = 81^2, the saturation " "index squared; both K_Pi have rank 6", ok2 and ok0 and pencil[2][0] == 6 and pencil[0][0] == 6) r_crit = sp.Rational(8, 15) - 4 * sp.sqrt(19) / 15 in_range = (-1 < float(r_crit) <= 0) shared_roots_out = all(not (-1 < float(rr) <= 0) for rr in (-8 - 4 * sp.sqrt(3), -8 + 4 * sp.sqrt(3))) check("R2.6 [E, THE CRITICAL TEMPERATURE] the compiler twin's extra " "factor has the root s* = (8 - 4 sqrt(19))/15 = %.4f INSIDE the " "physical Gibbs range s in (-1, 0] (a finite critical temperature " "g* = -log(1 + s*)), while the shared roots -8 +- 4 sqrt(3) lie " "OUTSIDE it: the COMPILER twin is the one whose RP kernel " "degenerates along the Gibbs flow, the alt twin never does -- " "the first ONE-PARAMETER-DYNAMICAL distinction of the bit " "(honest: a rank-drop marker, not a positivity selection)" % float(r_crit), in_range and shared_roots_out) # --- R2d: the critical mode identified --------------------------------------- Vc2 = Qc(2) * sp.diag(0, 1, 1) evs2 = sorted(Vc2.eigenvals().keys(), key=lambda e: abs(e)) Pi2 = sp.eye(3) for e in evs2[:-1]: Pi2 = Pi2 * (Vc2 - e * sp.eye(3)) / (evs2[-1] - e) K0c = sp.Matrix(7, 7, lambda i, j: sp.trace(theta(basis[2][i]) * basis[2][j])) K1c = sp.Matrix(7, 7, lambda i, j: sp.trace(Pi2 * theta(basis[2][i]) * basis[2][j])) K0c = (K0c + K0c.T) / 2 K1c = (K1c + K1c.T) / 2 s_star = sp.Rational(8, 15) - 4 * sp.sqrt(19) / 15 null = sp.simplify(K0c + s_star * K1c).nullspace() Xc = sp.zeros(3, 3) for k in range(7): Xc += null[0][k] * basis[2][k] Xc = sp.simplify(Xc) e3 = sp.Matrix([0, 0, 1]) top_rows_zero = all(sp.simplify(Xc[i, j]) == 0 for i in (0, 1) for j in range(3)) check("R2.7 [E, THE CRITICAL MODE IDENTIFIED] the compiler twin's " "kernel at the critical temperature is ONE-dimensional and " "structurally canonical: the null matrix X is 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, left eigenvalue 1) and is annihilated " "by the dominant projector (Pi X = 0): the Gibbs criticality " "lives exactly in the Ext-carrier slot -- the typed resonance " "with the alignment bit deepens to a shared carrier", len(null) == 1 and Xc.rank() == 1 and top_rows_zero and sp.simplify(Xc * e3 - e3) == sp.zeros(3, 1) and sp.simplify(Vc2 * Xc - Xc) == sp.zeros(3, 3) and sp.simplify(Pi2 * Xc) == sp.zeros(3, 3)) # --- R2e: the half-sided restrictions stay blind ------------------------------ LEVI = ((0, 0), (0, 1), (1, 0), (1, 1), (2, 2)) RAD = ((2, 0), (2, 1)) def sublattice(mats, pattern): out_idx = [k for k, (i, j) in enumerate(IDX7) if (i, j) not in pattern] Mcoef = sp.Matrix([[m[IDX7[r][0], IDX7[r][1]] for m in mats] for r in out_idx]) base = [] for vv in Mcoef.nullspace(): d = sp.lcm([sp.denom(x) for x in vv]) vv = sp.expand(vv * d) g = sp.gcd(list(vv)) vv = sp.Matrix([sp.cancel(x / g) for x in vv]) X = sp.zeros(3, 3) for k in range(7): X += vv[k] * mats[k] base.append(X) return base half_blind = True for c in (2, 0): Vc_ = Qc(c) * sp.diag(0, 1, 1) evs_ = sorted(Vc_.eigenvals().keys(), key=lambda e: abs(e)) Pi_ = sp.eye(3) for e in evs_[:-1]: Pi_ = Pi_ * (Vc_ - e * sp.eye(3)) / (evs_[-1] - e) res = {} for name, pattern in (("levi", LEVI), ("rad", RAD)): base = sublattice(basis[c], pattern) for sv in (sp.Rational(0), sp.Rational(-1, 2), sp.Rational(-63, 100), sp.Rational(-99, 100)): K = sp.Matrix(len(base), len(base), lambda i, j: sp.trace((sp.eye(3) + sv * Pi_) * theta(base[i]) * base[j])) evn = np.linalg.eigvalsh(np.array(sp.N((K + K.T) / 2).tolist(), float)) res[(name, sv)] = (int((evn > 1e-9).sum()), int((evn < -1e-9).sum())) if c == 2: ref = res else: if res != ref: half_blind = False expect = all(ref[("levi", sv)] == (3, 2) and ref[("rad", sv)] == (1, 1) for sv in (sp.Rational(0), sp.Rational(-1, 2), sp.Rational(-63, 100), sp.Rational(-99, 100))) check("R2.8 [E, honest negative -- the half-sided restrictions stay " "blind] restricting the Gibbs kernels to the LEVI sublattice " "(dim 5, inertia (3,2)) and to the RADICAL sublattice (dim 2, " "inertia (1,1)) gives IDENTICAL inertia on both twins across the " "whole sampled Gibbs range -- no positivity anywhere and no " "differential change: the RP-positivity selection resists every " "kinematic and half-sided design tried (full kernel, V-flow, " "Gibbs family, Levi/radical restriction); what remains is the " "genuinely INTERACTING construction (the v534 FK-quartet " "analogue), typed open", half_blind and expect) # --- R3: the excluded member ------------------------------------------------ mats1 = hermite_basis(Qc(1)) check("R3.1 [E, control] the excluded member q32 = 1 (the split point) " "has a %d-dimensional order (< 7): the v568 collapse reproduces " "in this construction -- the RP-form question does not even pose " "on the split geometry" % len(mats1), len(mats1) == 5) VERDICT = ("ANCHOR-VISIBLE" if not FAILS else "MIXED") print("\nVERDICT: %s -- spectrally blind (same inertia everywhere), " "integrally separated exactly through the anchor/ladder states " "(Smith slot 18 vs 2, factor 9 = N_fam^2); trace discriminant " "-81^2 on both" % VERDICT) print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or "")) print("elapsed: %.1f s" % (time.time() - T0)) print("--- DIAMOND.BIT.RPFORM.01 RP form on the twins: %d passed, %d failed ---" % (N_CHK - len(FAILS), len(FAILS))) return len(FAILS) if __name__ == "__main__": raise SystemExit(1 if run() else 0)