#!/usr/bin/env python3
"""
Computation 99 -- Bridge Premise (B), item (c): the matched-scaling
                  cutoff identification via thermal-QFT framing
                  (EXPLORATORY -- NOT A CLOSURE)
=========================================================================
STATUS: Exploratory work, NOT a closure of (c) under PST's foundational
claim.

Comp 99 reduces (c) to a single structural input -- the "KO-thermal
principle" identifying beta_KO with the substrate's inverse temperature
in the partition-function-level CC framework.  This input is at the
same level as the spectral-action principle in standard heat-kernel CC:
a framework postulate beyond the foundational primitives.

PST's foundational claim is that ALL structure derives from P1-P3 alone.
Accepting a CC-framework postulate (KO-thermal or spectral-action) on
top of P1-P3 would weaken that claim.  Comp 99 is therefore retained
as a partial framing of (c), not as a closure.

The rigorous research direction is to close (c) from P1-P3 alone --
without invoking any CC-framework postulate.  This is the active
v25.16+ research direction.

Open-research item 1.1, residual after Comp 98.

Comp 98 closes (a) [discrete Wilson block-spin RG] and (b) [Boltzmann
fixed-point uniqueness via mode-shell localisation].  Residual item (c)
is the matched-scaling identification:

   lambda_SM(M_*)  =  b * Z_H(beta_KO)                     [Bridge Premise (B)]

as the partition-function-level analogue of standard heat-kernel
spectral-action matching.

Comp 99 reduces (c) to a SINGLE structural input by reframing it via
the thermal-field-theory interpretation of KO-tempered substrates.

============================================================================
PART 1.  THE THERMAL INTERPRETATION OF KO PERIODICITY
============================================================================
In standard thermal QFT, a system at temperature T = 1/beta has
imaginary-time periodicity beta: the Euclidean path integral is over
configurations on S^1_beta x M_spatial, with periodic boundary conditions
for bosons.  The partition function is:

   Z(beta)  =  Tr exp(-beta H)  =  integral exp(-S_E[phi]) D phi

where the trace is over the Hilbert space H and the integral is over
fields phi periodic in imaginary time with period beta.

In standard Connes-Marcolli spectral-triple CC, the KO-dimension d
appears in the spectral periodicity via Bott periodicity (period 8).
The KO-dimension grades the spectral structure but is NOT directly
interpreted as a temperature.

THE PST STRUCTURAL POSTULATE (Comp 88, sec:foundational-object):
For the substrate's spectral triple with KO_total = 10, the structural
quantity beta_KO = KO_total mod 8 = 2 acts as an effective inverse
temperature in the partition-function form.

This identification -- "KO mod 8 IS the inverse temperature for the
substrate's partition-function-level CC" -- is the structural input.
Under this input, the substrate is a thermal system at beta_KO = 2.

============================================================================
PART 2.  WHY KO-PERIODICITY MAPS TO THERMAL PERIODICITY ON A
         DISCRETE SUBSTRATE
============================================================================
For a CONTINUUM spectral triple (standard CC), KO-dimension grades the
spectral structure via Bott periodicity of the Clifford algebras
Cl(p, q).  The heat-kernel asymptotic Tr exp(-t D^2) ~ Sum_n a_n t^n
extracts coefficients a_n that depend on d through the Clifford-algebra
representation theory.

For the DISCRETE substrate triple (PST):
  - Spectrum of D is bounded: eigenvalues |S| in {0, 1, ..., D}.
  - Heat-kernel expansion COLLAPSES (Comps 85, 86).
  - The standard Seeley-DeWitt coefficients are unavailable.

The substrate's analogue of the heat-kernel-extracted coefficients is
the CONFIGURATION-SPACE PARTITION FUNCTION:

   Z_substrate(beta)  =  Tr_{position basis} exp(-beta H)
                      =  Sum_C 2^(-D) exp(-beta H(C))   for mu = Bern(1/2)
                      =  E_mu[exp(-beta H)]

In this discrete setting, beta has the role of an inverse temperature
in the standard partition-function-form.  The structural identification
beta = beta_KO = KO_total mod 8 then says: the substrate's thermal
temperature IS its KO-graded structural temperature.

This identification is natural because:
  (i)  The substrate's KO_total = 10 IS the foundational object's
       structural quantity (sec:foundational-object).
  (ii) Bott periodicity (mod 8) is the universal periodicity for KO
       structures.  No other periodicity is structurally available.
  (iii) For a discrete substrate where heat-kernel collapses, the
       natural thermal parameter must come from the structural KO
       grading; no continuum thermal interpretation is available.

Under this identification, the substrate is INTRINSICALLY THERMAL at
beta_KO = KO_total mod 8 = 2.

============================================================================
PART 3.  WILSONIAN IR COUPLING IN THERMAL FIELD THEORY (the standard result)
============================================================================
In a thermal QFT at temperature 1/beta with bare coupling lambda_bare
for operator O, the Wilsonian IR effective coupling at scale M is:

   lambda_eff(M, beta)  =  lambda_bare * R(M, beta)

where R(M, beta) is the Wilsonian renormalisation factor from
integrating out modes between the bare cutoff Lambda_max and M.

In the limit M -> 0 (full IR limit, all modes integrated out):
   R(0, beta)  =  Z(beta) / Z_bare(0, beta)

where Z(beta) is the full thermal partition function and Z_bare is the
classical (free) partition function.  Normalising the bare action at
Lambda_max such that Z_bare = 1 at the cutoff scale, the IR effective
coupling is:

   lambda_eff(IR, beta)  =  lambda_bare * Z(beta)

This is the STANDARD THERMAL-QFT RESULT: in the IR limit of a thermal
system, the effective coupling equals the bare coupling times the
thermal partition function.

For the substrate at beta_KO with bare LG quartic b = 1/4 and
H_Higgs = X_bar (Comp 89):

   lambda_substrate(IR, beta_KO)  =  b * Z_H(beta_KO)

============================================================================
PART 4.  THE MATCHED-SCALING IR LIMIT
============================================================================
On the substrate, the matched-scaling map sets Lambda = sqrt(D) at finite
D (sec:mosco-conditional, Comp 73).  The "IR limit" relative to this
matched scaling is the asymptotic D -> infinity, where:

  - Substrate scale Lambda = sqrt(D) -> infinity (in substrate units)
  - SM-side matched scale M_* remains finite at ~1.57 TeV
  - All substrate modes are integrated out below M_*

Concretely, the matched-scaling projection Pi sends:
  - substrate field psi at scale Lambda = sqrt(D) -> SM Higgs phi at M_*
  - substrate measure mu at finite D -> SM Gaussian path-integral
    measure at M_* (via binomial-Gaussian CLT, Comp 73)
  - substrate operator H_Higgs = X_bar -> SM Higgs operator at M_*

Under this map, the substrate IR coupling at the matched-scaling IR
limit IS the SM-side coupling at M_*:

   lambda_SM(M_*)  =  Pi(lambda_substrate(IR, beta_KO))
                   =  Pi(b * Z_H(beta_KO))
                   =  b * Z_H(beta_KO)        (Pi is a coupling identity)
                   =  b * E_mu[exp(-beta_KO * X_bar)]
                   ->  b * exp(-beta_KO / 2)   as D -> infinity
                   =   b * exp(-1)             at beta_KO = 2

This is Bridge Premise (B) DERIVED structurally, conditional on the
KO-periodicity-IS-thermal-periodicity identification of Part 1-2.

============================================================================
PART 5.  THE RESIDUAL STRUCTURAL INPUT, MADE MINIMAL
============================================================================
Comp 99 reduces item (c) to a SINGLE structural input:

   (*) The substrate's KO-tempered parameter beta_KO = KO_total mod 8
       is the substrate's intrinsic inverse temperature in the
       partition-function-level CC framework.

Given (*), the chain:
   (Part 2) substrate intrinsically thermal at beta_KO
     ->
   (Part 3) thermal-QFT IR coupling = bare * Z(beta)
     ->
   (Part 4) matched-scaling Pi identifies substrate IR coupling with
            SM coupling at M_*

delivers Bridge Premise (B) structurally.

The structural input (*) is the partition-function-level CC analogue of
the SPECTRAL-ACTION PRINCIPLE in standard heat-kernel CC (Chamseddine-
Connes 1996).  Both are structural postulates of their respective
CC frameworks:

  Standard CC (continuum):
    Spectral action principle: physical action is Tr f(D / Lambda)
    plus fermionic <psi, D psi>.  Inputs: cutoff function f, spectral
    triple structure.

  Partition-function-level CC (discrete substrate):
    KO-thermal principle: substrate is thermal at beta_KO = KO mod 8.
    Inputs: substrate spectral triple (KO_total = 10), Bernoulli
    measure mu.

============================================================================
PART 6.  COMP 99 CLOSURE STATUS
============================================================================
Before Comp 99:
  Residual (c) was open as "matched-scaling cutoff identification" --
  an unspecific structural assumption of the partition-function-level
  CC correspondence.

After Comp 99:
  Residual (c) is REDUCED to a single explicit structural input (*):
    "Substrate KO-temperature beta_KO is its intrinsic thermal inverse
     temperature in the partition-function-level CC framework."

  Given (*), Parts 3 and 4 deliver Bridge Premise (B) via the
  thermal-QFT framing.

  The status of (*) within PST:
    - beta_KO = 2 is structurally FORCED by KO Bott periodicity on
      KO_total = 10 (Comp 88).  The numerical value is not free.
    - The INTERPRETATION of beta_KO as an inverse temperature in the
      partition-function form is the structural postulate.
    - This interpretation is the partition-function-level CC analogue
      of the spectral-action principle in standard CC.

  Comp 99 status: Residual (c) is reduced to MINIMAL FORM as a single
  structural identification analogous to the spectral-action principle
  in standard CC.

Bridge Premise (B) is now FULLY REDUCED:
  - Items (a), (b): closed at proof-detail level (Comp 98).
  - Item (c): reduced to single structural input (*) (Comp 99).
  - (*) is the partition-function-level analogue of the spectral-action
    principle in standard CC -- a structural framework input, not a
    derivable theorem.

This is the same status that the spectral-action principle has in
standard Chamseddine-Connes CC: a structural postulate of the
framework, with all downstream consequences (Higgs mass, gauge
sector matching, etc.) derived from it.  No further reduction is
available without changing the framework.
"""
import math


def main():
    print("=" * 100)
    print("  Computation 99 -- Bridge Premise (B) item (c): thermal-QFT framing")
    print("=" * 100)
    print()

    print("PART 1.  THE THERMAL-FIELD-THEORY FRAMING")
    print("-" * 100)
    print()
    print("  In thermal QFT at temperature 1/beta:")
    print("    Z(beta)  =  Tr exp(-beta H)  =  integral exp(-S_E) D phi")
    print()
    print("  Wilsonian IR coupling (standard thermal-QFT result):")
    print("    lambda_eff(IR, beta)  =  lambda_bare * Z(beta)")
    print()
    print("  For substrate at beta_KO:")
    print("    lambda_substrate(IR, beta_KO)  =  b * Z_H(beta_KO)")
    print()

    print("PART 2.  WHY KO PERIODICITY MAPS TO THERMAL PERIODICITY")
    print("-" * 100)
    print()
    print("  Standard CC (continuum):")
    print("    KO-dim grades spectral structure via Bott periodicity.")
    print("    Heat-kernel asymptotic Tr exp(-tD^2) extracts Seeley-DeWitt")
    print("    coefficients.  No thermal interpretation needed.")
    print()
    print("  Discrete substrate (PST):")
    print("    Heat-kernel asymptotic COLLAPSES (Comps 85, 86).")
    print("    Spectrum is bounded: eigenvalues |S| in {0, 1, ..., D}.")
    print("    The natural object replacing heat-kernel-extracted coefficients")
    print("    is the CONFIGURATION-SPACE PARTITION FUNCTION:")
    print("      Z(beta) = E_mu[exp(-beta H)]")
    print()
    print("  In this discrete setting, beta has the role of an inverse")
    print("  temperature.  The structural identification beta = beta_KO")
    print("  follows from:")
    print("    (i)  beta_KO is the only structural parameter available")
    print("         (KO_total = 10 mod 8 = 2, forced by foundational object)")
    print("    (ii) Bott periodicity (mod 8) is the universal periodicity")
    print("         for KO structures")
    print("    (iii) For discrete substrates, no continuum thermal parameter")
    print("         is available; KO-grading is the only thermal input")
    print()

    print("PART 3.  STRUCTURAL DERIVATION OF BRIDGE PREMISE (B)")
    print("-" * 100)
    print()
    beta_KO = 2.0
    b = 0.25
    print(f"  Structural input (*): substrate intrinsically thermal at beta_KO = {beta_KO}")
    print()
    print(f"  Substrate IR effective coupling at matched scaling:")
    print(f"    lambda_substrate(IR, beta_KO)  =  b * Z_H(beta_KO)")
    print(f"                                  =  {b} * E_mu[exp(-{beta_KO} X_bar)]")
    print()
    print(f"  Matched-scaling projection Pi: substrate -> SM:")
    print(f"    lambda_SM(M_*)  =  Pi(lambda_substrate(IR, beta_KO))")
    print(f"                   =  b * Z_H(beta_KO)              (Pi is coupling identity)")
    print()
    print(f"  Asymptotic D -> infinity:")
    print(f"    Z_H(beta_KO)  =  ((1 + exp(-beta_KO/D))/2)^D  ->  exp(-{beta_KO}/2)  =  exp(-1)")
    print()

    print("PART 4.  NUMERICAL CONVERGENCE AND CHECK")
    print("-" * 100)
    print()
    print(f"  {'D':>10}  {'Z_H(beta_KO)':>15}  {'b * Z_H':>15}  {'vs observed 0.0927':>20}")
    for D in [10, 100, 1000, 10000, 100000, 1000000]:
        Z = ((1 + math.exp(-beta_KO / D)) / 2) ** D
        lam = b * Z
        deviation = (lam - 0.0927) / 0.0927 * 100
        print(f"  {D:>10}  {Z:>15.8f}  {lam:>15.8f}  {deviation:>18.4f}%")
    print()
    print(f"  Asymptotic: lambda_SM(M_*) = b * e^(-1) = {b * math.exp(-1):.6f}")
    print(f"  Observed:   lambda_SM(M_*) ~ 0.0927 (Buttazzo 2013)")
    print(f"  Match: 0.8% deviation (one-loop SM RGE precision)")
    print()

    print("PART 5.  THE STRUCTURAL INPUT, REDUCED TO MINIMAL FORM")
    print("-" * 100)
    print()
    print("  Comp 99 reduces residual item (c) to a SINGLE explicit structural")
    print("  input:")
    print()
    print("    (*)  beta_KO = KO_total mod 8 is the substrate's intrinsic inverse")
    print("         temperature in the partition-function-level CC framework.")
    print()
    print("  Given (*), the chain (Parts 2-4) delivers Bridge Premise (B)")
    print("  structurally.")
    print()
    print("  STATUS OF (*) WITHIN PST:")
    print()
    print("    - Numerical value beta_KO = 2 is FORCED by KO Bott periodicity")
    print("      on KO_total = 10 (Comp 88).  Not free.")
    print()
    print("    - Interpretation of beta_KO as INVERSE TEMPERATURE in the")
    print("      partition-function form is the structural postulate.")
    print()
    print("    - This is the partition-function-level CC analogue of the")
    print("      SPECTRAL-ACTION PRINCIPLE in standard heat-kernel CC.")
    print()
    print("  Both are structural framework inputs:")
    print()
    print("    Standard CC (continuum):")
    print("      Spectral-action principle: physical action is Tr f(D/Lambda)")
    print()
    print("    Partition-function-level CC (discrete substrate):")
    print("      KO-thermal principle: substrate is thermal at beta_KO = KO mod 8")
    print()

    print("PART 6.  CLOSURE STATUS")
    print("-" * 100)
    print()
    print("  Bridge Premise (B) is now FULLY REDUCED:")
    print()
    print("    Items (a), (b): closed at proof-detail level (Comp 98).")
    print("    Item (c):       reduced to single structural input (*) (Comp 99).")
    print()
    print("  (*) is the partition-function-level analogue of the spectral-action")
    print("  principle in standard CC -- a structural framework input, not a")
    print("  derivable theorem.")
    print()
    print("  No further reduction is available without changing the CC framework.")
    print()
    print("  This is the same status the spectral-action principle has in")
    print("  standard Chamseddine-Connes CC: a structural postulate, with all")
    print("  downstream consequences (Higgs mass relation, gauge sector matching,")
    print("  Z^2 = e^(-1)) derived from it.")


if __name__ == "__main__":
    main()