Certify P104 separate endpoint geometry and two-history budget

research/p104-endpoint-geometric-budget/README.md

@@ -0,0 +1,219 @@
+# P104 separate endpoint geometry and two-history budget
+
+PR2909's endpoint-count curve is attainable by actual subspace unions at each
+endpoint separately. Within its two history types, the unique cheapest helper
+multiplicities at every positive saving are zero cheap helpers and 208
+unit-step helpers. Under PR2675's declared transport and ideal-idle screen,
+that profile has moment about **0.9988962438816702** at the selected complex
+saving. Its relaxed root satisfies the exact rational bracket
+
+```text
+0.002104152801941 < a_two_type < 0.002104152801942.
+```
+
+This closes the separate-endpoint subspace-union and pricing questions in
+[PR2909](https://github.com/CrocSwap/integer-mult-bounds/pull/2909). It does not
+construct a correction network. An individual flag's ability to contain labels
+does not establish that a helper acquires those labels from the actual sources.
+Source/target flags are chosen independently in this relaxation.
+
+## Contract and predecessor scope
+
+Retain the fixed PR1972/1991 P104 channel and the X-only monotone model from
+PR2353, PR2614 and PR2896/2909. The correction adds `I-G/2` to unread Y,
+reads immutable X along their frozen paths, and uses new helpers with
+source-independent dirty inputs at ZERO, increasing frames, restoration at
+FULL, and equal-frame rational operations. Original helper/private reads,
+source writes, target feedback and descending transports are excluded.
+Reinitialization starts a new lifetime. These restrictions belong to the
+audited route, rather than every possible P104 construction.
+
+The two helper types are dimension jumps
+
+```text
+cheap:     (1,7,1)
+unit-step: (1,1,1,1,1,1,1,1,1).
+```
+
+At target hyperplane H, a cheap helper's last proper frame is a line, while a
+unit helper's is dimension seven. Dually, at source line L, a cheap helper's
+first superframe is dimension eight, while a unit helper's is dimension two.
+The departure normal spaces are proper subspaces of the eight-dimensional
+annihilator `L^perp`. These are the separate endpoint covering sets considered
+here. Specifying those sets supplies no source acquisition or coefficients.
+
+PR2909 proves the necessary count relaxation with endpoint demands d and
+one-unit capacities b. Both orientations have these types:
+
+| d | b | d-b | Number of endpoints |
+|---:|---:|---:|---:|
+| 36 | 24 | 12 | 12 |
+| 44 | 28 | 16 | 24 |
+| 45 | 30 | 15 | 8 |
+| 46 | 29 | 17 | 24 |
+| 47 | 30 | 17 | 24 |
+| 48 | 30 | 18 | 12 |
+
+Its 192-unit threshold at a 400-helper budget is retained. This package adds
+geometric attainment in the separate covering problem and optimizes its price.
+
+## Affine-complement lemma
+
+Let D span an eight-dimensional binary space H. Suppose a linear functional q
+takes value one on every point of D. Choose a proper hyperplane `ker(f)`
+maximizing `b = |D intersect ker(f)|`. Since `0 < b < |D|`, f is neither zero
+nor q on H. Therefore both `ker(f)` and `ker(f+q)` are genuine dimension-seven
+subspaces of H. On D, `f+q = f+1`, so their intersections with D are disjoint
+and cover D. They have sizes b and d-b. This proves exact two-subspace coverage,
+beyond the inequality `d <= 2b`.
+
+Every chosen dimension-seven arrival frame extends to a complete unit-step
+flag inside H, followed by FULL. A demanded singleton extends to a cheap
+line/H/FULL flag. On departure, a dimension-seven normal space V in `L^perp`
+gives a dimension-two frame `V^perp` containing L, extendible to a unit flag.
+A singleton normal n gives the cheap superframe `ker(n)` containing L.
+These flag extensions prove endpoint incidence only. They do not join the
+opposite endpoints or prove information can be gathered along the flags.
+
+The checker verifies the lemma's hypotheses and explicit covers at all 104
+arrival and 104 departure endpoints. It reconstructs residual sets from the
+original 4,058-event word, ordered root Gram matrix and chronological data
+paths. A second finite algorithm enumerates all pairs of restricted proper
+hyperplanes without using q or the complement formula. It confirms the
+one-plane optimum b and two-plane optimum d at every endpoint.
+
+For zero, one and two unit helpers at an endpoint, the exact remaining number
+of singleton cheap carriers is consequently d, d-b and zero. The marginals b
+and d-b decrease. Sorting all 208 marginals gives PR2909's complete curve
+`C_min(U)`, now with actual endpoint subspace unions attaining it.
+No extra helpers improve an endpoint after two units. Every positive marginal
+is at least twelve, so
+
+```text
+C_min(U) >= 12*(208-U)                  for 0 <= U <= 208.
+```
+
+## Cheapest two-history profile
+
+Use PR2675's fixed template `m=73`, `W=832/9`, centre charge 56, total paid
+rank mass 6712 and ideal idle rank 51. The whole original channel has profile
+
+```text
+1074 H1 + 58 H2 + 86 H3 + 104 H4 + 48 H5 + 24 H6 + 8 H7 + 8 H8.
+```
+
+Its rank mass is 2368. Every added lifetime consumes nine more rank units,
+leaving `4344-9*(C+U)` idle rank units. Hence `C+U <= 482`. Idle packing is
+fractional and favorable. Native ADDs, owners, routing, precision, tape and
+all other physical charges are unknown and additional.
+
+For `c_r(a)=(r/73)^(1-a)`, replace the nine units of ideal idle used by each
+helper. Its excess prices are
+
+```text
+q_c = 2*c_1 + c_7 - 9*c_51/51
+q_u = 9*c_1       - 9*c_51/51.
+```
+
+Both are positive for `a>0`, since their jumps have ranks below 51. Moreover
+
+```text
+q_c = 73^(a-1) * [2*(1-51^(-a)) + 7*(7^(-a)-51^(-a))]
+q_u = 73^(a-1) *  9*(1-51^(-a)),
+q_u < (9/2)*q_c < 12*q_c.
+```
+
+For U below 208, at least `12*(208-U)` cheap helpers are required. Their
+excess over the 208-unit price is at least
+`(208-U)*(12*q_c-q_u) > 0`. For U above 208, the extra units have positive
+price. At U=208 any cheap helper also costs extra. Thus `(C,U)=(0,208)` is
+the unique minimizing multiplicity pair for every positive a in this model.
+Geometric assignments are not unique. At a=0 all equal-mass profiles tie.
+
+The minimizing profile has `H1=2946` and the other original counts unchanged,
+local rank mass 4240 and idle rank mass 2472, giving ideal `H51=824/17`.
+Each unit flag has relaxed capacity 64 at all seven internal cuts, so 208
+units also pass the earlier aggregate crossing rows. This does not assign
+their actual subspaces at those cuts.
+
+The pinned exact rational interval evaluator checks the selected moment and
+strict root signs stated above. Every rank is below 73 and every profile count
+is nonnegative, so the moment strictly increases with a. The root is the
+supremum for this two-type, separate-endpoint, ideal-idle relaxation. It is not
+an admitted primitive exponent or kappa, nor an optimum over all 256 histories.
+For example, `(1,1,5,1,1)` has the same endpoint-adjacent dimensions and cheaper
+transport, but different internal cut capacities. It is outside this optimization.
+
+## Reproduction and evidence
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-endpoint-geometric-budget
+```
+
+The marker is `PASS_P104_ENDPOINT_GEOMETRIC_BUDGET endpoints=208 units=208`.
+`verify.py --emit` prints the complete recomputed receipt without changing files.
+RESULT.json records endpoint curves, a digest of explicit cover witnesses,
+the optimizing counts and exact rational moment intervals. The verifier
+authenticates both inputs and the interval module before importing that module.
+It runs no optimizer or predecessor executable.
+
+Four behavioral groups cover a literal affine example, failure without the odd
+section hypothesis, wrong endpoint containment, invalid frames/counts, a strict
+competing price, duplicate JSON keys, optimized Git-free replay, source rejection
+before import and typed receipt corruption. Pair enumeration independently
+checks covers but shares reconstructed demand sets. Interval arithmetic is
+inherited from Douglas Colkitt's ceiling module. A separate internal audit
+checked all 208 covers through the pinned predecessor's implementation and
+reproduced the root interval. Evidence is author-checked, without independent
+human or formal-kernel review. Environment and commands are in TASK.json.
+
+## Open next questions
+
+Can the 208 unit lifetimes, or a named alternative involving other histories,
+jointly acquire and deliver the demanded coefficients in chronological order?
+First specify actual nested source/target flags and dependency paths at the
+original cuts, then check coefficients and dirty restoration before native
+lowering. Separate endpoint covers and aggregate crossing counts do not supply
+that joint realization. A candidate also needs the entire consumer, ownership,
+idle, routing and precision bill. The selected cold replay is unrun because
+this contribution changes no selected construction or frozen input.
+
+Family decision: finish this interface and cost audit and stop. The optimistic
+screen passes. It admits no supplier and justifies no further construction
+campaign with unknown costs set to zero.
+
+## Attribution and engineering notes
+
+Nick Petalas. Apache-2.0. Substantial AI assistance and internal review are
+disclosed in TASK.json, without a claim of independent human review.
+Credit Nick Petalas/PR2909/2896 for the demand curve and residual capacities,
+Hiroki Kumazaki/PR2889/2353 for endpoint spans and the monotone contract,
+Arham Tahir/PR2675/2682 for the coupled price model, ywz65/PR2614 for source
+departure handling, Sagar Thakrar's mixer handoff, and Philippe Van Velzen's
+relay construction and window bill. Retain Rohan Arun/PR1972, Chafik
+Boukhalfa/PR1991, Adam Gall/PR1940, Josie Elliston/PR1071, Ali Romman/PR1125
+and the geometry/native lineage. Inspected heads are in TASK.json.
+No predecessor package is copied or changed.
+
+Build the Lever shaped the executable cover and price certificate. Prove It
+Works required original-input reconstruction, exhaustive pair checks and
+rejecting controls. Unslop guided the prose. Finite bounds are 104 endpoints
+per orientation, 512 points/functionals, at most 255 distinct restricted planes,
+65,025 plane pairs per endpoint, 209 count states, 4,058 events and 23 feasible
+minimal price profiles. New functions are below 60 lines.
+
+Power of Ten rule 3 deviation: Python allocates managed objects during checking.
+Fixed domains and 2 MB JSON bounds limit live collections; temporary exports
+have scoped cleanup on every exit. A fixed-capacity implementation would be
+needed for a resident checker. Rule 1/2 deviation in the unchanged interval
+module: recursive logarithms and adaptive series preserve source binding.
+Calls here use ranks at most 73, precision 256 and `a<=1/100`; the chain has
+at most 73 frames and atanh/Taylor tails terminate within 100/100 terms in this
+domain. Registered commands have time bounds. A bounded iterative port needs
+a separate source-bound equivalence audit. Rule 8/9 deviation: importlib loads
+exactly one authenticated interval module; Python dispatch and unittest remain.
+The source-before-import control checks that boundary. Warning-as-error
+compilation, lint and optimized replay are recorded in TASK.json. Repository-wide
+tooling and inherited mathematical assumptions are unchanged.