7.6 KiB
Mass-Number Admissibility Closure Conjecture
Status: FORMALLY_STABLE_READY_FOR_PROOF_ENGINEERING
Canonical GCL: EQUATION/MASS_NUMBER/ADMISSIBILITY_CLOSURE/GEODESIC_METRIC
Notion canonical: https://app.notion.com/p/352375cc7bfc81aabfaec84b82f49394
Core doctrine
Mass is not distance. Mass becomes distance only through admissibility closure.
A mass-number field is not itself a metric space. It is a reality-local admissibility potential over candidates. Normalized reducibility phi converts admissible reduction into bounded confidence; compatibility-weighted divergence converts reducibility into pairwise translation cost; symmetrization and viability filtering produce an admissibility graph; shortest-path closure over that graph induces a pseudometric; quotienting zero-distance candidates yields a lawful metric space. Closure is achieved exactly when every candidate is promoted, connected, typed as residual, category-rescued, quarantined, or rejected.
Deterministic stochastic coarse-graining doctrine
Deterministic stochastic coarse-graining is signal, just not signal that can be aligned in the original coordinate frame.
A raw observer sees signal + noise. A naive denoiser treats noise as error and discards it. The Mass-Number Lens treats some residual noise as unaligned signal: structure that behaves stochastically at the current scale or basis, but may form invariant foci after coarse-graining, unfolding, and residual typing.
Canonical decomposition:
ObservedField
= AlignedSignal
+ MisalignedDeterministicStochasticSignal
+ TypedResidualNoise
Operational rule:
Residual noise is not promoted by default.
Residual noise becomes candidate signal only if deterministic coarse-graining produces stable invariant foci.
Collapsed doctrine:
Signal is what remains invariant under the right coarse-graining.
Noise is what has not yet found its admissible alignment map.
Conservation rule:
Mass cannot vanish into "noise".
It must become one of:
aligned signal,
unaligned/coarse-grained signal,
typed residual,
category-rescued branch,
quarantine,
or rejection.
Transition chain
M = admissibility potential
phi = normalized reducibility
delta = raw admissibility divergence
c = symmetrized admissibility edge cost
G_theta = viable admissibility graph
d_theta = shortest-path closure distance
X / ~0 = quotient metric space
Mass-number potential
For a candidate x in domain D under frame R:
M_D,R(x)
= [sum_i w_i,D * rho_i,D(x) * kappa_i,D(x) * alpha_i,D(x)]
/
[1 + T_D,R(x) + S_D,R(x) + L_D,R(x) + V_D,R(x) + O_D,R(x) + Delta_Drift_D,R(x)]
Interpretation:
Mass Number = Admissible Reduction / Residual Risk
M is a scalar potential. It is not a distance.
Mass-Number Lens and Foci
A scalar mass number can be unfolded through invariant-energy lenses:
MassNumberScalar M_D,R(x)
-> spectral energy field
-> Brownian / diffusion energy field
-> recurrence field
-> vibration-mode field
-> residual-risk field
-> n-space shape vector
-> Mass Number Foci
Mass Number Foci are higher-dimensional convergence basins revealed by the unfolding. They are not raw points. They are lens-formed concentration zones where spectral energy, Brownian energy, mode persistence, recurrence, and residual typing agree strongly enough to concentrate admissibility mass.
Short doctrine:
Mass is potential.
The lens forms foci.
Foci organize the forest.
Mass-Number Stochastic Conservation
When a mass number is unfolded through stochastic or residual fields, total admissibility mass must be accounted for across promoted foci, candidate foci, typed residuals, category-rescued branches, quarantines, and rejections.
M_before_unfold
≈ M_promoted_foci
+ M_candidate_foci
+ M_typed_residuals
+ M_category_misplaced
+ M_quarantined
+ M_rejected
+ epsilon_loss
with:
epsilon_loss <= tolerance
No unexplained mass leakage is allowed.
Normalized reducibility
phi_D,R(x)
= R_admissible_D,R(x)
/
[R_admissible_D,R(x) + R_residual_D,R(x)]
Required bound:
0 <= phi_D,R(x) <= 1
Canonical bounded divergence
Use the smoothed divergence:
delta(x,y)
= -ln(epsilon + (1 - epsilon) * K(x,y) * sqrt(phi(x) * phi(y)))
Proof-stability constraints:
0 < epsilon <= 1
0 <= K(x,y) <= 1
0 <= phi(x), phi(y) <= 1
Then:
0 <= delta(x,y) <= -ln(epsilon)
This avoids logarithmic singularities and gives finite bounded raw divergence.
Symmetrized edge cost
c(x,y)
= 1/2 * [delta(x,y) + delta(y,x)]
+ HandoffPenalty(x,y)
+ DriftPenalty(x,y)
Side conditions:
HandoffPenalty(x,y) >= 0
DriftPenalty(x,y) >= 0
penalties are symmetric or explicitly symmetrized
Admissibility graph
G_theta = (X, E_theta)
(x,y) in E_theta iff
c(x,y) < infinity
M_D,R(x) >= theta_min
M_D,R(y) >= theta_min
residuals are typed
Closure distance
d_theta(x,y)
= inf over paths p:x~>y of sum_{(u,v) in p} c(u,v)
If no admissible path exists, the candidates are disconnected unless a TypedResidual or adapter bridge creates a lawful edge.
Operational closure predicate
Closed_D,R(X) iff for all x in X,
Status(x) in {
Promoted,
Connected,
TypedResidual,
CategoryMisplaced,
Quarantined,
Rejected
}
Untyped residual drift is impossible after closure.
Lean proof roadmap
1. phi_bounded
prove 0 <= phi <= 1
2. compatibility_bounded
prove 0 <= K <= 1
3. raw_divergence_nonneg
prove delta(x,y) >= 0 from bounded log argument
4. sym_cost_nonneg
prove c(x,y) >= 0
5. sym_cost_symmetric
prove c(x,y) = c(y,x)
6. closure_pseudometric
prove shortest-path closure satisfies pseudometric laws
7. zero_distance_equivalence
define x ~0 y iff d_theta(x,y) = 0
8. quotient_closure_metric
prove the quotient by ~0 is a metric space
Target theorem names
theorem phi_bounded : 0 <= phi x ∧ phi x <= 1 := by
sorry
theorem raw_divergence_nonneg : 0 <= delta x y := by
sorry
theorem sym_cost_symmetric : c x y = c y x := by
sorry
theorem closure_pseudometric : PseudoMetricSpace X := by
sorry
theorem massNumber_admissibilityClosure_metric :
MetricSpace (AdmissibleQuotient X) := by
sorry
Additional proof targets for the lens layer:
theorem stochasticConservation_accounted :
accountedMass + epsilonLoss = initialMass := by
sorry
theorem coarseGrainedSignal_requiresInvariantFocus :
PromotedCoarseGrainedSignal x -> ExistsStableFocus x := by
sorry
Shell-mass side conjecture
In S3C / DIAT, shell mass S_n = a*b is not a distance. It is a throat or curvature weight: high shell mass marks representational ambiguity near the midpoint between adjacent perfect squares. It should weight adapter pressure, not replace d_theta.
Category-error rescue rule
CategoryMisplaced(x) iff
Var_D(M_D,R(x)) is high
and exists D',R' such that M_D',R'(x) >= theta_rescue
Low mass in one domain is not falsehood by itself. It may indicate wrong-frame evaluation.
Definition of Done
A candidate field is closed when no candidate remains untyped, every viable candidate is connected or promoted, every cross-domain mismatch is typed as residual or category-misplaced, and every unsafe candidate is quarantined or rejected.
For deterministic stochastic coarse-graining, a candidate residual field is closed when every residual component is assigned to aligned signal, unaligned/coarse-grained signal, typed residual, category-rescued branch, quarantine, or rejection, and the mass ledger balances within tolerance.