# Forward Foundation Equation Compiler Status: Draft v0.1 Date: 2026-05-09 Scope: trust boundary for equation atoms, foundation kernels, derivation receipts, and theorem/logogram labels Claim state: compiler contract and admission doctrine; not a theorem prover, benchmark result, or proof of external equations ## 1. Purpose This document defines the forward foundation equation compiler. The rule is: ```text No backward trust chain. Only forward admissible generation. ``` Human theorem labels, expert names, citation chains, equation names, and logogram names are routing hints only. They are not trust objects. A trusted equation object must be generated forward from the foundation kernel, closed under the declared transform, and accompanied by a receipt. The human-origin doctrine is: ```text Origin may inspire. Only closure admits. ``` The irreverent short form is: ```text No vibes-to-axioms pipeline without a receipt. ``` This does not claim that unusual historical, cultural, personal, mystical, countercultural, or altered-state origins invalidate an equation. It only says origin stories are metadata, not authority. Local generator: ```text 4-Infrastructure/shim/foundation_forward_equation_compiler.py ``` Current receipt: ```text shared-data/data/foundation_forward_equation_compiler/foundation_forward_equation_compiler_receipt.json ``` Human summary: ```text shared-data/data/foundation_forward_equation_compiler/foundation_forward_equation_compiler.md ``` ## 2. Foundation Kernel The foundation set is: ```text F0 = {O4, SD, MN, gamma_star, H_dV, Omega, Lambda, A} ``` | Symbol | Role | |---|---| | `O4` | four primitives: field, shear, packet, spectral | | `SD` | dimensional shell: projection, residual, closure | | `MN` | Mass Number metric pressure | | `gamma_star` | shortest lawful projection path | | `H_dV` | information horizon / Underverse boundary | | `Omega` | torsion, shear, and event correction | | `Lambda` | logogram substitution / callable abstraction atom | | `A` | admission gate: ACCEPT, HOLD, QUARANTINE | The foundation kernel is not a citation bundle. It is the root object from which equation atoms must compile. ## 3. Shell Equation The no-infinity shell is: ```text SD = L4(O4) + L3(Rg3) + chi0 + U4 + E_HD + U_under ``` Where: | Term | Meaning | |---|---| | `SD` | source object in full domain dimension | | `O4` | visible four-primitive projection | | `Rg3` | genus-3 residual shadow | | `chi0` | closure witness | | `U4` | unseen but potentially promotable reserve | | `E_HD` | high-dimensional projection energy tax | | `U_under` | failed, forbidden, entropy-bound, or non-promotable residue | If an equation cannot fit this shell, it is not promoted structure. It routes to `HOLD`, `QUARANTINE`, `U_under`, or `NaN0`. ## 4. Forward Derivation The compiler does not ask whether a theorem label has a prestigious citation chain. It asks whether the object can be generated from `F0`: ```text F0 -> E1 -> E2 -> E3 -> ... ``` Each equation atom is shaped as: ```text E_next = Compile(E_prev, transform_rule, constraints, residual) ``` Promotion requires: ```text Admit(E_next) = ACCEPT ``` Otherwise the result remains: ```text HOLD | QUARANTINE | U_under | NaN0 ``` ## 5. Equation Atom Contract Every generated equation atom carries: ```yaml equation_atom: identity: equation_id: semantic_key: canonical_equation: equation_hash: foundation: source_kernel: F0_forward_foundation_kernel parent_equations: transform_rule: dependency_hash: projection: O4: Rg3: chi0: U4: E_HD: Underverse: admissibility: domain_laws: dimensional_scaling: energy_budget: information_budget: closure_status: residual_policy: receipt: source_hash: equation_hash: dependency_hash: receipt_hash: decision: ``` This mirrors the Omindirection rule: ```text payload != glyph != rendered layout ``` For equations: ```text equation object != theorem label != citation chain != rendered math ``` ## 6. Admission Equation A derived equation is accepted only when: ```text ACCEPT(E) iff receipt_recomputes(E) and chi0(E) = 0 and residual_declared(E) and B(E) < B_max and E_HD_paid(E) ``` The stack may propose candidate equations. The foundation compiler decides only whether the object is closed, accounted, and receipted. ## 7. PASS / ADD / PAUSE / SUBTRACT To avoid clock skew and hidden accounting drift, the compiler uses the same four-gate deterministic loop as the reconstruction-core receipts: ```text PASS -> ADD -> PAUSE -> SUBTRACT ``` | Gate | Function | |---|---| | `PASS` | verify exact replay or payload closure plus hashes | | `ADD` | count deterministic costs: core, residual, receipt, protocol, dictionary, energy | | `PAUSE` | zero-delta logical event fence; wall-clock time is metadata only | | `SUBTRACT` | compute trust/compression deltas only after costs are sealed | Timestamps may appear in human logs, but they are excluded from receipt hashes and cannot affect admission. ## 8. Godel Gauntlet `Godels_Gauntlet` is the promotion/quarantine gate: ```text the stack may propose and defend, but may not promote itself without receipts ``` It blocks the suspicious pattern: ```text human label -> trusted theorem ``` and replaces it with: ```text human label -> routing hint F0 -> compiled atom -> receipt -> closure -> admission decision ``` ## 9. Claim Boundary This document does not claim that the foundation compiler proves external mathematics. It defines the local trust boundary: ```text trusted object = compiled, receipted, closed object ``` Everything else is a hint, fixture, negative control, or HOLD candidate. ## 10. Origin Metadata Historical origin is retained as metadata: ```yaml provenance: human_origin: metadata_only era_context: metadata_only institutional_prestige: metadata_only theorem_label: routing_hint_only aesthetic_elegance: routing_hint_only accepted_result: hold_until_forward_receipted formal_closure: admission_candidate ``` This distinguishes: ```text historical origin != formal admissibility ``` The stack may record that an equation came from a strange era, private notebook, philosophical program, dense internal notation, institutional seminar, or beautiful intuition. None of those facts can promote it. They only help route the candidate into the forward compiler. The fair version is: ```text Humans may discover; the compiler must admit. ``` ## 11. Derived Fixture Example The first small derived physics atom is: ```text shared-data/data/buoyancy_added_mass_mobius/buoyancy_added_mass_mobius_receipt.json ``` It records `lambda_BAM`, a Mass-Number Mobius compression of the early-time buoyancy added-mass equation: ```text lambda_BAM(MN_rho, C) = g * alpha_C * MN_rho / (1 + kappa_C * MN_rho) ``` This fixture is accepted only as: ```text ACCEPT_FIXTURE_WITH_BOUND_CORRECTION ``` It shows the intended pattern: an external equation can be a candidate, but the accepted object is the normalized equation atom, exact equivalence checks, inverse check, residual policy, and receipt. The fixture does not promote a new fluid theorem or broad experimental claim. ## 12. Mass Number Transform Registry The Mass Number transform registry is: ```text 4-Infrastructure/shim/mass_number_transform_registry.py ``` Current receipt: ```text shared-data/data/mass_number_transform_registry/mass_number_transform_registry_receipt.json ``` Human summary: ```text shared-data/data/mass_number_transform_registry/mass_number_transform_registry.md ``` Receipt hash: ```text b215abe8cca08253dd62a2c2e84ff1f90fbd8e7eb5b2bb02d60dec39bbea2b9c ``` It records exact algebraic transform kernels that compile repeated pair equations into: ```text MN(a,b) = (a-b)/(a+b) MN plus small transform opcode ``` Accepted exact-kernel opcodes: ```text MN MN_RATIO_INV MN_MOBIUS_LOAD MN_SPLIT MN_REDUCED MN_PAIR_PRODUCT MN_BLEND MN_REFLECT MN_TRANSMIT_POWER MN_BINARY_P MN_ELASTIC_1D ``` `MN_BINARY_ENTROPY` is recorded only as `HOLD_ANALYTIC` until log base, numeric precision, and approximation/error receipts are declared. Decision: ```text ACCEPT_REGISTRY_WITH_HOLD_ANALYTIC ``` This registry does not prove every domain equation named in its route surface. It admits only the exact algebraic identities checked by the local receipt. Domain-specific uses still need source equations, residual policy, and forward-foundation admission. ## 13. Cross-Domain Kernel Adapters The cross-domain kernel adapter registry is: ```text 4-Infrastructure/shim/cross_domain_kernel_adapter_registry.py ``` Current receipt: ```text shared-data/data/cross_domain_kernel_adapters/cross_domain_kernel_adapter_registry_receipt.json ``` Human summary: ```text shared-data/data/cross_domain_kernel_adapters/cross_domain_kernel_adapter_registry.md ``` Receipt hash: ```text a66552526d5213a8122ce8f1efa56137f70c707d991ac7fdc90dc83d970ac081 ``` The adapter equation is: ```text X_d = A_d[K_j(theta)] + R_d + chi0 same shape does not imply same law ``` Decision: ```text HOLD_CROSS_DOMAIN_WITH_ACCEPTED_KERNEL_ADAPTERS ``` This admits the doctrine, not broad domain truth. Reusing a Mass Number kernel across fluid mechanics, impedance boundaries, routing probability, two-body mechanics, expert blending, moving-sofa contact geometry, or seismic horizon inference is lawful only when the adapter has its own source, replay, residual, and closure receipts. The moving-sofa route remains `HOLD_CONTACT_TOPOLOGY` until corridor geometry, signed-distance convention, motion path replay, collision closure, and area accounting exist. The seismic-horizon route remains `HOLD_BOUNDARY_WITNESS` until boundary-wave data and material residual models are receipted. ## 14. Magnetic Derivative Kernels The magnetic derivative kernel probe is: ```text 4-Infrastructure/shim/magnetic_derivative_kernel_probe.py ``` Current receipt: ```text shared-data/data/magnetic_derivative_kernels/magnetic_derivative_kernel_receipt.json ``` Human summary: ```text shared-data/data/magnetic_derivative_kernels/magnetic_derivative_kernel.md ``` Receipt hash: ```text b4617a8ff31586250efafd13e3ed402535fcad5d564922599aaa3cb95134c7e3 ``` Decision: ```text HOLD_MAGNETIC_DOMAIN_WITH_ACCEPTED_FIXTURES ``` Accepted local fixtures: ```text d/dB [B^2/(2*mu)] = B/mu F_x = m*dB/dx F_B = q*cross(v,B) Gamma_mu = MN(mu2,mu1) ``` These are algebra/vector fixtures and adapter candidates only. Field equations such as Faraday induction, Ampere-Maxwell routing, Alfven-speed routes, susceptibility contrast, hysteresis, and material-response claims stay HOLD until unit systems, gauge/sign conventions, boundary conditions, source data, and residual policies are receipted. ## 15. Solids Physics Kernels The solids physics kernel probe is: ```text 4-Infrastructure/shim/solids_physics_kernel_probe.py ``` Current receipt: ```text shared-data/data/solids_physics_kernels/solids_physics_kernel_receipt.json ``` Human summary: ```text shared-data/data/solids_physics_kernels/solids_physics_kernel.md ``` Receipt hash: ```text 98501df5a36ddd8a103ff40e6c8973f93e5271df325dd43ebdbebe59e896defb ``` Decision: ```text HOLD_SOLIDS_DOMAIN_WITH_ACCEPTED_FIXTURES ``` Accepted local fixtures: ```text sigma = E*epsilon U = E*epsilon^2/2 = sigma^2/(2E) dU/depsilon = sigma G = E/(2*(1+nu)) K = E/(3*(1-2*nu)) E_eff = S/2*(1-MN(E1,E2)^2) Gamma_Z = MN(Z2,Z1) ``` These are local algebra fixtures and adapter candidates only. Elastic wave speed, plasticity, fracture, anisotropy, finite-element boundary value, geometry, and material-model claims stay HOLD until units, conventions, source data, boundary conditions, and residual policies are receipted. ## 16. Easy-Wins Route Map The cross-domain easy-wins route map is: ```text 4-Infrastructure/shim/cross_domain_easy_wins_route_map.py ``` Current receipt: ```text shared-data/data/cross_domain_easy_wins/cross_domain_easy_wins_route_map_receipt.json ``` Human summary: ```text shared-data/data/cross_domain_easy_wins/cross_domain_easy_wins_route_map.md ``` Receipt hash: ```text ac1fe2ca6ee469c046cdb5fc78cef9efca6a601aee2e5270ef4b8c5854bb1e2e ``` Decision: ```text ADMIT_ROUTE_MAP_HOLD_FIRST ``` The ranked route queue is: ```text circuits_impedance thermal_diffusion acoustics_waves probability_routing orbital_two_body chemistry_equilibrium optics_fresnel statistics_effect_size bio_expression_contrast geometry_contact ``` This is a planning receipt only. It ranks low-cost probes where exact local algebra can be checked before nonlinear, field, geometry, measurement, or material-law claims are touched.