#!/usr/bin/env python3 """ PROVENANCE: PROOF Computation 117 -- Bridge Premise (B), milestone M8(a): the embedding-norm selection is STRUCTURAL, not empirical ========================================================================= STATUS (v26.19+): M8(a) of the wave-function attack on Bridge Premise (B). After the v26.15 referee (P1, accepted), the open content of (B) is the substrate->SM-coupling matching, and one sub-caveat inside the substrate side was that the embedding-norm-vs-2pt-residue selection (Comp 114 / M5) "rests partly on the empirical finiteness of lambda_SM, not a theorem alone". This computation removes that empirical crutch. THE TWO READINGS (recap, Comp 114) ================================== The wave-function renormalisation is lambda_SM(M_*) = b * Z_phi^2. Two candidate substrate objects for Z_phi^2: (A) embedding norm Z_phi^2 = E_mu[exp(-beta_KO X_bar)] -> e^-1 (O(1)); (B) 2-pt residue Z_phi^2 = Var_mu(X_bar) = 1/(4D) -> 0. Comp 114 selected (A) and excluded (B) by noting (B) gives lambda_SM -> 0, "excluded by the finiteness of the observed quartic" -- an EMPIRICAL exclusion. THE STRUCTURAL ARGUMENT (M8a) ============================= The order parameter X_bar = (1/D) sum_a C_a is an INTENSIVE average. Its 2-pt residue is its variance, and that variance vanishes as 1/(4D) by the LAW OF LARGE NUMBERS (P1: the C_a are i.i.d., so the intensive average self-averages to its mean 1/2 with fluctuations O(1/sqrt(D))). This vanishing is a generic large-D effect, not a field-strength. A field-strength Z_phi is the normalisation relating the substrate field to the CANONICALLY-NORMALISED emergent field; it must be finite and non-zero (O(1)) for a canonical emergent field to exist at all. Reading (B) sets Z_phi^2 = (an intensive variance) -> 0, i.e. it identifies the field-strength with a quantity the LLN drives to zero. That is structurally inadmissible: a vanishing Z_phi^2 means lambda_SM -> 0 in the substrate limit D->inf -- a FREE emergent Higgs, no canonical interacting field. An intensive order parameter cannot play the role of a fundamental field whose propagator residue is its susceptibility. Reading (A) sets Z_phi^2 = the embedding norm (the tempered-vacuum amplitude of the order-parameter sector, Comp 100), which is O(1) -> e^-1 and DOES normalise a canonical emergent field, giving an interacting lambda_SM = b e^-1. The emergent EFT below M_* is the (interacting) Standard Model -- this is framework premise F4, a STRUCTURAL claim of PST, independent of the measured value of lambda_SM. An interacting emergent Higgs requires Z_phi^2 = O(1); the LLN excludes (B)'s vanishing susceptibility; so the field-strength is the O(1) embedding norm (A). The selection is forced by F4 + the LLN, NOT by the empirical value 0.0927. This computation verifies the numerics underlying the argument and states honestly what it does and does not remove. ========================================================================= """ import math def var_xbar(D, p=0.5): """2-pt residue reading: Var_mu(X_bar) = p(1-p)/D = 1/(4D) at p=1/2.""" return p * (1.0 - p) / D def embedding_norm(D, beta=2.0): """Embedding-norm reading: E_mu[exp(-beta X_bar)] = ((1+e^{-beta/D})/2)^D.""" return ((1.0 + math.exp(-beta / D)) / 2.0) ** D def main(): print("=" * 72) print("Computation 117: M8(a) -- embedding-norm selection is structural") print("=" * 72) print() b = 0.25 # bare PST quartic (doublet convention) e_inv = math.exp(-1.0) # ---- 1. the susceptibility vanishes by the LLN (intensive average) ---- print("1. THE 2-PT RESIDUE IS AN INTENSIVE VARIANCE -> 0 BY THE LLN") print("-" * 72) print(" X_bar = (1/D) sum_a C_a, C_a i.i.d. Bernoulli(1/2).") print(" Var(X_bar) = 1/(4D); D * Var -> 1/4 (pure self-averaging).") print(f" {'D':>7} {'Var(X_bar)':>14} {'D*Var':>8} {'lambda_SM=b*Var':>18}") for D in (10, 100, 1000, 10000): v = var_xbar(D) print(f" {D:>7} {v:>14.3e} {D*v:>8.4f} {b*v:>18.3e}") print(f" {'limit':>7} {0.0:>14.3e} {0.25:>8.4f} {0.0:>18.3e}") print() print(" Reading (B) Z_phi^2 = Var(X_bar) -> 0, so lambda_SM -> 0:") print(" a FREE emergent Higgs in the substrate limit. The vanishing") print(" is the law of large numbers, not a field-strength.") print() # ---- 2. the embedding norm is O(1) ---- print("2. THE EMBEDDING NORM IS O(1) -> e^-1") print("-" * 72) print(f" {'D':>7} {'E_mu[e^-2Xbar]':>16} {'lambda_SM=b*norm':>18}") for D in (10, 100, 1000, 10000): z = embedding_norm(D) print(f" {D:>7} {z:>16.6f} {b*z:>18.6f}") print(f" {'limit':>7} {e_inv:>16.6f} {b*e_inv:>18.6f}") print() print(" Reading (A) Z_phi^2 = embedding norm -> e^-1 (O(1)), so") print(" lambda_SM -> b e^-1 = 0.0920: an INTERACTING emergent Higgs.") print() # ---- 3. the structural distinction: variance vs amplitude ---- print("3. THE TWO OBJECTS ARE STRUCTURALLY DIFFERENT KINDS") print("-" * 72) print(" Var(X_bar) = connected 2-pt function of an INTENSIVE") print(" average; an O(1/D) self-averaging quantity") print(" (LLN); -> 0. Measures fluctuation SIZE.") print(" E_mu[exp(-bX_bar)] = the tempered-vacuum amplitude / MGF of") print(" the order-parameter sector (Comp 100);") print(" O(1); -> e^-1. Normalises the FIELD.") print() print(" For a FUNDAMENTAL scalar the field-strength equals the 2-pt") print(" residue -- but only because that residue is O(1). For an") print(" INTENSIVE order parameter the 2-pt residue is the variance,") print(" which the LLN sends to 0; it therefore cannot be a") print(" field-strength. The emergent Higgs is the PROJECTED field,") print(" normalised by the embedding norm (A), not a fundamental field") print(" with residue = susceptibility (B).") print() # ---- 4. assessment ---- print("=" * 72) print("ASSESSMENT: does M8(a) remove the empirical crutch?") print("=" * 72) print() print(" THE SELECTION, RESTATED STRUCTURALLY:") print(" - Reading (B) sets the field-strength to the intensive") print(" susceptibility, which -> 0 by the LLN (P1's independence);") print(" lambda_SM -> 0, a FREE emergent Higgs. A vanishing Z_phi^2") print(" defines no canonical emergent field.") print(" - Reading (A) sets it to the O(1) embedding norm; lambda_SM ->") print(" b e^-1, a canonical INTERACTING emergent Higgs.") print(" - Framework premise F4 (the emergent EFT below M_* is the") print(" interacting Standard Model) requires an interacting Higgs,") print(" hence Z_phi^2 = O(1), hence reading (A). This uses F4 + the") print(" LLN, NOT the measured value lambda_SM = 0.0927.") print() print(" WHAT THIS REMOVES:") print(" The Comp 114 exclusion of (B) said 'lambda_SM is finite") print(" (observed), so not (B)'. That empirical-VALUE dependence is") print(" removed: (B) is now excluded because it makes the emergent") print(" theory free in the substrate limit, contradicting F4 -- a") print(" structural premise, not a measurement. The value 0.0927 is no") print(" longer an input to the SELECTION (it remains the empirical") print(" TEST of the resulting prediction b e^-1 = 0.0920).") print() print(" WHAT THIS DOES NOT DO (honest residual):") print(" - It still rests on F4 (emergent EFT = interacting SM), a") print(" framework premise. F4 is structural, but it is a premise;") print(" M8(a) trades 'empirical value of lambda_SM' for 'F4', not for") print(" nothing.") print(" - It does not derive the matching equation lambda_SM(M_*) =") print(" b Z_phi^2 itself (Wilsonian threshold matching, F4-F7) from") print(" the SM side -- that remains the open step of (B). M8(a)") print(" sharpens the substrate-side field-strength selection only.") print() print(" NET: sub-caveat (a) is closed at the substrate-side level. The") print(" embedding-norm reading is forced by P1's LLN + F4, with the") print(" measured lambda_SM demoted from an INPUT (to the selection) to") print(" a TEST (of the prediction). The remaining open content of (B)") print(" is the matching equation to the SM-measured coupling (F4-F7)") print(" and the F11 all-orders rigour write-up.") if __name__ == "__main__": main()