IndisputableMonolith.Gravity.SevenGaps.StrainDynamicsKernelReach
Packages the recognition kernel's cost content (the exact premise set that forces the cost functional to be J) and proves its reach is strictly limited: reciprocal involution on ratios preserves cost but never forces cost-spending or residue collapse. Gravity and strain-dynamics workers cite it to separate forced J-uniqueness from any dynamical spending law. The argument is definitional packaging plus involution lemmas and two negative entailment theorems.
claimThe recognition kernel's cost content is the premise package (reciprocity, normalization, calibration, continuity, and the composition law on a single ratio) that forces the cost to be $J(x)=\frac12(x+x^{-1})-1$. The reciprocal map $x\mapsto x^{-1}$ is a continuous cost-preserving involution whose even iterates stay away from any forced spending step. Kernel cost content does not entail a cost-spending substrate or a derivation of residue; a nonempty cost-spending substrate and a collapse step sit outside that package.
background
Recognition Science forces the canonical cost $J(x)=\frac12(x+x^{-1})-1$ (equivalently $\cosh(\log x)-1$) from the Recognition Composition Law together with reciprocity, normalization, calibration, and continuity. That uniqueness is T5 in the forcing chain; the Cost and SymplecticAction modules record that $J$ is the symplectic action of the double-entry ledger.
This module sits in the SevenGaps gravity layer, downstream of DescentPrincipleUniversality: carrier dynamics is already reduced to steepest-descent on hinge-link strain, so the remaining question is what the bare recognition kernel actually knows about cost. Kernel cost content is defined as exactly the premise package consumed by the law that forces $J$, with every field a property of the cost of a single ratio.
The reciprocal involution $x\mapsto 1/x$ is the natural symmetry of that package. Cost spending and residue collapse are separate dynamical claims about the substrate; the module isolates them as objects that the kernel package does not entail.
proof idea
Definitional core: KernelCostContent is the structure of single-ratio cost axioms matching the T5 input; jcost_kernelCostContent witnesses that $J$ satisfies it. Reciprocal involution is introduced as the map $x\mapsto x^{-1}$, then shown continuous, cost-preserving, and of even iterate identity on the relevant domain, with a never-reaches lemma blocking forced arrival at a spending state. Two negative results close the reach argument: kernel cost content does not entail cost spending, and there is no kernel derivation of residue. CostSpendingSubstrate, its nonemptiness witness, and collapseStep are supplied as the external objects that would be needed for spending dynamics, not as consequences of the kernel package.
why it matters in Recognition Science
SevenGaps gravity work needs a clean cut between what the recognition kernel forces (unique $J$ via RCL and T5) and what strain dynamics must still assume (cost spending, residue collapse, descent on the substrate). This module supplies that cut: KernelCostContent is the precise interface to law_of_logic_forces_jcost, while the negative entailment theorems stop any silent smuggling of spending laws out of the kernel alone.
It feeds the broader strain-dynamics story opened in DescentPrincipleUniversality (metric choice cannot matter; carrier dynamics reduces to steepest descent on strain). No downstream used_by edges are recorded yet; the module is a leaf that pins the kernel's reach so later gravity theorems can cite spending and collapse as explicit extra structure rather than kernel corollaries. Framework landmarks: T5 J-uniqueness, RCL, and the ledger interpretation of $J$ as symplectic action.
scope and limits
- Does not prove T5 J-uniqueness; only packages the premises that uniqueness consumes.
- Does not derive cost spending or residue collapse from kernel cost content.
- Does not construct full strain-descent dynamics; that lives in DescentPrincipleUniversality and related modules.
- Does not claim the reciprocal involution is the only symmetry of the cost package.
- Does not fix numerical gravity constants or close any of the seven gaps by itself.
depends on (4)
declarations in this module (17)
-
structure
KernelCostContent -
theorem
jcost_kernelCostContent -
def
reciprocalInvolution -
theorem
reciprocalInvolution_continuous -
theorem
reciprocalInvolution_preserves_cost -
theorem
reciprocalInvolution_iterate_even -
theorem
reciprocalInvolution_never_reaches -
theorem
kernel_cost_content_does_not_entail_cost_spending -
theorem
no_kernel_derivation_of_residue -
structure
CostSpendingSubstrate -
theorem
costSpendingSubstrate_nonempty -
def
collapseStep -
def
halfwayStep -
def
collapse_substrate -
def
halfway_substrate -
theorem
residue_has_two_models -
theorem
no_unique_dynamics_from_residue