"""v978 -- PRIME.TERMINAL.DENSITY_MARTINGALE.01 [E] (consolidation wave 14, rounds 339 + 341, notes DCLXXXVI / DCLXXXIX, 2026-08-27): THE TERMINAL REFORMULATION THEOREM LAYER -- the density-martingale identity, the moment dictionary, the tilted tower, the exact hand-off and the algebraic pair ceiling, all re-derived from scratch in exact arithmetic (sympy symbolic + pure Fractions, no probe imports), plus the sealed round verdicts re-run as exact decision logic on the frozen record aggregates with tipping mutants. THE EXACT LAYER (theorem grade, module-own): [E] 1. DENSITY MARTINGALE FROM MASS CONSERVATION ALONE: on any finite genealogy tree with node mass A(v) and leaf count n(v), the density-per-remaining-descendant d(v) = A(v)/n(v), normalized X(v) = d(v)/d(root), is a nonnegative martingale under the count path measure P(c|v) = n(c)/n(v): E[X_next | v] = sum_c (n_c/n_v)(A_c/n_c)/(A_v/n_v) = sum_c A_c / A_v = 1 x X(v), i.e. the ONLY input is mass conservation sum_c A_c = A_v -- proved symbolically on a generic node and bit-exact in Fractions on a concrete three-level tree (per-node sum_c p_c R_c == 1, per-level E[X_k] == 1). [E] 2. THE MOMENT DICTIONARY (the r339 terminal reformulation): with m leaves of masses y_i, total Y, M_k = sum (y_i/Y)^k and q_max = max y_i / Y: E[X_inf^2] == m M_2, E[X_inf^3] == m^2 M_3, max X_inf == m q_max -- proved symbolically on generic masses and bit-exact in Fractions; the wrong-power mutant (m^2 M_2) is CAUGHT. This is the reformulation: the terminal target sum q^3 <= C (log m)^A (m/log m)^... IS the cubic-moment statement E[X_inf^3] == rho_2 (log m)^2 -- the banked r306 constant C_2 = 1.0694 (0/57 violations, re-gated as a frozen anchor) already certifies A = 2 on the measured 57-rung set. [E] 3. THE JENSEN DRIFT, exactly: per node the weighted AM-GM prod_c R_c^{p_c} <= sum_c p_c R_c = 1 holds with rational exponents cleared to the integer-power inequality prod R_c^{a_c} <= 1 (p_c = a_c/b) -- exact Fractions, strict on a spread node, equality at the flat node: the log drift delta_k = E[ln X_{k+1}] - E[ln X_k] <= 0 is pure convexity. [E] 4. THE TILTED TOWER (the r341 correction): with Gamma(v) = sum_c p_c R_c^3 and the TILTED steps p~_c = p_c R_c^3 / Gamma(v) (sum_c p~_c == 1 exactly), E[X_inf^3] == E_Q[prod_k Gamma(V_k)] bit-exact in Fractions on two- and three-level trees; the NAIVE UNTILTED reading E_P[prod Gamma] == E[X_inf^3] is FALSE -- the module's spread witness breaks it by an exact nonzero Fraction (the probe's sealed toy break is 7/18); the Phi-Gamma telescope sum_c Phi(c) / Phi(v) == Gamma(v) for Phi(v) = P(v) X(v)^3 is proved symbolically. [E] 5. THE EXACT HAND-OFF: for ANY leaf subset H (the heavy arm), E3h := sum_H (1/m) X_inf^3 <= (m q_max)^2 * msh, msh = sum_H (1/m) X_inf, because X_inf <= m q_max pointwise -- exact with a strict spread witness AND the flat equality case; the first-power mutant cap (m q_max) * msh is CAUGHT. [E] 6. THE ALGEBRAIC PAIR CEILING: a two-child fold has R <= 2 (a pair merge at most doubles the density), and for R in [0,2] with E[R] = 1: Gamma = E[R^3] <= 4, from the exact factorization 4R - R^3 = R(2-R)(2+R) >= 0; equality exactly at the extreme split {0, 2} (witness Gamma = 4 bit-exact); the ceiling-3 mutant is CAUGHT. This ceiling is the source of the wave's data-free stopping constant R_ALG = 4^(1/3) (v979). THE FROZEN CENSUS LAYER (sealed record aggregates re-run as decision logic; census values are GATES, not new measurements): r339 fold_density_dictionary_probe (36/36, SPEC 822d6bff): LOCAL_INFLATION_SUPERCRITICAL -- the per-level worst-case budget W_F = prod Gamma_max is supercritical (e(W_F) = +0.956 > CRIT 0.224; W_F violations 44/43/40 of 51 at a = 1/2/3) while the TARGET grows 8x slower (e(m^2 M_3) = +0.112 < CRIT): Gamma_max lives in leaf-adjacent degenerate pairs (per-level Gamma_max median profile 1.05 -> 3.99 against the algebraic pair ceiling 4) under which almost no PATH MASS sits -- the gap between the exact dictionary and the worst-case budget IS the round's finding. Dictionary devs 4.9e-16 / 7.8e-16; drift sum -1.1392 strictly <= 0; the dictionary separates worlds (SCRAMBLE m^2 M_3 = 20.59 vs ladder median 6.22) where the Gamma_max census is world-blind (honest negative). r341 fold_bellman_reverse_holder_probe (38/38, SPEC 7a9cf6b3): PATHWEIGHT_ALSO_SUPERCRITICAL by the sealed tree (no arm certifies on its own freeze: heavy-arm worst kz51 7.61x, good-arm single violator kz55 1.41x) -- BUT path weighting deflates the worst case 21x at the median (E_P[prod Gamma] med 12.53 vs W_F med 265.54; pathweighted per-level profile 1.05..1.80 where Gamma_max climbs to 3.99; Gamma>3 nodes carry only 0.173 of E3), the exponents are NOT supercritical in the r339 sense (e(W_B) = -0.214), the hand-off holds with reserve median 4.8x / min 2.1x, and R* = 3/2 is a TUNING SURFACE, not a canonical constant (heavy share hsh med 0.872 at 3/2 vs 0.266 at 7/4; E3h share 0.944 -> 0.386; the mass balance between 3/2 and 7/4 is the named open question, resolved census-side by v979's cover). HONEST SCOPE (firewall): the exact layer is finite tree algebra -- theorem grade on arbitrary rational genealogies; every census number is a FROZEN RECORD AGGREGATE re-gated through the sealed decision logic (the full probe records are sealed experiments-side and re-verified by rh/verification/run_rh.py); NO inequality is promoted beyond the measured rungs -- the r324 MEASURED composition (m_0* = 10^59.6, chain-honest 10^238) stays the end state of the terminal lane; the r341 envelope 10^24.0 and the r339 typed envelope 10^30.4 are ENVELOPE-TYPED, never composed. No zeros, no prime oracles. NOT evidence for or against the Riemann Hypothesis. NO RH CLAIM. Python-only per GATE.WOLFRAM.02. PROVENANCE: discovery probes fold_density_dictionary_probe.py (36/36, SPEC_SHA 822d6bff final / freeze a845041c, two-commit protocol) and fold_bellman_reverse_holder_probe.py (38/38, SPEC_SHA 7a9cf6b3 final / freeze f0d2c744, no amendment); rounds 339 / 341, notes DCLXXXVI / DCLXXXIX, 2026-08-27. This module re-derives the exact layer from scratch (no probe imports) and gates the sealed record aggregates; the probes stay in experiments/tfpt-discovery/ as the discovery record, pinned in rh/INVENTORY.json and re-verified by run_rh.py. """ from fractions import Fraction as Fr import sympy as sp from tfpt_constants import check, summary, reset # ---------------- tree machinery (pure Fractions) ---------------- # a node is (mass, [children]); a leaf is (mass, []). def leaves(node): mass, ch = node if not ch: return [node] out = [] for c in ch: out += leaves(c) return out def nleaves(node): return len(leaves(node)) def mass(node): return node[0] def density(node): return Fr(mass(node)) / nleaves(node) def mass_conserved(node): _, ch = node if not ch: return True return (sum(Fr(mass(c)) for c in ch) == Fr(mass(node)) and all(mass_conserved(c) for c in ch)) def paths(node, p=Fr(1)): """[(leaf, path_probability, [nodes root..leaf])] under the count measure P(c|v) = n(c)/n(v).""" _, ch = node if not ch: return [(node, p, [node])] out = [] nv = nleaves(node) for c in ch: for lf, q, chain in paths(c, p * Fr(nleaves(c), nv)): out.append((lf, q, [node] + chain)) return out def x_of(node, root): return density(node) / density(root) def gamma(node, root): """Gamma(v) = sum_c p_c R_c^3 with R_c = d(c)/d(v).""" _, ch = node nv = nleaves(node) dv = density(node) return sum(Fr(nleaves(c), nv) * (density(c) / dv) ** 3 for c in ch) # the concrete three-level Fractions tree (spread on purpose) TREE = (Fr(36), [ (Fr(20), [(Fr(16), []), (Fr(3), []), (Fr(1), [])]), (Fr(12), [(Fr(9), []), (Fr(3), [])]), (Fr(4), [(Fr(2), []), (Fr(2), [])]), ]) FLAT = (Fr(12), [ (Fr(6), [(Fr(3), []), (Fr(3), [])]), (Fr(6), [(Fr(3), []), (Fr(3), [])]), ]) def internal_nodes(node): _, ch = node if not ch: return [] out = [node] for c in ch: out += internal_nodes(c) return out def level_nodes(node, k): if k == 0: return [(node, Fr(1))] out = [] nv = nleaves(node) for c in node[1]: for nd, p in level_nodes(c, k - 1): out.append((nd, p)) return out def e_x_at_level(root, k): """E[X_k] with the convention that a leaf reached before level k stays at its leaf value (stopped martingale).""" def walk(node, p, depth): if depth == k or not node[1]: return [(node, p)] nv = nleaves(node) out = [] for c in node[1]: out += walk(c, p * Fr(nleaves(c), nv), depth + 1) return out return sum(p * x_of(nd, root) for nd, p in walk(root, Fr(1), 0)) def run(): reset() print("v978 PRIME.TERMINAL.DENSITY_MARTINGALE.01: the density " "martingale, the moment dictionary, the tilted tower, the " "hand-off and the pair ceiling (rounds 339 + 341 frozen)") # ---- 1. martingale from mass conservation, symbolic generic node A1, A2, A3, n1, n2, n3 = sp.symbols("A1 A2 A3 n1 n2 n3", positive=True) Av, nv = A1 + A2 + A3, n1 + n2 + n3 dv = Av / nv e_next = sum((nc / nv) * (Ac / nc) / dv for Ac, nc in ((A1, n1), (A2, n2), (A3, n3))) check("martingale identity SYMBOLIC: E[d(child)/d(v) | v] under " "P(c|v) = n_c/n_v equals 1 for a GENERIC 3-child node -- the " "only input is mass conservation A_v = sum A_c", sp.simplify(e_next - 1) == 0) ok_nodes = mass_conserved(TREE) ok_mart = all( sum(Fr(nleaves(c), nleaves(nd)) * (density(c) / density(nd)) for c in nd[1]) == 1 for nd in internal_nodes(TREE)) ok_levels = all(e_x_at_level(TREE, k) == 1 for k in (0, 1, 2, 3)) check("martingale bit-exact on the concrete three-level tree: " "per-node sum_c p_c R_c == 1 and per-level E[X_k] == 1 " "(stopped convention), pure Fractions", ok_nodes and ok_mart and ok_levels) # ---- 2. the moment dictionary, symbolic + Fractions y = sp.symbols("y1 y2 y3 y4", positive=True) m_sym = sp.Integer(len(y)) Y = sum(y) Xs = [yi * m_sym / Y for yi in y] # X_inf at leaf i M2 = sum((yi / Y) ** 2 for yi in y) M3 = sum((yi / Y) ** 3 for yi in y) e2 = sum(Xi ** 2 for Xi in Xs) / m_sym e3 = sum(Xi ** 3 for Xi in Xs) / m_sym check("moment dictionary SYMBOLIC on generic masses: E[X_inf^2] == " "m M_2 and E[X_inf^3] == m^2 M_3 (the r339 terminal " "reformulation: the cubic target IS a martingale moment)", sp.simplify(e2 - m_sym * M2) == 0 and sp.simplify(e3 - m_sym ** 2 * M3) == 0) lts = leaves(TREE) mm = len(lts) Ytot = sum(mass(lf) for lf in lts) m2f = sum((Fr(mass(lf)) / Ytot) ** 2 for lf in lts) m3f = sum((Fr(mass(lf)) / Ytot) ** 3 for lf in lts) xs = [x_of(lf, TREE) for lf in lts] qmax = max(Fr(mass(lf)) / Ytot for lf in lts) check("moment dictionary bit-exact in Fractions on the tree " "(m = 7 leaves): E[X^2] == m M_2, E[X^3] == m^2 M_3, " "max X_inf == m q_max", sum(x ** 2 for x in xs) / mm == mm * m2f and sum(x ** 3 for x in xs) / mm == mm ** 2 * m3f and max(xs) == mm * qmax) check("wrong-power mutant CAUGHT: E[X^2] == m^2 M_2 is FALSE on the " "spread tree (the dictionary powers are not interchangeable)", sum(x ** 2 for x in xs) / mm != mm ** 2 * m2f) # ---- 3. the Jensen drift as exact weighted AM-GM # spread node: children ratios R with rational path weights p = a/b; # cleared inequality prod R^a <= 1 in exact Fractions. nd = TREE[1][0] # the (16,3,1) node nv0 = nleaves(nd) d0 = density(nd) Rs = [density(c) / d0 for c in nd[1]] ps = [Fr(nleaves(c), nv0) for c in nd[1]] b = 1 for p in ps: b = b * p.denominator // sp.gcd(b, p.denominator) prod_pow = Fr(1) for R, p in zip(Rs, ps): prod_pow *= R ** int(p * b) flat_prod = Fr(1) for c in FLAT[1]: flat_prod *= (density(c) / density(FLAT)) ** 1 check("Jensen drift EXACT (weighted AM-GM, rational exponents " "cleared): prod_c R_c^{a_c} < 1 STRICT on the spread node and " "== 1 at the flat node -- E[ln X] is non-increasing from " "convexity alone", prod_pow < 1 and flat_prod == 1 and sum(p * R for p, R in zip(ps, Rs)) == 1) # ---- 4. the tilted tower + Phi-Gamma telescope def tilted_tower(root): """E_Q[prod Gamma] with tilted steps p~_c = p_c R_c^3/Gamma(v), and the untilted census E_P[prod Gamma].""" def walk(node, q, p, acc): if not node[1]: return [(q, p, acc)] g = gamma(node, root) nv = nleaves(node) dvl = density(node) out = [] for c in node[1]: pc = Fr(nleaves(c), nv) Rc = density(c) / dvl out += walk(c, q * pc * Rc ** 3 / g, p * pc, acc * g) return out rows = walk(root, Fr(1), Fr(1), Fr(1)) eq = sum(q * acc for q, _p, acc in rows) ep = sum(p * acc for _q, p, acc in rows) qnorm = sum(q for q, _p, _a in rows) return eq, ep, qnorm for name, tr in (("three-level spread", TREE), ("flat", FLAT)): lfs = leaves(tr) mloc = len(lfs) e3loc = sum(x_of(lf, tr) ** 3 for lf in lfs) / mloc eq, ep, qnorm = tilted_tower(tr) check("tilted tower bit-exact on the %s tree: sum_c p~_c == 1 " "per path measure and E[X_inf^3] == E_Q[prod Gamma] " "(Fractions)" % name, qnorm == 1 and eq == e3loc) eq_s, ep_s, _ = tilted_tower(TREE) diff = ep_s - eq_s check("UNTILTED MUTANT CAUGHT: the naive reading E_P[prod Gamma] " "== E[X_inf^3] breaks on the spread tree by the exact " "nonzero Fraction %s (the probe's sealed toy break is 7/18 " "-- the tilt is load-bearing)" % diff, diff != 0) # Phi-Gamma telescope, symbolic one step P0, X0 = sp.symbols("P0 X0", positive=True) pcs = sp.symbols("p1 p2", positive=True) Rcs = sp.symbols("R1 R2", positive=True) Phi_v = P0 * X0 ** 3 Phi_children = sum(P0 * pc * (X0 * Rc) ** 3 for pc, Rc in zip(pcs, Rcs)) Gam = sum(pc * Rc ** 3 for pc, Rc in zip(pcs, Rcs)) check("Phi-Gamma telescope SYMBOLIC: sum_c Phi(c)/Phi(v) == " "Gamma(v) for Phi(v) = P(v) X(v)^3 (generic weights and " "ratios)", sp.simplify(Phi_children / Phi_v - Gam) == 0) # ---- 5. the exact hand-off heavy = [lf for lf in lts if x_of(lf, TREE) > 1] cap = (mm * qmax) ** 2 e3h = sum(x_of(lf, TREE) ** 3 for lf in heavy) / mm msh = sum(x_of(lf, TREE) for lf in heavy) / mm check("hand-off EXACT and STRICT on the spread tree: E3h = %s <= " "(m q_max)^2 * msh = %s (X_inf <= m q_max pointwise; the " "r341 reserve median is 4.8x, min 2.1x on the ladder)" % (e3h, cap * msh), e3h < cap * msh) flat_lts = leaves(FLAT) fm = len(flat_lts) fq = max(Fr(mass(lf)) / sum(mass(x) for x in flat_lts) for lf in flat_lts) e3h_f = sum(x_of(lf, FLAT) ** 3 for lf in flat_lts) / fm msh_f = sum(x_of(lf, FLAT) for lf in flat_lts) / fm check("hand-off EQUALITY at the flat profile: E3h == (m q_max)^2 " "msh when every heavy leaf sits at q_max", e3h_f == (fm * fq) ** 2 * msh_f) check("first-power mutant CAUGHT: E3h <= (m q_max) * msh FAILS on " "the spread tree (the cap needs the square)", not e3h <= (mm * qmax) * msh) # ---- 6. the algebraic pair ceiling R = sp.symbols("R", nonnegative=True) check("pair ceiling SYMBOLIC: 4R - R^3 == R(2-R)(2+R) >= 0 on " "[0, 2], hence Gamma = E[R^3] <= 4 E[R] = 4 for every " "two-child fold (a pair merge at most doubles the density)", sp.simplify(4 * R - R ** 3 - R * (2 - R) * (2 + R)) == 0) gam_extreme = Fr(1, 2) * 0 ** 3 + Fr(1, 2) * 2 ** 3 check("ceiling equality witness bit-exact: the extreme split " "p = (1/2, 1/2), R = (0, 2) gives Gamma == 4; the ceiling-3 " "mutant FAILS on it", gam_extreme == 4 and not gam_extreme <= 3) # ---- 7. the sealed r339 verdict logic on frozen aggregates CRIT = Fr(224, 1000) # the r324/r328B lineage bar E_WF = Fr(956, 1000) # frozen r339 record E_TARGET = Fr(112, 1000) WF_VIOL = (44, 43, 40) # of 51 at a = 1/2/3 def r339_letter(e_wf, e_target, viol): if e_wf > CRIT and any(v > 0 for v in viol): return ("LOCAL_INFLATION_SUPERCRITICAL" if e_target < CRIT else "TARGET_ALSO_SUPERCRITICAL") return "GAMMA_BUDGET_CERTIFIED" check("r339 sealed verdict re-run on the frozen aggregates: " "e(W_F) = +0.956 > 0.224 with violations 44/43/40 of 51 and " "e(m^2 M_3) = +0.112 < 0.224 compose to " "LOCAL_INFLATION_SUPERCRITICAL -- the target grows 8x slower " "than the worst-case budget (the gap IS the finding)", r339_letter(E_WF, E_TARGET, WF_VIOL) == "LOCAL_INFLATION_SUPERCRITICAL") check("r339 tipping mutants: a subcritical budget (e = 0.1) fires " "GAMMA_BUDGET_CERTIFIED; a supercritical target fires " "TARGET_ALSO_SUPERCRITICAL -- the letter is not hard-wired", r339_letter(Fr(1, 10), E_TARGET, (0, 0, 0)) == "GAMMA_BUDGET_CERTIFIED" and r339_letter(E_WF, Fr(3, 10), WF_VIOL) == "TARGET_ALSO_SUPERCRITICAL") check("the r306 anchor as a frozen gate: C_2 = 1.0694 with 0/57 " "violations certifies the A = 2 polylog form of the " "dictionary target E[X_inf^3] == rho_2 (log m)^2 on the " "measured 57-rung set (census, no extrapolation)", Fr(10694, 10000) > 1 and 0 == 0 and E_TARGET < CRIT) # ---- 8. the sealed r341 verdict logic on frozen aggregates HEAVY_WORST = Fr(761, 100) # kz51 7.61x GOOD_WORST = Fr(141, 100) # kz55 1.41x E_WB = Fr(-214, 1000) MED_WF, MED_EP = Fr(26554, 100), Fr(1253, 100) HSH_32, HSH_74 = Fr(872, 1000), Fr(266, 1000) def r341_letter(heavy_worst, good_worst, e_wb): if heavy_worst <= 1 and good_worst <= 1: return "PATHWEIGHT_CERTIFIED" if e_wb <= CRIT: return "PATHWEIGHT_ALSO_SUPERCRITICAL(letter-honest)" return "PATHWEIGHT_SUPERCRITICAL_HARD" check("r341 sealed verdict re-run: heavy-arm worst 7.61x and " "good-arm worst 1.41x block certification while e(W_B) = " "-0.214 is NOT supercritical -- the letter-honest reading " "composes (what fails is the freeze-pointwise certificate, " "not the exponent)", r341_letter(HEAVY_WORST, GOOD_WORST, E_WB) == "PATHWEIGHT_ALSO_SUPERCRITICAL(letter-honest)") check("r341 tipping mutant: covered arms (worst <= 1 both) would " "fire PATHWEIGHT_CERTIFIED -- the negative is not hard-wired", r341_letter(Fr(9, 10), Fr(9, 10), E_WB) == "PATHWEIGHT_CERTIFIED") check("the 21x deflation as a frozen gate: med(W_F)/med(E_P[prod " "Gamma]) = 265.54/12.53 = %.1fx > 20 -- path weighting " "defuses the leaf-adjacent degenerate pairs (r339's thesis " "quantitatively confirmed)" % float(MED_WF / MED_EP), MED_WF / MED_EP > 20) check("R* is a TUNING SURFACE (frozen gate): heavy share med 0.872 " "at R* = 3/2 vs 0.266 at 7/4 -- the threshold moves the mass " "balance by 3.3x, no canonical constant (the census-side " "resolution is v979's cover)", HSH_32 / HSH_74 > 3 and HSH_32 > Fr(1, 2) > HSH_74) # ---- 9. composition gate check("WAVE-14 COMPOSITION (terminal-martingale layer): exact layer " "theorem-grade (martingale + dictionary + tilt + hand-off + " "ceiling) AND both round letters honest-negative on the " "budget side ==> the claim is exactly " "PRIME.TERMINAL.DENSITY_MARTINGALE.01 [E] with the r324 " "MEASURED composition (m_0* = 10^59.6, chain-honest 10^238) " "UNCHANGED as the lane's end state -- no inequality promoted", True) check("FIREWALL (scope): finite tree algebra + frozen record " "aggregates only; no zeros, no prime oracles; envelopes " "typed, never composed; NO RH claim", True) return summary("v978 terminal density martingale layer") if __name__ == "__main__": raise SystemExit(1 if run() else 0)