# Hodge-to-RSP factor obstruction: source and scope review v0.1 14 October 2026 UTC. Development — no registry entry. Physical promotion: 0. Reviewer: Codex `source_program_audit` parallel AI thread. This is a separate AI reasoning/scope audit. It is not human peer review, empirical evidence, custodian promotion, or CRL-2b eligibility. No original scientific source or registry status is changed. ## Objects and source boundary U_delta=exp(delta A) denotes the closed finite-dimensional native skew Hodge evolution sampled at a declared internal interval delta. It is not identified with the separately typed primitive local edge transports U_e, the C1 frame transfer, or a physical clock. F denotes the complete deterministic RSP-1 register transition, including its cached latent receipt, stock, bank/cursor, branch, fixed integrity and response bits. The RSP ranking potential V is nonnegative integer valued and strictly decreases whenever F changes complete state. It is a termination device, not a derived physical energy. Original HD latent witnesses and RSP token stock remain different objects. C0593 supplies lawful close/hold response maps conditional on source receipts; C0926 supplies HD latent/integrity readouts; E01/E03 preserve the missing complete native producer and projection boundary. None supplies the proposed semiconjugacy. ## 1. Finite-image surjective-factor theorem Assume a nonempty source domain X, a surjective U:X→X, a decoder phi:X→S, and exact semiconjugacy phi U=F phi. Let M=phi(X), assumed finite. Then F(M)=phi(U(X))=phi(X)=M. Thus F restricted to M is a permutation. Every state in M lies on a finite F-cycle. Strict decrease of V on a changed state rules out every nontrivial cycle: traversing such a cycle would give V(s)>V(s). Consequently F(s)=s for every s in M. This theorem requires neither continuity nor a uniform settling bound. A two-sided invariant source domain for an invertible Hodge flow provides the needed surjectivity. Mere forward invariance does not generally provide it. ## 2. Uniform-settling variant The finite-image hypothesis can instead be replaced by one common integer N such that F^(N+1)(s)=F^N(s) for every s in M. Surjectivity gives F^N(M)=M, while the uniform bound puts F^N(M) inside Fix(F). Therefore M is already fixed. Equivalently, for any x choose y with U^N y=x. Then phi(x)=F^N(phi(y)) is fixed. This version requires no finite-image or continuity assumption. Unrestricted RSP preparations have arbitrary nonnegative stock and finite but arbitrarily long banks. Their per-preparation settling bound does not imply one common N for the whole family. Do not use this variant without an explicit uniform bound. ## 3. Finite-order sampling theorem If U^m=id on X, exact semiconjugacy gives F^m(phi(x))=phi(x). Strict potential descent forces every decoded state to be fixed. This version works with infinite target/image and requires no decoder regularity. For the unit-metric filled tetrahedron the active frequencies equal 2 and the harmonic mode is fixed, so exp(pi A)=I. If delta/pi=p/q in lowest terms, U_delta^q=I. In particular delta=pi/4 has order four. Irrational delta/pi removes finite order on an active orbit; it does not remove recurrence. ## 4. Recurrent orbit and stable complete-state decoder Assume the orbit of x has arbitrarily late forward returns U^(n_k)x→x with n_k→infinity. Assume complete-state decoding is stable at x: sufficiently close returning orbit points decode exactly to phi(x). A continuous map into a discrete complete register space satisfies this requirement. If any F-iterate starting from phi(x) changes, its potential strictly drops. Subsequent iterates cannot equal phi(x), because their potential stays smaller. But semiconjugacy and stable returns give F^(n_k)(phi(x))=phi(U^(n_k)x)=phi(x) for sufficiently large k. Contradiction. Therefore that decoded orbit is fixed. Every finite-dimensional closed skew Hodge sample is recurrent. Its active phases belong to a compact torus; simultaneous phase approximation supplies arbitrarily late powers near identity. Irrational glued-carrier frequency ratios exclude a nonzero exact common period, not approximate recurrence. They do not evade this theorem. The stability hypothesis must apply to COMPLETE decoded RSP state, not Theta alone. Theta may remain constant while stock, latent cache or queue changes; stability of that projection supplies no complete-state return. ## 5. Conservation limitation Closed skew Hodge evolution preserves its declared norm. An identification of total available RSP stock with that invariant norm cannot reproduce a consumed-token transition. This is a conditional mismatch, not an unconditional impossibility of any stock decoder. Token count is not presently support norm or physical energy. Local resource can decline while support transfers into a separately modeled environment; HD coordinates and R_e=c1,e²(c0,i²+c0,j²) can vary while total norm is constant. RSP's potential likewise is not a source-derived energy. Any claimed stock/support identification needs a typed carrier map and balance law. ## 6. Exceptions and permitted weaker constructions - A forward-only restriction of an invertible U can fail surjectivity. On a nonrepeating irrational orbit, a discontinuous decoder can label a finite transient followed by a fixed register state. Exact semiconjugacy on that restricted orbit does not imply stable decoding or a robust measurement bridge. - Infinite decoded image plus unbounded preparation-dependent settling times evades both the finite-image and uniform-bound arguments. This does not evade the recurrent stable-decoder theorem when that hypothesis holds. Pure abstract surjectivity alone is insufficient for infinite targets: U(n)=n+1 on integers, phi(n)=max(-n,0), F(s)=max(s-1,0) is an explicit infinite-image factor with strict integer descent off zero. - A finite prepared window can embed a declared RSP trajectory in distinct samples of a Hodge orbit. The equality phi(U^n x)=F^n(s0) then holds only for the checked window. Extending it to all future times can fail at recurrence. Choosing decoder labels from the already computed RSP trace establishes a representation, not an independently predictive bridge. - A reversible cyclic history embedding represents an arbitrary finite trajectory, but its return edge violates exact absorbing factorization. Its reversible operator is not automatically a native simplicial Hodge generator; incidence, metric and spectrum require separate realization. - Extra unbounded clock/history state or fresh reservoirs change the source system and can break the closed recurrent assumptions. They must be independently sourced, rather than silently included in the native carrier. - Dissipation, export, feedback or reset can permit settling in an open model. A finite closed enlarged environment remains recurrent, so permanent robust absorbing behavior is not established merely by finite-window capture. - Approximate finite-window correspondence, ensemble readout or selected-preparation agreement remains a separate possible programme with explicit errors, preparations, observation rules and clocks. ## Disposition Exact mathematical obstruction conditional on the stated factor assumptions. No source-derived Hodge-to-stock realization, native event generator, empirical score or physical clock has been supplied. Recommended wording: RSP-1 is a finite prepared source-interface candidate; nontrivial exact realization as a robust autonomous factor of closed recurrent Hodge evolution is obstructed under the declared hypotheses. Finite-window and open-system realization remain separate source programmes. Physical promotion: 0.